7.7. Bivariate Probit Models for Correlated Binary Responses
In clinical trial analysis, the toxicity and efficacy responses usually occur together and it may be useful to assess them together. However, in practice the assessment of toxicity and efficacy is sometimes separated. For example, determining the maximum tolerated dose (MTD) is based on toxicity alone and then efficacy is evaluated in Phase II trials over the predetermined dose range. Obviously, the fact that the two responses from the same patient are correlated will introduce complexity into the analysis. But if we study these outcomes simultaneously, more information will be gained for future trials and treatment effects will be understood more thoroughly. In drug-response relationship, the correlation between efficacy and toxicity can be negative or positive depending on the therapeutical area. Two commonly used models, the Gumbel model (Kotz et al, 2000) and the bivariate binary Cox model (Cox, 1970), have been introduced to incorporate the two dependent outcomes, toxicity and efficacy, when both of them are dichotomous. In those two models, we needs to model the probabilities of different outcome combinations separately. When both outcomes are binary, the total number of combinations is four, but when outcomes contain more than two categories, the number of unknown parameters may increase dramatically. Here we propose a bivariate probit model which incorporates the correlated responses naturally via the correlation structure of the underlying bivariate normal distribution; see Lesaffre and Molenberghs (1991). When the number of responses is more than two, the multivariate probit model may be used in a similar fashion.
Let Y = {0 or 1} denote the efficacy response and U = {0 or 1} denote the toxicity response in a clinical trial. Here 0 indicates no response and 1 indicates a response. Let d denote the dose of a drug and
Assume that Z1 and Z2 follow bivariate normal distribution with zero mean and variance-covariance matrix
where ρ may be interpreted as the correlation measure between toxicity and efficacy for the same patient. This set-up may be viewed as a standardization of the observed responses Y and U since under the natural scale the mean and variance vary from study to study. After simple transformation, the correlated responses follow the standard bivariate normal distribution,
where θ1, θ2 are unknown parameters and f1(x) and f2(x) contain the covariates of interest. In this section, we will study a simple linear model defined as follows (refer to Fedorov, Dragalin, and Wu, 2006).
In this case, the efficacy and toxicity response rates (p1. and p.1, respectively) can be expressed as the marginals of the bivariate normal distribution,
where Φ(z) is the cumulative distribution function of the standard normal distribution. Note that p11, p1. and p.1 uniquely define the joint distribution of Y and U, i.e., p10 = p1. - p11, p01 = p.1 - p11 and p00 = 1 - p1. - p.1 + p11.
Assume that the {yi,ui}'s are independent for different i's. Then the likelihood function for {Y,U} is given by
The information matrix of a single observation is
where
ϕ(v) denotes the probability density function of the standard normal distribution, ϕ2 = f(θ1Tf,θ2Tf, ρ) denotes the probability density function of bivariate normal distribution with mean θ1Tf1 and θ2Tf2, variance 1 and correlation coefficient ρ.
7.7.1. Example 8: A Bivariate Probit Model for Correlated Binary Responses (Efficacy and Toxicity)
Consider the problem of constructing the D-optimal design for the following model:
Here x denotes the dose of the experimental drug with the dose range given by χ = [0,1]. Figure 7.12 depicts the response probabilities for
This setting represents a scenario which is often encountered in clinical trial applications. The probabilities of efficacy and toxicity responses, p.1 and p1., are both increasing as the dose increases. On the other hand, probability of "positive" response p10 (i.e., probability of positive efficacy and no toxicity) increases at the beginning. Then, at a certain dose level it begins to decrease (the probability of having a positive efficacy response without any side effects is low at high dose levels).
Figure 7-12. Bivariate probit model, probability of positive response (solid line), probability of efficacy response (dashed line) and probability of toxicity response (dotted line)
Program 7.8 constructs the D-optimal design for the described bivariate probit model by calling the %OptimalDesign4 macro. The initial design includes four equally-spaced doses in the dose range χ = [0,1] (POINTS variable) and the doses are assumed to be equally weighted (WEIGHTS variable). The grid consists of 401 equally-spaced points (GRID variable) and the PARAMETER variable specifies the true values of θ and ρ. The information matrix and sensitivity function are computed using the %infoele, %info and %infod macros. The complete version of Program 7.8 is provided on the book's companion Web site.
Example 7-8. D-optimal design for the bivariate probit model with correlated binary responses (Design and algorithm parameters)
* Design parameters;
%let points={0 0.333 0.667 1};
%let weights={0.25 0.25 0.25 0.25};
%let grid=do(0,1,1/400);
%let parameter={-0.9 10 −1.2 1.6 0.5};
* Number of parameters;
%let paran=5;
* Algorithm parameters;
%let convc=1e-9;
%let maximit=1000;
%let const1=1;
%let const2=1;
%let cmerge=5;
%OptimalDesign4; |
Example. Output from Program 7.8
Initial design
Weight X
0.250 0.000
0.250 0.333
0.250 0.667
0.250 1.000
Optimal design
Weight X
0.451 0.000
0.356 0.180
0.194 1.000 |
Output 7.8 displays the initial and D-optimal designs. The locally D-optimal design is a three-point design with unequal weights. Two optimal doses are on the boundaries of the dose range (x1* = 0, x2* = 1) and the other optimal dose lies in the middle (x3* = 0.18). Almost half of the patients are assigned to the lowest dose and about 20% of patients are assigned to the highest dose. Figure 7.13 depicts the initial and optimal designs as well as the sensitivity funtions.