In this chapter, we re-analyze several examples from Chapter 4 now using all the data, not just the completers, under an ignorability assumption for the missing data. We use the same full data models and priors as in Chapter 4 and refer to reader to Chapter 4 for the objectives of each analysis. We also introduce another analysis using the CTQ II smoking cessation data, Data Example 1.3, to illustrate a joint model.
7.1 Re-analysis of GH study under MAR (cont. of Example 4.1)
We redid the analysis previously done in Example 4.1 now using all of the data under an assumption of MAR.
Results and Comparison with completers only analysis
Re-show some results from Chapter 4?? Posterior means and 95%
credible intervals for the mean parameters are given in Tables 7.1 and 7.2. Comparing these results to the complete case results in Tables 4.3 and 4.4, we notice narrower credible intervals under ignorability; this is expected as we are now using all the data. By comparing the posterior means of the means in treatment 1, we can see the potentially strong bias from the completers only analysis (e.g., the posterior mean of the month 12 mean was 88 (79,98) in the completers only analysis vs. 81 (72,90) in the MAR analysis). This is related to the fact that at baseline, there were large differences between the mean of completers and the mean of all the subjects, compare Tables 4.3 and 7.1.
Posterior probabilities that each of the pairwise differences are greater than zero are given in Table 7.3. The posterior probabilities were fairly similar in the common Σ model versus the different Σ model. However, the posterior probability for treatment 1 versus treatment 3 changed substantially from .82 to .92. In addition, the posterior probabilities under ignorability showed differences from the completers only analysis (cf: Table 4.5). For example, the posterior probabilities for the difference
163
164 DATA EXAMPLES: IGNORABILITY Table 7.1 transpose; 1 decimal place Posterior means and 95 % credible intervals of the mean parameters for each treatment assuming a common Σ across treatments
Time treatment1 treatment2 treatment3 treatment4
0 69.39(61.59,77.31) 68.46(60.91,76.05) 65.85(58.21,73.54) 65.21(57.63,72.79) 6 82.21(73.08,91.25) 66.11(57.79,74.44) 81.31(72.79,89.79) 62.17(53.76,70.62) 12 81.29(72.56,89.98) 65.13(57.30,73.04) 72.73(64.74,80.76) 62.68(54.65,70.77)
Table 7.2 Posterior means and 95 % credible intervals of the mean parameters for each treatment assuming different Σ for each treatment
Time treatment1 treatment2 treatment3 treatment4
0 69.40(61.03,77.71) 68.44(61.31,75.59) 65.89(57.51,74.28) 65.22(57.59,72.84) 6 81.91(69.57,93.78) 66.10(58.27,73.87) 81.35(72.57,90.08) 62.15(54.75,69.55) 12 79.37(66.34,91.60) 64.95(57.73,72.28) 72.77(65.20,80.32) 62.88(55.61,70.15)
between treatment 1 and 3 changed from .99 and .97 from the completers only analysis to .92 and .82 for the MAR analysis.
Model Fit
We used DIC I to compare the fit of the common and different Σ models.
The results, which appear in Table 7.4, supported the common Σ model.
To assess the absolute fit of the common Σ model, we used posterior predictive checks using the statistics described in Example 3.18. The p-values for the three correlations for treatment 3 ranged from .94 to .97.
In addition, the correlations between baseline and month 6 and baseline and month 12 for treatment 2 were overestimated with p-values of .94
Table 7.3 Posterior probabilities that each of the pairwise differences at month 12 is greater than zero.
µ(1)3 − µ(2)3 µ(1)3 − µ(3)3 µ(1)3 − µ(4)3 µ(2)3 − µ(3)3 µ(2)3 − µ(4)3 µ(3)3 − µ(4)3
common Σ 1.00 0.92 1.00 0.091 0.67 0.96
Σ by Tx 0.97 0.82 0.98 0.069 0.66 0.97
independence 1.00 0.98 1.00 0.068 0.52 0.94
RE-ANALYSIS OF GH STUDY UNDER MAR (CONT. OF EXAMPLE 4.1) 165 Table 7.4 DIC for the common Σ model and model with Σ differing for each
of the four treatments. For the DIC, the values in parentheses are the values when setting θ = (β, Σ).
model DIC D¯ PD
common Σ 2783 (2668) 2761 (2650) 22 (18) different Σ 2787 (2671) 2745 (2643) 41 (28)
Table 7.5 Posterior means and 95 % credible intervals of the mean parameters for each treatment under the independence model.
Time treatment1 treatment2 treatment3 treatment4
0 69.38(59.96,78.75) 68.46(61.44,75.43) 65.89(58.20,73.61) 65.20(58.40,71.98) 6 86.85(75.48,98.19) 66.22(58.62,73.74) 81.58(73.08,90.10) 62.04(54.47,69.58) 12 88.38(76.03,100.60) 63.32(55.14,71.50) 72.45(63.67,81.25) 62.96(54.75,71.18)
and .92. The magnitudes of these p-values suggest the common Σ model does not adequately model the correlation structure.
Comparison with diagonal covariance matrices
To further illustrate the importance of modelling dependence under MAR, we fit several independence models. We present the results of the model that set Σ(T xi) = σ2(T xi)I3. add Ip to the notation at end of book The DIC for this model was 2956 and the effective num-ber of parameters was 16. Based on the DIC, this model fit considerably worse than the dependence models (cf: Table 7.4). Table 7.5 gives pos-terior means and 95% credible intervals for the means at each time for each treatment. The largest difference from the dependence models (cf:
Tables 7.1 and 7.2) can be seen in the month 12 mean for treatment 1. Under the independence model, this mean is 7-9 units higher than under the dependence models. This large difference is due to the inde-pendence model not using the observed values at month 0 and month 6 to ’fill-in’ the month 12 values for the dropouts for inference. This dif-ference also led to larger (but incorrect) posterior probabilities for the differences between month 12 means for treatment 1 and the other three treatments.
Conclusions
The more parsimonious model that assumed a common Σ across treat-ments was preferred in this analysis (as measured by the DIC and PPL).
However, posterior predictive checks on the common Σ model suggested
166 DATA EXAMPLES: IGNORABILITY Table 7.6 Posterior mean of covariance matrices by treatment. The entries on the main diagonal are the innovation variances and below the main diagonal are the GARP.
some problems. We only compared models that assumed a common Σ versus a distinct Σ for each of the four treatments. It may be that a co-variance model somewhere between these extremes may provide a better fit. The covariance models in Section 6.? are well suited to address this but are inefficient to fit in WinBUGS at this time. Table 7.6 presents the treatment specific GARP and IV parameters. The innovation variances for months 6 and 12 were almost twice as large in treatment 1 versus the other treatments. In addition, several of the GARP in treatment 4 were considerably smaller than their counterparts for the other treatments.
We refer the interested reader to Pourahmadi, Daniels, and Park (2007) and Daniels (2006) for detailed exploration of such models for this data.
7.2 Analysis of schizophrenia clinical trial under MAR (cont.
of Example 4.2)
Results and Comparison with completers only analysis
Figure 7.1 plots the posterior means of the trajectories for each treatment over the 6 weeks of the trial. The BPRS for the medium and high dose individuals started to go back up by week 6 unlike under the completers only analysis. This is related to the fact that those dropping out were more likely to have been doing poorly under their respective treatments.
Tables 7.7 and 7.8 are of primary interest for inference as they contain summary information from the posterior on the change from baseline to 6 weeks; in particular, they contain the change from baseline for all four treatments and the differences in the change from baseline among the
ANALYSIS OF SCHIZOPHRENIA CLINICAL TRIAL UNDER MAR (CONT. OF EXAMPLE 4.2) 167 Table 7.7 transpose and 1 decimal Posterior means of change from
base-line to week 6 for all four treatments under the random effects model (R), unstructured model (U) and independence model (I).
Low Medium High Standard
MAR (R) -5.52(-12.20, 1.26) -13.87(-19.49, -8.20) -12.01(-17.84,-6.02) -13.12(-18.65, -7.45) MAR (U) -3.98( -9.23, 1.47) -13.90(-18.54, -9.23) -11.25(-16.06, -6.38) -12.35(-16.84, -7.72) MAR (I) -12.00(-17.68, -6.34) -15.30(-20.22,-10.40) -18.46(-23.60,-13.31) -16.22(-21.17,-11.26)
treatments, respectively. All the changes from baseline were significantly different from zero (credible intervals excluded zero) except for the low dose treatment; this was significant in the completers only analysis. The low dose arm also exhibited the smallest improvement with a posterior mean of −2.9 and a 95% credible interval of (−15.2, 10.1). In addition, note that the change from baseline using only the completers was larger (in absolute value) than the corresponding analysis here under MAR.
Table 7.8 indicates none of the changes from baseline were significantly different between the four treatments as the credible intervals for all the differences covered zero. In addition, the posterior means were very different than the completers only analysis and credible intervals for the differences were wider.
Alternative covariance structures
For comparison to the covariance structure induced by the random effects model, we also fit a directly specified normal model with an unstructured covariance matrix differing by treatment, Σ(Tx), an unstructured matrix constant across treatments, Σ, and an independence covariance matrix, σ2I. Table 7.9 gives the DIC values for the random effects model and other three models. The unstructured covariance models fit much better than the random effects model and the unstructured covariance model which differed across treatments fit better than the one that was con-stant. All the correlated models fit much better than the independence models.
There were small differences between the treatment-specific changes from baseline between the random effects and unstructured models (see Ta-ble 7.7). However, the independence model had large differences from the dependence models showing considerably more extreme changes (more negative). For the pairwise differences among the treatments (Table 7.8), the independence model again had large differences both in posterior means and in credible intervals. In terms of ’significant’ differences, the unstructured model gave significant differences (credible interval
exclud-168 DATA EXAMPLES: IGNORABILITY Table 7.8 Pairwise differences of the changes among the four treatments under the random effects model (R), unstructured model (U) and independence model (I). tranpose and get rid of diff notation
diff12 diff13 diff14 diff23 diff24 diff34
MAR (R) 8.35(-0.24,17.00) 6.49(-2.25,15.20) 7.60(-1.01,16.29) -1.86(-10.04,6.18) -0.74(-8.68,7.18) 1.12(-6.90, 9.07) MAR (U) 9.92( 3.07,16.88) 7.27( 0.34,14.27) 8.37( 1.57,15.25) -2.65( -9.24, 3.94) -1.55(-8.06, 4.84) 1.10(-5.42, 7.64) MAR (I) 3.30(-4.20,10.79) 6.46(-1.22,14.14) 4.22(-3.33,11.79) 3.17( -3.97,10.27) 0.92(-6.03, 7.89) 0.84(-7.01, 8.72)
Table 7.9 DIC for schizophrenia data. be consistent with columns in DIC tables and notation
model D¯ Pd DIC
Random effects 6520.84 15.12 6535.95 Σby Tx 6261.10 84.89 6345.99 common Σ 6328.97 33.34 6362.30 Indep 7282.54 13.02 7295.57
ing zero) for the comparisons between treatment 1 and the other three treatments, while the random effects model did not.
Conclusion
There were no significant differences in the change from baseline between the treatments similar to the completers only analysis. However, the point estimates and corresponding estimates of uncertainty were very different between the MAR and completers only analysis as expected.
7.3 Analysis of CTQ I using all the data under MAR (cont.
of Example 4.3)
Need to fill in this analysis: Joe Models
We will focus on the MTM’s for re-analysis under MAR. Serial correla-tion is modeled only for the weeks following the quit date. We now con-sider two different formulations for the marginal mean, the first (Model 1) assuming an separate unstructured mean for each treatment (as in Example 4.3) is the following redundent wiht chapter 4???,
logit(µij) = (1 − Zj)α∗+ Zj(β∗0+ β1∗Xi),
ANALYSIS OF CTQ I USING ALL THE DATA UNDER MAR (CONT. OF EXAMPLE 4.3) 169
Figure 7.1 Posterior means of the trajectories for each of the four treatments for the random effects model.
and the second (Model 2), an unstructured mean with a constant treat-ment effect,
logit(µij) = (1 − Zj)α∗+ Zj(β0∗+ β∗1Xi),
where superscript ∗ denotes parameters from the marginal rather than conditional distribution. For j = 6, . . . , 12, serial correlation follows
logit(φij) = ∆ij+ γiYi,j−1;
170 DATA EXAMPLES: IGNORABILITY Table 7.10 Parameters from fitting the MTM(1) model in Chapter 4 to longi-tudinal smoking cessation data from CTQ I.
Parameter Posterior mean 95% credible interval
MTM(1) β0 -.82 (-1.2,-.51)
β1 .34 (-.09,.77)
φ0 4.6 (4.2, 5.2)
φ1 0.68 (-0.10, 1.4)
Table 7.11 Comparison of dependence parameters from Models 1 and 2 fit to longitudinal smoking cessation data from CTQ I.
Parameter Posterior mean 95% credible interval
Model 1 φ0 4.6 (4.0, 5.1)
φ1 0.92 (0.04, 1.8)
Model 2 φ0 4.6 (4.1, 5.2)
φ1 0.87 (–0.04, 1.8)
correlation is allowed to differ by treatment group by setting γi= φ0+ φ1Xi.
Priors are specified as in Data example 4.3.
Results and Comparison with completers only analysis
We first compare the models fit in Chapter 4 to the completers only to the models fit here using all the data under MAR. Table 7.11 contains the results under MAR. The posterior means and credible intervals for the dependence parameters are similar to the completers only analysis (cf: Table 4.9). However, the posterior means of the marginal mean pa-rameters show some differences with the treatment effect on the quit rate (β1) now smaller, but still not significant.
NEED DIC
Alternative dependence structures
As a comparison, we also fit models with the same marginal mean struc-ture, but under independence, φ0 = φ1 = 0. Table 7.12 contains sum-maries of the posterior distributions of the marginal mean parameters.
The treatment effect is no larger than when assuming dependence and the credible interval now excludes zero, (.27, .68). Clearly, ignoring the
ANALYSIS OF WEEKLY SMOKING AND WEIGHT CHANGE FROM CTQ II USING ALL THE DATA UNDER MAR 171 Table 7.12 Parameters from fitting the model in Chapter ?? under
indepen-dence to the longitudinal smoking cessation data from CTQ I.
Parameter Posterior mean 95% credible interval
MTM(0) β0 -.80 (-.95,-.66)
β1 .47 (.27,.68)
dependence here results in a incorrect determination of a statistically significant treatment effect.
Plots of the marginal means at each time under Models 1 and 2 for the MTM(1)’s and independence models appear in Figures ?? and ??. (need these as individual figure files). need these plots
Conclusions fill in
7.4 Analysis of weekly smoking and weight change from CTQ II using all the data under MAR
intro on objectives from Chapter 1 Model
We will use the joint model for binary and continuous longitudinal data introduced in Example 6.15. Here Si= (Si1, . . . , SiT)0is the longitudinal vector of binary quit status and Yi= (Yi1, . . . , YiT)0 is the longitudinal vector of weekly weight gain with T = 6. Xy is a quadratic polynomial in time with an interaction with treatment and Xz is composed of a separate mean for each time and an interaction with treatment. We fit models allowing dependence between the two processes, Byz 6= 0 (or equivalently, Σyx 6= 0), independence between the processes, Byx = 0, and independence within and between processes (diagonal Σ).
Priors
The vector of regression coefficients, β was given an improper uniform prior. Parameterizing Σ as (Σzz, Bzy, Σ?yy), we placed a (proper) uni-form prior on the correlation matrix, Σzz, an (improper) uniform prior on the (autoregressive) coefficient matrix Bzy, and a flat prior on Σ?yy. See Liu and Daniels (2007) for verification that this specification gives a proper posterior.
Posterior sampling
172 DATA EXAMPLES: IGNORABILITY Table 7.13 DIC for CTQ II analysis. add in effective number of params etc
model DIC Joint 2746 B= 0 2843 Indep 3776
The key to posterior sampling in this model is moving between two forms of the likelihood in deriving the full conditional distributions. The stan-dard form for the multivariate likelihood is given in Example ??. Given this form and the priors specified in the previous section, the full condi-tional distribution for β can easily be derived as a multivariate normal distribution. To derive the full conditional distributions of Bzyand Σ?yy, which will be multivariate normal and inverse Wishart, respectively, the multivariate normal likelihood can be factored into p(y | z)p(z). For full details on the forms and derivations of the full conditional distributions for these models, see Liu and Daniels (2007) .
This model could be fit in WinBUGS. However, the sampling algorithm would be very inefficient as WinBUGS would not be able to move be-tween the two forms of the likelihood in constructing the full conditionals as given above.
Model Fit
To compute the DIC I, we use Monte Carlo integration and re-weighting as discussed in Chapter 3 for the multivariate probit model. Table 7.13 gives the DIC values for the three models fit. Clearly, the joint model which allows dependence between the smoking and weight gain process provides the best fit.
We also conducted posterior predictive checks using the techniques out-lined in Section 6.6. In particular, at each iteration of the sampling algorithm, we computed test statistics (for each treatment and week) as the differences in quit rates and weight gains for the replicated full data at each iteration and the completed data set at each iteration us-ing the observed data and the missus-ing data sampled durus-ing the data augmentation step. define better using T similar to paper These statistics were chosen to examine consistent over- or underestimation of the mean for each process at each measurement time. The posterior pre-dictive p-values were defined as the probabilities that these differences were bigger than zero; these are given in Table 7.14 remove table??.
ANALYSIS OF WEEKLY SMOKING AND WEIGHT CHANGE FROM CTQ II USING ALL THE DATA UNDER MAR 173 Table 7.14 Posterior predictive p-values for CTQ II analysis.
Tx Response Weeks
1 2 3 4 5 6
NE Quit rate .63 .54 .31 .22 .54 .44 Weight .48 .42 .52 .42 .54 .47 E Quit rate .45 .44 .45 .30 .41 .26 Weight .40 .36 .42 .60 .31 .41
Table 7.15 Posterior means of weekly quit rates for Exercise (E) and No Ex-ercise (NE) treatment arms with 95% credible intervals for each of the three models considered.
Joint B= 0 indep observed
NE
.39 (.34,.45) .39 (.34,.45) 0.38 (0.31,0.46) .39 .50 (.44,.55) .50 (.44, .55) 0.5 (0.43,0.56) .49 .55 (.49,.66) .55 (.49, .60) 0.56 (0.48,0.64) .57 .53 (.47, .59) .53 (.47, .59) 0.56 (0.48,0.64) .56 .53 (.47, .59) .53 (.47, .59) 0.53 (0.45,0.62) .53 .46 (.40,.52) .46 (.40,.52) 0.46 (0.40,0.52) .46 E
.45 (.39,.51) .45 (.39,.51) 0.47 (0.40,0.54) .44 .37 (.32, .43) .37 (.32,.43) 0.38 (0.30,0.47) .38 .44 (.38,.49) .44 (.38,.49) 0.45 (0.39,0.51) .44 .42 (.37,.48) .42 (.37,.48) 0.47 (0.40,0.55) .46 .46 (.40, .52) .46 (.40, .52) 0.50 (0.43,0.57) .49 .38 (.32.,.43) .38 (.32, .43) 0.43 (0.36,0.51) .42
All the p-values were between .2 and .6 providing no evidence of lack of fit.
Inference on questions of interest
Table 7.15 shows the posterior means of the quit rates under the full joint model, the model which assumes B = 0 (independence between weight change and smoking, and complete independence, Σ = I.
The reason for the small difference between the model which assumes B= 0 and the full joint model is the fact that the dependence here is
174 DATA EXAMPLES: IGNORABILITY Table 7.16 Posterior means of difference in quit rates with 95% credible in-tervals at week 12 for each of the three models considered. add one more decimal place
Model Posterior mean 95% credible interval
joint .08 (.08,.08)
B= 0 .08 (.08,.08)
indep .03 (.02, .04)
observed .04
much stronger within each process than between the processes; for ex-ample, the correlations between the quit process over time are all greater than .5, while they are all less than .2 for the correlations between the quit and weight process. See Liu and Daniels (2007) for a more detailed exploration of the dependence structure and a discussion addressing the inference on the relationship between smoking and weight between the two treatments.
Table 7.16 contains the treatment difference at the final week with 95%
credible intervals. Clearly, the independence model underestimates the treatment difference (as does the inference based on the (unmodelled) observed data). This is due to the fact that under ignorability, the values filled in for the missing responses using data augmentation were more often smokers than quitters.
Conclusions
Surprisingly, the quit rate was significantly lower on the exercise treat-ment at the end of the trial. The inclusion of the auxiliary longitudinal process weight gain had little effect on the conclusions in this example.
Point out HERS example in Chapter 4 did MAR implicitly;
reference simulation in Li’s paper???
CHAPTER 8
Models under nonignorable dropout
8.1 Introduction
We will discuss three classes of models defined in terms of specifying the joint distribution f (r, y | ω). However, before we do this, we fac-tor this joint distribution into two pieces: the first can be identified by the observed data without any modelling assumptions but the second cannot,
p(r, y | ω) = p(ymis| yobs, r, ω1)p(yobs, r | ω2). (8.1) The first piece, might be called the ’extrapolation’ distribution and is not identified without modelling assumptions. Many of the approaches discussed in this chapter will implicitly identify this extrapolation distri-bution either directly or indirectly and lead to non-distinct parameters ω1 and ω2. If the model lends itself to the factorization given above with distinct ω1and ω2, this will provide a convenient platform for sen-sitivity analysis, the topic of Chapter 9. The parameters ω1 are then not identified by the observed data without modelling assumptions. We provide a formal definition next.
Definition 8.1. Nonparametrically non-identifiable (NNI) parameters A parameter ω? is parametrically identifiable if it is a non-constant function of ω1, the parameters of the extrapolation distribution.
2
We will discuss such NNI parameters and give examples in specific model settings throughout this chapter.
8.2 Selection models 8.2.1 Background and history
A natural way to deal with incomplete data is to first specify a full data response model and then append to it a model for the selection
mech-175
176 MODELS UNDER NONIGNORABLE DROPOUT anism, i.e., how observations are selected into (or out of) the sample.
This corresponds to factoring full data model as
p(r, y | ω) = p(r | y, ψ(ω))p(y | θ(ω)), (8.2) and then specifying each component separately. This is the same fac-torization used to classify missing data, as discussed in Chapter 5 and most closely corresponds to the complete data setting where the full data response model is specified directly and then, for incomplete data, the missing data (selection) mechanism is appended to form the full data model. The direct specification of the full data response model facili-tates interpretation of covariate effects. In addition, the missing data mechanism directly relates to the assumptions of MCAR, MAR, and MNAR. Other approaches to specify the full data model do not share these convenient features (see Sections 7.3 and 7.4).
The typical strategy to model the two components, until recently, was to specify a parametric model for each and to assume that ψ and θ are
The typical strategy to model the two components, until recently, was to specify a parametric model for each and to assume that ψ and θ are