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**COMMON CONCEPTS IN STATISTICS**

**M.Tevfik Dorak, MD, PhD**

*Please use this address next time***: http://www.dorak.info/mtd/glosstat.html**** **

*See also ***Common Terms in Mathematics**;** ****Statistical Analysis in HLA & Disease Association
Studies**;

**Epidemiology** (incl. **Genetic Epidemiology Glossary**)

**For more LINKS, see the end of this
page**

** [Please note that the
best way to find an entry is to use the Find option from the Edit menu, or CTRL
+ F]**

**Absolute
risk**:
Probability of an event over a period of time; expressed as a cumulative
incidence like 10-year risk of 10% (meaning 10% of individuals in the group of
interest will develop the condition in the next 10 year period). It shows the
actual likelihood of contracting the disease and provides more realistic and
comprehensible risk than **relative risk**/**odds ratio**.

**Addition
rule**:
The probability of any of one of several mutually exclusive events occurring is
equal to the sum of their individual probabilities. A typical example is the
probability of a baby to be homozygous or heterozygous for a Mendelian
recessive disorder when both parents are carriers. This equals to 1/4 + 1/2 =
3/4. A baby can be either homozygous or heterozygous but not both of them at
the same time; thus, these are mutually exclusive events (see also **multiplication
rule**).

**Adjusted
odds ratio**:
In a multiple logistic regression model where the response variable is the
presence or absence of a disease, an odds ratio for a binomial exposure
variable is an adjusted odds ratio for the levels of all other risk factors
included in a **multivariable model**. It is also possible to calculate the
adjusted odds ratio for a continuous exposure variable. An adjusted odds ratio
results from the comparison of two strata similar at all variables except
exposure (or the marker of interest). It can be calculated when stratified data
are available as contingency tables by **Mantel-Haenszel test**.

**Affected
Family-Based Controls (AFBAC) Method**: One of several family-based association
study designs (**Thomson, 1995**). This one uses
affected siblings as controls and examines the sharing between two affected family
members. The parental marker alleles not transmitted to an affected child or
never transmitted to an affected sib pair form the so-called affected
family-based controls (AFBAC) population. See also **HRR** and **TDT**
and **Genetic Epidemiology**.

**Age-standardized
rate**:
An age-standardized rate is a weighted average of the age-specific rates, where
the weights are the proportions of a standard population in the corresponding
age groups. The potential confounding effect of age is removed when comparing
age-standardized rates computed using the same standard population (from the **Glossary** of **Disease Control Priorities Project**.)

**Alternative
hypothesis**:
In practice, this is the hypothesis that is being tested
in an experiment. It is the conclusion that is reached when a null hypothesis
is rejected. It is the opposite of null hypothesis, which states that there is
a difference between the groups or something to that effect.

**Analysis of molecular variance (AMOVA)**: A
statistical (analysis of variance) method for analysis of molecular genetic
data. It is used for partitioning diversity within and among populations using
nucleotide sequence or other molecular data. AMOVA produces estimates of
variance components and F-statistic analogs (designated as phi-statistics). The
significance of the variance components and phi-statistics is tested using a
permutational approach, eliminating the normality assumption that is
inappropriate for molecular data (**Excoffier, 1992**). AMOVA can be performed on **Arlequin**.
For examples, see **Roewer, 1996**; **Stead, 2003**; **Watkins, 2003**); see also **AMOVA Lecture Note** (**EEB348**).

**ANCOVA**: See **covariance models**.

**ANOVA** (analysis of
variance): A test for significant differences between multiple means by
comparing variances. It concerns a normally distributed response (outcome)
variable and a single categorical explanatory (predictor) variable, which
represents treatments or groups. ANOVA is a special case of multiple regression
where indicator variables (or orthogonal polynomials) are used to describe the
discrete levels of factor variables. The term analysis of variance refers not
to the model but to the method of determining which effects are statistically
significant. Major assumptions of ANOVA are the homogeneity of variances (it is
assumed that the variances in the different groups of the design are similar)
and normal distribution of the data within each treatment group. Under the null
hypothesis (that there are no mean differences between groups or treatments in
the population), the variance estimated from the within-group (treatment)
random variability (**residual sum of squares** = RSS) should be about the
same as the variance estimated from between-groups (treatments) variability (**explained
sum of squares** = ESS). If the null hypothesis is true, mean ESS / mean RSS
(variance ratio) should be equal to 1. This is known as the **F test** or
variance ratio test (see also **one-way** and **two-way ANOVA**). The
ANOVA approach is based on the partitioning of sums of squares and degrees of
freedom associated with the response variable. ANOVA interpretations of main
effects and interactions are not so obvious in other regression models. An
accumulated ANOVA table reports the results from fitting a succession of
regression models to data from a factorial experiment. Each main effect is
added on to the constant term followed by the interaction(s). At each level an
F test result is also reported showing the extra effect of adding each variable
so it can be worked out which model would fit best. In a two-way ANOVA with
equal replications, the order of terms added to the model does not matter,
whereas, this is not the case if there are unequal replications. When the
assumptions of ANOVA are not met, its non-parametric
equivalent **Kruskal-Wallis test** may be used (a review by ; a tutorial on **ANOVA**, and** ANOVA posttest; online calculators for ANOVA (1),
(2),
(3)** and **(4;**
for analysis of summary data). See also **MANOVA**.** **

**Arithmetic
mean**:
M = (x_{1} + x_{2} + .... x_{n}) / n (n = sample size).

**Association**: A statistically significant correlation
between an environmental exposure or a biochemical/genetic marker and a disease
or condition. An association may be an artifact (random error-chance, bias,
confounding) or a real one. In population genetics, an association may be due
to **population stratification**, **linkage disequilibrium**, or direct
causation. A significant association should be presented together with a
measure of the strength of association (**odds ratio**, **relative risk**
or **relative hazard** and its 95% confidence interval) and when appropriate
a measure of potential impact (**attributable risk,** **prevented fraction,
attributable fraction/etiologic fraction**).

**Assumptions**: Certain conditions
of the data that are required to be met for validity of a statistical test. **ANOVA**
generally assumes normal distribution of the data within each treatment group,
homogeneity of the variances in the treatment groups, and independence of the
observations. In **regression analysis**, main assumptions are the normal
distribution of the response variable, constant
variance across fitted values, independence of **error terms**, and the
consistency of underlying hazard rate over time (**proportionality assumption**) in **Cox
Proportional Hazard Model****s**. See **StatNotes**: **Testing of Assumptions**.

**Asymptotic**: Refers to a curve
that continually approaches either the x or y axis but does not actually reach
it until x or y equals infinity. The axis so approached is the asymptote. An
example is the **normal distribution curve**.

**Asymptotically
unbiased**:
In point estimation, the property that the bias approaches zero as the sample
size (N) increases. Therefore, estimators with this property improve as N
increases. See also **bias**.

**Attributable
risk (AR)**:
Also called excess risk or risk difference. A measure of potential impact of an
association. It quantifies the additional risk of disease following exposure
over and above that experienced by individuals who are not exposed. It shows
how much of the disease is eliminated if no one had the risk factor
(unrealistic). The information contained in AR combines the **relative risk**
and the risk factor prevalence. The larger the AR, the greater the effect of
the risk factor on the exposed group. See also **prevented fraction**, **Walter, 1978**; **PowerPoint presentation on AR**; and **Attributable Risk Applications in Epidemiology**.
For online calculation, see **EpiMax Table Calculator**.

**Attributable
fraction (etiologic fraction)**: It shows what proportion of disease in the
exposed individuals is due to the exposure.

**Balanced
design**:
An experimental design in which the same number of observations is taken for
each combination of the experimental factors.

**Bartlett’s
test**:
A test for homogeneity of variance.

**Bayesian
inference**:
An inference method radically different from the classical frequentist approach
which takes into account the prior probability for an event. Established as a
new method by **Reverend Thomas Bayes**. See a slide
presentation on **Introduction
to Bayesian Statistics**; **+Plus**:
**Bayesian Statistics Explained**; **Bayesian Revolution in Genetics**;
**BMJ Collection on Bayesian Statistics**;
**MathWorld**: **Bayesian Analysis**; **Books**; **Partition Software for Online Bayesian Analysis**; **Bayesian Calculator**; **A Bayesian Perspective on Interpreting Statistical
Significance (InStat)** & Bayesian Calculator.

**Bayes'
method in genetic counseling**: This method uses available additional information
to modify risks calculated purely by Mendelian probabilities. It combines prior
and conditional probabilities to give joint and posterior probabilities of
unknown events. See **Bayesian
Methods in Genetic Risk Calculation, Bayesian Analysis and Risk Assessment in Genetic Counseling
and Testing** and **A Unified Approach to Bayesian Risk Calculations**.

**Bernoulli
distribution**
models the behavior of data taking just two distinct values (0 and 1).

**Bias**: An estimator for a
parameter is unbiased if its expected value is the true value of the parameter.
Otherwise, the estimator is biased. It is the quantity E = (q -hat) - q. If the estimate of q is the same as actual but unknown q, the estimate is unbiased (as in
estimating the mean of normal, binomial and Poisson distributions). If bias
tends to decrease as n gets larger, this is called **asymptotic unbiasedness**.
See reviews on epidemiologic meaning of bias: **Bias in Clinical Trials**,** Bias & Confounding in Molecular Epidemiology**,
**Bias of Ascertainment** in **Complex Disease Genetics**, and **Bias and Confounding Lecture
Note**.

**Binary
(dichotomous) variable**: A discrete random variable that can only take two
possible values (success or failure).

**Binomial
distribution**:
The binomial distribution gives the probability of obtaining exactly *r* successes in *n* independent trials, where there are
two possible outcomes one of which is conventionally called success (**Binomial
Distribution**; **Online
Binomial Test** for observed vs expected value; **Binomial Probability Calculator**).

**Blocks**: Homogeneous
grouping of experimental units (subjects) in experimental design. Groups of
experimental units that are relatively homogeneous in that the responses on two
units within the same block are likely to be more similar than the responses on
two units in different blocks. Dividing the experimental units into blocks is a
way of improving the accuracy between treatments. Blocking will minimize
variation between subjects that is not attributable to the factors under study.
Blocking is similar to matching in two-sample tests or stratification to
control for confounding.

**Blocking**: When the available
experimental units are not homogeneous, grouping them into blocks of
homogeneous units (stratification) will reduce the experimental error variance.
This is called blocking where differences between experimental units other than
those caused by treatment factors are taken into account. This is like
comparing age-matched groups (blocks) of a control group with the corresponding
blocks in the patients group in an investigation of the side effects of a drug
as age itself may cause differences in the experiment. Block effects soak up
the extra, uninteresting and already known
variability in a model. Blocking is preferable to randomization when the
factors that might affect the outcome are known.

**Bonferroni Correction**: This is a multiple comparison
technique used to adjust the a error level. See also **HLA
and Disease Association Studies**, **Online
Bonferroni Correction**, **GraphPad
Guide to Multiple Comparisons & a commentary by Perneger, 1998**).

**Bootstrap**: An application of resampling statistics. It
is a data-based simulation method used to estimate variance and bias of an estimator
and provide confidence intervals for parameters where it would be difficult to
do so in the usual way (**Online Resampling Book**).

**Canonical**: Something that has been reduced to its
simplest form.

**Carryover effect**: Any effect of a drug that lasts
beyond the period of treatment. This is a worry in drug trials with **crossover
design** and the reason for the washout period between treatments.

**Case-control study**: A design preferred over cohort
studies for relatively rare diseases in which cases with a disease or exposure
are compared with controls randomly selected from the same study base. This
design yields **odds ratio** as opposed to **relative risk** from cohort
studies. See **Case-control Studies Chapter** in **Epidemiology
for the Uninitiated**.

**Causal relationship**: It does not matter how small it
is, a *P* value does not signify causality. To establish a causal
relationship, the following non-statistical evidence is required: consistency
(reproducibility), biological plausibility, dose-response, temporality (when
applicable) and strength of the relationship (as measured by odds
ratio/relative risk/hazard ratio). See **Hills's criteria of causality**; **Seven Common Errors in Statistics**;
and Causality by DR Cox, JR Stat Soc A 1992;155:291 (**JSTOR-UK link**). The original reference for Hill's criteria is Hill AB: The
environment and disease: association or causation. Proc R Soc Medicine
1965;58:295-300.

**Categorical
(nominal) variable**:
A variable that can be assigned to categories. A non-numerical (**qualitative**)
variable measured on a (discrete) **nominal** scale such as gender, drug treatments,
disease subtypes; or on an **ordinal** scale such as low, median or high
dosage. A variable may alternatively be **quantitative** (**continuous**
or **discrete**). See **GraphPad
QuickCalc**: **Categorical
Data Analysis**.

**Censored
observation**:
Observations that survived to a certain point in time before dropping out from
the study for a reason other than having the outcome of interest (lost to
follow up or not enough time to have the event). Thus, censoring is simply an
incomplete observation that has ended before time-to-event. These observations
are still useful in **survival analysis**.

**Central limit theorem**: The means of a relatively large
(>30) number of random samples from any population (not necessarily a normal
distribution) will be approximately
normally distributed with the population mean being their mean and variance being
the (population variance / n). This approximation will improve as the sample
size (the number of samples) increases. See **Mathematical Basis**; **QuickTime Demonstration**;** JAVA Demonstration**; **Simulation**.

**Chi-squared distribution**: A distribution derived from the
**normal distribution**. Chi-squared (C^{2}) is
distributed with v degrees of freedom with mean = v and variance = 2v. (**Chi-squared
Distribution**, **a Lecture
on Chi-Squared Significance Tests**).

**Chi-squared test**: The most commonly used test for
frequency data and goodness-of-fit. In theory, it is nonparametric but because
it has no parametric equivalent, it is not classified as such. It is not an
exact test and with the current level of computing facilities, there is not
much excuse not to use Fisher’s exact test for 2x2 contingency table analysis
instead of Chi-squared test. Also for larger contingency tables, the G-test
(log-likelihood ratio test) may be a better choice. The Chi-square value is
obtained by summing up the values (residual^{2}/fit) for each cell in a
contingency. In this formula, residual is the difference between the observed
value and its expected counterpart and fit is the expected value. See **Statistical
Analysis in HLA and Disease Association Studies** for assumptions
and restrictions of the Chi-squared test. (**Tables
of critical values of t, F and Chi-square**).

**Cochran's Q Test**: A nonparametric test examining
change in a dichotomous variable across more than two observations. If there are two
observations, **McNemar's test** should be used.

**Coefficient
of determination (R ^{2})**: See

**Coefficient
of variation**:
It is a measure of spread for a set of data. It is a measure of variation in
relation to the mean. Calculated as standard deviation divided by the mean
(x100). (**Online Calculator for Coefficient of Variation and Other
Descriptive Statistics**).

**Cohort effect**: The tendency for persons born in certain
years to carry a relatively higher or lower risk of a given disease. This may
have to be taken into account in case-control studies.

**Concomitant
variable**:
See **covariance models**.

**Conditional
(fixed-effects) logistic regression**: The conditional logistic regression (CLR)
model is used in studies where cases and controls can be matched (as pairs)
with regard to variables believed to be associated with the outcome of
interest. The model creates a likelihood that conditions on the matching
variable. It is the preferred method for the analysis of nested case-control
studies when matching is done at the individual level (there may be more than
one control per case). In economic analysis, it is called fixed-effects logit
for panel data. See **Preisser & Koch, 1997**.

**Confounding
variable**:
A variable that is associated with both the outcome and the exposure variable.
A classic example is the relationship between heavy drinking and lung cancer.
Here, the data should be controlled for smoking as it is related to both
drinking and lung cancer. A positive confounder is related to exposure and
response variables in the same direction (as in smoking); a negative confounder
shows an opposite relationship to these two variables (age in a study of
association between oral contraceptive use and myocardial infarction is a
negative confounder). The data should be stratified before analyzing it if
there is a confounding effect. **Mantel-Haenszel** test is designed to
analyze stratified data to control for a confounding variable. Alternatively, **a multivariable regression model** can
be used to adjust for the effects of confounders. See **Bias &
Confounding Lecture Note**; a review by **Greenland, 2001**.

**Conservative
test**:
A test where the chance of type I error (false positive) is reduced and type II
error risk is increased. Thus, these tests tend to give larger *P* values
compared to non-conservative (liberal) tests for the same comparison.

**Contrast**: A contrast is
combinations of treatment means, which is also called the main effect in **ANOVA**.
It measures the change in the mean response when there is a change between the
levels of one factor. For example, in an analysis of three different
concentrations of a growth factor on cell growth in cell culture with means m_{1}, m_{2}, m_{3}, against a control
value (m_{o}) without any growth
factor, a contrast would be:

q = m_{o} - 1/3 (m_{1}+ m_{2}+ m_{3})

The important point is that the coefficients sum to zero (1/1 - 1/3 - 1/3 - 1/3). If the value of the contrast (q) is zero or not significantly different from zero, there is no main effect, i.e., the combined growth factor mean is not different (positive or negative) from the no growth factor mean.

**Cook
statistics**:
A diagnostic *influence* statistics in regression analysis designed to
show the influential observations. **Cook's distance** considers the
influence of the *i* th value on all n fitted values and not on the fitted
value of the *i *th observation. It yields the shift in the estimated
parameter from fitting a regression model when a particular observation is
omitted. All distances should be roughly equal; if not, then there is reason to
believe that the respective case(s) biased the estimation of the regression
coefficients. Relatively large Cook statistics (or Cook's distance) indicates
influential observations. This may be due to a high **leverage**, a large **residual**
or their combination. An **index plot** of residuals may reveal the reason
for it. The leverages depend only on the values of the explanatory variables
(and the model parameters). Cook statistics depends on the residuals as well. Cook statistics may not be very satisfactory in
binary regression models. Its formula uses the standardized residuals but the
modified Cook statistics uses the **deletion residuals**.

**Correlation Coefficient**: See **Pearson's correlation
coefficient (r)**,
**Spearman’s rank correlation (rho)** and **Multiple
regression correlation coefficient (R ^{2})**. (

**Correspondence
analysis**:
In population genetics, a complementary analysis to
genetic distances and dendrograms. It displays a global view of the relationships
among populations (**Greenacre MJ, 1984; Greenacre & Blasius, 1994; Blasius & Greenacre, 1998**). With its
visual output, it supplements more formal inferential analyzes. This type of
analysis tends to give results similar to those of dendrograms as expected from
theory (**Cavalli-Sforza & Piazza, 1975**), but is
more informative and accurate than dendrograms especially when there is
considerable genetic exchange between close geographic neighbors (**Cavalli-Sforza et al. 1994**). Cavalli-Sforza
et al concluded in their enormous effort to work out the genetic relationships among
human populations that two-dimensional scatter plots obtained by correspondence
analysis frequently resemble geographic maps of the populations with some
distortions (**Cavalli-Sforza et al. 1994**). Using the same
allele frequencies that are used in phylogenetic tree construction, **correspondence analysis** can be performed on **ViSta**
(**v7.0), VST, Statistica, SAS **but most conveniently** **on
MultiVariate Statistical Package (**MVSP**). Link to **a
Tutorial**; **Course Notes**; **StatSoft
Textbook**: **Correspondence
Analysis Chapter**.

**Cox
proportional hazards model**: A regression method described by D.R. Cox (J Royal Stat
Soc, Series B 1972;34:187-220; **JSTOR-UK**) for modeling survival times
(for significance of the difference between survival times, **log-rank test **is
used). It is also called proportional hazards model because it estimates the
ratio of the risks (**hazard ratio **or **relative hazard**). As in any
regression model, there are multiple predictor variables (such as prognostic
markers whose individual contribution to the outcome is being assessed in the
presence of the others) and the outcome variable (e.g.,
whether the patients survived five years, or died during follow-up, etc). The
model assumes that the underlying hazard rate (rather than survival time) is a
function of the independent variables and consistent over time (**proportionality assumption**, i.e. the
survival functions of the groups are approximately parallel). There is no
assumption for the shape and nature of the underlying survival function.
Cox's regression model has been the most widely used
method in survival data analysis regardless of whether the survival time is
discrete or continuous and whether there is censoring (**Lee & Go, 1997**). Cox regression uses the
**maximum likelihood method**
rather than the **least squares method** (**Online
Cox Proportional Hazards Survival Regression**; a **Superlectures** on survival analysis; **Cox's proportional hazards model**; an example
of **Model Building**).

**Covariate**: Generally used to
mean explanatory variable, less generally an additional explanatory variable is
of no interest but included in the model to adjust the statistical model. More
specifically, it denotes an explanatory variable which is unaffected by
treatments and has a linear relationship to the response. The intention is to
produce more precise estimates of the effect of the explanatory variable of
main interest. In the analysis, a model is first fitted using the covariate.
Then the main explanatory variable is added and its additional effect is
assessed statistically. Whether the use of a covariate is wise (i.e., whether
it has a statistically significant influence) can be judged by checking its
effect on the residual (error) mean square (variance). If the addition of
covariate reduces it remarkably, it will improve the analysis. See also **covariance
models**.

**Covariance
(covariation)**:
It is a measure of the association between a pair of variables: the expected
value of the product of the deviation of two random variables from their
respective means. It is also called a measure of ‘linear dependence’ between
the two random variables. If the two variables are independent (no linear
correlation), then their covariance is zero but for non-zero values covariance
is unstandardized (unlike **correlation coefficient**); there is no limit to
possible values. Because of this, it is difficult to compare covariances. A
negative value means that for small values of X, there are large values of Y
(inverse association). It is calculated as the mean sum-of-products using each
x_{i} and y_{i} values, their means
m_{x} and m_{y}, and (n-1). (The
covariance standardized to lie between -1 and +1 is **Pearson’s correlation
coefficient**.) See **Wikipedia**:
**Covariance**;
**NetMBA
Statistics**: **Covariance**.

**Covariance
models**:
Models containing some quantitative and some qualitative explanatory variables,
where the chief explanatory variables of interest are qualitative and the
quantitative variables are introduced primarily to reduce the variance of the
error terms. [Models in which all explanatory variables are qualitative are
called analysis of variance -**ANOVA**- models.] Analysis of covariance
-ANCOVA- combines features of ANOVA and regression. It augments the ANOVA model
containing the factor effects with one or more additional quantitative
variables that are related to the response variable. The intention is to make
the analysis more precise by reducing the variance of the error terms. Each
continuous quantitative variable added to the ANOVA model is called a **concomitant
variable** (and sometimes also covariates). If qualitative variables are
added to an ANOVA model to reduce error variance, the model remains to be
ANOVA. By adding extra variables, the results are said to be controlled or
adjusted for the additional variables (like age or sex).

Apart from the above use of the term, analysis of covariance is more generally used for almost any analysis assessing the relationship between a response variable and a number of explanatory variables. In a multiple regression model, additional variables, which are known not to have any effect on the response variable, such as age and sex, are sometimes added to the model to adjust the response for these variables (age and sex in this case). Such variables are called confounders (or covariates). When the response is normally distributed, this is the preferred method over a simple t-test when the two groups compared differ, say, in their age and sex distribution (or any other confounding variable). The result is then controlled (or adjusted) for age and sex. When such adjustments are made, the regression coefficient for the significant effect variable will be (most probably) different from the one obtained from a univariable model involving only that variable (say the effect of a disease on pulse rate as compared to healthy controls).

**Cramer’s
coefficient of association** **(C)**: Also known as contingency coefficient. While
Chi-squared is used to determine significance of an association (and varies by
sample size for the same association), Cramer’s C is a measure of association
varying from 0 (no association) to 1 (perfect association) that is independent
of the sample size. However, it seldom reaches its upper limit. It allows
direct comparison of the degree of association between different contingency
tables. It is calculated directly from the Chi-squared value and the total
sample size as (C^{2}/C^{2}+N) ^{½}.

**Cramer’s
V**: A
measure of the strength association for any size of contingency tables. It can
be seen as a correction of the Chi-squared value for sample size. The
transformation of the chi-squared value provides a value between 0 and 1 for
relative comparison of the strength of the association. For a 2x2 table,
Cramer's V is equal to the **Phi coefficient**. Cramer’s V is most
useful for large contingency tables. It can also be used as a global linkage
disequilibrium value for multiallelic loci (See **GOLD-Disequilibrium Statistics**;
Online **Cramer’s
V calculation**.)

**Crossover design**: A clinical trial design during
which each subject crosses over from receiving one treatment to another one.

**Cross-sectional data**: Data collected at one point in
time (as opposed to longitudinal/cohort data for example). See **Cross-Sectional Studies Chapter** in **Epidemiology
for the Uninitiated**.

**Degrees
of freedom (df)**:
The number of independent units of information in a sample used in the
estimation of a parameter or calculation of a statistic. In the simplest
example of a 2x2 table, if the marginal totals are fixed, only one of the four
cell frequencies is free to vary and the others will be dependent on this value
not to alter the marginal totals. Thus, the df is only 1. Similarly, it can
easily be worked out that in a contingency table with r rows and c columns, the
df = (r-1)(c-1). In parametric tests, the idea is slightly different that the n
bits of data have n degrees of freedom before we do any statistical
calculations. As soon as we estimate a parameter such as the mean, we use up
one of the df, which was initially present. This is why in most formulas, the
df is (n-1). In the case of a two-sample t-test with n_{1} and n_{2}
observations, to do the test we calculate both means. Thus, the df = (n_{1 }+
n_{2 }- 2). In linear regression, when the linear equation y = a + bx
is calculated, two parameters are estimated (the intercept and the slope). The
df used up is then 2: df = (n_{1 }+ n_{2 }- 2). Non-parametric
tests do not estimate parameters from the sample data, therefore, df do not
occur in them.

In simple
linear regression, the df is partitioned similar to the total sum of squares (**TSS**).
The df for TSS is N-k. Although there are n deviations, one df is lost due to
the constraint they are subject to: they must sum to zero. TSS equals to **RSS**
+ **ESS**. In one-way ANOVA, the df for RSS is N-2 because two parameters
are estimated in obtaining the fitted line. ESS has only one df associated with
it. This is because the n deviations between the fitted values and the overall
mean are calculated using the same estimated regression line, which is associated
with two df (see above). One of them is lost because the of the constraint that
the deviations must sum to zero. Thus, there is only one df associated with
ESS. Just like TSS = RSS + ESS, their df have the same relationship: N-1 =
(N-2) + 1.

**Deletion
(or deleted) residual**:
A modified version of the standardized residual, which uses an estimate of s^{2} from a regression in
which point *i* has been deleted. It is used in calculation of the
modified **Cook's distance** instead of standardized residuals. Deletion
residuals are also known as **likelihood residuals**.

**Descriptive
statistics**:
Summary of available data. Examples are male-to-female ratio in a group;
numbers of patients in each subgroup; the mean weight of male and female
students in a class, etc. Only when the distribution is symmetric, mean and
standard deviation can be used. Otherwise (as in survival data), mean and
percentiles/range should be used to describe the data. See GraphPad guide to **Interpreting Descriptive Statistics**.

**Deviance**: A measure for
judging the degree of matching of the model to the data when the parameter
estimation is carried out by maximizing the likelihood (as in GLMs). The deviance
has asymptotically a Chi-squared distribution with df equal to the difference
in the number of parameters in the two models being compared. The total
deviance compares the fit of the saturated model to the null model, thus,
expresses the total variability around a fitted line which can be decomposed to
explained and unexplained (error) variability. The *residual deviance* in
a GLM analysis of deviance table corresponds to RSS in an ANOVA table, *and
regression deviance* corresponds to ESS. The residual deviance measures how
much fit to the data is lost (in likelihood terms) by modeling compared to the
saturated model. This will be a small value if the model is good (and it will
be zero for the saturated model containing all main effects and all
interactions). The regression deviance measures how much better is the model
taking into account the explanatory variables compared to the simplest model
ignoring all of them and only containing a constant (the mean of the
responses). This measurement is again made in terms of log-likelihood and the
bigger the regression deviance the better fits the model (i.e., the regression
effect is strong). In likelihood terms, the *residual* deviance can be
expressed as follows:

*D* = -2 [*ln* L_{c}
- *ln* L_{s}] or *D* = -2 *ln* [L_{c} / L_{s}]

where L_{c}
is the likelihood of the current model, and L_{s} is the likelihood of
the saturated model.

Similarly,
the *regression* deviance can be expressed as:

*D* = -2 [*ln* L_{c}
- *ln* L_{n}] or *D* = -2 *ln* [L_{c} / L_{n}]

where L_{n} is the likelihood of
the null model. The bigger the regression deviance the better the model
including this particular variable.

For
purposes of assessing the significance of an independent variable, the value of
*D* with an without the independent variable is compared (note the nested
character of the two sets). This is called the **deviance difference**. If a
variable is dropped, the residual deviance difference, if a variable is added,
the regression deviance difference is compared with the C^{2} distribution.

**Deviance
difference**:
In generalized linear modeling, models that have many explanatory variables may
be simplified, provided information is not lost in this process. This is tested
by the difference in deviance between any two nested models being compared:

*G* = *D* (for the
model without the variable) - *D* (for the model with the variable)

The
smaller the difference in (residual) deviance, the smaller the impact of the
variables removed. This can be tested by the C^{2} test.

In the
simplest example (in simple linear regression), if the log-likelihood of a model
containing only a constant term is L_{0} and the model containing a
single independent variable along with the constant term is L_{1},
multiplying the difference in these log-likelihoods by -2 gives the deviance
difference, i.e., *G* = -2 (L_{1} - L_{0}). *G* statistics
(**likelihood ratio test**) can be compared with the C^{2} distribution on df =
1, and it can be decided whether the model with the independent variable has a
regression effect (if *P* < 0.05, it has). The same method can be used
to detect the effect of the interaction by adding any interaction to the model
and obtaining the regression deviance. If this deviance is significantly higher
than the one without the interaction in the model, there is interaction [the
coefficient for the interaction, however, does not give the **odds ratio**
in **logistic regression**].

**Discrete
variable**:
A variable of countable number of integer outcomes. Examples include -**ordinal
multinomial**- several prognostic outcomes (such as poor, median and good) as
a function of treatment modalities, stage of the disease, age etc., or -**multinomial**-
people’s choices of hospitals (hospital A, B or C) as a function of their
income level, age, education etc. A discrete variable may be **binomial**:
diseased or non-diseased in a cohort or case-control study. The nature of the
outcome variable as discrete or continuous is crucial in the choice of a
regression model (see the **generalized linear model**).

**Dummy
variables**:
A binary variable that is used to represent a given level of a **categorical
variable**. In genetic data analysis, for example, it is created for each
allele at a multiallelic locus. The most common choices as dummy variables are
0 and 1. For each level of the variable, 1 denotes having the feature and 0
denotes all other levels. Also called indicator variables. If an indicator
variable is used, the regression coefficient gives the change per unit compared
to the reference value. Creating dummy variables on Stata: **Stata** & **UCLA Statistics**: **STATA Dummy Variables**.

**Dunn's
Test**:
This test is used when a difference between the groups is found in a non-parametric
ANOVA test. Dunn's test is a **post hoc test** that makes pairwise
(multiple) comparisons to identify the different group. See **GraphPad Prism Guide**: **K-W and Dunn's Test** (for an example,
see **Online K-W Test**; if the link does not work:
go to **Stats ToolBoxs Homepage** and choose
Kruskal-Wallis OWAV from Frequency Tables. Dunn's Q scores are presented at the
end of the analysis for pairwise difference tests. Another example is presented
in an **ANOVA PPT Presentation**).

**Dunnett's
test**:
When ANOVA shows a significant difference between groups, if one of the groups
is a control (reference) group, Dunnett's Test is used as a ** post hoc
test**. This multiple comparison test can be used to determine the
significant differences between a single control group mean and the remaining
treatment group means in an analysis of variance setting. It is one of the
least conservative

**Ecological
study**:
Analyses based on data grouped to the municipal, provincial or national level. See **Ecological Studies Chapter** in **Epidemiology
for the Uninitiated**.

**Ecological
fallacy: **The
aggregation bias, which is the unfortunate consequence of making inferences for
individuals from aggregate data. It results from thinking that relationships
observed for groups necessarily hold for individuals. The problem is that it is
not valid to apply group statistics to an individual member of the same group.
See an essay on **Ecological Fallacy**.

**Edwards' test**: A statistical test
for seasonality that looks for a one-cycle sinusoidal deviation from the null
distribution (see **Westerbreek et al, 1998**).

**Effect modification**: The situation in which a
measure of effect changes over values of another variable (the association
estimates are different in different subpopulations of the sample). The
relative risk or odds ratio associated with exposure will be different
depending on the value of the effect modifier. For example if in a disease
association study, the odds ratios are different in different age groups or in
different sexes, age or sex are effect modifiers. Effect modification is highly
related to statistical interaction in regression models. If where an exposure
decreases risk for one value of the effect modifier and increases risk for
another value of effect modifier, this is called crossover. See **Thompson, 1991** and *Effect Modification*
in __ Encyclopedia of
Biostatistics__.

**Eigenvalues** (latent values): In multivariate statistics,
eigenvalues
give the variance of a linear function of the variables. Eigenvalues measure
the amount of the variation explained by each principal component (PC) and will
be largest for the first PC and smaller for the subsequent PCs. An eigenvalue
greater than 1 indicates that PCs account for more variance than accounted by
one of the original variables in standardized data. This is commonly used as a
cut-off point for which PCs are retained.

**EM
Algorithm**:
A method for calculating maximum likelihood estimates with incomplete data. E
(expectation)-step computes the expected values for missing data and M
(maximization)-step computes the maximum likelihood estimates assuming complete
data. It was first used in genetics (**Ceppellini R et al, 1955**) to estimate
allele frequency for phenotype data when genotypes are not fully observable
(this requires the assumption of HWE and calculation of expected genotypes from
phenotype frequencies). For a brief overview, see **ARC CIGMR**:**
EM
Algorithm**. **Arlequin** implements EM algorithm in
haplotype construction and frequency analysis.

**Empirical P value**: A

**Empirical rule**: In variables normally distributed, 68% of
the data values are within 1SD of the mean; 95% are within 2SD of the mean; and
99.7% (nearly all) are within 3SD of the mean.

**Epidemiologic flaws and fallacies**: Beware
of confounders, selection bias, response bias, variable observer, Hawthorne
effect (changes caused by the observer in the observed values), diagnostic
accuracy bias, **regression to the mean**, significance Turkey,
nerd of nonsignificance, cohort effect, **ecological fallacy**, Berkson bias (selection
bias in hospital-based studies) and others (discussed in M Michael III et al. **Biomedical Bestiary**. Little, Brown and
Company, 1984; and in **Bias & Confounding in Molecular Epidemiology**).

**Epi-Info**: An epidemiologic data management and analysis package freely
available from **the CDC website**. Originally a DOS program,
the latest version (**3.3.2**) is designed for Microsoft Windows
95-XP (release date: Febr 9, 2005). There are also tutorials available online: **CDC**,
**Nebraska University**, **Dalhousie
University** and **Henry Ford Health Systems**.

**Error
terms**:
Residuals in regression models. Shown as W_{i} or e_{i}. Their expected
value is zero, thus, they vary around zero with a variance equal to s^{2}: N(0, s^{2}). They are assumed
to be normally distributed, have equal variance for all fitted values, and independent.
Normality is a reasonable assumption in many cases. The assumption of equal
variance implies that every observation on the dependent variable contains the
same amount of information. The impact of heterogeneous variances is a loss of
precision of estimates compared to the precision that would have been realized
if the heterogeneous variances had been taken into account. Transformation of
the dependent variable may help to homogenize the unequal variances. Correlated
errors are most frequent in time sequence data and they also cause the loss in
precision in the estimates. See also **residuals**.

**Explained
(regression) sum of squares (ESS)**: The measure of between treatments sum of
squares (variability) in ANOVA. If the means of treatment groups are different,
the ESS would be greater than RSS to yield a high variance ratio. The bigger
the ESS, the better explained the data by the model.

**Exploratory
data analysis**:
An initial look at the data with minimal use of formal mathematics or
statistical methods, but more with an informal graphical approach. Scatter
plots, correlation matrices and contingency tables (for binary data) can be
used to get an initial idea for relationships between explanatory variables
(for collinearity) or between an explanatory variable(s) and a response
variable(s) (correlation). In ANOVA, normality can be checked by box-plots. It
gives some indication of which variables should be in the model and which one
of them should be put into the model first, and whether linear relationship is
adequate.

**Exponential
distribution**:
The (continuous) distribution of time intervals between independent consecutive
random events like those in a Poisson process.

**Exponential
family**:
A family of probability distributions in the form

f(x) = exp {a(q )b(x) + c(q ) + d(x)}

where q is a parameter and a, b, c, d are
known functions. This family includes the **normal distribution**, binomial
distribution, Poisson distribution and gamma distribution as special cases.

**Exposure**: In an epidemiologic
study, exposure may represent an environmental exposure, an intervention or the
presence of a marker (biomarker/genetic marker).

**Factor**: A categorical
explanatory variable with small number of **levels** such that each item
classified belongs to exactly one level for that category. If the factor is
'sex', the levels are 'male' and 'female'; if the factor is 'drug received',
the levels are 'drug A', 'drug B', 'drug C', etc. A set of factor levels, uniquely
defining a single treatment, is called a **cell**. A cell may have just one
observation (no replication) or multiple observations (replications).

**Factorial
experiments**:
In some data, the explanatory (predictor) variables are all categorical (i.e., **factors**)
and the response variable is quantitative. When there are two or more
categorical predictor variables, the data are called **factorial**. The
different possible values of the factors are often assigned numerical values
known as **levels**.

**Factorial
analysis of variance**:
An analysis in which the treatments differ in terms of two or more factors
(with several levels) as opposed to treatments being different levels of a
single factor as in one-way ANOVA.

**False
discovery rate**:
**Link** to a calculator (for 2x2
tables)..

**F
distribution**:
A continuous probability distribution of the ratio of two independent random variables,
each having a Chi-squared distribution, divided by their respective degrees of
freedom. The commonest use is to assign *P* values to mean square ratios
(variance ratios) in ANOVA. In regression analysis, the F-test can be used to
test the joint significance of all variables of a
model. (**Tables of critical values of t, F and Chi-square**).

**Fisher's exact test**: An exact significance test to
analyze 2x2 tables for any sample size. It is a misconception that it is
suitable only for small sample sizes. This arises from the demanding
computational procedure for large samples, which is no longer an issue. It is
the only test for a 2x2 table when an expected number in any cell is smaller
than 5 (**Online Fisher's Test (1)**;
**(2)**;
**Calculator 3** in **Clinical
Research Calculators** at **Vassar**). For an exact test for larger contingency tables,
see **Vassar Online** or download **RxC**
by Mark Miller.

**F test**: The F test for
linear regression tests whether the slope is significantly different from 0,
which is equivalent to testing whether the fit using non-zero slope is
significantly better than the null model with 0 slope. See also **mean squares**.

**Gambler's
ruin**:
A classical topic in probability theory. It is a game of chance related to a
series of Bernoulli trials. There are variations of the game theory associated
with problems of the random walk and sequential sampling.

**Game
theory**:
The theory of contests between two or more players under specified sets of
rules. The statistical aspect is that the game proceeds under a chance scheme
such as throwing a die.

**Gaussian
distribution**:
Another name for the **normal distribution** (**GraphPad Gaussian Distribution Calculator**).

**G Statistics**: An application of the **log-likelihood
ratio statistics** for the hypothesis of independence in an *r* x *c*
contingency table. It can also be used to test goodness-of-fit. The G-test should
be preferred over Chi-squared test when for any cell in the table, ½ O-E½ > E.
The Chi-squared distribution is usually poor for the test statistics G^{2}
when N/rc is smaller than five (preferable to the Chi-squared test in
Hardy-Weinberg Equilibrium (HWE) test as long as this condition is met). **HyperStat** and **StatXact** perform
G statistics (**Online G
Statistics**).

**General
linear model**:
A group of linear regression models in which the response variable is
continuous and normally distributed, the response variable values are predicted
from a linear combination of predictor variables, and the linear combination of
values for the predictor variables is not transformed (i.e., there is no **link
function** as in **generalized linear models**). Linear multiple
regression is a typical example of general linear models whereas simple linear
regression is a special case of **generalized linear models** with the
identity link function.

**Generalized
linear model (GLM)**:
A model for linear and non-linear effects of continuous and categorical
predictor variables on a discrete or continuous but not necessarily normally
distributed dependent (outcome) variable. (Note that in the **general linear
model**, the dependent (outcome) variable should be normally distributed).
Normal, binary (or linear logistic; when the outcome variable is a proportion),
binomial or Poisson (when the outcome variable is a count), exponential and
gamma (when the outcome variable is continuous and non-negative) models are
different versions of generalized linear models. Particular types of models
arise by specifying an appropriate ** link function**, variance and
distribution. For example, normal linear regression corresponds to an identity
link function, constant variance and a normal distribution. Logistic regression
arises from a logit link function and a binomial distribution (the variance of
the response (npq) is related to its mean (np): variance = mean (1 - (mean/n)).
Loglinear models are used for binomial or Poisson counts. Standard techniques
for analyzing censored survival data, such as the

**Genetic distance**: A measurement of genetic
relatedness of populations. The estimate is based on the number of allelic
substitutions per locus that have occurred during the separate evolution of two
populations. Link to a lecture on **Estimating
Genetic Distance** and **GeneDist:
Online Calculator of Genetic Distance**. The
software **Arlequin**,
**PHYLIP**, **GDA**, **PopGene** and **SGS** are
suitable to calculate population-to-population genetic distance from allele
frequencies. See **Basic
Population Genetics**.

**Genetic
Distance Estimation by PHYLIP**: The most popular (and free) phylogenetics
program **PHYLIP** can be used to estimate genetic
distance between populations. Most components of PHYLIP can be run **online**. One component of the package **GENDIST** estimates genetic distance
from allele frequencies using one of the three methods: Nei's, Cavalli-Sforza's
or Reynold's (see papers by **Nei et al, 1983**, **Nei M, 1996** and a **lecture note** for more information on
these methods). GENDIST can be run **online** using the default options (**Nei's genetic distance**) to obtain
genetic distance matrix data. The PHYLIP program **CONTML** estimates phylogenies from
gene frequency data by maximum likelihood under a model in which all divergence
is due to genetic drift in the absence of new mutations (Cavalli-Sforza's
method) and draws a tree. The program comes as a freeware as part of PHYLIP or
this program can be run **online** with default options. If new
mutations are contributing to allele frequency changes, Nei's method should be
selected on GENDIST to estimate genetic distances first. Then a tree can be
obtained using one of the following components of PHYLIP: **NEIGHBOR** also draws a phylogenetic
tree using the genetic distance matrix data (from GENDIST). It uses either
Nei's "**Neighbor Joining Method**," or the
**UPGMA** (**u**nweighted **p**air **g**roup
**m**ethod with **a**rithmetic mean; average linkage clustering) method.
Neighbor Joining is a distance matrix method producing an unrooted tree without
the assumption of a clock (UPGMA does assume a clock). NEIGHBOR can be run **online**. Other components of PHYLIP
that draw phylogenetic trees from genetic distance matrix data are **FITCH** / **online** (does not assume evolutionary
clock) and **KITSCH** / **online** (assumes evolutionary clock).

**Geometric
mean**:
G = (*x*_{1}.*x*_{2}...*x*_{n})^{1/}^{n} where n is the
sample size. This can also be expressed as antilog ((1/n) S log *x*), which means the
antilog of the mean of the logs of each value.

**Half-normal
plot**:
A diagnostic test for model inadequacy or revealing the presence of outliers.
It compares the ordered residuals from the data to the expected values of
ordered observations from a normal distribution. While the full-normal plots
use the signed residuals, half-normal plots use the absolute values of the
residuals. Outliers appear at the top right of the plot as distinct points, and
departures from a straight line mean that the model is not satisfactory. It is
appropriate to use a half-normal plot only when the distribution is symmetrical
about zero because any information on symmetry will be lost.

**Haplotype
Relative Risk method**:
This method uses non-inherited parental haplotypes of affected persons as the
control group and thus eliminates the risks and bias associated with using
unrelated individuals as controls in case-control association studies, as well
as the higher cost (see **Falk & Rubinstein, 1987**; **Knapp, 1993**; **Terwilliger & Ott, 1992**).

**Hardy-Weinberg
equilibrium (HWE)**: In an infinitely large
population, gene and genotype frequencies remain stable as long as there is no
selection, mutation, or migration. For a bi-allelic locus where the gene
frequencies are p and q: p^{2}+2pq+q^{2 }= 1 (see **Hardy-Weinberg parabola**). HWE should be assessed
in controls in a case-control study and any deviation from HWE should alert for
genotyping errors (**Lewis, 2002**) unless there are biological
reasons for any deviation (see **Ineichen & Batschelet, 1975** for the
effect of natural selection on Hardy-Weinberg equilibrium). (__ Online HWE Analysis__;

**Harmonic
mean**:
Of a set of numbers (y_{1} to y_{n}), the harmonic mean is the
reciprocal of the arithmetic mean of the reciprocal of the numbers [H = N /
(1/(y_{1} + y_{2} + .... y_{n}))]. The harmonic mean is
either smaller than or equal to the arithmetic mean. It is a measure of
position.

**Hazard
function**
(instantaneous failure rate, conditional failure, intensity, or force of
mortality function): The function that describes the probability of failure
during a very small time increment (assuming that no failures have occurred
prior to that time). Hazard is the slope of the survival curve – a measure of
how rapidly subjects are having the event (dying, developing an outcome etc).

**Hazard Rate**: It is a time-to-failure function used in survival analysis.
It is defined as the probability per time unit that a case that has survived to
the beginning of the respective interval will fail in that interval.
Specifically, it is computed as the number of failures per time units in the
respective interval, divided by the average number of surviving cases at the
mid-point of the interval.

**Hazard
Ratio (Relative Hazard)**: Hazard ratio compares two groups differing in
treatments or prognostic variables etc. If the hazard ratio is 2.0, then the
rate of failure in one group is twice the rate in the other group. The
computation of the hazard ratio assumes that the ratio is consistent over time,
and that any differences are due to random sampling. Before performing any tests of hypotheses to compare survival curves,
the **proportionality of hazards assumption**
should be checked (and should hold for the validity of **Cox's proportional hazard
models**). (See also **Log-rank test**).

**Hetereoscedastic
data**:
Data that have non-constant (heterogeneous) variance across the predicted
values of y. In this case, residual graph will show varying variability across
the fitted values. This is a regression diagnostic problem and should be fixed
by transforming the data. See also **Homoscedasticity**.

**Heuristics**: A term in computer
science that refers to guesses made by a program to obtain approximately
accurate results. Frequently used in phylogenetics and computational biology.

**Hierarchical
model**:
In linear modeling, models which always include all the lower-order
interactions and main effects corresponding to any interaction they include.

**Historical
fallacy**:
The mistake of assuming that an association observed in **cross-sectional data**
will be similar to that observed in longitudinal data or vice versa.

**Homoscedasticity
(homogeneity of variance):** Normal-theory-based tests for the equality of population
means such as the t-test and analysis of variance, assume that the data come
from populations that have the same variance, even if the test rejects the null
hypothesis of equality of population means. If this assumption of **homogeneity
of variance** is not met, the statistical test results may not be valid. **Heteroscedasticity**
refers to lack of homogeneity of variances.

**Hotelling's
T ^{2} test**:
This is a generalization of Student's t-test for multivariate data. Designed to
provide a global significance test for the difference between two groups with
simultaneously measured multiple dependent/outcome variables and multiple
explanatory/independent variables. It can also be used for one group with
simultaneously measured multiple dependent outcome variables (another test
similar to Hotelling's T

**Hypergeometric
distribution**:
A probability distribution of a discrete variable generally associated with
sampling from a finite population without replacement. An example may be that
given a lot with 25 good units and five faulty. The probability that a sample
of five will yield not more than one faulty item follows a hypergeometric
distribution.

**Index
plot**:
An index plot plots each **residual**, **leverage**, or **Cook's distance**
against the corresponding observation (row) number (*i* or index) in the
dataset. In many cases, the row number corresponds to the order in which the
data were collected. If this is the case, this would be similar to plotting the
residuals (or another diagnostic quantity) against time. The index plot is a
helpful diagnostic test for normal linear and particularly generalized linear
models. Both outliers and influential points can be detected by the index plot.
It is particularly useful when the data is in time order so that pattern in the
residuals, etc. over time can be detected. If a residual index plot is showing
a trend in time, then they are not independent (violation of a major assumption
of linear regression).

**Inferential
statistics**:
Making inferences about the population from which a sample has been drawn and
analyzed.

**Influential
points**:
Observations that actually dominate a **regression analysis** (due to high **leverage**,
high **residuals** or their combination). The method of ordinary **least
squares** gives equal weight to every observation. However, every observation
does not have equal impact on the **least squares** results. The slope, for
example, is influenced most by the observations having values of the
independent variable farthest from the mean. An observation is influential if
deleting it from the dataset would lead to a substantial change in the fit of
the **generalized linear model**. High-leverage points have the potential to
dominate a regression analysis but not necessarily exert an influence (i.e., a
point may have high leverage but low influence as measured by **Cook
statistics**). Cook statistics is used to determine the influence of a data
point on the model.

**Interaction**: If the effect of
one factor depends on the level of another factor, the two factors involved are
said to interact, and a contrast involving all these levels is called their
interaction. Factors A and B interact if the effect of factor A is not
independent of the level of factor B. For example, when there are two main
effects on a response variable, if their combined effect is higher than the sum
of their main effects due to a bonus (say, the effects of a kind of exercise
and a kind of diet on blood lipid levels), they have an interaction (meaning a
simple additive model is not sufficient to account for the observed data and a
multiplicative term must be added). Briefly, interaction is a deviation from
additivity. Also, there would be an interaction between the factors sex and
treatment if the effect of treatment was not the same for males and females in
a drug trial. Interaction is closely linked with **effect modification** in
epidemiology (see **Genetic Epidemiology Glossary**; **Wikipedia**: **Statistical Interaction**).

**Intercept**: In linear
regression, the intercept is the mean value of the response variable when the
explanatory variable takes the value of zero (the value of y when x=0).

**Interpolation**: Making deductions
from a model for values that lie between data points. Making deductions for
values beyond the data points is called extrapolation and the results are not
valid.

**Interquartile
range (dQ)**:
dQ is a measure of spread and is the counterpart of the standard deviation for
skewed distributions. dQ is the distance between the upper and lower quartiles
(Q_{U}-Q_{L}).

**Interval
variable**
(equivalent to **continuous variable**): A quantitative variable measured on
a scale with constant intervals (like days, milliliters, kilograms, miles so
that equal-sized differences on different parts of the scale are equivalent)
where the zero point and unit of measurement are arbitrary. When temperature is
measured on two scales, Fahrenheit and Centigrade, the zero points in these two
scales do not correspond, and a 10% increase in Fahrenheit (from 50^{0}
to 55^{0}) is not a 10% increase in the corresponding Centigrade scale
(10^{o} to 12.8^{o} = 2.8%); these two measurements cannot be
mixed or compared. For estimation of correlation coefficients, data should be
interval type (See also **ratio variable **and **variable**).

**Kolmogorov-Smirnov
two-sample test**:
A non-parametric test applicable to continuous frequency distributions. It is
considered to be the equivalent of the C^{2}-test for quantitative data and has greater power than
the G-statistics or C^{2}-test for goodness of
fit especially when the sample size is small. It can be used to compare two independent
groups. The test is based on differences between two cumulative relative
frequency distributions (it compares the distributions not the parameters).
Thus, the **Kolmogorov-Smirnov** test is also sensitive to differences in
the general shapes of the distributions in the two samples such as differences
in dispersion, skewness. Its interpretation is similar to that of the Wald-Wolfowitz runs test. **Online
Kolmogorov-Smirnov “One-Sample” Test** at **Vassar**.

**Kruskal-Wallis test **(One-way ANOVA by ranks): It is
one of the non-parametric tests equivalent to one-way **ANOVA** that are
used to compare *multiple* (k > 2) *independent* samples. This
test assesses the hypothesis that the different samples in the comparison were
drawn from the same distribution or from distributions with the same median. It
can be used to analyze ordinal variables. It is an extension of the **Mann-Whitney
(U) test **(for two independent samples). The interpretation of the
Kruskal-Wallis test is identical to that of one-way ANOVA, but is based on
ranks rather than means (**Online K-W test**; if the link does not work:
go to **Stats ToolBoxs Homepage** and choose
Kruskal-Wallis OWAV from Frequency Tables).

**Kurtosis**: Kurtosis
is a measure of whether the data are peaked or flat in its distribution
relative to a normal distribution (whose kurtosis is zero). Positive kurtosis
indicates a ‘peaked’ distribution and negative kurtosis indicates a ‘flat’
distribution (data sets with high kurtosis have a distinct peak near the mean
and decline rapidly; data sets with low kurtosis tend to have a flat top near
the mean rather than a sharp peak) (**Definition of Kurtosis and Skewness**;** Online Skewness-Kurtosis Calculator**). See also **skewness**.

**Large sample effect**: In large samples, even small or
trivial differences can become statistically significant. This should be
distinguished from biological/clinical importance.

**Least
squares method**:
A method of fitting a straight line or curve based one minimization of the sum
of squared differences (residuals) between the predicted and the observed
points. Given the data points (x_{i}, y_{i}), it is possible to fit a straight line
using a formula, which gives the y=a+bx. The gradient of the straight line b is
given by [S(x_{i} - m_{x})(y_{i}-m_{y})] / [(S(x-m_{x}))^{2}], where m_{x} and m_{y} are the means for x_{i} and y_{i}. The intercept a is
obtained by m_{y} - bm_{x}. See **Wikipedia**: **Least Squares**.

**Leverage
points**:**
**In regression analysis, these are the observations that have an extreme
value on one or more explanatory variable. The leverage values indicate whether
or not X values for a given observation are outlying (far from the main body of
the data). A high leverage value indicates that the particular observation is
distant from the centre of the X observations. High-leverage points have the
potential to dominate a regression analysis but not necessarily influential. If
the residual of the same data point and **Cook's distance** are also high,
then it is an influential point. See also **influential points**.

**Likelihood**: The probability of
a set of observations given the value of some parameter or set of parameters.
For example, the likelihood of a random sample of n observations (x_{1}
to x_{n}) with probability distribution f(x; q ) is given by: L = P f(x_{i}; q _{0}). This function, which applies
equally to continuous density and discrete mass functions, is the basis of
maximum likelihood estimation.

**Likelihood
distance test (likelihood residual, deletion residual)**: This test is based
on the change in deviance when one observation is excluded from the dataset. It
uses the difference between the log-likelihood of the complete dataset and the
log-likelihood when a particular observation is removed. A relatively large
difference indicates that the observation involved is an outlier (poorly fitted
by the model).

**Likelihood
ratio test**:
A general purpose test of hypothesis H_{o} against an alternative H_{1}
based on the ratio of two likelihood functions one derived from each of H_{o}
and H_{1.} The statistics l
is given by l = -2 *ln* (L_{H0
}/ L_{H1}) has approximately a C^{2} distribution with df equal to the difference in the number of parameters in the two hypotheses. One application
of this test is the **G-test**, which is used in categorical data analysis
as a goodness-of-fit or independence test (the tests statistics has a
Chi-squared distribution).

**Linear expression**: A **polynomial** expression
with the degree of **polynomial** being 1. It will be something like,
f(x)=2x^{1}+3, but not x^{2}+2x+4.

**Linear
logistic model**:
A linear logistic model assumes that for each possible set of values for the
independent (X) variables, there is a probability p that an event (success)
occurs. Then the model is that Y is a linear combination of the values of the X
variables: Y = b_{o} + b_{1}*X_{1} + b_{2}*X_{2} + b_{3}*X_{3} + … b_{k}*X_{k}, where Y is the
logit transformation of the probability p. Logistic in statistical usage stems
from logit and has nothing to do with the military use of the word which means
the provision of material.

**Linear
regression models**:
In the context of linear statistical modeling, 'linear' means linear in the
parameters (coefficients), not the explanatory variables. The explanatory
variables can be transformed (say, x^{2}), but the model will still be
linear if the coefficients remain linear. When the overall function (Y) remains
a sum of terms that are each an X variable multiplied by a coefficient, the
function Y is said to be linear in the coefficients.
A non-linear model is different in that it has a non-constant slope (a tutorial
on **Simple Linear Regression**; see
also **Vassar College**; Excel macro for **Linear
Correlation & Regression**).

**Linkage
disequilibrium**:
Also called gametic association, which is more appropriate. It means increased
probability for two or more alleles to be on the same chromosome at the
population level. In a population at equilibrium, haplotype frequency is
obtained by multiplying the allele frequencies x2.
When there is linkage disequilibrium, the observed frequency (say by family
analysis or sperm typing) is different from the expected frequency. The
difference gives the D value (D for difference), which can be tested for significant
difference from 0 by a 2x2 table analysis (for two alleles). The D value
can be negative or positive. Linkage disequilibrium can derive from population
admixture, tight linkage or elapse of insufficient time for the population to
reach equilibrium. A classic example in immunogenetics is the HLA-A1-B8-DR3
haplotype which shows significant linkage disequilibrium extending over 6.5 Mb.
Software for LD estimation: **Genetic Data Analysis**,
**EH**,** ****2LD****, MLD, **

**Link
function**:
A particular function of the expected value of the response variable used in modeling
as a linear combination of the explanatory variables. For example, logistic
regression uses a *logit* link function rather than the raw expected
values of the response variable; and Poisson regression uses the *log*
link function. The response variable values are predicted from a linear
combination of explanatory variables, which are connected to the response
variable via one of these link functions. In the case of the general linear
model for a single response variable (a special case of the generalized linear
model), the response variable has a normal distribution and the link function
is a simple identity function (i.e., the linear combination of values for the
predictor variable(s) is not transformed).

**LOD
Score**:
Stands for the logarithm of odds. It is a statistical measure of the likelihood
that two genetic markers occur together on the same chromosome and are
inherited as a single unit of DNA (co-segregation). The LOD score serves as a
test of the null hypothesis of free recombination versus the alternative
hypothesis of linkage. Determination of LOD scores requires pedigree analysis
and a score of >3 is traditionally taken as evidence for linkage. Linkage is
between two genetic loci but not alleles. An example is the linkage between the
hemochromatosis gene (HFE) and HLA-A. This means that within the same family
all affected subjects will have the same HLA-A allele but not necessarily a
particular one, i.e., there will be no recombination between HFE and HLA-A. LOD
score has nothing to do with **linkage disequilibrium**.

**Log 0** is undefined. If we
need to use a log transformation but some data values are 0, the usual way to
get round this problem is to add a small positive quantity (such as 1/2) to all
the values before taking logs.

**Log-rank
test (of equality across strata)**: A non-parametric test of significance for
the difference between two survival curves (for categorical data). It is a
special application of the **Mantel-Haenszel test**. It can be adjusted for
confounders (not preferable to **Cox proportional hazard regression** which
is a semi-parametric model), or performed for trend (for details and parametric
alternatives, see **Lee & Go, 1997**). It was developed by Mantel and Haenszel as an
adaptation of the **Mantel-Haenszel ****C**^{2}** test**. The other commonly used nonparametric
tests for comparison of two survival distributions are the Gehan's Generalized
Wilcoxon test (**Gehan** **1965a** & **1965b**). The log-rank
test is appropriate for survival distributions whose hazard functions are
proportional over time, i.e. the two survival curves do not cross (**proportionality assumption**). Otherwise, the Gehan's Wilcoxon test is recommended. The
Wilcoxon statistic puts more weight on early deaths compared to the log-rank (**Lee & Go, 1997**).
See **UCLA Stata**:** Survival Analysis & Log-rank Test**; **BMJ Statistics Notes**: **Logrank Test**.

**Log
transformation**:
This transformation pulls smaller data values apart and brings the larger data
values together and closer to the smaller data values (shrinkage effect). Thus,
it is mostly used to shrink highly positively skewed data.

**Logistic
(binary) regression**:
A statistical analysis most frequently models the relationship between a
dichotomous (binary) outcome variable (such as diseased or healthy; dead or
alive; relapsed or not relapsed), and a set of explanatory variables of any
kind (such as age, HLA type, blood pressure, kind of treatment, disease stage
etc). It can also be used when the outcome variable is polytomous (several
categories of the prognosis; including ordinal response 'ordinal logistic
regression' or 'proportional odds ratio model'), and when there are several
outcome variables (multinomial logistic regression - a special class of
loglinear models). Analysis of data from case-control studies via logistic
regression can proceed in the same way as cohort
studies. See **Logistic Regression Lecture Note**, **Online
Logistic Regression**,** Logistic Regression with SPSS,** **STATA** and **SAS**; **Power Calculation for Logistic Regression (including
Interaction)**.

**Logit
transformation**:
The logit (or logistic) transformation Y of a probability p of an event is the
logarithm of the ratio between the probability that the event occurs (p) and
that the event does not occur (1-p): Y = ln (p/(1-p)). Thus it is a transformation
of a binary (dichotomous) response variable. The logit transformation of p is
also known as the *log odds* of p, since it is the logarithm of the **odds**.
There are other link functions for binary response variables.

**Loglinear
model**:
Multinomial data (from contingency tables) can be fitted by using a generalized
linear model with a Poisson response distribution and a log link function. The
resulting models for counts in the cells of a contingency table are known as
loglinear models in which the logarithm of the expected value of a count
variable is modeled as a linear function of parameters. Loglinear models try to
model all important associations among all variables. In this respect, they are
related to **ANOVA** models for quantitative data. Loglinear modeling allows
more than two discrete variables to be analyzed and interactions between any
combination of them to be identified. Associations between variables in
log-linear models are analogous to interactions in **ANOVA** models. The aim
is to find a small model, which achieves a reasonable fit (small residual).
Data sets with a binary response (outcome) variable and a set of explanatory
variables that are all categorical can be modeled either by **logistic
regression** or by loglinear modeling. More on **Loglinear Models **and **Online
Loglinear Test** at **Vassar College**.

In
loglinear modeling, the counts in the cells of the contingency table are
treated as values of the response variable rather than either of the
categorical variables defining the rows and columns. A loglinear model relates
the distribution of the counts to the values of the explanatory variables (rows
and columns), and tests for the presence of an interaction between them.
Whether the interaction term is necessary is tested by fitting two models, one
without and one with the interaction term, and using the difference between the
regression deviance values for the two models (as well as the difference in
df). If the SP obtained by the C^{2}-test is small, the
interaction term should be in the model. This corresponds to a significant
difference in Fisher's exact test or the C^{2}-test. For a 2x2 table, the difference between the
regression deviances in the two models is the same as the residual deviance in
the model with no interaction term. In loglinear modeling, it makes no sense if
any main effect is omitted.

In loglinear modeling (as in other GLMs), the saturated model, which includes all interactions, fits the data exactly, and the fitted values are exactly equal to the observed values (the residual deviance is zero). The residual deviance of any other model can be used to test how worse it is compared to the saturated model. A small SP arising from a high residual deviance would mean that it does not fit better than the saturated model (the alternative model is rejected). If the associated SP is not small, then the model is not significantly worse than the saturated model (i.e., the exact fit).

An important constraint on the choice of models to fit, i.e., which terms and which interactions to include, is that whether any marginal totals (row or column) for combinations of terms are fixed in advance. If this applies to any combination of terms, their interaction must be included (the main effects and this particular interaction make up the null model for such data). When only the overall sample size is fixed in advance, there is no constraint on the models that can be fit.

When there are more than two variables, whether any interaction can be omitted from the model can be tested by fitting a model containing all the interactions of a particular order (say, second-order interactions). Then, new models omitting each interaction are fitted. Each new model's regression deviance is compared with the regression deviance of the model containing all interactions. If the resulting difference in regression deviances (and df) results in a large SP, that interaction can be omitted (it is not significantly different but has fewer terms; or it does not significantly contribute to the model).

The loglinear model can be used to find a conditional probability involving the factors in the contingency table for one of the factors chosen as a response variable. When the factor chosen as a response variable is binomial, logistic regression can be used to analyze the same data (a binomial response variable and categorical explanatory variables). Logistic regression substitutes the loglinear model with equal results as long as the fitted loglinear model includes the main effects of the response variable and a saturated model for all the explanatory variables.

**Longitudinal
data**:
Data collected over a period of time as in cohort studies. These data are
usually analyzed by using **survival analysis** techniques. See **Longitudinal Studies Chapter** in **Epidemiology
for the Uninitiated**.

**Mann-Whitney
(U) test**:
A non-parametric test for comparing the distribution of a continuous variable
between two *independent* groups. It is analogous to the independent
two-sample t-test, so that it can be used when the data are not normally
distributed. The Wilcoxon signed ranks T-test for independent samples is
another non-parametric alternative to the t-test in
this context (for *paired* samples, **Wilcoxon matched pairs signed rank
test** should be used) (online **Mann-Whitney
Test**). See also **Non-parametric tests on two independent
samples in XLStat
and IBM-SPSS**.

**Mantel-Haenszel
****C**^{2}** test** (also called Cochran-Mantel-Haenszel
(CMH) test): Test for a null hypothesis of no overall relationship in a series
of 2x2 tables for stratified data derived either from a cohort or a
case-control study. It allows analysis of **confounding** and gives an **adjusted
odds ratio** or relative risk. It can be used on categorical or categorized
continuous data. The test is only valid when the variance of observed data is ³ 5. It is inappropriate when the
association changes dramatically across strata (heterogeneity is usually tested
by Breslow-Day test). It is, however, applicable for sparse data sets for which
asymptotic theory does not hold for G^{2}. The test statistics, M^{2},
has approximately a Chi-squared distribution with df = 1 (see a review by **Stefano & Ezio, 2007**; online **Mantel-Haenszel Test** for a single table). Mantel & Haenszel’s 1959 JNCI
paper is now a **citation classic**.

**Markov Chain Monte Carlo** (**MCMC, random walk Monte
Carlo) methods**: See **Wikipedia**: **MCMC**; **MCMC Applet**;
**Markov Chain Simulation Applets**; **Buffon's
Needle Applet**; **Monte
Carlo Methods Links**; **MCMC Tests of Genetic Equilibrium (Lazzeroni & Lange,
1997)**; **Markov Chain Monte Carlo in Practice (Book) **and**
Virtual Labs:
Markov Chain** (requires **MathPlayer**).
See also **Metropolis-Hastings algorithms**.

**Maximum
likelihood**:
This method is a general method of finding estimated values of parameters. It yields
values for the unknown parameters, which maximize the probability of obtaining
the observed values. The estimation process involves considering the observed
data values as constants and the parameter to be estimated as a variable, and
then using differentiation to find the value of the parameter that maximizes
the likelihood function. First a likelihood function is set up which expresses
the probability of the observed data as a function of the unknown parameters.
The maximum likelihood estimators of these parameters are chosen to be those
values, which maximize this function. The resulting estimators are those, which
agree most closely with the observed data. This method works best for large
samples, where it tends to produce estimators with the smallest possible
variance. The maximum likelihood estimators are often
biased in small samples (see **maximum
likelihood**). Another method for point estimation is the **method of
moments**.

**McNemar's
test**:
A special form of the Chi-squared test used in the analysis of paired (not
independent) proportions. This non-parametric test compares two correlated
dichotomous responses and finds its most frequent use in situations where the
same sample is used to find out the agreement (concordance) of two diagnostic
tests or difference (discordance) between two treatments. If the pairs of data
points are the measurements on two matched people (such as affected and
unaffected siblings) in a case-control study, or two measurements on the same
person, the appropriate test for equality of proportions is the McNemar's test.
It can be used to assess the outcome of two treatments applied to the same
individuals or the significance of the agreement between two detection methods
of a physical sign. If there are more than two periods of
data collection (such as pretest, posttest and follow-up), **Cochran's Q test**
should be used (**Online
McNemar's Test (1)** **(2)
(3) (4)**.** **

**Mean** (or average): A
measure of location for a batch of data values; the sum of all data values
divided by the number of elements in the distribution. Its accompanying measure
of spread is usually the **standard deviation**. Unlike the **median**
and the **mode**, it is not appropriate to use the
mean to characterize a skewed distribution (see also **standard deviation**)
(**Online Calculator for Mean**).

**Mean
squares**:
A sum of squares divided by its associated df is a mean square. In **ANOVA**,
the regression (explained) mean square is **ESS**/k-1, and the residual
(error) mean square is **RSS**/N-k. Note that the mean squares are not
additive, i.e., they do not add up to **TSS**/N-1. Importantly, the residual
(error) mean square is an unbiased estimator of the variance (s^{2}) in ANOVA. The
regression (explained) mean square equals to the variance only when the slope
(b) is zero. Their ratio (mean square ratio = regression MS / residual MS) is
therefore provides a test for the null hypothesis that b = 0. Large F values
support the alternative hypothesis that the slope is not zero. This is the
basis of the **F test** in **ANOVA**.

**Measurement
type****:**** **The data may be measured in the following
scales: nominal, ordinal, interval or ratio scales (known as Stevens'
typology). The scale of the measurements may be (other than nominal scale
measurements) either continuous or discrete, and either bounded or unbounded.

**Measures of association**: These measures include
the **Phi coefficient of association**, **Cramer's contingency coefficient
(C) and V**, Kendall's tau-b and (Stuart's) tau-c, Somers' D (a modification
of Kendall's tau-b), Yule's Q, gamma, Spearman's rank correlation coefficient
(rho), Pearson's correlation coefficient, lambda (symmetric and asymmetric),
uncertainty coefficients (symmetric and asymmetric), Guttman's coefficient of
predictability (lambda)/Goodman-Kruskal's lambda, Goodman-Kruskal's gamma and
Goodman-Kruskal's tau (concentration coefficient). See also **Measures of Effect-Size & Association**; **Measures of Association for Cross-tabulations**;
Measures on **SAS**, **STATA**, **SYSTAT**;
**Ennis, 2001**; **Morton,
2001**.

**Measures
of central tendency**:
These are parameters that characterize an entire distribution. These include **mode**,
**median** and **mean**.

**Median**: Another measure of
location just like the **mean**. The value that divides the frequency
distribution in half when all data values are listed in order. It is
insensitive to small numbers of extreme scores in a distribution. Therefore, it
is the preferred measure of central tendency for a skewed distribution (in
which the mean would be biased) and is usually paired with the **interquartile
range (dQ)** as the accompanying measure of spread. See Martin Bland's page
for calculation of **confidence intervals for a median**.

**Median
test**:
This is a crude version of the **Kruskal-Wallis ANOVA **in that it assesses
the difference in samples in terms of a contingency table. The number of cases
in each sample that fall above or below the common median is counted and the
Chi-square value for the resulting 2xk samples contingency table is calculated.
Under the null hypothesis (all samples come from populations with identical
medians), approximately 50% of all cases in each sample are expected to fall
above (or below) the common median. The median test is particularly useful when
the scale contains artificial limits, and many cases fall at either extreme of
the scale. In this case, the median test is the most appropriate method for comparing samples (**Online Median Test**).

**Meta-analysis**: A systematic approach yielding an overall
answer by analyzing a set of studies that address a related question. This
approach is best suited to questions, which remain unanswered after a series of
studies. Meta-analysis provides a weighted average of the measure of effect
(such as odds ratio). The rationale is to increase the power by analyzing the
sets of data. The selection of studies to include in a meta-analysis study is
the main problem with this approach. **Funnel Plot** is an informal method to assess
the effect of publication bias in this context. See also **Introduction to Meta-Analysis** by the
Cochrane Collaboration; **Meta-Analysis in Epidemiology** by Stroup et
al (2000); **Methods for Meta-Analysis in Medical Research**** **by AJ Sutton; **Introduction to Meta-Analysis** by
Borenstein et al (2009), and **Online Meta-Analysis Tests**.

**Metropolis-Hastings
algorithms**:
These algorithms are a class of Markov chains which are commonly used to
perform large scale calculations and simulations in Physics and Statistics. See
**Metropolis-Hastings Applet**. See **Markov
Chain Monte Carlo methods**.

**Mode**: The observed value that occurs with the greatest frequency. The
mode is *not* influenced by small numbers of extreme values.

**Model building**: The traditional approach to statistical
model building is to find the most parsimonious model that still explains the
data. The more variables included in a model (overfitting), the more likely it
becomes mathematically unstable, the greater the estimated standard errors
become, and the more dependent the model becomes on the observed data. Choosing
the most adequate and minimal number of explanatory variables helps to find out
the main sources of influence on the response variable, and increases the
predictive ability of the model. Ideally, there should be more than 10
observations for each variable in the model. The usual procedures used in
variable selection in regression analysis are: univariate analysis of each
variable (using C^{2} test), stepwise method (backward or forward elimination of
variables; using the deviance difference), and best subsets selection. Once the
essential main effects are chosen, interactions should be considered next. As
in all model building situations in biostatistics, biological considerations
should play a role in variable selection.

**Monte Carlo trial**: Studying a complex relationship
difficult to solve by mathematical analysis by means of computer simulations.
An online book on **Resampling
Statistics**, and software (**CLUMP**, downloadable from **clump22.zip**) to do Monte Carlo statistics
for case-control association studies.

**Multicolinearity**: In multiple
regression, two or more X variables are colinear if they show strong linear
relationships. This makes estimation of regression coefficients impossible. It
can also produce unexpectedly large estimated standard errors for the
coefficients of the X variables involved. This is why an exploratory analysis
of the data should be first done to see if any collinearity among explanatory
variables exists. Multicolinearity is suggested by non-significant results in
individual tests on the regression coefficients for important explanatory
(predictor) variables. Multicolinearity may make the determination of the main
predictor variable having an effect on the outcome difficult.

**Multiple regression**: To quantify the
relationship between several independent (explanatory) variables and a
dependent (outcome) variable. The coefficients (a, b_{1} to b_{i}) are estimated by
the **least squares** method, which is equivalent to **maximum likelihood**
estimation. A multiple regression model is built upon three major assumptions:

1. The response variable is normally distributed,

2. The residual variance does not vary for small and large fitted values (constant variance),

3. The observations (explanatory variables) are independent.

Multiple regression is the prototype for **general
linear models** because the response variable should be normally distributed
and there is no **link function**, whereas, simple linear regression is a special
case for **generalized linear models**. The extension of multiple regression
to multivariate data analysis is called canonical correlation (**Online Multiple Regression**; **Reference Guide on Multiple Regression**).

**Multiple
regression correlation coefficient (R ^{2} - R-squared)**: R

**Multiplication
rule**:
The probability of two or more statistically independent events all occurring
is equal to the product of their individual probabilities. If the probability
of having trisomy 21 is *a*, and trisomy 6 is *b*, assuming
independence of these two events, for a baby the probability of having both
trisomies is (*a* x *b*). One of the most critical errors of judgment
in the use of independence assumption relates to a court case in the UK (**Watkins, 2000**). (See also **addition
rule**.)

**Multivariable
analysis**:
As opposed to univariable analysis, statistical analysis performed in the
presence of more than one explanatory variable to determine the relative
contributions of each to a single event is (or should be) called multivariable
analysis (in practice, however, it is called univariate and multivariate
analysis more frequently). It is a method to simultaneously assess
contributions of multiple variables or adjust for the effects of confounders.
Multiple linear regression, multiple logistic regression, proportional hazards
analysis are examples of multivariable analysis, which has no similarity whatsoever to **multivariate analysis** (see also
**Peter TJ, 2009**). See a review on **Multivariable Methods by MH Katz**
(book on **Multivariable Analysis by MH Katz**).

**Multivariate
analysis**:
Methods to deal with more than one related 'outcome/dependent variable' (like
two outcome measures from the same individual) simultaneously with adjustment for
multiple confounding variables (covariates). When there is more than one
dependent variable, it is inappropriate to do a series of univariate tests. **Hotelling's
T ^{2}** test is used when there are two groups (like cases and controls)
with multiple dependent measures (may be more than two), and multivariate
analysis of variance (

**Multivariate analysis of variance (MANOVA)**: An extension of **Hotelling's
T ^{2}** test to more than two groups with related 'multiple' outcome measures.
Groups are compared on all variables simultaneously as a global test (rather
than one-by-one as ANOVA does). See also

**Natural
(raw) residuals**:
The difference between the observed (Y* _{i}*) and fitted values (Y

**Negative
predictive value**: Probability
of a true negative as in a person identified healthy by a test is really free
from the disease (see also **positive predictive value)**.

**Nested
model**:
Models that are related where one model is an extension of the other.

**Nominal
variable**:
A qualitative variable defined by mutually exclusive unordered categories such
as blood groups, races, sex etc. (see also **ordinal variable**).

**Nonlinear
Regression**:
Regression analysis in which the fitted (predicted) value of the response
variable is a nonlinear function of one or more X variables. **GraphPad Guide to Nonlinear Regression**, **Introduction to Nonlinear Regression**,
**A GraphPad Practical Guide to Curve Fitting**.

**Nonparametric
methods**
(distribution free methods): Statistical methods to analyze data from
populations, which do not assume a particular population distribution. **Mann-Whitney
U test**, **Kruskal-Wallis test** and **Wilcoxon's (T) test** are
examples. Such tests do not assume a distribution of the data specified by
certain parameters (such as mean or variance). For example, one of the
assumptions of the Student's t-test and ANOVA is normal distribution of the
data. If this is not valid, a non-parametric equivalent must be used. If a
wrong choice of test has been made, it does not matter very much if the sample
size is large (a non-parametric test can be used where a parametric test might
have been used but a parametric test can only be used when the assumptions are
met). For a small sample size, non-parametric tests tend to give a larger *P*
value. In general, parametric tests are more robust, more complicated to compute and have greater power efficiency. Parametric
tests compare parameters such as the mean in t-test and variance in F-test as
opposed to non-parametric tests that compare distributions. Nonparametric
methods are most appropriate when the sample sizes are small. In large (e.g., *n*
> 100) data sets, it makes little sense to use nonparametric statistics
(reviews of non-parametric tests by **Whitley & Ball, 2002** and **Bewick, 2004**; a tutorial on **Parametric
vs Nonparametric Methods**; **Review
of Nonparametric Tests** in **Intuitive
Biostatistics; Nonparametric Tests for Ordinal Data at Vassar**).

**Normal distribution** (Gaussian distribution) is a
model for values on a continuous scale. A normal distribution can be completely
described by two parameters: mean (m) and variance (s^{2}). It is
shown as C ~ N(m, s^{2}). The
distribution is symmetrical with mean, mode, and median all equal at m. In the
special case of m = 1 and s^{2} = 1, it is called the standard normal distribution. See **Normal
Distribution (1)**, **(2)**,** (3) **& **(4)**.

**Normal
probability plot of the residuals**: A diagnostic test for the assumption of
normal distribution of residuals in linear regression models. Each residual is
plotted against its expected value under normality. A plot that is nearly
linear suggests normal distribution of the residuals. A plot that obviously
departs from linearity suggests that the error distribution is not normal.

**Null
model**:
A model in which all parameters except the intercept are 0. It is also called
the intercept-only model. The null model in linear regression is that the slope
is 0, so that the predicted value of Y is the mean of Y for all values of X.
The F test for the linear regression tests whether the slope is significantly
different from 0, which is equivalent to testing whether the fit using non-zero
slope is significantly better than the null model with 0 slope.

**Number needed to treat**: The reciprocal of the reduction in
absolute risk between treated and control groups in a clinical trial. It is
interpreted as the number of patients who need to be treated to prevent one
adverse event. See also **Number Needed to Treat (NNT) Guide by Bandolier (1)
(2)**; **Interpreting
Diagnostic Tests;** **NNT Calculator**; **GraphPad
NNT Calculator**;** EpiMax Table Calculator**;** Evidence-based
Medicine Toolbox** and articles
by **Cook & Sackett, 1995**; **Wu, 2002**; **Barratt,
2004**.

**Odds**: The odds of a
success is defined as the ratio of the probability of a success to the
probability of a failure (p/(1-p)). If a team has a probability of 0.6 of
winning the championship, the odds for winning is 0.6/(1-0.6) = 3:2. Similarly,
the odd in a case-control study is the frequency of the presence of the marker
divided by the frequency of absence of the marker (in cases or controls
separately). The link function logit (or logistic) is the log_{e} of
the odds.

**Odds
multiplier**:
In logistic regression, b = log_{e}
(odds ratio), thus exp b = odds ratio. For a
continuous (explanatory) variable, exp b
is called the odds multiplier and corresponds to the odds ratio for a unit
increase in the explanatory variable. The odds multiplier of the coefficient is
the odds ratio for its level relative to the reference level. If x increases
from a to b by c, the odds multiplier becomes exp (cb). The resulting value shows the
proportional change in the odds associated with x = b relative to x = a. It
follows that for binary variables where x can only get values of 0 and 1, exp b = odds ratio.

**Odds
ratio (OR)**:
Also known as relative odds and approximate relative risk. It is the ratio of
the odds of the risk factor in a diseased group and in a non-diseased (control)
group (the ratio of the frequency of presence / absence of the marker in cases
to the frequency of presence / absence of the marker in controls). The
interpretation of the OR is that the risk factor increases the odds of the
disease ‘OR’ times. OR is used in retrospective case-control studies (**relative
risk** (RR) is the ratio of proportions in two groups which can be estimated
in a prospective -cohort- study). These two and **relative hazard** (or **hazard
ratio**) are measures of the strength/magnitude of an association. As opposed
to the *P* value, these do not change with the sample size. OR and RR are
considered interchangeable when certain assumptions are met, especially for
large samples and rare diseases. Odds ratio is calculated as ad/bc where
a,b,c,d are the entries in a 2x2 contingency table (hence the alternative
definition as the cross-product ratio). In logistic regression, the coefficient
b corresponds to the log_{e} of the odds ratio. There are
statistical methods to test the homogeneity of odds ratios (**Online
Odds-Ratio & 95% CI Calculation**; **Odds
Ratio-Relative Risk Calculation **(**Calculator
3**) in **Clinical Research Calculators** at **Vassar**).

**Offset**: A fixed, already
known regression coefficient included in a **generalized linear model** (which
does not have to be estimated).

**Omnibus test**: If the chi-square
test has more than one degree of freedom (larger than 2x2 table), it is called
an ‘omnibus’ test, which evaluates the significance of an overall hypothesis
containing multiple sub-hypotheses (these multiple sub-hypotheses then need to
be tested using follow up tests).

**One-way
ANOVA**:
A comparison of several groups of observations, all of which are independent
and subjected to different levels of a single treatment (such as cells exposed
to different dosage of a growth factor). It may be that different groups were
exposed to the same treatment (different cell types exposed to a new agent).
The main interest focuses on the differences among the means of groups.

**Ordinal
variable**:
An ordered (ranked) **qualitative/categorical** variable. The degree of HLA
matching (one, two, three or four antigen matching in two loci) in transplant
pairs, or HLA sharing in parents (one-to-four shared antigens) are ordinal
variables although the increments do not have to be equal in magnitude (see **interval**
and **ratio variables**). An ordinal variable may be a categorized
quantitative variable. When two groups are compared for an ordinal variable, it
is inappropriate to use ordinary Chi-squared test but the **trend test** or
its equivalents must be used.

**Outcome
(response, dependent) variable**: The observed variable, which is shown on y
axis. A statistical model shows this as a function of predictor variable(s).

**Outlier**: An extreme observations
that is well separated from the remainder of the data. In regression analysis,
not all outlying values will have an influence on the fitted function. Those
outlying with regard to their X values (high **leverage**), and those with Y
values that are not consistent with the regression relation for the other
values (high **residual**) are expected to be influential. The test the
influence of such values, the **Cook statistics** is used.

**Overdispersion**: Dispersion is a
measure of the extent to which data are spread about an average. Overdispersion
is the situation that occurs most frequently in Poisson and binomial regression
when variance is much higher than the mean (normally, it should be similar). It
is evident with a high (>2) residual mean deviance (which should normally be
around one) and the presence of too many outliers. The reasons for
overdispersion may be outliers, misspecification of the model, variation
between the response probabilities and correlation between the binary
responses. It distorts standard error and confidence interval estimations. In
the analysis, overdispersion may be taken into account by estimating a
dispersion parameter.

**Overfitting**: In a **multivariable
model**, having more variables than can be justified from sample size. The
statistical rule of thumb is to have at least ten subjects for each variable
investigated.

**Overmatching**: When cases and
controls are matched by an unnecessary non-confounding variable, this is called
overmatching and causes underestimation of an association. For example,
matching for a variable strongly correlated with the exposure but not with the
outcome will cause loss of efficiency. Another kind of overmatching is matching
for a variable which is in the causal chain or closely linked to a factor in
the causal chain. This will also obscure evidence for a true association.
Finally, if a too strict matching scheme is used, it will be very hard to find
controls for the study. See **BMJ Statistics Notes**: **Matching**.

**Parameter**: A numerical
characteristic of a population specifying a distribution model (such as normal or
Poisson distribution). This may be the mean, variance, degrees of freedom, the
probability of a success in a binomial distribution, etc.

**Parsimonious**: The simplest
plausible model with the fewest possible number of variables.

**Pearson's
correlation coefficient (r)**: A measure of the strength of the 'linear' relationship
between two quantitative variables. A major assumption is the normal
distribution of variables. If this assumption is invalid (for example, due to
outliers), the non-parametric equivalent **Spearman's rank correlation**
should be used. The** r** represents C^{2} obtained from the 2x2 table, corrected for the
total sample size. It can then be calculated as ±(C^{2}/N)^{1/2}.
This formula is equivalent to covariance divided by the product of the standard
deviations of the two variables. The **correlation coefficient**, r, can
take any value between -1 and +1; 0 meaning no "linear" relationship
(there may still be a strong non-linear relationship). It is the absolute value
of r showing the strength of relationship. An associated *P* value can be
computed for the statistical significance (a small *P*
value does not necessarily mean a strong relationship). The square of the r is
r^{2} (r-squared or **coefficient of determination**) which corresponds
to the variance explained by the correlated variable (see **GraphPad Guide to Correlation Parameters and Interpretation of r**). R^{2} is also
used in regression analysis (see **multiple regression correlation coefficient**;
**Online Correlation & Regression Calculators** at **Vassar College**; **Excel Macro for Linear Correlation & Regression)**.

**Pharmacoepidemiology**: Application of epidemiological
reasoning, methods and knowledge to the study of the uses and effects
(beneficial and adverse) of drugs in human populations. A relatively new field
in epidemiology becoming more closely related to pharmacogenetics. See a review
on statistical analysis of pharmacoepidemiological case-control studies (**Ashby, 1998**).

**Phi coefficient**: A measure of association of two
variables calculated from a contingency table as (X^{2} / N) ^{1/2}.
Its value varies between 0 (no association) and 1 (strongest association) for
2x2 tables where it is an accurate statistics (for larger tables, **Cramer’s V**
is more accurate). In a way, it is a corrected Chi-squared value for the number
of observations. See **Calculator 3** in **Clinical
Research Calculators** at **Vassar**.

**Poisson distribution**: The probability distribution of
the number of (rare) occurrences of some random event in an interval of time or
space. Poisson distribution is used to represent distribution of counts like
number of defects in a piece of material, customer arrivals, insurance claims,
incoming telephone calls, or alpha particles emitted. A transformation that
often changes Poisson data approximately normal is the square root. See** ****Poisson
Distribution**** (QuickTime)**; **GraphPad Poisson Probability Calculator**.

**Poisson
regression**:
Analysis of the relationship between an observed count with a Poisson
distribution (i.e., outcome variable) and a set of explanatory variables. In
general it is appropriate to fit a Poisson model to the data if the sample size
is > 100 and the mean for the occurrence of the event is <0.10xN.

**Polynomial**: A sum of multiples
of integer powers of a variable. The highest power in the expression (n) is the
degree of the polynomial. When n=1, for example, f(x)=2x^{1}+3, this is
a linear expression. If n=2, it is quadratic (for example, x^{2} + 2x +
4); if n=3, it is cubic, if n=4, it is quartic and if n=5, it is quintic.

**Polytomous
variable**:
A variable with more than two levels. If there are two levels it is called
dichotomous (as in the most common form of **logistic regression**).

**Population**: The population is
the universe of all the objects from which a sample could be drawn for an
experiment.

**Population attributable risk**: The proportion of a
disease in a specified population attributable to a specific factor (such as a
genetic risk factor).

**Population stratification (substructure)**: An^{ }example
of 'confounding by ethnicity' in which the co-existence of different disease
rates and allele frequencies within population sub-sections lead to an
association between the two at a whole population level. Differing allele
frequencies in ethnically different strata in a single population may lead to a
spurious association or mask an association by artificially modifying allele
frequencies in cases and controls when there is no real association (for this
to happen, the subpopulations should differ not only in allele frequencies but
also in baseline risk to the disease being studied). Case-control association
studies can still be conducted by using genomic
controls (**Devlin, 1999**; **Pritchard, 1999**) even when population
stratification is present. The software **STRUCTURE
& STRAT** or **ADMIXMAP** can be used to analyze
case-control data with genomic control. See **Cardon
& Palmer, 2003 **for an example of spurious association
due to population stratification. See also **Genetic Epidemiology**.

**Positive
predictive value**:
Probability of a true positive as in a person identified as diseased by a test
is really diseased (see also **negative predictive value**).

**Post
hoc test**:
A test following another one. The most common example is performing multiple
comparisons between groups when the overall comparison between groups shows a
significant difference. For example, when an **ANOVA** analysis yields a
small *P* value, *post hoc* tests (such as Newman-Keuls, Duncan's or **Dunnett's**
tests) are done to narrow down exactly which pairs differ most significantly
(similarly, **Dunn's test** is done in a non-parametric **ANOVA**
setting) (**GraphPad
Post ANOVA Test Calculator**). In genetic association
studies, multiple comparisons are justified only when performed as a *post
hoc* test following a significant deviation in overall gene/marker
frequencies (see **HLA and Disease Association
Studies**).

**Power of a statistical test**: See **Statistical Power**.

**Predictor (explanatory, independent) variable**: The
variable already in hand
in the beginning of an experiment or observation and whose effect on an outcome
variable is being modeled.

**Predictive
value**:
The probability that a person with a positive test if affected by the disease
(positive predictive value) or the probability that a person with a negative
test does not have the disease (negative predictive value). Estimation requires
sensitivity, specificity and disease prevalence.

**Prevented
fraction**:
The amount of a health problem that actually has been prevented by a prevention
strategy in real world.

**Probability**: The ratio of the number
of likely outcomes to the number of possible outcomes.

**Probability
density function**:
When a curve is used to model the variation in a population and the total area
between the curve and the x-axis is 1, then the function that defines the curve
is a probability density function.

**Probability
distribution function**:
A function which gives for each number x, the probability that the value of a
continuous random variable X is less than or equal to x. For discrete random
variables, the probability distribution function is given as the probability
associated with each possible discrete value of the variable.

**Probability
vector**:
Any vector with non-negative entries whose sum is equal to 1.0. See **Wikipedia**.

**Proportional
odds ratio**:
When the response variable in an ordered/ordinal logistic regression model has
more than two ordered response categories, odds ratio obtained for each
category is called a proportional odds ratio. See **UCLA Stata**:
**Ordinal Logistic Regression**; **Lecture note on Logistic Regression**.

*P*** value (SP =
significance probability)**: The *P* value gives the probability that the null
hypothesis is correct; therefore, if it is a small value (like <0.05), null hypothesis
is rejected. More technically, it is the probability of the observed data or
more extreme outcome would have occurred by chance, i.e., departure from the
null hypothesis when the null hypothesis is true. In a genetic association
study, the *P* value represents the probability of error in accepting the
alternative hypothesis (or rejecting the null hypothesis) for the presence of
an association. For example, the *P *level of 0.05 (i.e., 1/20) indicates
that assuming there was no relation between those variables whatsoever (the
null hypothesis is correct), and we were repeating experiments like ours one
after another, we could expect that approximately in every 20 replications of
the experiment, there would be one in which the relation between the variables in question would be equal to or more extreme
than what has been found. In the interpretation of a *P* value, it is
important to know the accompanying measure of association and the
biological/clinical significance of the significant difference. However small,
a *P* value does not indicate the size of an effect (odds ratio/relative
risk/hazard ratio do). A *P* value >0.05 does not necessarily mean lack
of association. It does so only if there is enough power to detect an
association. Most statistical nonsignificance is due to lack of power to detect
an association (poor experimental design). Both 'p' and 'P' are used to
indicate significance probability but the international standard is *P*
(capital italic). See **Interpreting Statistical P Values**;

**Qualitative**: Qualitative (**categorical**)
variables define different categories or classes of an attribute. Examples are
gender, blood groups or disease states. A qualitative (categorical) variable
may be **nominal** or **ordinal**. When there are only two categories, it
is termed binary (like sex, dead or alive).

**Quantitative**: Quantitative
variables are variables for which a numeric value representing an amount is
measured. They may be discrete (for example, taking values of integers) or
continuous (such as weight, height, blood pressure). If a quantitative variable
is categorized, it becomes an **ordinal variable**.

**R ^{2}
(R-squared)**:
See

**Random sampling**: A method of
selecting a sample from a target population or study base using simple or
systematic random methods. In random sampling, each subject in the target
population has equal chance of being selected to the sample. Sampling is a
crucially important point in selection of controls for a case-control study. By
randomization, systematic effects are turned into error (term), and there is an
expected balancing out effect: known and unknown factors that might influence
the outcome are assigned equally to the comparison groups. One disadvantage of
randomization is generation of a potentially large error term. This can be
avoided by using a **block design**. See **Wikipedia**:
**Random Sampling**.

**Randomized
(complete) block design**: An experimental design in which the treatments in each
block are assigned to the experimental units in random order. Blocks are all of
the same size and each treatment appears in the same number of times within each
block (usually once). A different level of the factor is assigned to each
member of the block randomly. The data can be analyzed using the paired t-test
(when there are two units per block) or by randomized block ANOVA (in blocks of
any size). The results are substantially more precise than a completely
randomized design of comparable size. In studies with a block design, more
assumptions are required for the model: no interactions between treatments and
blocks, and constant variance from block to block.

**Ratio
variable**:
A quantitative variable that has a zero point as its origin (like 0 cm = 0
inch) so that ratios between values are meaningfully defined. Unlike the **interval
variables**, which do not have a true zero point, the ratio of any two values
in the scales is independent of the unit of measurement. For example, 2/12 cm
has the same ratio as the corresponding values in inch (but the same cannot be
said for 2/12 Celsius and 2/12 Fahrenheit which are interval variables).

**Receiver
operating characteristics (ROC) curve analysis**: Also called
discrimination statistics. See **ROC** in **Clinical Research Calculators** and **Difference
Between the Areas Under Two ROC Curves** at **Vassar, ROC101** by Tom Fawcett; **ROC Analysis by Obuchowski NA, 2005**; **Cook NR, 2007**. See also the **Supplementary Data File** for **Mamtani,
2006** for the use of **Stata in
ROC analysis**.

**Regression
diagnostics**:
Tests to identify the main problem areas in regression analysis: normality,
common variance and independence of the error terms; outliers, influential data
points, collinearity, independent variables being subject to error, and
inadequate specification of the functional form of the model. The purpose of
the diagnostic techniques is to identify weaknesses in the regression model or
the data. Remedial measures, correction of errors in the data, elimination of
true outliers, collection of better data, or improvement of the model, will
allow greater confidence in the final product. See also **error terms**, **residuals**
(including **likelihood distance test**), **leverages** and **Cook
statistics**.

**Regression
modeling**:
Formulating a mathematical model of the relationship between a response
(outcome, dependent) variable, Y, and a set of explanatory (predictor,
independent, regressor) variables, x. Depending on the characteristics of the
variables, the choice of model can be simple linear regression, multiple
regression, logistic (binary) regression, Poisson regression, etc. In any
regression problem, the key quantity is the mean value of the outcome variable,
given the value of the independent variable(s). This quantity is called the
conditional mean and expressed as "E (Y½x)"
where Y is the response (outcome), x is the explanatory (predictor) variable.
The question is whether the variable(s) in question tells us more about the
outcome variable than a model that does not include that variable. In other
words, whether the coefficient of the variable(s) is zero and the outcome is
equal to a constant (which is the mean for Y) or not. The aim of model building
is to arrive at a meaningful (say, biologically relevant) and parsimonious
model that explains the data. The model may be linear if the parameters are
linear, or nonparametric if the parameters are not linear. No matter how strong
is the statistical relationship between x and Y, no
cause-and-effect pattern is necessarily implied by the regression models. See **Regression Applet**.

**Regression towards the mean**: See the explanation and a **simulation
**at** Rice Virtual Lab in Statistics**; and a **Lecture Note**.

**Relative
risk (RR)**:
Also known as **risk ratio**. The RR shows how many times more or less the
individuals with the risk factor are likely to get the disease relative to
those who do not have the risk factor. RR gives the strength of association in
prospective cohort studies. It cannot be estimated in retrospective
case-control studies, and its use to describe the cross-product ratio (as
frequently done in HLA association studies) is inappropriate. See **Calculator
3** in **Clinical Research Calculators** at **Vassar**. See also **odds
ratio**.

**Repeated
measures design**:
In this design, the same experimental unit is subjected to the different
treatments under consideration at different points in time. Each unit,
therefore, serves as a block. If for example, two different treatments and
placebo treatment are applied to the same patient sequentially, this is a
repeated measures design. See also **cross-over design**.

**Resampling
statistics**:
Data-based simulation procedures that sample (with replacement) repeatedly from
observed data to generate empirical estimates of results that would be expected
by chance. Examples include **bootstrapping** and permutation tests. See
also **Online Resampling Book**.

**Residuals**: Residuals reflect
the overall badness-of-fit of the model. They are the differences between the
observed values of the outcome variable and the corresponding fitted values
predicted by the regression line (the vertical distance between the observed
values and the fitted line). In a regression analysis, a large residual for a
data point indicates that the data point concerned is not close to its fitted
value. If there are too many large (standardized) residuals either the model
fitted is not adequate or there is **overdispersion** of the data. Ideally,
the residuals should have constant variance along the line. This can be checked
by a normal probability plot of the residuals. In the plot of residuals against
the explanatory variable (or the fitted values), there should not be any
pattern if the assumption of constant variation is met, i.e., residuals do not
tend to get larger as the variable values get larger or smaller (see also **likelihood
distance test**).

**Residual
plot**:
A graph that plots residuals against fitted values. It is used to check equal
variance assumption of the error terms in linear regression. Residual analysis
for logistic regression is more difficult than for general linear regression
models because the responses Y* _{i}* can take only the values 0
and 1. Consequently, the residuals will not be normally distributed. Plots of
residuals against fitted values or explanatory variables will be uninformative.
Residual plots are generally unhelpful for

**Residual
(error) sum of squares (RSS)**: The measure of within treatment groups sum of
squares (variability) in ANOVA. It is the deviation around the fitted
regression line. The sum of squared differences between each observed Y value
(Y* _{i}*) and the fitted Y value (Y

**Regression
(explained) sum of squares (ESS)**: The measure of between treatment groups sum
of squares (variability) in ANOVA. It is the deviation of fitted regression
value around mean. The sum of squared differences between each fitted Y value
(Y* _{i}* -hat) and the overall mean of the Y values equals to the
explained (regression) sum of squares. The sum of ESS and RSS gives the total
sum of squares

**Risk
ratio (relative risk)**:
The risk ratio is the percentage difference in classification between two
groups obtained as the ratio of two risks or proportions. For example, the
proportion of people recovering after stroke with one treatment equals 0.10,
while the proportion after a different treatment equals 0.16. The risk ratio
equals 0.625 (0.10/0.16); 37.5% ((1-0.625)*100 or (0.16-0.10)/0.16) fewer
patients treated by the first method recover. The risk ratio takes on values
between zero ('0') and infinity. One ('1') is the neutral value and means that
there is no difference between the groups compared. See also **relative risk**.

**Robustness**: A statistical test
or procedure is robust when violation of assumptions has little effect on the
results. Student's t-test, for example, is robust against departures from
normality.

**R
project for statistical computing**: R is a language
and environment for statistical computing and graphics which can be seen as a
different implementation of the S language. R and a comprehensive set of
programs written for a variety of statistical analysis are all available as
Free Software. See the **R Project Website** & **List
of Contributed R Packages**.

**Sample
size determination**:
Mathematical process of deciding how many subjects should be studied (at the
planning phase of a study). Among the factors to consider are the incidence or
prevalence of the condition, the magnitude of difference expected between cases
and controls, the power that is desired and the allowable magnitude of type I error (pre-determined significance probability). **Sample size calculator****, ****sample size calculation**.

**SAS **(Statistical Analysis System): A comprehensive computer software system
for data processing and analysis. It can be used for almost any type of
statistical analysis. Produced by **SAS Institute**. See **SAS Learning
Resources (UCLA)**; **SAS Tutorials**; **Getting Started with SAS Enterprise Guide (Free Online
Course)**; **SAS e-Learning**; **SAS Genetic Software** and **Genetic Data Analysis**.

**Saturated model**: A model
that contains as many parameters as there are data points. This model contains
all main effects and all possible interactions between factors. For categorical
data, this model contains the same number of parameters as cells and results in
a perfect fit for a data set. The (residual) deviance is a measure of the
extent to which a particular model differs from the saturated model.

**Scales of measurement**: The type of data is always one
of the following four scales of measurement: nominal, ordinal, interval, or
ratio. Each of these can be discrete or continuous.

**Schoenfeld residual test**: One of the diagnostic tests to
check the proportionality assumption (covariates are time independent) in
proportional hazard modeling. A variation is the use of scaled Schoenfeld
residuals (see **Tests of Proportionality in SAS, Stata, R and SPLUS**).

**Sensitivity**: Sensitivity is the proportion of true
positives that are correctly identified by a diagnostic test. Those that
produce few false negatives have higher sensitivity. See also **specificity**
(**Sensitivity and Specificity by Altman & Bland. BMJ 1994**),
**Interpreting
Diagnostic Tests** and **DAG-STAT**.

**Sign test**: A test based on the probabilities of
different outcomes for any number of pluses and minuses, i.e., observations
below or above a prespecified value. The sign test can be used to investigate
the significance of the difference between a population median and a specified
value for it, or between the observed sex/transmission ratio and the 50:50
expected value. It can also be used for paired data. This time, the differences
between the pairs will be either negative or positive, and the smaller of the
two total negatives or positives plus the total number of pairs will form the
test statistics. For example, when the total number is 20, if the number for
the less frequent sign is 5 or smaller, *P* < 0.05 (two-tailed). A sign
test in disguise is **McNemar's test**, which is used for paired data for
dichotomous response.

**Simple
linear regression model**: The linear regression model for a normally distributed
outcome (response) variable and a single predictor (explanatory) variable. The
straight line models the mean value of the response variable for each value of
the explanatory variable. The major assumption is constant variation of
residuals along the fitted line which points out that the model is equally good
across all x values. The null hypothesis stating that the explanatory variable
has no effect on the response (in other words, the slope of the fitted line is
zero) can be tested statistically. The two main aims of regression analysis are
to predict the response and to understand the relationships between variables. As
in all linear models, the error term (shown as W_{i} or e_{i}) is additive (as
opposed to multiplicative, i.e., y_{i} = a + bx_{i} + e_{i}) and independent,
and they are assumed to have a normal distribution. As an exception, the simple
linear regression is a special case for **generalized linear models**.

**Skewness**: The degree of (lack
of) asymmetry about a central value of a distribution. A distribution with many
small values and few large values is positively (right) skewed (long tail in
the distribution curve or stemplot is to the right); the opposite (left tail)
is negatively (left) skewed. The measures of location median, midinterquartile
range (midQ) and midrange decrease in this order for a left-skewed
distribution. (**Definition of Kurtosis and Skewness**; **Online Skewness-Kurtosis Calculator**; see also **kurtosis**).

**Sparseness**: A contingency table
is sparse when many cells have small values. When N is the total sample size,
and r and c are the number of rows and columns, N / rc is an index of
sparseness. Smaller values refer to more sparse tables. Sparse tables often
contain zero values (empty cell).

**Spearman's
rank correlation**:
A non-parametric **correlation coefficient** **(rho)** that is calculated by computing the **Pearson's
correlation coefficient** **(r)** for the association between the ranks
given to the values of the variables involved. It is used for ordinal data and
interval/ratio data. It is * not* appropriate to take the square of
Spearmen’s correlation coefficient rho to obtain

**Specificity**: Specificity is the proportion of true
negatives that are correctly identified by the test. Those that produce few
false positives have higher specificity. See also **sensitivity** (**Sensitivity and Specificity by Altman & Bland, BMJ 1994**),
**Interpreting
Diagnostic Tests** and **DAG-STAT**.

**Square
root transformation**:
Usually used for highly positively skewed data, but especially in transforming
Poisson counts to normality.

**Standard
deviation**:
Like **variance**, the standard deviation (SD) is a measure of spread
(scatter) of a set of data. Unlike variance, which is expressed in squared
units of measurement, the SD is expressed in the same units as the measurements
of the original data. It is calculated from the deviations between each data
value and the sample mean. It is the square root of the variance. For different
purposes, n (the total number of values) or n-1 may be used in computing the
variance/SD. If you have a SD calculated by dividing by n and want to convert
it to a SD corresponding to a denominator of n-1, multiply the result by the
square root of n/(n-1). If a distribution's SD is greater than its **mean**,
the mean is inadequate as a representative measure of central tendency. For
normally distributed data values, approximately 68% of the distribution falls
within ± 1 SD of the mean,
95% of the distribution falls within ±
2 SDs of the mean, and 99.7% of the distribution falls within ± 3 SDs of the mean (empirical rule).
SD should not be confused with the **standard error of the mean (SEM)**,
which is the SD of the sampling distribution of a statistics and quantifies how
accurately the mean is known (See **Normal Distribution**; **Online
Calculator for Standard Deviation**).

**Standard
error**:
The standard error (SE) or as commonly called the standard error of the mean
(SEM) is a measure of the extent to which the sample mean deviates from the
true but unknown population mean. It is the **standard deviation** (SD) of
the random sampling distribution of means (i.e., means of multiple samples from
the same population). As such, it measures the precision of the statistic as an
estimate of a population. The (estimated) SE/SEM is dependent on the sample
size. It is inversely related to the square root of the sample size:

(estimated)
SE = SD / (N)^{1/2}

The true value of the SE can only be calculated if the SD of the population is known. When the sample SD is used (as almost always), it is an estimate and should be called estimated standard error (ESE). When the sample size is relatively large (N ³ 100), the sample SD provides a reliable estimate of the SE.

**Standard
residual**:
The standardized **residual** value (observed minus predicted divided by the
square root of the residual **mean square**).

**Stata**: A powerful
statistical package particularly useful for epidemiologic and longitudinal data
management and analysis. It is mainly a command driven program produced by **Stata
Corporation**. See the list of **Stata Capabilities**, **Stata Starter Kit** with **Learning Modules** by **UCLA**; **Tutorial by University of Essex**; **Tutorial by Princeton University**; **Stata Highlights by Notre Dame University**;** Tutorial by Carolina Population Center**; **Stata Refresher by Syracuse University**; **Genetic Data
Analysis on Stata** and **Stata
Programs for Genetic Epidemiologists**.

**Statistical Power**: The probability that a test
will produce a significant difference at a given significance level is called the
power of the test. This is equal to the probability of rejecting the null
hypothesis when it is untrue, i.e., making the correct decision. It is 1 minus
the probability of a type II error. The true differences between the populations
compared, the sample size and the significance level chosen affect the power of
a statistical test. Ideally, power should be at least 0.80 to detect a
reasonable departure from the null hypothesis. See power calculators: **PS: Power and Sample Size Calculation**; **G*Power 3 (User Guide)**;
**General Statistical Calculators Including a Power
Calculator (UCLA)**; **Russ
Lenth's Power Calculator (JAVA-based)**; **Statistical Power Calculator for Frequencies**;** Retrospective Power Calculation**; **General
Power Calculator**; **Power Calculation for Logistic Regression (including
Interaction)**; **Genetic
Power Calculator**; **Power for
Genetic Association Analyses (PGA) @ NCI**; **Quanto** (sample size and power
calculation for association studies of genes, gene-environment or gene-gene
interactions).

**Stepwise
regression model**:
A method in multiple regression studies aimed to find the best model. This
method seeks a model that balances a relatively small number of variables with
a good fit to the data by seeking a model with high R^{2}_{a}
(the most parsimonious model with the highest percentage accounted for). The
stepwise regression can be started from a null or a full model and can go
forward or backward, respectively. At any step in the procedure, the
statistically most important variable will be the one that produces the
greatest change in the log-likelihood relative to a model lacking the variable.
This would be the variable, which would result in the largest likelihood ratio
statistics, *G* (a high percentage accounted for gives an indication that
the model fits well). See also **multiple regression correlation coefficient -
R ^{2}**.

**Stochastic
model**:
A probability model that includes chance events in the form of random measurement
error or uncertainty. In a deterministic model, however, random error is
inconsequential or nonexistent. See **Wikipedia**:
**Stochastic Modeling**.

**Stratum** (plural strata):
When data are stratified according to its characteristics, each subgroup is a
stratum.

**Student's
t-test**:
A parametric test for the significance between means (**two-samples t-test**) or between a mean and a hypothesized value (**one-sample
t-test**). One assumption is that the observations must be normally
distributed, and the ratio of variances in two samples should not be more than
three. If the assumptions are not met, there are non-parametric equivalents of
the t-test to use (see for example, **Wilcoxon's Test**). It is
inappropriate to use the t-test for multiple comparisons as a ** post hoc
test**. The t-test for independent samples tests whether or not two means are
significantly different from each other but only if they were the only two
samples taken (

**Subgroup analysis**: Analysis of subgroups of a
sample either because of a prior hypothesis (gender or age-specific
effect/association) or as a fishing expedition / data dredging. This practice
increases type I error rates. See a commentary by **Sleight,
2000**.

**Survival Analysis**: See **Superlectures** on ‘Survival Analysis’; **A Primer on Survival Analysis (J Nephrol 2004)**;
**Tutorials on Survival Analysis in Br J Cancer 2003: Part I - II - III - IV**; ‘**Understanding Survival Curves**’ at NMDP
website; **Survival Curves** and **Comparing
Survival Curves** in **Intuitive
Biostatistics**; **BMJ Statistics Notes**: **Time to Event (Survival) Data -** **Survival Probabilities (the Kaplan-Meier method)
- Logrank Test**; Survival
Analysis by **STATA** and **SAS**; **Power Calculator for Survival Outcomes**, **PS: Power and Sample Size Calculation**. **Online
survival analysis** at **Vassar College**. For a comprehensive review, see **Lee & Go, 1997**.

**Survival function**: A time to
failure function that gives the probability that an individual survives past a
time point (does not experience an event like death, metastasis, conception
etc)). Where the event is death, the value of the survival function at time T
is the probability that a subject will die at some time greater than T. The
survival function always has a value between 0 and 1 and is nonincreasing.

**Synergism**: A joint effect of
two treatments being greater than the sum of their effects when administered
separately (positive synergism) or the opposite (negative synergism).

**Theta (****q****)**: Used to denote
recombination fraction (in statistical genetics).

**Transformations
(ladder of powers)**:
Transformation deals with non-normality of the data points and non-homogeneous
variance. The power transformations form the following ladder: ..., *x*^{-2},
*x*^{-1}, *x*^{-1/2}, log *x*, *x*^{1/2}
**;** *x*^{1}, *x*^{2}, *x*^{3},
..... Provided *x* > 1, powers below 1 (such as *x*^{1/2}
or log *x*) reduce the high values relative to the low values as in positively
skewed data, whereas, powers above 1 (such as *x*^{2}) have the
opposite effect of stretching out high values relative to low ones, as in
negatively skewed data. All power transformations are monotonic when applied to
positive data (they are either increasing or decreasing, but not first
increasing and then decreasing, or vice versa). The **square root
transformation** often renders Poisson data approximately normal.

**Transmission
Disequilibrium Test**
(TDT): A family-based study to compare the proportion of alleles transmitted
(or inherited) from a heterozygous parent to a disease-affected child. Any
significant deviation from 0.50 in transmission ratio implies an association (**Spielman, 1993** & **1994**).

**Treatment**: In experiments, a
treatment is what is administered to experimental units (explanatory
variables). It does not have to be a medical treatment. Fertilizers in
agricultural experiments; different books and multimedia methods in teaching;
and chemotherapy of bone marrow transplantation in the treatment of leukemia
are examples of treatments in regression analysis.

**Trend
test for counts and proportions**: A special application of the Chi-squared
test (with a different formula) for ordinal data tabulated as a 2xk table. It
should be used when the intention is not just to compare the differences
between the two groups but to see whether there is a consistent trend towards
decrease or increase in the difference between the groups. An example is the
association of parental HLA sharing (sharing one-to-four antigens in two loci)
with fetal loss in a case-control study (those with recurrent miscarriages and
normal fertile couples). In genetic studies, the additive model analysis is
done by trend test for counts in cases and controls (2 columns) of three
genotypes (3 rows): wildtype homozygous (0 variant allele), heterozygote (1
variant allele) and variant homozygote (2 variant alleles). A frequent
application is the analysis of dose-response relationships in toxicology and
pharmacology. The Chi-squared test for trend has one degree of freedom. The
associated *P* value obtained by the Chi-squared for trend test (1 df) is
always smaller than the corresponding *P* value of an ordinary Chi-squared
test (2 df) for departure if indeed there is a trend. The
trend test for counts and proportions is called Cochrane-Armitage trend test.
Alternative tests for the analysis of trend are **Wilcoxon-Mann-Whitney test **or
the t-test with use of ordered scores, and the **Jonckheere-Terpstra test** as
a non-parametric test for ordered data (see **Trend for Binomial Outcome** in the manual of
Epi Info;** ****Epi Info Freeware for Trend Test** (Trend Test in
StatCalc or **Open-Epi Online**); **InStat**
fully functional demo version for Trend Test; **Trend Tests in Stata**).

**t-statistics**: Defined as
difference of sample means divided by standard error of difference of sample
means (see **Student's t-test**).

**Two-way
ANOVA**:
This method studies the effects of two factors (with several levels) separately
(main effect) and, if desired, their effect in combination (interaction).

**Type I
error**:
If the null hypothesis is true but we reject it this is an error of first kind
or type I error (also called a
error). This results in a false positive finding.

**Type II
error**:
If the null hypothesis is accepted when it is in fact wrong, this is an error
of the second kind or type II error (also called b error). This results in a false negative result.

**Unreplicated
factorial**:
A single replicate of a 2^{k} design (where each of k factors of
interest has only two levels).

**Variable**: Some characteristic
that varies among experimental units (subjects) or from time to time. A
variable may be **quantitative** or **categorical**. A quantitative
variable is either **discrete** (assigning meaningful numerical values to
observations: number of children, dosage in mg) or **continuous** (such as
height, weight, temperature, blood pressure; also called **interval variable**).
A categorical variable is either **nominal** (assigning observations to
categories: gender, treatment, disease subtype, groups) or **ordinal**
(ranked variables: low, median, high dosage). Conventionally, a random variable
is shown by a capital letter, and the data values it takes by lower case
letters.

**Variance**: The major measure
of variability for a data set. To calculate the variance, all data values,
their mean, and the number of data values are required. It is expressed in the
squared unit of measurement. Its square root is the **standard deviation**.
It is symbolized by s^{2} for a population and
*S*^{2} for a sample (**Online
Calculator for Variance and Other Descriptive Statistics**).

**Variance
ratio**:
Mean square ratio obtained by dividing the mean square (regression) by mean
square (residual). The variance ratio is assessed by the F-test using the two
degrees of freedom (k-1, N-k).

**Wald
test**:
A test for the statistical significance of a regression coefficient. It is
obtained by comparing the maximum likelihood estimate of the slope parameter
(expected b_{1}) to an estimate of
its standard error. The resulting ratio (*W*), under the hypothesis that b_{1} = 0, will follow a
standard normal distribution. The two-tailed *P* value will be found from
the Z table corresponding to *P* ( ç
Z ç > *W*). It
is not more reliable than the **likelihood ratio test** (**deviance
difference**).

**Welch-Satterthwaite
t-test:**
The Welch-Satterthwaite t-test is an alternative to the pooled-variance t-test,
and is used when the assumption that the two populations have equal variances
seems unreasonable. It provides a t statistic that asymptotically approaches a
t-distribution, allowing for an approximate t-test to be calculated when the
population variances are not equal (**Online
Welch's Unpaired t-test**; **t-Test
Assuming Unequal Sample Variances** at **Vassar**).

**Wilcoxon matched pairs signed rank T-test**: A
non-parametric significance test analogous to paired t-test. Most suitable for

**William's correction **(for** G statistics**)**:**
This is equivalent to Yates' continuity correction for Chi-squared test but used
in likelihood ratio (G) statistics for 2x2 tables (**Online G
Statistics**** **with William's correction).

**Woolf-Haldane analysis**: A method first described by
Woolf and later modified by Haldane for the analysis of 2x2 table and relative
incidence (**relative risk**) calculation. It is the preferred method for
relative risk calculation when one of the cells has a zero using the formula:
RR = (2a+1)(2d+1) / (2b+1)(2c+1). Since it is a modification of the
cross-product ratio, it should be called **odds ratio**. For details and
references, see **Statistical Analysis in HLA and
Disease Association Studies**.

**Yates's
correction**:
The approximation of the Chi-square statistic in small 2x2 tables can be
improved by reducing the absolute value of differences between expected and
observed frequencies by 0.5 before squaring. This correction, which makes the
estimation more conservative, is usually applied when the table contains only
small observed frequencies (<20). The effect of this correction is to bring
the distribution based on discontinuous frequencies nearer to the continuous
Chi-squared distribution. This correction is best suited to the contingency tables
with fixed marginal totals. Its use in other types of contingency tables (for
independence and homogeneity) results in very conservative significance
probabilities. This correction is no longer needed since exact tests are
available.

**Z score**: The Z score or
value expresses the number of standard errors by which a sample mean lies above
or below the true population mean. Z scores are standardized for a distribution
with mean = 0 and standard deviation = 1 (**Corresponding P values for Z**;

**Major
Resources in Biostatistics**

**Armitage P & Colton T. ****Encyclopedia of
Biostatistics**. **Volumes 1-8. John Wiley & Sons,
2005**

**Bland M. ****An Introduction to Medical Statistics****. 3rd
Edition. Oxford Medical Publications, 2000**

**Campbell MJ & Machin D. ****Medical Statistics: A Common Sense Approach****.
Wiley, 2002**

**Daly LE & Bourke GJ. ****Interpretation
and Uses of Medical Statistics****. 5th
Edition. Blackwell Scientific Publications, 2000**

**Motulsky H. ****Intuitive Statistics****. OUP,
1995**

**Norman GR & Streiner DL. ****PDQ Statistics****. BC
Decker, 1997**

**Rosner B. ****Fundamentals
of Biostatistics****. 5th Edition. Duxbury Press,
1999**

**Sokal RR, Rohlf FJ. ****Biometry****. 3rd Edition.
WH Freeman & Company, 1994**

**Zar JH. ****Biostatistical
Analysis****. 4th Edition. Prentice Hall,
1998**

**Elston, Olson & Palmer. ****Biostatistical Genetics and Genetic Epidemiology****.
Wiley, 2002**

**Internet
Links**

**Basic Biostatistics Concepts and Tools **

**JHSPH Open Courses: Statistics for Laboratory Scientists I & II **

**Medical
Statistics Series (14 articles) by Ball, Bewick & Cheek in Critical Care,
2001-2005 **

**Extensive Epidemiology
and Biostatistics Links** ** Understanding the Fundamentals of Epidemiology**

**Epidemiology & BioStatistics Super Lectures
Epidemiology-ResearchEasy **

**Centre for Evidence-based Medicine **

**Clinical Epidemiology & Evidence-Based Medicine
Glossary (1)** **(2)** **Evidence-based
Practice**

**Glossary of Statistical Terms (Berkeley) **

**A
Glossary for Multilevel Analysis **

**Epidemiology - Biostatistics Board Review**

**Online
Handbook of Biological Statistics (PDF) Essential Statistics in Biology**

** Commonly Used Statistical Tests Interpreting
Diagnostic Tests (DAG-STAT)**

**Rice
Virtual Lab in Statistics (including simulations)
Statistic Simulations**

**StatPrimer: Statistics for Public Health Practice
StatNotes: An Online Statistics Textbook
Statistical Associates e-Books**

** Improving
Medical Statistics Medical Statistics Misadventures
Introductory Biostatistics (e-Medicine) **

**Downloadable Statistical Books / Papers
**

**ONLINE STATISTICAL ANALYSIS**

**A Compilation of Online
Analyses**

**Concepts and
Applications of Inferential Statistics**** & ****Online Statistics Site
(Vassar)**: **TOC**
**(IE
users)**

**Clinical
Research Calculators** at **Vassar **

**Statistical Online Computational Resource (SOCR)
**

**EasyCalculation Online Calculators**

**EpiMax Table Calculator Evidence-based
Medicine Toolbox OpenEpi-Epidemiologic
Calculators **

**Globally Accessible Statistical Procedures - GASP
**

**The Chinese University
of Hong Kong Statistical Tools Pages** **GraphPad
QuickCalcs** **PHYLIP
Online **

**StatCrunch Online
Statistics****
****Online Statistical
Analysis****
****HyperStat Statistics Online**** ****Wessa**** **

** ****Online LD Analysis** **Genotype2LDBlock (online) **

**Simple SNP Data Analysis (incl HWE): SNPStats & HWA**

**Partition for Online Bayesian Analysis
Free Statistics on the Web **

**TEXTBOOKS **

**Epidemiology Textbook Open-Epi Book **

** ****Statistics at Square One**
** Statistics Notes**: **BMJ** **CMAJ** ** Radiology **

**SCOR: Probability
& Statistics eBOOK & Educational Materials **

**Reference Guide on Statistics (&
Glossary) Data Analysis BriefBook (Contents)**

**SticiGui (Statistics Tools for Internet &
Classroom Instruction with a Graphical User Interface) **

**Introductory Statistics (DW Stockburger) **

**StatsDirect Help STATISTICA Glossary
Statistical Tables **

**Learning
by Simulations WISE (Web Interface for Statistical Education):
Tutorials**

**SMART (Explorapedia of Statistical &
Mathematical Techniques) **

**InStat Guide to Choosing
the Right Test** **The Prism Guide to
Interpreting Statistical Results** **Resampling **

**Applying
the Right Statistics: Analyses of Measurement Studies by M Bland
Seven Common Errors**

**Statistics to Use****
AS/A2
Mathematics & Statistics Modules **

**Multimedia Statistics**** ****Statistical
Animal Models**** STATA (Tutorial) SAS (e-Learning) **

** ****JMP**
**JMP
GENOMICS** **WINKS GENSTAT
R
S-PLUS
NCSS
PASS
DAG-STAT**

**SPSS Help-Tutorial-Coach & Training SYSTAT
SigmaStat ****STATISTICA****
**

**InStat**** ****Epi Info** **ViSta
6.4 MVSP LISREL
StatsDirect
**

**PAST (Paleontological Statistics Software Package for
Education and Data Analysis) (Hammer, 2001) **

**Statistics
Software Discussion List Subscription Services **

**Animated Glossary ****Virtual
Laboratories in Probability and Statistics**

**Statistical Terms**** ****Statistics Glossary (1)**** (2) MV
Stat Glossary Statistics.Com
Glossary **

**Discussion Groups: MedStats
AllStat
**

**Links to Mathematics & Statistics Sites****
**

** **

*Address for bookmark***: ****http://www.dorak.info/mtd/glosstat.html**** **

** **

*Last updated on 21 October 2012
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