1. SJS SDI_101 Design of Statistical Investigations Stephen Senn 10 Random Effects

2. SJS SDI_102 More Than one Random Term So far only one term has been considered to be random –disturbance term It is possible to have models in which more than one source of variation is taken to be random We now consider such models

3. SJS SDI_103 The Example of Clinical Trials So far we have always taken patient effects to be fixed Suppose however we ran a parallel group trial Some patients have one treatment some have another Patients and treatments are confounded If we treat patient effects as fixed, cannot estimate treatments

4. SJS SDI_104 Solution We treat patient effects as random This is done implicitly in parallel group trials as follows ij is effect for patient j of treatment group i. If we declare this to be random we can form a new disturbance term as follows

5. SJS SDI_105 So What? We do not even need to model this explicitly We just have a model in which we say response = treatment + noise, without worrying about what terms the noise is made up of. However for more complicated designs such distinctions may be useful

6. SJS SDI_106 Cross-over Trials We shall now take a simple example AB/BA cross-over We shall, however ignore period effects to make it even simpler Just consider the following –Treatment effects –Patient effects –Within-patient error

7. SJS SDI_107 Model Here all the i and ij terms are assumed independent of each other.

8. SJS SDI_108 Consequences The variance covariance structure of the Y ij then has the following block diagonal form

9. SJS SDI_109 Variance-Covariance Matrix Here is is assumed that measurements in successive rows of Y are on the same patient

10. SJS SDI_1010 Alternative Representation

11. SJS SDI_1011 Estimation 1 If we now write this as a linear model with only one error term, we must now have We can no longer use ordinary least squares but must use generalised least squares instead.

12. SJS SDI_1012 Estimation 2 We are not going to cover the details of GLS. However, as it turns out, the estimator here is exactly the same as in the model in which we treat the patient effects as fixed rather than random. This equivalence does not generally hold. It does hold for certain balanced designs

13. SJS SDI_1013 Illustration Exp_5 This experiment was an AB/BA cross-over We previously analysed this using a fixed effects model for the patient effect We now analyse this treating the patient effect as random To do this we use the SPlus lme function (lme = linear mixed effect)

14. SJS SDI_1014 Exp_5 Random Effects Analysis with SPlus > fit4 <- lme(pef ~ treat, random = ~ 1 | patient) > summary(fit4) Linear mixed-effects model fit by REML Data: NULL AIC BIC logLik 271.8428 276.5551 -131.9214 Random effects: Formula: ~ 1 | patient (Intercept) Residual StdDev: 66.24667 28.70328 Fixed effects: pef ~ treat Value Std.Error DF t-value p-value (Intercept) 295.7692 20.02402 12 14.77072 <.0001 treat 45.3846 11.25835 12 4.03120 0.0017

15. SJS SDI_1015 Question In this balanced case, the estimator is the same for the random effect model as for the fixed effect model Show that the variance is the same whether patient is treated as a fixed or a random effect

16. SJS SDI_1016 An Example Where this Equivalence Does not Apply In Exp_5, treating the patient effect as fixed or random produces the same result This is not the case for all designs We now consider an example where this does not apply Exp_12, an incomplete blocks design, is a case in point

17. SJS SDI_1017 Exp_12 Analysis with SPlus Random effects: Formula: ~ 1 | patient numeric matrix: 1 rows, 2 columns. (Intercept) Residual StdDev: 0.7055782 0.2263807 Fixed effects: FEV1 ~ treat Value Std.Error DF t-value p-value (Intercept) 2.021375 0.1568938 23 12.88372 <.0001 treatF24 0.034457 0.0946403 22 0.36409 0.7193 treatP -0.492521 0.0890777 22 -5.52912 <.0001 Compare these with fixed effects solution Value Std. Error t value Pr(>|t|) treatF24 0.0388 0.0955 0.4059 0.6888 treatP -0.5029 0.0897 -5.6051 0.0000

18. SJS SDI_1018 Notes The estimates are no longer the same The variances are (logically) no longer the same either The variances for the random effects approach are (slightly) smaller

19. SJS SDI_1019 Fixed effects Any effect we nominate as fixed has to be eliminated when estimating any other effect If we nominate patient as fixed then patient must be eliminated in estimating the treatment effect In Exp_12 each patient effect appears twice. –Once in period one, once in period 2 A patient effect can only be eliminated by forming the difference between period 1 and period 2 –Analysis uses such differences

20. SJS SDI_1020 Random Effects An effect that is random does not have to be eliminated –on average it is zero Nominating an effect as random increases the range of possible unbiased estimators The minimum variance estimator may or may not be the same as in the fixed effects case

21. SJS SDI_1021 A Further Source of Information If the patient effect is random, the totals for patients vary randomly from patient to patient These totals do not all reflect the same effects By comparing F12 + F24 with F12 + P we can estimate the difference between F24 and P This is a further source of information In general referred to as inter-block information This has been recovered by S-PLUS

22. SJS SDI_1022 Other Sorts of Random Effects In example considered main effect of block was random More unusual is to have a true random treatment effect (but this can happen) Quite common is to have block by treatment interactions that are considered random We consider an example of the former in the next lecture We consider an example of the latter here

23. SJS SDI_1023 Exp_14 Shumaker and Metzler Trial to compare two formulations of phenytoin –T = test, R = reference So-called bioequivalence study Four period cross-over Each of 26 subjects received each formulation twice

24. SJS SDI_1024 Data from Shumaker and Metzler, 1998 Drug Information Journal,32,1063-1072 Area under the concentration time curve (AUC) for Phenytoin. Data are log-transformed

25. SJS SDI_1025 Subject by Formulation Interaction Heuristic Explanation We can estimate the treatment effect twice independently for each subject –for example by comparing period 2 and 1 and period 3 and 4 We can see whether these estimates differ more between subjects than within This enables us to estimate whether there is an interaction

26. SJS SDI_1026

27. SJS SDI_1027 Exp_14 Two Fits > aov.1 <- aov(lAUC ~ Subject + Formulation) > summary(aov.1) Df Sum of Sq Mean Sq F Value Pr(F) Subject 25 7.747866 0.3099146 90.50657 0.0000000 Formulation 1 0.002529 0.0025288 0.73851 0.3928048 Residuals 77 0.263665 0.0034242 > aov.2 <- aov(lAUC ~ Subject * Formulation) > summary(aov.2) Df Sum of Sq Mean Sq F Value Pr(F) Subject 25 7.747866 0.3099146 82.32023 0.0000000 Formulation 1 0.002529 0.0025288 0.67172 0.4161950 Subject:Formulation 25 0.067898 0.0027159 0.72141 0.8112634 Residuals 52 0.195767 0.0037647

28. SJS SDI_1028 Different Philosophy MS residuals in first fit includes subject by treatment interaction MS residuals in second does not Hence F test in first uses such variation from subject to subject to assess precision of overall treatment estimate Second F test excludes it

29. SJS SDI_1029 Questions In this example the MS for residuals is actually higher when the subject by formulation interaction is fitted Is this phenomenon to be expected in general? What does it imply? Can you think of an explanation in this case?