# Parametric Shared Frailty Models - A Case Study for Recurrent Data

Parametric Shared Frailty Models - A Case Study for Recurrent Data

• Use case name - Treatment Side Effect.

• Actors - Public health investigator and practitioner.

• Preconditions - A cleaned data set available based on survival time, side effect status, and predictors to be adjusted.

• Description - Patrick, a public health investigator, is investigating a data sample that includes 20 participants. These participants are recruited in a study on a mild side effect that is potentially caused by a new treatment. The treatment designer claims that there would be no differences between males and females, regarding the side effect. Patrick would like to evaluate such a hypothesis. The variables that are included in the data sample are listed as follows:

– patID: ID number to identify a unique participant.

– endTime: Survival time (in days) of the side effect, following a treatment, which is measured from the start of a treatment to either a side effect reported or censoring within 60 days.

– sideEffect: Side effect status, status = 0 if censored and status = 1 if the mild sided effect is reported.

– age: Participant’s age at the research period.

– female: female = 0 if male and female = 1 if female.

Multiple treatments might apply, which results in the multiple records of recurrence times that are measured for a certain participant. The start time is always 0 for each record, which is omitted in the data sample. Patrick is interested in visualizing the survival and hazard functions to draw a comparison between a male and a female by controlling their age and frailty. He is aware that those treatments that are administered to the same participant are more correlated. By assuming that the survival time follows a Weibull distribution, Patrick decides to build a parametric shared-frailty survival model in SPSS Statistics to account for the treatment dependence for the same participant.

• Syntax-

```
DATA LIST FREE
/patID(F5.0) endTime(F5.0) sideEffect(F2.0) age(F5.2) female(F2.0) .
BEGIN DATA .
1 45 0 38.00 0
2 26 1 20.00 1
3 58 0 53.00 0
4 31 1 37.00 1
4 24 0 37.00 1
4 50 0 37.00 1
5 20 1 51.00 0
5 38 1 51.00 0
6 30 0 35.00 1
7 22 1 58.00 1
8 53 1 29.00 1
8 49 1 29.00 1
9 25 0 45.00 0
9 25 0 45.00 0
10 27 0 33.00 1
11 34 1 21.00 1
11 40 0 21.00 1
11 49 0 21.00 1
12 42 1 26.00 0
13 25 0 40.00 0
14 21 1 52.00 0
14 32 1 52.00 0
15 56 0 28.00 1
15 34 0 28.00 1
16 30 0 41.00 0
16 29 0 41.00 0
17 25 1 27.00 0
18 26 1 54.00 1
18 36 1 54.00 1
19 27 0 39.00 0
20 58 1 22.00 1
20 54 0 22.00 1
20 43 1 22.00 1
END DATA.
SURVREG RECURRENT endTime WITH age BY female
/MODEL SUBJECT=patID FRAILTY=GAMMA DISTRIBUTION=WEIBULL
/ESTIMATION HCONVERGE=1e−12 PCONVERGE=0 FCONVERGE=0
/STATUS VARIABLE=sideEffect FAILURE=1 RIGHT=0
/FUNCTIONPLOT SURVIVAL HAZARD PLOTBY(female) .
```

**Synopsis: **

The syntax that is specified by Patrick designates endTime as a single dependent time variable. The procedure automatically assumes that the start time is 0 for each record. The variables age and female are modeled as a covariate and a factor, respectively. The recurrence survival times are assumed to follow a Weibull distribution. The unobserved frailty term is assumed to follow a Gamma distribution, and its variance component is modeled. Regarding the outputs, the Model Summary table provides the procedure and model information. The Case Processing Summary table gives a comprehensive listing of the failure/censoring status and also those cases that are excluded from the analysis.

In Patrick’s data sample, all the records are valid and included in the analysis. By comparing the log likelihood with that of the corresponding model without the frailty component, the shared-frailty model fails to reach a significant level (p-value = 0.168). Patrick is wondering if it is necessary to include a shared-frailty term in the model. The estimated acceleration factor of a male participant is 1.017, which is obtained by calculating exponent of the estimated regression coefficient 0.017 of [female = 0.0]. Its associated 95% confidence interval is (.688, 1.504). These results suggest that a male individual has almost the same acceleration factor as a female individual with the same age and frailty. On the population level, Patrick plots the unconditional survival and hazard curves separately for males and females who are evaluated at the sample mean of age (37.45 years old).

Patrick confirms that, for any fixed value of survival time, a male and a female on average are supposed to have the same survival probability. Interestingly, despite a unimodal shape that is shown in the unconditional hazard chart, Patrick discovers that within a period of 60 days the population hazard is actually increasing. This behavior might imply the existence of the frailty effect. To further investigate the side effect caused by the treatments, Patrick may continue with a model without the frailty component and compare behavior of males and females. In addition, he may consider following up with the participants for a period longer than 60 days to collect more data.