Robustness of Bayesian D-optimal design for the logistic mixed model against misspecification of autocorrelation

Abstract

In medicine and health sciences mixed effects models are often used to study time-structured data. Optimal designs for such studies have been shown useful to improve the precision of the estimators of the parameters. However, optimal designs for such studies are often derived under the assumption of a zero autocorrelation between the errors, especially for binary data. Ignoring or misspecifying the autocorrelation in the design stage can result in loss of efficiency. This paper addresses robustness of Bayesian D-optimal designs for the logistic mixed effects model for longitudinal data with a linear or quadratic time effect against incorrect specification of the autocorrelation. To find the Bayesian D-optimal allocations of time points for different values of the autocorrelation, under different priors for the fixed effects and different covariance structures of the random effects, a scalar function of the approximate variance–covariance matrix of the fixed effects is optimized. Two approximations are compared; one based on a first order penalized quasi likelihood (PQL1) and one based on an extended version of the generalized estimating equations (GEE). The results show that Bayesian D-optimal allocations of time points are robust against misspecification of the autocorrelation and are approximately equally spaced. Moreover, PQL1 and extended GEE give essentially the same Bayesian D-optimal allocation of time points for a given subject-to-measurement cost ratio. Furthermore, Bayesian optimal designs are hardly affected either by the choice of a covariance structure or by the choice of a prior distribution.

Appendix: Derivation for the relative efficiency Eq. (14)

To compare designs we compute their efficiencies using the concept of equivalent sample size (see Atkinson et al. 2007, page 152 and Berger and Wong 2009, page 37). Let ${\text { Var}}\left( {\hat{\varvec{\beta }}_{\xi _s}} \right)$ and ${\text {Var}}\left( {\hat{\varvec{\beta }}_{\xi _q } } \right)$ be the variance–covariance matrices of $\hat{\varvec{\beta }}$ for the design $\xi _s$ with $s$ time points and the design $\xi _q$ with $q$ time points, respectively and $N_{s}$ and $N_{q}$ are the number of subjects for the design $\xi _s$ and $\xi _q$, respectively. For the D-criterion and a given model with $p$ parameters, the RE of design $\xi _s$ compared to design $\xi _q$ is given by:

$$\begin{aligned} \hbox {RE}\left( {\xi _s ;\xi _q } \right) =\frac{N_s }{N_q }\left[ {\frac{\det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _s } } \right) } \right] ^{-1}} \right\} } \right. }{\det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _q } } \right) } \right] ^{-1}} \right\} } \right. }} \right] ^{\frac{1}{p}}. \end{aligned}$$

(15)

where the two determinants in (15) are both based on one subject only, and the factor $N_{s}/N_{q}$ takes into account the sample size per design.

This RE (15) can be rewritten as follows:

$$\begin{aligned} \hbox {RE}\left( {\xi _s ;\xi _q } \right)&= \frac{N_s }{N_q }\exp \left\{ {\log \left\{ {\left[ {\frac{\det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _s } } \right) } \right] ^{-1}} \right\} } \right. }{\det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _q } } \right) } \right] ^{-1}} \right\} } \right. }} \right] ^{\frac{1}{p}}} \right\} } \right\} . \nonumber \\&= \frac{N_s }{N_q }\exp \left\{ {\frac{\log \det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _s } } \right) } \right] ^{-1}} \right\} -\log \det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _q } } \right) } \right] ^{-1}} \right\} } \right. } \right. }{p}} \right\} .\nonumber \\ \end{aligned}$$

(16)

Rewriting $N_{s}$ and $N_{q}$ in terms of cost ratio $k$ and number of time points for the same total cost using the cost function Eq. (13), i.e., $N_s =\frac{C}{C_2 \left( {k+s} \right) }$ and $N_q =\frac{C}{C_2 \left( {k+q} \right) }$, we obtain

$$\begin{aligned} \hbox {RE}\left( {\xi _s ;\xi _q } \right) =\frac{k+q}{k+s}\left[ {\exp \left\{ {\frac{\log \det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _s } } \right) } \right] ^{-1}} \right\} -\log \det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _q } } \right) } \right] ^{-1}} \right\} } \right. } \right. }{p}} \right\} } \right] .\nonumber \\ \end{aligned}$$

(17)

This RE (17) is for locally optimal design, i.e., for given parameter values. By generalizing this to Bayesian design, the RE of design $\xi _s$ compared to design $\xi _q$ with prior distribution $\pi$ for $\varvec{\beta }$ becomes as follows:

$$\begin{aligned}&\!\!\!\!\!\hbox {RE}\left( {\xi _s ;\xi _q (\pi )\left| \pi \right. } \right) \nonumber \\&\!=\!\frac{k+q}{k+s}\left[ {\exp \left\{ {\frac{E_\beta \log \det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _s } } \right) } \right] ^{-1}} \right\} \!-\!E_\beta \log \det \left\{ {\left. {\left[ {\hbox {Var}\left( {\hat{\varvec{\beta }}_{\xi _q } } \right) } \right] ^{-1}} \right\} } \right. } \right. }{p}} \right\} } \right] .\nonumber \\ \end{aligned}$$

(18)

Thus, using the Bayesian D-optimality criterion (12), the RE will be:

$$\begin{aligned} \hbox {RE}\left( {\xi _s ;\xi _q (\pi )\left| \pi \right. } \right) =\frac{k+q}{k+s}\left[ {\exp \left\{ {\frac{\phi _\mathrm{D} \left( {\xi _s \left| \pi \right. } \right) -\phi _\mathrm{D} \left( {\xi _q \left| \pi \right. } \right) }{p}} \right\} } \right] . \end{aligned}$$

(19)

When the ratio $\frac{k+q}{k+s}$ is one, that is, if either $q=s$ or the cost ratio $k$ is very large, this RE (19) becomes the same as the RE given by Chaloner and Larntz (1989).