Now, I feel it’s time to update a little bit these old notes with new material. This is motivated by my new interest toward bioinformatics and medical imaging that are both facing the n << p and p-value correction problems.
Multiple comparisons arise when one wants to compare more than two things at the same time (i.e. for the same dataset, in the same explanatory framework). Why do we have to adapt the usual test statistics? Because these multiple tests (of hypothesis) are generally not carried out in an independant way, thus leading to an inflation of the overall type I error rate. In other words, when faced with say 10 comparisons on the same data, the probability of falsely detecting a significant difference becomes greater than the “classical” 5% level. For a general introduction to multiple comparison procedure, the reader may be interested in (6). Other references are (5,7) and (8) (especially, Chapter 5) but the latter two are a bit more tricky…
Actually, there are two main packages that can handle multiple comparison, both relying on a different view of the type I error:
multcomp(R vignette), for “Simultaneous tests and confidence intervals for general linear hypotheses in parametric models, including linear, generalized linear, linear mixed effects, and survival models.”
multtest(see Dudoit and van der Laan(7), and a series of talks here: 1, 2, 3), for “Non-parametric bootstrap and permutation resampling-based multiple testing procedures for controlling the family-wise error rate (FWER), generalized family-wise error rate (gFWER), tail probability of the proportion of false positives (TPPFP), and false discovery rate (FDR). Single-step and step-wise methods are implemented. Tests based on a variety of t- and F-statistics (including t-statistics based on regression parameters from linear and survival models) are included. Results are reported in terms of adjusted p-values, confindence regions and test statistic cutoffs. The procedures are directly applicable to identifying differentially expressed genes in DNA microarray experiments.”
Interestingly, the former is attached to the Social Sciences Task View while the latter is on Bioconductor.
Also, I should mention the
Design (see its
contrast() function) package by F. Harrell, the npmc package for non parametric multiple comparisons, but see R Site Search, with keyword = “multiple comparison”.
For the purpose of the illustration, we will use the data from the Cholesterol study (included in the
A clinical study was conducted to assess the effect of three formulations of the same drug on reducing cholesterol. The formulations were 20mg at once (1time), 10mg twice a day (2times), and 5mg four times a day (4times). In addition, two competing drugs were used as control group (drugD and drugE). The purpose of the study was to find which of the formulations, if any, is efficacious and how these formulations compare with the existing drugs.
The main function is called
glht() and its description indicates that it provides general linear hypotheses and multiple comparisons for parametric models, including generalized linear models, linear mixed effects models, and survival models. Well, it covers quite interesting models for a biostatistician.
First, let’s try a basic ANOVA model on the Choslesterol data.
library(multcomp) data(cholesterol) lm1 <- aov(response ~ trt, data=cholesterol) summary(lm1)
This yields the following ANOVA table:
Df Sum Sq Mean Sq F value Pr(>F) trt 4 1351.37 337.84 32.433 9.819e-13 *** Residuals 45 468.75 10.42 --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
As can be seen from the resulting call to the
aov() function (Recall that
aov() is just a wrapper for the more general
lm() function), the treatment main effect appears significant (the p-value is largely <.001). This is not very surprising given the picture of the data shown in the figure below.
However, what can be said about the paired difference between consecutive formulations (20mg at once, 10mg twice a day, 5mg four times a day), and how do they compare to already in-use drugs (
We may have a look at regression coefficient for each treatment, considering 1time as the baseline. This is done with
summary.lm(lm1), and we get the results shown below:
Call: aov(formula = response ~ trt, data = cholesterol) Residuals: Min 1Q Median 3Q Max -6.541770 -1.967203 -0.001575 1.890143 6.600830 Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 5.782 1.021 5.665 9.78e-07 *** trt2times 3.443 1.443 2.385 0.0213 * trt4times 6.593 1.443 4.568 3.82e-05 *** trtdrugD 9.579 1.443 6.637 3.53e-08 *** trtdrugE 15.166 1.443 10.507 1.08e-13 *** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 Residual standard error: 3.227 on 45 degrees of freedom Multiple R-squared: 0.7425, Adjusted R-squared: 0.7196 F-statistic: 32.43 on 4 and 45 DF, p-value: 9.819e-13
This is quite informative but it just says that all treatment differ from the baseline which is usually not what we’re interested in. Here, the Intercept represents the mean response for 20mg at once.
with(cholesterol, mean(response[as.numeric(trt)==1]))  5.78197
We might be interested in which pairs of means (without consideration for the internal data structure, that is 3 different dosages + 2 classical drugs) are significatively different. This calls for something the Tukey HSD procedure which gives
lm1.hsd <- TukeyHSD(lm1) lm1.hsd
Results can be given both numerically,
Tukey multiple comparisons of means 95% family-wise confidence level Fit: aov(formula = response ~ trt, data = cholesterol) $trt diff lwr upr p adj 2times-1time 3.44300 -0.6582817 7.544282 0.1380949 4times-1time 6.59281 2.4915283 10.694092 0.0003542 drugD-1time 9.57920 5.4779183 13.680482 0.0000003 drugE-1time 15.16555 11.0642683 19.266832 0.0000000 4times-2times 3.14981 -0.9514717 7.251092 0.2050382 drugD-2times 6.13620 2.0349183 10.237482 0.0009611 drugE-2times 11.72255 7.6212683 15.823832 0.0000000 drugD-4times 2.98639 -1.1148917 7.087672 0.2512446 drugE-4times 8.57274 4.4714583 12.674022 0.0000037 drugE-drugD 5.58635 1.4850683 9.687632 0.0030633
opar <- par(mar=c(5,8,4,2),las=1) plot(lm1.hsd) abline(v=0,lty=2,col=”gray50”) par(opar)
Now, we see that 7 pairs of difference of means out of 10 are different from 0 at the 5% level. Lower bound of 95% CI is always >1.
lm1.hsd2 <- glht(lm1, linfct = mcp(trt = “Tukey”)) summary(lm1.hsd2)
Simultaneous Tests for General Linear Hypotheses Multiple Comparisons of Means: Tukey Contrasts Fit: aov(formula = response ~ trt, data = cholesterol) Linear Hypotheses: Estimate Std. Error t value p value 2times - 1time 0 3.443 1.443 2.385 0.138169 4times - 1time 0 6.593 1.443 4.568 0.000352 *** drugD - 1time 0 9.579 1.443 6.637 < 1e-04 *** drugE - 1time 0 15.166 1.443 10.507 < 1e-04 *** 4times - 2times 0 3.150 1.443 2.182 0.205103 drugD - 2times 0 6.136 1.443 4.251 0.000993 *** drugE - 2times 0 11.723 1.443 8.122 < 1e-04 *** drugD - 4times 0 2.986 1.443 2.069 0.251209 drugE - 4times 0 8.573 1.443 5.939 < 1e-04 *** drugE - drugD 0 5.586 1.443 3.870 0.003047 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1 (Adjusted p values reported -- single-step method)
As can be seen, we obtain the same result. There is also a plot method that displays difference of means in a more convenient way (see
The main interface to
glht() is the option
linfct, which stands for linear function in the (G)LM framework. Any contrast can be passed directly (as a vector or matrix) to glht() or one can choose among those already constructed with the contrMat() function. Available contrasts coding are: Dunnett, Tukey, Sequen, AVE, Changepoint, Williams, Marcus, McDermott, UmbrellaWilliams, GrandMean. They all are referenced from Bretz et al.(4).
As for R default contrast coding scheme (i.e. treatment contrast), the default contrasts are Dunnett contrast, that is all k-1 comparisons are done with reference to the baseline, taken as the first level of the unordered factor.
As an illustration, let’s consider Dunnet contrast coding:
n <- c(10,20,30,40) names(n) <- paste(“dose”, 1:4, sep=”“) contrMat(n)
|group2 – group1||-1||1||0||0|
|group3 – group1||-1||0||1||0|
|group4 – group1||-1||0||0||1|
One would get a similar contrast matrix by calling
contr.treatment(4). Note that such contrasts are not orthogonal to the Intercept and thus are not pure contrasts in standard LM Theory. Also note that some software doesn’t take the first level as the baseline, but instead choose the last one. It is the case for SAS at least. If we’re interested in getting the same result as SAS, we can either relevel the factor, specify which level to consider as the baseline (option base in
contr.treatment(), or simply use the wrapper function
The table below summarizes the different contrasts coding scheme in the
multcomp package. Contrasts are expressed in rows and are labelled
All pairwise comparisons are allowed with Tukey contrasts. In contrast, for what are called Sequence (aka “Sequen”) contrasts we are “slicing” between consecutive pairwise treatment.
Another class of useful contrasts is the Grand Mean model.
Here, we take a different approach to the correction for Type I error rate inflation. As explained in Dudoit, S. and van der Laan(7) (Section 2.8), we only consider the control of Type I error under the true parameter values (or the true generating distribution, in the case of simulated data). Furthermore, this is not a strong control over the error rate (ER) as the proposed methodology doesn’t control the supremum of the Type I ER over parameter values satisfying all 2M possible subsets of null hypotheses.
mt.plot() are by far the most useful routines for computing and displaying adjusted p-values. Let’s look at the example given in
library(multtest) data(golub) smallgd <- golub[1:100,] classlabel <- golub.cl res1 <- mt.maxT(smallgd,classlabel) rawp <- res1$rawp[order(res1$index)] procs <- c(“Bonferroni”,”Holm”,”Hochberg”,”SidakSS”,”SidakSD”,”BH”,”BY”) res2 <- mt.rawp2adjp(rawp,procs)
To appreciate the differences between the different methods in a more convenient, we can plot the (sorted) adjusted p-values as a function of the number of rejected hypotheses. Again, following the on-line help, this can be done easily as follows:
allp <- cbind(res2$adjp[order(res2$index),],res1$adjp[order(res1$index)]) dimnames(allp)29 <- “maxT” procs <- dimnames(allp)2 procs[7:9] <- c(“maxT”,“BH”,“BY”) allp <- allp[,procs] cols <- c(1:4,”orange”,”brown”,”purple”,5:6) ltypes <- c(3,rep(1,6),rep(2,2)) mt.plot(allp, teststat, plottype=“pvsr”, proc=procs, leg=c(80,0.4), lty=ltypes, col=cols, lwd=2)
False Discovery Rate (FDR) is tightly linked to the preliminary example used when talking about the multtest package. In fact, when we plotted the adjusted p-values against the number of rejected hypotheses, we were already talking about the proportion of correct rejection of the null hypothesis. Here, “correct” means that the significant tests of hypothesis really reflects the reality, that is the true population parameters.
- Feise, R.J. (2002). Do multiple outcome measures require p-value adjustment? BMC Medical Research Methodology, 2, 8.
- Westfall, P.H., Tobias, R.D., Rom, D., Wolﬁnger, R.D., and Hochberg, Y. (1999). Multiple Comparisons and Multiple Tests Using the SAS System. Cary, NC: SAS Institute Inc., page 153.
- Westfall, P.H. (2005). Advances in Multiple Comparisons and Multiple Tests using the SAS System. Sugi 25, Paper 264.
- Bretz, F., Genz, A. and Hothorn, L.A. (2001), On the numerical availability of multiple comparison procedures. Biometrical Journal, 43(5), 645-656.
- Hothorn, T., Bretz, F. and Westfall, P. (2008). Simultaneous Inference in General Parametric Models. Department of Statistics: Technical Reports, Nr. 19.
- Toothaker, L. (1993). Multiple Comparison Procedures. Sage University Paper, 89.
- Dudoit, S. and van der Laan, M.J. (2008). Multiple Testing Procedures with Applications to Genomics. Springer.
- Christensen, R. (2002). Plane Answers to Complex Questions, The Theory of Linear Models (3rd Ed.). Springer.
- Falissard, B. (2005). Comprendre et utiliser les statistiques dans les sciences de la vie (2nd Ed.). Masson.