Parametric Model for Interval Recurrent Events Data

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

The RELIABILITY Procedure

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.