Binary Response

TheToenaildataset, which is another classic illustration of GLMM (see, e.g. Verbeke and Molenberghs 2009), is available inR through the package HSAUR2 and contains measures of an experiment contrasting two competing oral antifungal treatments against toenail infection[De Backer et al., 1998]. The total number of patients (378 individuals) were randomly assigned to one of two treatments: a dose of 250mg of terbinafineper day for one group and 200mg of itraconazolefor the others. Over the course of sevenvisits, scheduled at weeks 0, 4, 8, 12, 24, 36, and 48 of the treatment, the degree of separation of the nail plate from the nail bed was measured. The outcome is the consequence of the dichotomization of the measures taken, which were coded in two levels moderate or severe and none or mild.

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prevalence

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Figure 1.5: Evolution of the average infection rate over thevisitsin theToenaildataset.

This data is often analyzed by means of a longitudinal mixed logit GLMM with a random intercept per subject as in:

log pij

1−p_ij =β₀ +β₁T_ij +β₂v_ij+β₃(T v)_ij+σu_i, (1.49) where T_ij represents the variable treatment, v_ij is the number of the visit, (T v)ij

the multiplicative interaction between treatment and time, and u_i denotes the standard normal random intercept. In our simulations, we generate values of vij from centered values of the sequence {1, . . . , n_i} and T_ij from the vector T = [0^T_n/2,1^T_n/2]^T. The values of the parameters come from a fit of the data using model (1.49) and are listed on Table 1.3.

Parameter β₀ β₁ β₂ β₃ σ Values 1.4252 −1.012×10⁻³ -0.8044 -0.2351 4.73 Table 1.3: Parameters in the second set of simulation studies.

Similarly to the study conducted in the LMM context (Section 1.6.1), we test the procedures not only with respect to the increase in the number of observational units,

but also with respect to increases in cluster size. We have thus allowed the cluster size to increase in three possible values, n_i = 7,14,21, for every scenario of number of clusters, n = 300,600. Overall, in each of our M = 1000 simulated samples, a total of B = 1000 bootstrap replications were generated.

Bootstrap Approximation of the Sampling Distribution

On a first stage, and as we did for the LMM in Section 1.6.1, we contrast the Average Bootstrap Distribution, obtained from the pool of replications over all the samples, against the sampling distribution of the estimators. For this part of the study, we consider the following schemes:

• Parametric Bootstrap (PB), as described in Section 1.4.2. We feel important to point out that the bootstrap samples of have been generated based on Laplace-Approximated MLE of the model parameters, while the replicates have also been obtained using the Laplace-Approximated Log-Likelihood;

• Random Weighted Laplace Bootstrap (RWLB) : with weights drawn from an expo-nential E(1) distribution.

• Random Effect Bootstrap using the suggestion based on Shao and Tu [2012], as in equation (1.20) in four versions, (REB.CM.V1, REB.CM.V1, REB.EP.V1, REB.EP.V2), determined according to:

– The predictions of Random Effects that were resampled: Conditional Modes (CM) orEmpirical Predictor (EP) approximated with Monte Carlo integration, – Whether a reflation step was considered: V1 denoting resampling of non-reflated predictions and V2 incorporating the reflation step, using the pro-cedure by Carpenter et al. [2003].

We consider the estimation obtained with the AGQ method using 50 quadrature points as our benchmark ML estimate in order to minimize the effects of the approximation error of the Laplace method in our conclusions, implying that in the following Q-q plots we contrast√

n(θ^b∗_k−θ^b_k^AGQ) against√

n(θ^bAGQ_k −θ_k) fork∈ {1, . . . , p+s}. As we have pointed out in the section pertaining the study in the LMM context, the ML sports some small-sample bias, yet other inferential methods that could potentially feature less bias such as e.g. the Simulated Method of Moments (SMM) are not as readily implemented as the AGQ. On the other hand, we expect that the elevated number of clusters (n = 300 in the basic scenario) added to the fact that we are comparing differences between replicates and estimates, helps to dinimish the effects of the bias in our conclusions, leaving the effects of the approximation error as the only issue in need of control.

In Figure1.6 we report the comparisons of these distributions for β_k, the fixed effect parameters, when n = 300. In this plot we can see that when n_i = 7 PB and RWLB approximate the distribution of estimates for fixed effects β_k reasonably well, aside from some isolated observations in the upper and lower tails and a little underestimation of the variance for β₀, β₁ and β₃, divergences that get reduced as the cluster size increases.

These remarks can be extended to the case when n= 600, an observation that transpires from Figure 1.7. On the other hand, all the versions of the REB (displayed in grey) fail heavily to represent this asymptotic distribution.

1.6. Simulation Study 27

n_i = 7 n_i = 14 n_i = 21

1.6. Simulation Study 29 As for σ², Figure 1.8 shows that the approximation by the bootstrap for moderate cluster sizes has a negative deviation with respect to the sampling distribution in both schemes, as evidenced by the negative shift of the Q-q plots with respect to the identity line in the first column, i.e. when ni = 7. That said, the RWLB shows less deviation compared to the PB throughout all the comparisons, an effect that can arguably be attributed again to the effects of the random weighting on the approximation error of the LALL. Moreover, it is apparent that this deviation does not reduce with increases in n but with increases in cluster size n_i, as seen by the reduction of the shift when n_i = 14 and n_i = 21. One could even say that both schemes experience more deviation when the number of clusters grows, which again comes back to the approximation error of the LALL e.g. O_p(n⁻²_i n) in a simple Logit GLMM in Ogden[2016].

n_i = 7 n_i = 14 n_i = 21

n = 300

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n = 600

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RWLB PAB REB

Figure 1.8: Q-Q plots of√

n(σb^2∗−σb²) vs. √

n(σb²−σ²) the Mixed Logit Model.

Bootstrap Confidence Intervals

We have also tested the quality of the BCI based on these two methods for a series of levels, on the grounds of their effective CR throughout the different scenarios. Table 1.4 shows such a comparison for the nominal level of 1 −α = 0.95, where it is apparent that the BCI for β_k, k ∈ {0, . . . ,3} based on both methods have comparable CR for all combinations of n and n_i. Overall, however, we can remark that there is a reduction of CR in both methods for most parameters when there are increments in the number of clusters, especially when the cluster sizes are modest, while the main improvements come from cluster size increases. The only case when this effect doesn’t hold is when the cluster size is set at n_i = 21 and arguably comes from the effects of the random weighting on the approximation error exposed previously. On the other hand, BCI based on the RWLB for σ²₂ show systematically a better coverage than the PB counterparts for modest cluster sizes, although they are still far from the nominal level. Again, this behaviour doesn’t translate to the case when n_i = 12, where the PB mantains a slight edge over RWLB.

These remaks remain valid when other nominal levels are considered, as it transpires from Tables 1.8 and 1.7 in the Appendix.

n= 300 n= 600

RWLB PB RWLB PB

β₀ 0.8670 0.9020 0.8730 0.9090 β₁ 0.9490 0.9650 0.9480 0.9680 ni = 7 β2 0.9430 0.9010 0.9070 0.8040 β₃ 0.9270 0.9430 0.9420 0.9550 σ² 0.7400 0.3210 0.5480 0.0560 β₀ 0.9230 0.9370 0.9160 0.9370 β₁ 0.9630 0.9710 0.9610 0.9740 ni = 14 β2 0.9440 0.9410 0.9360 0.9300 β₃ 0.9410 0.9570 0.9450 0.9520 σ² 0.8710 0.8230 0.8320 0.6990 β₀ 0.9390 0.9420 0.9350 0.9350 β₁ 0.9670 0.9670 0.9690 0.9730 n_i = 21 β₂ 0.9470 0.9470 0.9550 0.9550 β₃ 0.9460 0.9510 0.9470 0.9540 σ² 0.9240 0.9310 0.9310 0.9330

Table 1.4: Coverage ratios : 1−α= 0.95 in the Mixed Logit Study