Latent variable indirect response joint modeling of a continuous and a categorical clinical endpoint

Abstract

Informative exposure–response modeling of clinical endpoints is important in drug development. There has been much recent progress in latent variable modeling of ordered categorical endpoints, including the application of indirect response (IDR) models and accounting for residual correlations between multiple categorical endpoints. This manuscript describes a framework of latent-variable-based IDR models that facilitate easy simultaneous modeling of a continuous and a categorical clinical endpoint. The model was applied to data from two phase III clinical trials of subcutaneously administered ustekinumab for the treatment of psoriatic arthritis, where Psoriasis Area and Severity Index scores and 20, 50, and 70 % improvement in the American College of Rheumatology response criteria were used as efficacy endpoints. Visual predictive check and external validation showed reasonable parameter estimation precision and model performance.

Appendices

Appendix

The lognormal latent variable model derivation and a NONMEM code to implement the conditional approach are given below.

Lognormal latent variable model for categorical data

For the ACR variable defined in METHODS, let L(t) be the latent variable and γ_k, k = 1, 2, 3 be the thresholds such that

$${\text{ACR}} \le {\text{k}} \Leftrightarrow {\text{L}}\left( {\text{t}} \right) < {{\upgamma}}_{{\text{k}}}$$

Assuming the latent L(t) as positive and modeled as

$${\text{L}}\left( {\text{t}} \right) = {\text{M}}\left( {\text{t}} \right) \, {{exp}}({{\upsigma \upvarepsilon }})$$

where ε ~ N(0, 1) and σ is the error standard deviation. Then

$$\begin{aligned} {\text{prob}}\left[ {{\text{ACR}} \le {\text{k}}} \right] & = {\text{prob}}[{\text{L}}\left( {{t}} \right) < {{\upgamma}}_{{\text{k}}} ] = {\text{prob}}\{ {\text{log}}\left[ {{\text{L}}\left( {\text{t}} \right)} \right] < {\text{log}}({{\upgamma}}_{{{k}}} )\} \\ & = {\text{prob}}\{ {{\upvarepsilon}} < ({\text{log}}({{\upgamma}}_{{\text{k}}} ) - {\text{log}}\left[ {{\text{M}}\left( {\text{t}} \right)} \right])/{{\upsigma}}\} = {{\Phi}}\{ {\text{(log}}({{\upgamma}}_{{\text{k}}} ) - {\text{log}}\left[ {{\text{M}}\left( {\text{t}} \right)} \right])/{{\upsigma}}\} \\ \end{aligned}$$

Write β_k = log(γ_k)/σ, then:

$${{\Phi}}^{{ - {{1}}}} \{ {\text{prob}}\left[ {{\text{ACR}} \le {\text{k}}} \right]\} = {\text{(log}}({{\upgamma}}_{{\text{k}}} ) - {\text{log}}\left[ {{\text{M}}\left( {\text{t}} \right)} \right])/{{\upsigma}} = {{\upbeta}}_{{\text{k}}} - {\text{log}}\left[ {{\text{M}}\left( {\text{t}} \right)} \right]/{{\upsigma}}$$

Note here that if log[M(t)] contains no proportional constant parameters, then σ = 1 may not be assumed. In this form, it is convenient to incorporate placebo effect multiplicatively as M(t) = f_p(t) f_d(t). Let R₀ = f_d(0), this leads to

$$\begin{aligned} \Phi^{ - 1} \{ {\text{prob}}{\text{[ACR}} \le {\text{k}}]\} & = \beta_{k} - { \log }\left[ {{\text{f}}_{\text{p}} \left( {\text{t}} \right){\text{ f}}_{\text{d}} \left( {\text{t}} \right)} \right]/\sigma = {{\beta }}_{\text{k}} - { \log }\left[ {{\text{R}}_{ 0} {\text{f}}_{\text{p}} \left( {\text{t}} \right){\text{ f}}_{\text{d}} \left( {\text{t}} \right)/\text{R}_{0} } \right]/\sigma \\ & = {{\upbeta}}_{{{k}}} - {\text{log}}\left( {{\text{R}}_{{\text{0}}} } \right)/{{\upsigma}} - {\text{log}}\left[ {{\text{f}}_{{\text{p}}} \left( {\text{t}} \right)} \right]/{{\upsigma}} - {\text{log}}\left[ {{\text{f}}_{{\text{d}}} \left( {\text{t}} \right)/{\text{R}}_{{\text{0}}} } \right]/{{\upsigma}} \\ \end{aligned}$$

Write α_k = β_k − log(R₀)/σ, g_p(t) = −log[f_p(t)]/σ, D_e = 1/σ, and g_d(t) = f_d(t)/R₀, the above becomes:

$${{\Phi}}^{{ - {{1}}}} \{ {\text{prob}}\left[ {\text{{ACR}} \le {\text{k}}} \right]\} = {{\upalpha}}_{{\text{k}}} + {\text{g}}_{{\text{p}}} \left( {\text{t}} \right) - {\text{D}}_{{\text{e}}} {\text{log}} \left[ {\text{g}}_{{\text{d}}} \left( {\text{t}} \right) \right]$$

(16)

where g_d(0) = 1. As in Eq. 2, BSV may be modeled at the intercept level with η ~ N(0, ω²), The placebo model g_p(t) may still be empirically chosen, e.g., as the reduced model Eq. 4. A convenient choice of the drug model term g_d(t) is the class of the baseline-normalized IDR models [6, 19]

$${\text{dR}}_{2} /{\text{dt}} = {\text{k}}_{\text{in}} [1 + {\text{H}}_{1} \left( {\text{t}} \right)\left] {{\text{ }} - {\text{ k}}_{\text{out}} } \right[1 + {\text{H}}_{2} \left( {\text{t}} \right)]{\text{ R}}_{2} ,{\text{ k}}_{\text{in}} = {\text{k}}_{\text{out}} ,{\text{ R}}_{2} \left( 0 \right) = 1$$

(17)

where H₁(t) and H₂(t) indicate the corresponding inhibitory/stimulator term in the IDR model.

The lognormal latent variable model (Eq. 16) differs from the normal latent variable model (Eqs. 2–3) in that the last term as the drug effect is included in the model under a log-transformation. Both models have the same number of parameters, which can be seen, e.g., when a Type I IDR model is used, by comparing Eqs. 16–17 with Eqs. 6–7.

NONMEM Code and Data Format

An illustrative NONMEM code is given below along with the data format used. The dataset has ACR and PASI observations on the same line, and a variable FLGDVJ is used to indicate whether any of ACR and PASI observations is missing.

Table 4 Illustrative NONMEM data format