7.5. Models with a Bounded Response (Beta Models
The beta regression model provides another example where the information matrix can be calculated in a closed form.
7.5.1. General Model
In clinical trials, investigators often need to deal with ordinal variables containing many categories. The Beta regression model has been shown to be a good choice to analyze this type of response; see Wu, Fedorov and Propert (2005).
Figure 7-6. Left panel: Initial (open circles) and optimal (closed circles) designs. Right panel: Sensitivity functions for the initial (dashed curve) and optimal (solid curve) designs.
In this section we consider an example from a randomized, double-blind, placebo-controlled, parallel group clinical trial. Each patient was randomized to one of four treatment arms, with 0, 2, 4, and 8 dose units, respectively. The observations were taken at baseline and the end of the study at 6 months. The response is a severity measure score that varies between 0 and 1 on the on the normalized scale, with hundreds of levels. Higher scores indicate higher degrees of severity of the illness.
After investigating the data on the descriptive level, we observed that the median score has a slightly decreasing trend when dose increases. But the difference may not be big enough to be statistically and/or clinically significant. However, the 8 dose units group has a smaller variance than the other groups. Even among the other three treatment groups, the estimated population variances are different from one group to another. These facts indicate that, although the drug did not reduce the overall mean response, it might be effective in some subjects.
Let yij denote the response rate from patient j under dose level xi and assume that yij follows the Beta distribution with parameters p(xi) and q(xi). We can model p(xi) and q(xi) as a function of the dose x:
Note that
where B(p,q) = Γ(p)Γ(q)/Γ(p + q) and Γ(p) = ∫0∞ tp−1 e−t dt is the Gamma function. Let θ = (αT,βT), denote p(xi) and q(xi) as pi and qi, respectively, and consider the Digamma or Trigamma functions (Abramowitz and Stegun, 1972):
The information matrix μ(xi,θ) of a single observation at point xi is given by
The information matrix of N experiments, with ni independent observations at dose xi, admits presentation (7.2).
Here we consider a simple example with α = (α1,α2)T, β = (β1,β2)T and f(x) = Ψ(x) = (1,x)T. In that case the individual information matrix may be written as follows:
where 
is the Kronecker product.
Figure 7.7 displays the fitted Beta densities for each treatment group with
The figure shows that, as the dose level increases, the mean response becomes smaller and thus we see evidence of a drug effect. Note also that the variance of the response variable clearly changes with the average response level and the highest dose group has the smallest variance.
Figure 7-7. Distribution of simulated responses in each treatment group
Unlike quantal dose-response models discussed in Section 7.2 or continuous logistic models from Sections 7.3–7.4, the presentation (7.16) for the information matrix of the Beta regression model cannot be factorized. As a result, the general formulas (7.6) and (7.7) have to be implemented in this case.
7.5.2. Example 5: Beta Regression Model
To construct a D-optimal design for the clinical trial example discussed above, we use the following Beta regression model:
Program 7.5 computes a D-optimal design for the Beta regression model by calling the %OptimalDesign2 macro. The initial design contains 6 equally spaced doses between 0 and 8 dose units, i.e., χ = [0,8]. The true value of the parameter vector is θ = (4,−0.49,3.9,0.15)T.
The overall optimal design algorithm in this problem remains the same; however, unlike the previous examples, the information matrix is calculated directly, without using partial derivatives of the response function. Therefore, in this example (and subsequent examples in this chapter) the user does not need to specify the %deriv1 or %deriv2 macros. The direct calculation of the information matrix relies on the %Info, %infoele and %indinfo macros. The analytical form of the information matrix is specified in the %info macro. The %infoele macro is needed to store the individual information matrices based on the values of covariates on the grid. Elements of the information matrices are stored in different vectors based on the position in the matrix. Note that the information matrix is a symmetric matrix so only the upper (lower) triangular matrix is needed when storing it. For example, the ELEO14 vector contains the elements in the first row, fourth column of all the information matrices. The %indinfo macro outputs the individual information matrix for a given candidate point (note that the MDERIV macro variable is not used in this program and can be set to any value). The complete SAS code for the updated macros is provided on the book's companion web site.
Example 7-5. D-optimal design for the beta regression model (Design and algorithm parameters)
* Design parameters;
%let points=do(0,8,8/5);
%let weights=repeat(1/6,1,6);
%let grid=do(0,8,8/200);
%let parameter={4 −0.49 3.9 0.15};
* Number of parameters;
%let paran=4;
* Algorithm parameters;
%let convc=1e-9;
%let maximit=1000;
%let const1=2;
%let const2=1;
%let cmerge=5;
%OptimalDesign2; |
Example. Output from Program 7.5
Initial design
Weight X
0.167 0.000
0.167 1.600
0.167 3.200
0.167 4.800
0.167 6.400
0.167 8.000
Optimal design
Weight X
0.491 0.000
0.071 4.280
0.438 8.000 |
Output 7.5 shows the 6-point initial and 3-point optimal designs. Note that most of the weight in this D-optimal design is assigned to the two points on the boundaries of the dose range (placebo and 8 dose units). The middle point receives very little weight (only 7% of the patients will be allocated to this dose). Figure 7.8 displays the initial and optimal designs and their sensitivity functions.
Figure 7-8. Left panel: Initial (open circles) and optimal (closed circles) designs. Right panel: Sensitivity functions for the initial (dashed curve) and optimal (solid curve) designs.
