Byar (1980) provides data for recurrences of bladder tumors in patients in a clinical trial. Figure 16.51 is a partial listing of data for 86 patients, of which 48 were given a placebo and 38 were treated with the drug Thiotepa. The data are here grouped into one month intervals.
Figure 16.51: Partial Listing of the Bladder Tumor Data
Obs | Group | Age | Age1 | N | R |
---|---|---|---|---|---|
1 | Placebo | 0 | 1 | 48 | 0 |
2 | Placebo | 1 | 2 | 47 | 0 |
3 | Placebo | 2 | 3 | 46 | 1 |
4 | Placebo | 3 | 4 | 46 | 4 |
5 | Placebo | 4 | 5 | 46 | 7 |
6 | Placebo | 5 | 6 | 45 | 0 |
7 | Placebo | 6 | 7 | 45 | 2 |
8 | Placebo | 7 | 8 | 45 | 4 |
9 | Placebo | 8 | 9 | 44 | 1 |
10 | Placebo | 9 | 10 | 44 | 2 |
11 | Placebo | 10 | 11 | 44 | 4 |
12 | Placebo | 11 | 12 | 42 | 2 |
13 | Placebo | 12 | 13 | 42 | 1 |
14 | Placebo | 13 | 14 | 42 | 4 |
15 | Placebo | 14 | 15 | 42 | 1 |
16 | Placebo | 15 | 16 | 41 | 1 |
17 | Placebo | 16 | 17 | 41 | 5 |
18 | Placebo | 17 | 18 | 41 | 4 |
19 | Placebo | 18 | 19 | 41 | 4 |
20 | Placebo | 19 | 20 | 38 | 1 |
The following SAS statements fit a non-homogeneous Poisson process model with a power intensity function to the interval recurrence
data. Some patients were lost to follow-up in each month, so the number of patients observed changes from month to month.
The variable N
provides the number of patients available at the beginning of each month and assumed to be observed throughout the month.
The variable R
is the number of recurrences of tumors in each month. Age
represents the number of months after randomization into the trial (starting with month 0), and Age1=Age+1
is the end of a month. The variable Group
represents the treatment, either Placebo or Thiotepa. The MODEL statement requests a maximum likelihood fit of the model
with Group
as a classification variable. The MCFPLOT statement requests a plot of the fitted model and nonparametric estimates of the
mean cumulative function for each group.
proc reliability data=Tumor; distribution nhpp(pow); freq R; nenter N; class Group; model (Age Age1) = Group; mcfplot(Age Age1) = Group / fit=Model; run;
The resulting maximum likelihood parameter estimates are shown in Figure 16.52.
Figure 16.52: Power Model Parameter Estimates for the Bladder Tumor Data
NHPP-Power Parameter Estimates | |||||
---|---|---|---|---|---|
Parameter | Estimate | Standard Error | Asymptotic Normal | ||
95% Confidence Limits | |||||
Lower | Upper | ||||
Intercept | 23.5802 | 3.1567 | 17.3932 | 29.7671 | |
Group | Placebo | -4.3826 | 3.4873 | -11.2175 | 2.4523 |
Group | Thiotepa | 0.0000 | 0.0000 | 0.0000 | 0.0000 |
Shape | 1.1682 | 0.0960 | 0.9945 | 1.3723 |
Nonparametric estimates of the mean cumulative function are plotted as points, and the fitted model is plotted as the solid line in Figure 16.53. Pointwise parametric confidence intervals are plotted by default when the fit=Model option is used.
Figure 16.53: Mean Cumulative Function Plot for the Bladder Tumor Data