Parametric Shared Frailty Models - A Case Study

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  • Use case name: Leukemia Remission Analysis

  • Actors: Public health investigator and practitioner

  • Preconditions: A cleaned data set available based on survival time, status, and predictors, needs to be considered.

  • Description: Patricia, a public health researcher, is investigating a data sample that includes 22 participants who are recruited in a study on the treatment of leukemia. The variables that are included in the data sample are listed as follows:

    – id: Participant ID number.

    – time: Remission survival time in weeks.

    – status: Failure status, status = 0 if censored and status = 1 if failure.

    – gender: A potential confounder, gender = 0 if male and gender = 1 if female.

    – group: The exposure of interest, group = 0 if traditional treatment and group = 1 if new treatment.

    – logWBC: Another potential confounder, logarithm of white blood cell counts.

Patricia is interested in visualizing the survival and hazard functions to compare the remission under two different treatments. Moreover, Patricia would like to build a regression model to quantitatively assess the new treatment effect. One of Patricia’s choices might be Cox proportional hazards models. Aware of a small data sample, Patricia expects that a semi-parametric Cox regression model might lead to a survival or hazard function plot as step functions. To produce some charts more consistent to theoretical curves, Patricia turns to an alternative approach and considers parametric survival models. By assuming that the survival time follows a parametric distribution and that model effects are proportional to survival times, Patricia decides to build an acceleration failure time (AFT) model in SPSS Statistics.

Syntax:

 
DATA LIST FREE
/id(F5.0) time(F5.0) status(A2) gender(A2) group(A2) logWBC(F5.2). 
BEGIN DATA .
1    6   1    1    1   2.37 
2    7   1    0    1   4.52 
3    9   1    0    1   3.73 
4   11   1    1    1   4.63 
5   12   1    1    1   3.01 
6   20   1    1    1   2.85 
7   20   1    0    1   3.58 
8   24   1    1    1   2.29 
9    7   0    0    1   2.61 
10   9   0    0    1   3.23 
11  14   0    1    1   2.80 
12   1   1    0    0   2.74 
13   2   1    1    0   5.01 
14   2   1    0    0   4.89 
15   3   1    1    0   4.42 
16   3   1    0    0   4.07 
17   7   1    0    0   4.35 
18   9   1    1    0   2.47 
19   9   1    0    0   3.31 
20   9   1    1    0   3.06 
21  11   1    1    0   1.50 
22  12   1    1    0   3.94 
END DATA. 
SURVREG AFT time BY group(0) gender(0) WITH logWBC
/MODEL DISTRIBUTION=WEIBULL
/STATUS VARIABLE=status
/ESTIMATION HCONVERGE=1e−12 FCONVERGE=0 PCONVERGE=0 
/FUNCTIONPLOT SURVIVAL HAZARD PLOTBY(group).

Synopsis: The syntax that is specified by Patricia designates time as a single dependent time variable. The variables group and gender are modeled as factors and logWBC a covariate. By specifying (0) after group and gender, Patricia designates “traditional treatment” and “male” as the last categories to be modeled. A Weibull distribution is specified for the /MODEL subcommand, which is also a default setting.

Regarding the summary tables, the Model Summary table provides the procedure and model information. The Case Processing Summary table gives a comprehensive listing of the failure or censoring status and those cases that are excluded from the analysis. In Patricia’s data sample, all the participants are associated with a failure status and included in the analysis.

The estimated acceleration factor of group is 1.862, which is obtained by exponentiating the estimated regression coefficient 0.622 of the group variable. Its associated 95% confidence interval is (1.251, 2.773). These results suggest that, given that all the other predictors are held constant, the median survival time is increased by a factor of 1.862 for those participants who received the new treatment compared to the ones who received the traditional treatment. Usually, an acceleration factor greater than one might indicate a beneficial exposure (new treatment) to survival. Patricia is also aware that, for an AFT model, the acceleration factor keeps the same for any other quantile of survival time, not limited to the median, or the 50th percentile.

The survival and hazard curves are plotted separately on a chart, for the two treatments with gender and logWBC. These curves are evaluated at mean or proportion values that are shown in the Predictor Values for Plot table. As expected, Patricia observes smooth curves even with a modest sample size of 22. An obvious separation can be observed between the treatments in both survival and hazard plots. Patricia confirms that, for any fixed value of survival time, the horizontal distance is almost doubled, which is consistent to the estimated acceleration factor. To further examine some other predictors such as gender, Patricia can substitute PLOTBY(gender) for PLOTBY(group) to analyze and visualize the difference between males and females.