ABSTRACT
The likelihood is the probability of observing the sample of data obtained in the trial, and is, given these data, a function of a set of specified parameters. For BE testing, the parameters of interest are the formulation, period, and sequences effects and any within-or between-subject variances. In trials where subjects get repeated exposure to a formulation, i.e., where the design includes sequences such as RTTR and TRRT, it is possible to estimate σ2BT , σ
2 BR, the between-subject variances of T and
R, respectively, and the within-subject variances of T and R, σ2WT , σ 2 WR,
respectively. The method of maximum likelihood (ML) determines the parameter estimates to be those values of the parameters that give the maximum of the likelihood. Restricted maximum likelihood estimation (REML) is a form of ML estimation that uses an iterative procedure where within each iteration there are two steps. A simplified description of REML is as follows. Using a first guess or estimate of the parameter values, the procedure keeps the values of the variance parameters fixed and estimates the formulation, period, and sequence effects. This is the first step. The residuals from this model are then calculated and used to reestimate the variance parameters. This is the second step. These steps are repeated until the values of the parameters do not change from one iteration to the next. The ‘Restricted’ in the name of the method arises because within each step, one set of parameters is fixed while the other set is estimated by maximizing the likelihood under the restriction imposed by the fixed set of parameters. The usefulness of REML is that it can be used to estimate the between-
and within-subject variances. The estimates, so obtained, are informative for the interpretation of the data, particularly when bioequivalence between T and R is not demonstrated. A second, and less important, property of REML is that it can be used when the data set is incomplete, i.e., when a complete set of logAUC or logCmax is not obtained from each subject. We illustrated such an analysis in Chapter 3. There, it will be recalled, the REML results were very similar to the results obtained from an analysis that used just those subjects that had a complete set of values. For more information on the properties of REML when the trial has a relatively small number of subjects, see [237] Chapter 6. When a trial fails to show bioequivalence it is of interest to determine
which factors (i.e., a difference in formulation means, unexpectedly high variability, or both) led to such a circumstance, and REML models may of
data. The use of REML models is also important in the context of individual and population bioequivalence, and we review the application of such methods in the next chapter. REML has quite a long history [327], [181], [259], [47] and has been
particulary useful for the analysis of repeated measurements [237], [307], [435], [250]. Readers interested in application in the bioequivalence setting should see [332], [335], and [457]. Obviously REML estimation cannot be done by hand. SAS code to
perform these analyses is given in the following box. For standard two, three, and four-period designs such as those found
in Examples 3.1, 3.2, 4.5, and 4.6 (i.e., those where no formulation adminstration is replicated), analysis code may be found on the website in exam1.sas - exam4.sas, respectively. Some proc mixed code for Example 4.5 exam3.sd2 is included here for illustration purposes:
proc mixed data=my.exam3 method=reml ITDETAILS maxiter=200; class sequence subject period formula; model lnauct=sequence period formula /ddfm=KENWARDROGER; random subject(sequence); estimate ’T-R’ formula -1 0 1 /cl alpha=0.10; estimate ’T-S’ formula 0 -1 1 /cl alpha=0.10;
run;
Kenward and Roger’s [246] denominator degrees of freedom are specified to ensure the correct degrees of freedom are used and that a good estimate of the standard error of µˆT − µˆR is obtained. Estimates relevant to ABE testing may be found in Table 5.1 for AUC,
Cmax, and T 1 2 on the log scale. Note that Tmax was not analyzed using
a log-transformation; thus differences expressed for Tmax are on the original scale in hours. It will be recalled that while Example 3.1 demonstrated bioequiva-
lence, Example 3.2 did not (see Chapter 3) due to reasons discussed later in this chapter. In Example 4.5, formulation T was not equivalent to R nor S with results indicative of a potentially bioinequivalent new formulation. In Example 4.6, bioequivalence was demonstrated at both high and low doses of drug product. We have provided two additional data sets exam5.sd2 and exam6.sd2
on the website, that were obtained from trials that used the sequences (RTTR/TRRT) and (RTRT/TRTR), respectively. FDA-recommended
respectively. The proc mixed code for exam5 is given in the box below. The AUC and Cmax data for exam6 are given in Table 5.4, where we consider other interesting features of these data.
