DEL IMPUESTO SOBRE SUCESIONES Y DONACIONES ÍNDICE GENERAL

We analyze the azadirachtin data presented in 2.6 with a view to comparing dose ef- fects on both larval and pupal mortality of whiteflies. In particular, we seek to compare the mortalities associated with the different doses separately and simultaneously for both developmental stages. Once more we pursue either of two modeling strategies: a combina- tion of two stage-specific binomial GLMs for larval and pupal mortalities, or alternatively a joint GEE approach. In addition we illustrate the parameterization issue with MMM that was described in 3.5.2.

Comparing multiple doses at multiple stages: We fit a joint binomial model for numbers of dead and alive whitefly larvae and pupae with GEE using the logit link and an AR(1) working correlation. Our independent variables are the combinations of developmental stages and dose levels, the culture substrates, and their interaction. The single plants are taken as clusters. A Wald-type test for sequentially added model terms

Tunnel 2: Top − Middle Tunnel 1: Top − Middle Glasshouse: Top − Middle Tunnel 2: Top − Bottom Tunnel 1: Top − Bottom Glasshouse: Top − Bottom Tunnel 2: Middle − Bottom Tunnel 1: Middle − Bottom Glasshouse: Middle − Bottom Top: Tunnel 2 − Tunnel 1 Top: Tunnel 2 − Glasshouse Top: Tunnel 1 − Glasshouse Middle: Tunnel 2 − Tunnel 1 Middle: Tunnel 2 − Glasshouse Middle: Tunnel 1 − Glasshouse Bottom: Tunnel 2 − Tunnel 1 Bottom: Tunnel 2 − Glasshouse Bottom: Tunnel 1 − Glasshouse

−5.0 −2.5 0.0 2.5

Estimated Difference of log(Number) of Whiteflies GEE (naive) GEE (robust) MMM

Figure 55: Greenhouse whitefly data: 95% simultaneous confidence intervals for asymptotic all- pairwise comparisons of environments per plant part and all-pairwise comparisons of plant parts per environment simultaneously based on a joint GEE (with robust or naive covariance estimation) and combined marginal GLMs (MMM).

yields a p-value of 0.560 for the interaction term, thus we can leave it out with confidence, and the model is simplified to main effects only.

In an alternative approach, we fit two separate quasibinomial GLMs with logit link func- tion and overdispersion parameter φ to the larval and pupal data, respectively. The models include main effects for dose and substrate as well as an interaction term of the

1 0.85 0.71 0.89

0.85 1 0.79 0.88

0.71 0.79 1 0.81

0.89 0.88 0.81 1

GEE (robust covariance)

1 0.90 0.81 0.72

0.90 1 0.90 0.80

0.81 0.90 1 0.90

0.72 0.80 0.90 1

GEE (naive covariance)

1 0.71 0.61 0.64

0.71 1 0.71 0.84

0.61 0.71 1 0.75

0.64 0.84 0.75 1

Multiple marginal models

Figure 56: Epileptic seizures data: correlation matrices of test statistics for many-to-one com- parisons of treatments per time point (except baseline).

8 weeks 6 weeks 4 weeks 2 weeks

−0.25 0.00 0.25 0.50 0.75 1.00

Estimated Difference of log(Rate) of Seizures

z−tests + Bonferroni GEE (naive) GEE (robust) MMM

Figure 57: Epileptic seizures data: 95% simultaneous confidence intervals for asymptotic com- parisons of treatments per time point (except baseline) based on a joint GEE (with robust or naive covariance estimation), combined marginal GLMs (MMM), and Bonferroni-adjusted z-tests per time point.

two. The interactions turn out to be nonsignificant in the analysis-of-deviance F -tests (p-values of 0.683 and 0.364) and are thus removed to simplify the models. The main effect of substrate is significant with both modeling strategies, but having said that, we focus our interpretation on the comparisons of dose levels separately and simultaneously for larvae and pupae.

The estimated mortality differences on the logit link (i.e., log odds ratios) for pairwise comparisons of neem doses separately for larvae and pupae are displayed in Table 14 along with SE estimates and adjusted p-values. When comparing GEE and MMM, the estimated differences, SEs, and p-values are just slightly different for larval mortality but there is substantial disagreement for pupae.

Pupae: 2ml/kg − 1.5ml/kg Pupae: 2ml/kg − 1ml/kg Pupae: 1.5ml/kg − 1ml/kg Larvae: 2ml/kg − 1.5ml/kg Larvae: 2ml/kg − 1ml/kg Larvae: 1.5ml/kg − 1ml/kg 0 1 2 3

Estimated log(Odds Ratio) of Death MCTs + Bonferroni GEE MMM

Figure 58: Azadirachtin data: 95% simultaneous confidence intervals for asymptotic all-pairwise comparisons of doses per developmental stage based on a joint GEE (with robust covariance estimation), combined marginal GLMs (MMM), and Bonferroni-adjusted MCTs per stage.

We observe that uncertainty of estimation is considerably larger for pupae than larvae, and it also tends to increase with dosage. As concerns larval mortality, both 1.5 and 2 ml/kg are found to be significantly superior to the manufacturer-recommended dose of 1 ml/kg (p < 0.001), and the SCI boundaries are far away from the point of no effect. By contrast, such clear effects cannot be detected for pupal mortality: only the difference between 1 and 2 ml/kg is significant—and only with the GEE-based procedure (p = 0.011) but not with MMM (p = 0.069). We see this also from the corresponding 95% SCIs in Figure 58: the discrepancy between methods is minor for comparisons of larval mortality but noticeable for comparisons of pupal mortality. One explanation for this may be that sample sizes are too small for the longitudinal MCTs. Nonetheless, the correlation matrices of test statistics look overall very similar (Figure 59).

Table 14: Simultaneous inference for the azadirachtin data: estimated log odds ratios of death, standard errors, and adjusted p-values for all-pairwise stage-wise comparisons of dose levels.

Est(GEE) Est(MMM) SE(GEE) SE(MMM) p(GEE) p(MMM) p(Bonf) Larvae: 1.5 vs. 1 ml/kg 0.880 0.875 0.228 0.223 <0.001 <0.001 <0.001 Larvae: 2 vs. 1 ml/kg 1.581 1.582 0.287 0.292 <0.001 <0.001 <0.001 Larvae: 2 vs. 1.5 ml/kg 0.701 0.706 0.310 0.338 0.113 0.163 0.132 Pupae: 1.5 vs. 1 ml/kg 0.235 0.253 0.350 0.279 0.947 0.853 1.000 Pupae: 2 vs. 1 ml/kg 1.324 1.368 0.429 0.558 0.011 0.068 0.085 Pupae: 2 vs. 1.5 ml/kg 1.089 1.115 0.499 0.600 0.135 0.260 0.293

Parameterization with multiple marginal GLMs: We have broached in 3.5.2 that parameterization of the marginal models is an issue with finite sample sizes and can have substantial impact on the covariance and SE estimates because the observed Fisher information may be widely different from the expected one due to estimation problems. We can parameterize the quasi-binomial GLMs for larval and pupal mortalities in var- ious ways. Table 15 lists four parameterizations of the stage-specific GLMs along with

1 0.29 −0.46 −0.01 −0.06 −0.05 0.29 1 0.71 −0.03 −0.40 −0.32 −0.46 0.71 1 −0.03 −0.33 −0.26 −0.01 −0.03 −0.03 1 0.19 −0.54 −0.06 −0.40 −0.33 0.19 1 0.72 −0.05 −0.32 −0.26 −0.54 0.72 1 GEE 1 0.16 −0.52 −0.00 −0.06 −0.06 0.16 1 0.76 −0.05 −0.37 −0.31 −0.52 0.76 1 −0.05 −0.27 −0.23 −0.00 −0.05 −0.05 1 0.09 −0.38 −0.06 −0.37 −0.27 0.09 1 0.89 −0.06 −0.31 −0.23 −0.38 0.89 1 Multiple marginal GLMs

Figure 59: Azadirachtin data: correlation matrices of test statistics for all-pairwise comparisons of dose levels per developmental stage.

Table 15: Simultaneous inference for the azadirachtin data: four different options of parameter- izing the quasi-binomial GLMs for larval and pupal mortalities. The effects are log odds of death, and differences are on the logit scale.

β1 β2 β3 β4

a) effect with 1 ml/kg difference of 1.5 vs. 1 ml/kg difference of 2 vs. 1 ml/kg effect of sand b) effect with 1.5 ml/kg difference of 1.5 vs. 1 ml/kg difference of 2 vs. 1.5 ml/kg effect of sand c) effect with 2 ml/kg difference of 2 vs. 1 ml/kg difference of 2 vs. 1.5 ml/kg effect of sand

d) effect with 1 ml/kg effect with 1.5 ml/kg effect with 2 ml/kg effect of sand

the interpretations of the single parameters. Our longitudinal MCT procedure can be conducted with any of these parameterizations, but they lead to different results for the azadirachtin data.

Table 16: Simultaneous inference for the azadirachtin data: standard error estimates and ad- justed p-values with different parameterizations of the quasi-binomial GLMs in the MMM approach for larval and pupal mortalities.

a) b), d) c) Stage Comparison SE p SE p SE p Larvae 1.5 vs. 1 ml/kg 0.223 < 0.001 0.216 < 0.001 0.237 0.001 2 vs. 1 ml/kg 0.292 < 0.001 0.290 < 0.001 0.296 < 0.001 2 vs. 1.5 ml/kg 0.338 0.163 0.318 0.121 0.318 0.127 Pupae 1.5 vs. 1 ml/kg 0.279 0.853 0.282 0.856 0.442 0.970 2 vs. 1 ml/kg 0.558 0.068 0.510 0.037 0.576 0.087 2 vs. 1.5 ml/kg 0.600 0.261 0.606 0.268 0.606 0.279

The SE estimates and adjusted p-values associated with the treatment differences of interest are printed in Table 16. Note that they are identical for parameterizations b) and d), which we also used for Table 14, but considerably different from a) and c). This should make clear that SE estimates and p-values from multiple marginal GLMs are to be treated with caution as the experiment’s sample size is probably insufficient.

6

Extensions and Alternatives

The methods studied in this thesis can be expanded in various directions, and we spotlight some ideas here: using ratios rather than differences of means (6.1); including further sources of “repeatedness” e.g., multiple endpoints in addition to repeated measurements over time (6.2); and employing profile likelihood test statistics for discrete data to better cope with small sample sizes (6.3). As possible alternatives to our methods, we discuss in brief multiple contrast rotation tests to speed up computation with larger dimensions (6.4), and nonparametric rank tests for longitudinal setups (6.5).

6.1

Ratios

It may be desirable for various applications to estimate effects and SCIs on a percentage scale. To this end our longitudinal MCTs could be modified so that SCIs for ratios instead of differences of treatments means per occasion, or occasion means per treatment, are computed.

A generic method for constructing a CI around a ratio of normal variates is suggested by the theorem of Fieller (1954)5. Based on this result, Zerbe et al. (1982) and Young et al. (1997) developed SCIs for fixed- and mixed-effects model parameters, respectively. Dilba et al. (2004 2006) extended these works to MCTs for ratios so that simultaneous tests and SCIs are available for arbitrary sets of linear contrasts involving ratios.

Ratio-type MCTs for multivariate Gaussian data were considered in Hasler and Hothorn (2012) and Hasler and B¨ohlendorf (2013). In similar fashion it should be a simple task to put the puzzle together and build a longitudinal MCT for Gaussian endpoints that yields SCIs for ratios of interest, in the framework of a joint LMM or multiple occasion-specific linear models.