r is not an unbiased estimate of ρ. A relatively unbiased estimate is  the adjusted correlation coefficient :

Adjusted R2 is

where p is the total number of regressors in the linear model (not counting the constant term), and n is the sample size.

Adjusted R2 can also be written as

\bar R^2 = {1-{SS_\text{res}/df_e \over SS_\text{tot}/df_t}}

where dft is the degrees of freedom n– 1 of the estimate of the population variance of the dependent variable, and dfe is the degrees of freedom np – 1 of the estimate of the underlying population error variance.

The principle behind the adjusted R2 statistic can be seen by rewriting the ordinary R2 as

R^{2} = {1-{\textit{VAR}_\text{err} \over \textit{VAR}_\text{tot}}}


Assumption Problem Test Solution
Predictors are Independent Multicollinearity Tolerance, VIF Drop Variables, Mean Center, Combine variables into index
Errors are statistically independent autocollinearity Durbin-Watson Test, plot residuals over time Adjust data
Distribution of residuals is normal, random heteroscedasticity,
plot residuals against predicted y, quantile comparison plot, graph residuals in a boxplot, histogram transform y and/or x
Errors are not “unusual” (outliers) large outliers, influential points quantile comparion plot, test leverage and influence, Cook’s Distance, added variable plots, graph studentized residuals vs. predicted y remove outliers
Linear relationship is the correct function non-linear relationship added-variable plot, component and residual plot transform predictors, add terms


R (packages gee, geepack and multgee) are for Generalized estimating equation GEE

A generalized estimating equation (GEE) is used to estimate the parameters of a generalized linear model with a possible unknown correlation between outcomes.

Parameter estimates from the GEE are consistent even when the covariance structure is misspecified, under mild regularity conditions.

The focus of the GEE is on estimating the average response over the population (“population-averaged” effects) rather than the regression parameters that would enable prediction of the effect of changing one or more covariates on a given individual.

GEEs are usually used in conjunction with Huber-White standard error estimates, also known as “robust standard error” or “sandwich variance” estimates.

In the case of a linear model with a working independence variance structure, these are known as “heteroskedasticity consistent standard error” estimators.

GEE unified several independent formulations of these standard error estimators in a general framework.

GEEs belong to a class of semiparametric regression techniques because they rely on specification of only the first two moments.

Under correct model specification and mild regularity conditions, parameter estimates from GEEs are consistent.

They are a popular alternative to the likelihood–based generalized linear mixed model which is more sensitive to variance structure specification.

They are commonly used in large epidemiological studies, especially multi-site cohort studies because they can handle many types of unmeasured dependence between outcomes.


multinomial association cran – Google Search

An R Package for Hierarchical Multinomial Marginal Models

Package ‘multgee’   Marginal Models For Correlated Ordinal Multinomial Responses

Package ‘MultinomialCI’

Package ‘CoinMinD’    Simultaneous Confidence Interval for Multinomial Proportion


Tests of independence / dependence

coin: A Computational Framework for Conditional Inference

oneway_test two- and K-sample permutation test
wilcox_test Wilcoxon-Mann-Whitney rank sum test   ( tests for ordered categorical data)
normal_test van der Waerden normal quantile test
median_test Median test
kruskal_test Kruskal-Wallis test
ansari_test Ansari-Bradley test
fligner_test Fligner-Killeen test
chisq_test Pearson’s χ 2 test
cmh_test Cochran-Mantel-Haenszel test   (Independence in general two- or three-dimensional contingency tables can be tested)
lbl_test linear-by-linear association test     ( tests for ordered categorical data)
surv_test two- and K-sample logrank test ( tests for ordered categorical data)
maxstat_test maximally selected statistics
spearman_test Spearman’s test
friedman_test Friedman test
wilcoxsign_test Wilcoxon-Signed-Rank test ( tests for ordered categorical data)
mh_test marginal homogeneity test (Maxwell-Stuart).

coin: A Computational Framework for Conditional Inference

Package ‘coin’

Chapter Conditional Inference

Order-restricted Scores Test for the Evaluation of Population …


Exact tests

In statistics, an exact (significance) test is a test where all assumptions, upon which the derivation of the distribution of the test statistic is based, are met as opposed to an approximate test (in which the approximation may be made as close as desired by making the sample size big enough). This will result in a significance test that will have a false rejection rate always equal to the significance level of the test. For example an exact test at significance level 5% will in the long run reject true null hypotheses exactly 5% of the time.

So when the result of a statistical analysis is said to be an “exact test” or an “exact p-value”, it ought to imply that the test is defined without parametric assumptions and evaluated without using approximate algorithms.

Implementation of Barnard’s exact test using R


Package ‘Exact’ – Cran    Unconditional exact tests for 2×2 contingency tables

Package ‘Barnard’     implements the barnardw.test function for performing Barnard’s unconditional test of superiority. This is a more powerful alternative of Fisher’s exact test for 2×2 contingency tables.


Freeman-Halton extension of Fisher’s exact test to compute the (two-tailed) probability of obtaining a distribution of values in a 3×3 contingency table, given the number of observations in each cell.



Exact/permutation version of Jonckheere-Terpstra test (Jonckheere-Terpstra test to test for ordered differences among classes)

R Package ‘clinfun’




Non-parametric tests



Implementations of Generalized Mann-Whitney Type Tests
The package provides nonparametric tools for the comparison of several groups/treatments when
the number of variables is large. The tools are the following.

Package ‘gMWT’


Exact statistics


The formulae for confidence interval:

y ^ ± t α / 2 , n 2 M S E 1 / n + ( x x ¯ ) 2 ( x i x ¯ ) 2

\( \hat y \pm t_{\alpha/2, n-2} \sqrt{MSE} \sqrt{1/n + \frac{(x-\bar x)^2}{\sum (x_i - \bar x)^2}} \)

and prediction interval:

y ^ ± t α / 2 , n 2 M S E 1 + 1 / n + ( x x ¯ ) 2 ( x i x ¯ ) 2

\(\hat y \pm t_{\alpha/2, n-2} \sqrt{MSE} \sqrt{1 + 1/n + \frac{(x-\bar x)^2}{\sum (x_i - \bar x)^2}}\)

Confidence interval is an estimate of an interval in which mean of  observations will fall when x=xi
In its formula
1/n + \frac{(x-\bar x)^2}{\sum (x_i – \bar x)^2}
Tends to 0.

Prediction interval is an estimate of an interval in which future individual observations will fall when x=xi
In its formula
1 + 1/n + \frac{(x-\bar x)^2}{\sum (x_i – \bar x)^2}
tends to 1

That means that the confidence interval for the mean of the outcomes at xi gets smaller as sample size grows. (as Central limit Theorem would suggest) which means that by increase of the sample size our estimate for the average (mean) outcome for xi gets better.

\({ \lim_{n->infinity}{CI = \hat y}}\)


But the dispersion of the distribution of y|xi “the probability of an individual outcome” at xi, Doesn’t change very much because central limit theorem is related to central tendencies not to individual behavior or outcomes. Therefore the prediction interval doesn’t change very much.

Individual behavior remains uncertain no matter how much you increase your sample size 😉

\( \lim_{n->\infty} PI = \hat y \pm t_{\frac{\alpha} {2}_{df=n-2}} \sqrt{MSE} \)



lim n > i n f i n i t y P I = y ^ ± t α / 2 , n 2 M S E