Causal Effect Estimation and Dose Adjustment in Exposure-Response Relationship Analysis

Abstract

Determining causal exposure effects is often a challenging task even with randomized clinical trials. Confounding factors may cause bias in: (1) the pharmacokinetic exposure-response relationship and (2) dose-response relationship when dose-adjustment depends on potential responses. Dose adjustment often happens in clinical trials either designed for therapeutic dose monitoring, or spontaneously due to, for example, adverse events. It makes causal effect inference difficult since it often relates to potential response. On the other hand, dose adjustment in some trials such as the randomized concentration controlled (RCC) trials are designed to reduce confounding bias in exposure-response relationship. We review different types of dose-adjustment mechanisms and their impact on causal effect estimation with a number of dose-exposure and exposure response models. Following the concept of sequential randomization and approaches for missing data analysis, we examine a number of approaches for causal effect estimation including the classical joint modeling based on joint likelihood functions and instrumental variable and control function methods. We explore simplified approaches for joint modeling with sequential randomization conditional on potentially confounded subject effects and alternatives to the joint modeling approaches. Performance of these approaches in typical practical scenarios was assessed with a simulation study.

Appendix

This section provides a SAS program to fit the following joint model:

$$\displaystyle\begin{array}{rcl} \log (c_{\mathit{ij}})& =& \theta \log (d_{\mathit{ij}}) + v_{i} + e_{\mathit{ij}}, \\ y_{\mathit{ij}}& =& \mathit{Emax}/(1 + \mathit{EC}_{50}/c_{\mathit{ij}}) + u_{i} +\varepsilon _{\mathit{ij}},{}\end{array}$$

(9.29)

where the first one is the power model (9.2) and the second one is known as Emax model. u _i and v _i are correlated, hence u _i is a confounding factor. For simplicity no other random effect is included. This model cannot be fitted with SAS proc MIXED or GLIMMIX due to nonlinearity in the Emax model.

In dataset “joint” below, one variable rij contains both the exposure and response variables as two records identified by an indicator ind. When ind = "pk", rij = log(c _ij ) and logdose = log(d _ij ) in the power model. Otherwise (when ind = "resp"), rij = y _ij in the Emax model, and logcij = log(c _ij ) in the power model. In the program, variable i is the subject identifier, siguv is the covariance between u _i and v _i , and sigp and sigr are var(e _ij ) and $\text{var}(\varepsilon _{\mathit{ij}})$, respectively.

proc nlmixed data=simu qpoints=6;

  parms sigu=1 sigv=1 sigr=1 sigp=1 sige=1 theta=1 emax=1

     ec50=1;

  bounds sigu sigv sigr sigp sige >0;

   if ind="pk" then do;

      pred=theta*logdose +vi; g=sigp;

   end;

   else if ind="resp" then do;

      pred=emax/(1+ec50/exp(logcij))+ui; g=sigr;

   end;

  model rij~normal(pred,g);

  random ui vi~normal([0,0],[sigu,siguv,sigv]) subject=i;

run;

The program is illustration only. Adjustments on starting parameter values and options in the procedure is often necessary to fit real data. The program can be adapted to fit other types of response by specifying the likelihood function.