7.6. Models with a Bounded Response (Logit Link)
The Beta regression model takes care of the bounded variable naturally (by definition). In this section we introduce an alternative way to model the dose-response trial example from Section 7.5, using the linear regression model framework. After a simple transformation, the responses are not bounded and can be treated as normally distributed variables. Furthermore, to address the relationship between variance and dose level, one can model the variance as a function of the dose.
7.6.1. One-Dimensional Case
The model with a single drug can be written as
where the variance function is defined as follows
Further, Y is the response varying from A to B, x denotes the dose and the error term is normally distributed, i.e., ε ∼ N(0,1).
In this example we take A = 0 and B = 1. Note that this model provides another example of models with unknown parameters in the variance function (compare with (7.11)). Thus formulas (7.13) and (7.14) can be applied here, with
The information matrix of a single observation is
It follows from this equation that an optimal design will depend only on the values of θ3 and θ4.
7.6.2. Example 6: A Bounded-Response Model with a Logit Link and Unknown Parameters in the Variance Function
Program 7.6 calls the %OptimalDesign2 macro to find the D-optimal design for the bounded-response model defined earlier in this subsection. We need to change the %infoele macro that computes the information matrix and remove the %devpsi macro. The other macros do not change compared to Example 5. The complete SAS code of Program 7.6 can be found on the book's companion web site.
Note that, as in Program 7.5, the initial design includes six equally spaced points in the dose region χ = [0,8]. The parameter verctor is given by θ = (1,−0.5,1,−0.6)T and the fitted normal densities for each treatment group are displayed in Figure 7.9 (the densities are shown on a normal scale, i.e., ln(1 − y)/y). As in Figure 7.7, we see that, as the dose level increases, the mean response (disease severity) decreases and the variance decreases as well.
Figure 7-9. Distribution of the response variable in each treatment group
Example 7-6. D-optimal design for the bounded-response model with a logit link and unknown parameters in the variance function (Design and algorithm parameters)
* Design parameters;
%let points=do(0,8,8/5);
%let weights=repeat(1/6,1,6);
%let grid=do(0,8,8/400);
%let parameter={-1 0.5 −1 −0.6};
* Number of parameters;
%let paran=4;
* Algorithm parameters;
%let convc=1e-9;
%let maximit=1000;
%let const1=1;
%let const2=1;
%let cmerge=3;
%OptimalDesign2; |
Example. Output from Program 7.6
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
0.307 0.000
0.228 4.720
0.465 8.000 |
Output 7.6 displays the initial and D-optimal designs. The optimal design is a three-point design with two points on the boundaries and one design point inside the design region. Comparing this optimal design with the design shown in Output 7.5, it is easy to see that the middle dose (4.720) is not much different from the optimal middle dose computed from the Beta regression model (4.280); however, it receives a much greater weight. The initial and optimal designs as well as the associated sensitivity functions are depicted in Figure 7.10.
Figure 7-10. Left panel: Initial (open circles) and optimal (closed circles) designs. Right panel: Sensitivity functions for the initial (dashed curve) and optimal (solid curve) designs.
7.6.3. Models with a Bounded Response (Combination of Two Drugs)
Under the same dose-response set-up as in Section 7.5, we now consider the case of a two-drug combination: Drug A and Drug B. The daily dose of each drug ranges from 0.1 mg to 10 mg. Due to safety restrictions in drug development, often it is not safe to assign patients to the treatment with both drugs at the highest dose. In this example, we require that the total dose of the two drugs cannot exceed 10.1 mg. Note that the resulting design region is not square, it is a triangle.
Let x and z denote the daily doses of Drugs A and B, respectively. The corresponding model is
where the variance function is given by
It can be shown that the information matrix of a single observation is
where i= 1,...,n, j = 1,...,n,
Here n is the number of possible doses of each drug. Note that, similar to the one-dimensional case considered in Example 6, the optimal design depends only on the parameters of the variance function, i.e., θ7, θ8 and θ9.
7.6.4. Example 7: Two Design Variables (Two Drugs)
Program 7.7 computes the D-optimal design for the introduced model. This program calls the %OptimalDesign3 macro that supports a two-dimensional version of the optimal design algorithm defined in Section 7.1.5. The design parameters are specified in the %DesignParameters macro and then they are passed to the %OptimalDesign3 macro. The initial design (POINTS and WEIGHTS variables) includes 10 equally-weighted points that are evenly distributed across the triangular design region:
Example 7-7. D-optimal design for the bounded-response model with two design variables (Design and algorithm parameters)
%macro DesignParameters;
* Initial design: Design points;
* Four equally spaced doses for each drug;
dosen=4;
* Total number of points;
dosem=dosen*(dosen+1)/2;
points=j(dosem,2,0);
k=0;
do i=1 to dosen;
do j=1 to dosen-i+1;
k=k+1;
points[k,1]=0.1+(i-1)*9.9/(dosen-1);
points[k,2]=0.1+(j-1)*9.9/(dosen-1);
end;
end;
* Initial design: Weights;
* Equally weighted points;
weights=t(repeat(1/10,1,10));* Grid;
* 26 equally spaced points on each axis;
gridn=26;
gridm=gridn*(gridn+1)/2;
grid=j(gridm,2,0);
k=0;
do i=1 to gridn;
do j=1 to gridn-i+1;
k=k+1;
grid[k,1]=0.1+(i-1)*9.9/(gridn-1);
grid[k,2]=0.1+(j-1)*9.9/(gridn-1);
end;
end;
* Parameter values;
para={1 1 1 1 1 1 1 0.6 0.4};
%mend DesignParameters;
* Number of parameters;
%let paran=9;
* Algorithm parameters;
%let convc=1e-9;
%let maximit=500;
%let const1=1;
%let const2=1;
%let cmerge=3;
%OptimalDesign3; |
Example. Output from Program 7.7
Initial design
Drug Drug
Weight A B
0.10 0.10 0.10
0.10 0.10 3.40
0.10 0.10 6.70
0.10 0.10 10.0
0.10 3.40 0.10
0.10 3.40 3.40
0.10 3.40 6.70
0.10 6.70 0.10
0.10 6.70 3.40
0.10 10.0 0.10
Optimal design
Drug Drug
Weight A B
0.17 0.10 0.10
0.14 0.10 2.87
0.20 0.10 10.0
0.14 2.48 0.10
0.14 3.66 5.64
0.21 10.0 0.10 |
Figure 7-11. Left panel: two-dimensional grid for the optimal design problem involving a combination therapy (dots) and initial design (open circles). Right panel: optimal design (closed circles)
The grid (GRID variable) consists of 351 points in the design region. Figure 7.11 (left panel) depicts the selected grid (dots) and initial design (open circles). The parameter vector (PARAMETER variable) is given by
The information matrix and sensitivity function are defined using the %infoele, %indinfo and %info macros. The complete version of Program 7.7 is given on the book's companion web site.
Output 7.7 displays the initial and D-optimal designs. The right panel of Figure 7.11 provides a graphical summary of the optimal design (closed circles). It is worth noting that five of the six points in the optimal design lie on the boundary of the design region. The corresponding combination therapies involve the minimum dose (0.1 mg) of at least one of the two drugs. There is only one "non-trivial' combination with 3.7 mg of Drug A and 5.6 mg of Drug B.