Extending the latent variable model for extra correlated longitudinal dichotomous responses

Abstract

Since generalized nonlinear mixed-effects modeling methodology of ordered categorical data became available in the pharmacokinetic/pharmacodynamic (PK/PD) literature over a decade ago, pharmacometricians have been increasingly performing exposure–response analyses of such data to inform drug development. Also, as experiences with and scrutiny of these data have increased, pharmacometricians have noted fewer transitions (or greater correlations) between response values than predicted by the model. In this paper, we build on the latent variable (LV) approach, which is convenient for incorporating pharmacological concepts such as pharmacodynamic onset of drug effect, and present a PK/PD methodology which we term the multivariate latent variable (MLV) approach. This approach uses correlations between the latent residuals (LR) to address extra correlation or a fewer number of transitions, relative to if the LR were independent. Four approximation methods for handling dichotomous MLV data are formulated and then evaluated for accuracy and computation time using simulation studies. Some analytical results for models linear in the subject-specific random effects are also presented, and these provide insight into modeling such repeated measures data. Also, a case study previously analyzed using the LV approach is revisited using one of the MLV approximation methods and the results are discussed. Overall, consideration of the simulation and analytical results lead us to some conclusions we feel are applicable to many of the models and situations frequently encountered in analysis of such data: the MLV approach is a flexible method that can handle many different extra correlated data structures and therefore can more accurately predict the number of transitions between response values; incorrect modeling of the population covariances by implementing an LV model when extra correlation exists is not likely to (and in many cases does not) influence accuracy of the population (marginal) mean predictions; adequate prediction of the population mean probabilities achieves adequate predictions of the population variances, regardless of the correct specification of the population covariances—that is, if the LV model accurately predicts the means in the presence of extra correlation, it will accurately predict the variances; the between subject random effects component to the model describe the marginal covariances in responses—not the marginal variances as with continuous-type data. From these conclusions we make a general statement that it may not be necessary to model the extra correlation in every case using the MLV model, which requires technical implementation with currently available commercially or publically available software. The LV model may be sufficient for answering many of the typical questions arising during drug development. The MLV approach should be considered however if prediction or simulation of individual level data is an objective of the analysis.

Appendices

Appendices

Appendix 1: Stochastic integral approximation

$$ \Upsigma = \left[ {\begin{array}{llll} 1 & {\rho_{12} } & \cdots & {\rho_{1n} } \\ {\rho_{12} } & 1 & {} & {\rho_{2n} } \\ \vdots & {} & \ddots & \vdots \\ {\rho_{1n} } & {\rho_{2n} } & \cdots & 1 \\ \end{array} } \right] = S^{T} S = \left[ {\begin{array}{llll} {s_{11} } & 0 & \cdots & 0 \\ {s_{12} } & {s_{22} } & {} & 0 \\ \vdots & {} & \ddots & \vdots \\ {s_{1n} } & {s_{2n} } & \cdots & {s_{nn} } \\ \end{array} } \right]S $$

$$ s_{11} = 1;\,s_{ll} = \sqrt {1 - \sum\limits_{k = 1}^{l - 1} {s_{kl}^{2} } } ;\,s_{lm} = \left( {\rho_{lm} - \sum\limits_{k = 1}^{l - 1} {s_{kl} s_{km} } } \right)s_{ll}^{ - 1} $$

$$ \begin{aligned} P\left( {Y_{l} = y_{l} \left| {Y_{1} = y_{1} } \right., \ldots ,Y_{l - 1} = y_{l - 1} } \right) & = P\left( {\varepsilon_{l} \bar{ \le }q_{l} \left| {\varepsilon_{1} \bar{ \le }q_{1} } \right., \ldots ,\varepsilon_{l - 1} \bar{ \le }q_{l - 1} } \right) \\ & \approx Q_{{l\left| {1, \ldots ,l - 1} \right.}} = \left( { - 1} \right)^{{\left( {1 - y_{l} } \right)}} \times \tfrac{1}{M}\sum\limits_{j = 1}^{M} \Upphi \left( {\left( {q_{l} - \sum\limits_{k = 1}^{l - 1} {s_{kl} e_{kj}^{ * } } } \right)s_{ll}^{ - 1} } \right) \\ \end{aligned} $$

where $e_{kj}^{ * } = \Upphi^{ - 1} \left( {1 - y_{k} + \left( { - 1} \right)^{{\left( {1 - y_{k} } \right)}} u_{kj}^{ * } \Upphi \left( {\left( { - 1} \right)^{{\left( {1 - y_{k} } \right)}} \left( {q_{k} - \sum\limits_{g = 1}^{k - 1} {s_{g,k} e_{gj}^{ * } } } \right)s_{k,k}^{ - 1} } \right)} \right)$ and $u_{kj}^{ * }$ is a random draw from U(0,1).

Appendix 2: Gaussian quadrature

Borrowing some of the quantities from Appendix 1,

$$ \begin{aligned} P\left( {Y_{l} = y_{l} \left| {Y_{1} = y_{1} } \right., \ldots ,Y_{l - 1} = y_{l - 1} } \right) = P\left( {\varepsilon_{l} \bar{ \le }q_{l} \left| {\varepsilon_{1} \bar{ \le }q_{1} } \right., \ldots ,\varepsilon_{l - 1} \bar{ \le }q_{l - 1} } \right) \\ \approx Q_{{l\left| {1, \ldots ,l - 1} \right.}} = \left( { - 1} \right)^{{\left( {1 - y_{l} } \right)}} \times \Upphi \left( {\left( {q_{l} - \sum\limits_{k = 1}^{l - 1} {s_{kl} e_{kl}^{IOV} } } \right)s_{ll}^{ - 1} } \right) \\ \end{aligned} $$

where $e_{kl}^{IOV} = \Upphi^{ - 1} \left( {1 - y_{k} + \left( { - \Upphi \left( {\eta_{kl}^{IOV} } \right)} \right)^{{\left( {1 - y_{k} } \right)}} \Upphi \left( {\left( { - 1} \right)^{{\left( {1 - y_{k} } \right)}} \left( {q_{k} - \sum\limits_{g = 1}^{k - 1} {s_{g,k} e_{gl}^{IOV} } } \right)s_{k,k}^{ - 1} } \right)} \right)$ and η ^IOV _kl is a N(0,1) random effect. Note that in this case the e’s and the η’s have two subscript variables to imply that a different sequence of random effects was used for each conditional probability.

Appendix 3: Binary variable method

From Drezner and Wesolowsky

$$ \begin{aligned} K\left( {q_{1} ,q_{2} ;\rho_{12} } \right)&=& \rho_{12} \left[ {5\phi_{2} \left( {q_{1} ,q_{2};\left( {1 - \sqrt {{3 \mathord{\left/ {\vphantom {3 5}} \right.\kern-\nulldelimiterspace} 5}} } \right){{\rho_{12} }\mathord{\left/ {\vphantom {{\rho_{12} } 2}} \right.\kern-\nulldelimiterspace} 2}} \right) + 8\phi_{2} \left( {q_{1},q_{2} ;{{\rho_{12} } \mathord{\left/ {\vphantom {{\rho_{12} } 2}}\right. \kern-\nulldelimiterspace} 2}} \right)} \right. + {{\left.{5\phi_{2} \left( {q_{1} ,q_{2} ;\left( {1 + \sqrt {{3\mathord{\left/ {\vphantom {3 5}} \right.\kern-\nulldelimiterspace} 5}} } \right){{\rho_{12} }\mathord{\left/ {\vphantom {{\rho_{12} } 2}} \right.\kern-\nulldelimiterspace} 2}} \right)} \right]} \mathord{\left/{\vphantom {{\left. {5\phi_{2} \left( {q_{1} ,q_{2} ;\left( {1 +\sqrt {{3 \mathord{\left/ {\vphantom {3 5}} \right.\kern-\nulldelimiterspace} 5}} } \right){{\rho_{12} }\mathord{\left/ {\vphantom {{\rho_{12} } 2}} \right.\kern-\nulldelimiterspace} 2}} \right)} \right]} {18}}} \right.\kern-\nulldelimiterspace} {18}} + \Upphi \left( { - q_{1} }\right)\Upphi \left( { - q_{2} } \right) + \Upphi \left( {q_{1} }\right) + \Upphi \left( {q_{2} } \right) - 1 \\ \end{aligned} $$

K(q _i1, q _i2; ρ₁₂) ≈ Φ(q _i1, q _i2; ρ₁₂) and ρ₁₂ is the correlation between ε ₁ and ε ₂.

$$ H_{lm} = \left( { - 1} \right)^{{y_{l} }} \left( { - 1} \right)^{{y_{m} }} K\left( {q_{l} ,q_{m} ,\rho_{lm} } \right) + y_{m} \left( { - 1} \right)^{{1 - y_{l} }} \Upphi \left( {q_{l} } \right) + y_{l} \left( { - 1} \right)^{{1 - y_{ml} }} \Upphi \left( {q_{m} } \right) + \frac{{y_{l} + y_{m} }}{{\left( { - 2} \right)^{{y_{l} y_{m} }} }} $$

where K(·) is defined above. Let $M_{l} = \Upphi \left({\left({- 1} \right)^{{1 - y_{l}}} q_{l}} \right)$ and $C_{ll} = M_{l} - M_{l}^{2}$ and $C_{lm} = H_{lm} - M_{l} M_{m}$, then

$$ \begin{aligned} P\left( {Y_{l} = y_{l} \left| {Y_{1} = y_{1} } \right., \ldots ,Y_{l - 1} = y_{l - 1} } \right) = P\left( {\varepsilon_{l} \bar{ \le }q_{l} \left| {\varepsilon_{1} \bar{ \le }q_{1} } \right., \ldots ,\varepsilon_{l - 1} \bar{ \le }q_{l - 1} } \right) \\ \approx Q_{{l\left| {1, \ldots ,l - 1} \right.}} = M_{l} + \left[ {C_{1l} , \ldots ,C_{l - 1,l} } \right]\left[ {\begin{array}{llll} {C_{11} } & \cdots & {C_{1,l - 1} } \\ \vdots & \ddots & \vdots \\ {C_{1,l - 1} } & \cdots & {C_{l - 1,l - 1} } \\ \end{array} } \right]^{ - 1} \left[ {\begin{array}{ll} {1 - M_{1} } \hfill \\ {\quad \vdots } \hfill \\ {1 - M_{l - 1} } \hfill \\ \end{array} } \right] \\ \end{aligned} $$

Based on preliminary evaluation (not shown) the 3 point approximation of K which can be programmed with 1 line of code has acceptable accuracy compared to the 5 or 10 point approximation.

Appendix 4: Mendell–Elston

This approximation is defined recursively. We slightly modify the notation of Rice et al. for clarity and construct a general likelihood.

$$ \begin{aligned} P\left( {Y_{l} = y_{l} \left| {Y_{1} = y_{1} } \right., \ldots ,Y_{l - 1} = y_{l - 1} } \right) = P\left( {\varepsilon_{l} \bar{ \le }q_{l} \left| {\varepsilon_{1} \bar{ \le }q_{1} } \right., \ldots ,\varepsilon_{l - 1} \bar{ \le }q_{l - 1} } \right) \\ \approx Q_{{l\left| {1, \ldots ,l - 1} \right.}} = \Upphi \left( {\left( { - 1} \right)^{{\left( {1 - y_{l} } \right)}} Z_{{l\left| {0,1, \ldots ,l - 1} \right.}} } \right) \\ \end{aligned} $$

where $Z_{{l\left| {0,1, \ldots ,j} \right.}} = \left( {Z_{{l\left| {0,1, \ldots ,j - 1} \right.}} - a_{{j\left| {0,1, \ldots ,j - 1} \right.}} r_{{jl\left| {0,1, \ldots ,j - 1} \right.}} } \right)\sigma_{{l\left| {0,1, \ldots ,j} \right.}}^{ - 1}$, $a_{{j\left| {0,1, \ldots ,j - 1} \right.}} = {{\left( { - 1} \right)^{{y_{j} }} \phi \left( {Z_{{j\left| {0,1, \ldots ,j - 1} \right.}} } \right)} \mathord{\left/ {\vphantom {{\left( { - 1} \right)^{{y_{j} }} \phi \left( {Z_{{j\left| {0,1, \ldots ,j - 1} \right.}} } \right)} {\Upphi \left( {( - 1)^{{\left( {1 - y_{j} } \right)}} Z_{{j\left| {0,1, \ldots ,j - 1} \right.}} } \right)}}} \right. \kern-\nulldelimiterspace} {\Upphi \left( {( - 1)^{{\left( {1 - y_{j} } \right)}} Z_{{j\left| {0,1, \ldots ,j - 1} \right.}} } \right)}}$, $\sigma_{{k\left| {0,1, \ldots ,i} \right.}} = \sqrt {1 - r_{{ik\left| {0,1, \ldots ,i} \right. - 1}}^{2} a_{{i\left| {0,1, \ldots ,i} \right. - 1}} \left( {a_{{i\left| {0,1, \ldots ,i} \right. - 1}} - Z_{{i\left| {0,1, \ldots ,i} \right. - 1}} } \right)}$, and $r_{{mn\left| {0,1, \ldots ,j} \right.}} = \left( {r_{{mn\left| {0,1, \ldots ,j - 1} \right.}} - r_{{jm\left| {0,1, \ldots ,j - 1} \right.}} r_{{jn\left| {0,1, \ldots ,j - 1} \right.}} a_{{j\left| {0,1, \ldots ,j} \right. - 1}} \left( {a_{{j\left| {0,1, \ldots ,j - 1} \right.}} - Z_{{j\left| {0,1, \ldots ,j - 1} \right.}} } \right)} \right)\sigma_{{m\left| {0,1, \ldots ,j} \right.}}^{ - 1} \sigma_{{n\left| {0,1, \ldots ,j} \right.}}^{ - 1}$.

Note that $Z_{l\left| 0 \right.} = q_{l}$, $r_{lm\left| 0 \right.} = \rho_{lm}$.

Appendix 5: Calculation of a marginal covariance

$$ \begin{aligned} COV\left( {Y_{i} ,Y_{j} } \right) = E\left( {Y_{i} Y_{j} } \right) - E\left( {Y_{i} } \right)E\left( {Y_{i} } \right) = E_{\eta } \left( {E\left( {Y_{i} Y_{j} \left| \eta \right.} \right)} \right) - E_{\eta } \left( {E\left( {Y_{i} \left| \eta \right.} \right)} \right)E_{\eta } \left( {E\left( {Y_{j} \left| \eta \right.} \right)} \right) \\ = E_{\eta } \left( {\Upphi_{2} \left( {q_{i} ,q_{j} ,\rho \left| \eta \right.} \right)} \right) - E_{\eta } \left( {\Upphi \left( {q_{i} \left| \eta \right.} \right)} \right)E_{\eta } \left( {\Upphi \left( {q_{j} \left| \eta \right.} \right)} \right) \\ \approx \tfrac{1}{M}\sum\limits_{k = 1}^{M} {\Upphi_{2} \left( {q_{i} ,q_{j} ,\rho \left| {\eta_{k}^{(s)} } \right.} \right)} - \left[ {\tfrac{1}{M}\sum\limits_{k = 1}^{M} {\Upphi \left( {q_{i} \left| {\eta_{k}^{(s)} } \right.} \right)} } \right]\left[ {\tfrac{1}{M}\sum\limits_{k = 1}^{M} {\Upphi \left( {q_{j} \left| {\eta_{k}^{(s)} } \right.} \right)} } \right] \\ \end{aligned} $$

The PROBBNRM function in SAS was used to compute $\Upphi_{2} \left( {q_{i} ,q_{j} ,\rho \left| {\eta_{k}^{(s)} } \right.} \right)$ and η ^(s) _k was sampled from a N(0, $\hat{\Upomega }$) distribution

Computation of the marginal variance

$$ V\left( {Y_{i} } \right) = E\left( {Y_{i}^{2} } \right) - E\left( {Y_{i}^{{}} } \right)^{2} = \tfrac{1}{M}\sum\limits_{k = 1}^{M} {\Upphi \left( {q_{i} \left| {\eta_{k}^{(s)} } \right.} \right)\left[ {1 - \tfrac{1}{M}\sum\limits_{k = 1}^{M} {\Upphi \left( {q_{i} \left| {\eta_{k}^{(s)} } \right.} \right)} } \right]} $$

where $E\left( {Y_{i}^{2} } \right) = E\left( {Y_{i} } \right)$ and $E\left( {Y_{i} } \right) = P\left( {Y_{i} } \right).$