1 General Purpose
In survival studies often time to first
outpatient clinic check instead of time to event is measured.
Somewhere in the interval between the last and current visit an
event may have taken place. For simplicity such data are often
analyzed using the proportional hazard model of Cox (Chaps.
56 and 57). However, this analysis is not
entirely appropriate here. It assumes that time to first outpatient
check is equal to time to relapse. Instead of a time to relapse, an
interval is given, in which the relapse has occurred, and so this
variable is somewhat more loose than the usual variable time to
event. An appropriate statistic for the current variable would be
the mean time to relapse inferenced from a generalized linear model
with an interval censored link function, rather than the
proportional hazard method of Cox.
2 Schematic Overview of Type of Data File
3 Primary Scientific Question
This chapter is to assess whether an
appropriate statistic for the variable “time to first check” in
survival studies would be the mean time to relapse, as inferenced
from a generalized linear model with an interval censored link
function.
4 Data Example
In 51 patients in remission their
status at the time-to-first-outpatient-clinic-control was checked
(mths = months).
|
Time to 1st check (month)
|
Result relapse 0 = no
|
Treatment modality 1 or 2 (0 or 1)
|
|
11
|
0
|
1
|
|
12
|
1
|
0
|
|
9
|
1
|
0
|
|
12
|
0
|
1
|
|
12
|
0
|
0
|
|
12
|
0
|
1
|
|
5
|
1
|
1
|
|
12
|
0
|
1
|
|
12
|
0
|
1
|
|
12
|
0
|
0
|
The first ten patients are above. The
entire data file is entitled “chapter59intervalcensored”, and is in
extras.springer.com. Cox regression was first applied. Start by
opening the data file in SPSS statistical software.
5 Cox Regression
For analysis the statistical model Cox
Regression in the module Survival is required.
Command:
-
Analyze….Survival….Cox Regression….Time : time to first check….Status : result….Define Event….Single value: type 1….click Continue….Covariates: enter treatment….click Categorical….Categorical Covariates: enter treatment….click Continue….click Plots….mark Survival….Separate Lines for: enter treatment….click Continue….click OK.
Variables in the equation
|
B
|
SE
|
Wald
|
df
|
Sig.
|
Exp(B)
|
|
|---|---|---|---|---|---|---|
|
Treatment
|
.919
|
.477
|
3.720
|
1
|
.054
|
2.507
|
The above table is in the output. It
shows that treatment is not a significant predictor for relapse. In
spite of the above Kaplan-Meier curves, suggesting the opposite,
the treatments are not significantly different from one another
because p > 0,05. However, the analysis so far is not entirely
appropriate. It assumes that time to first outpatient check is
equal to time to relapse. However, instead of a time to relapse an
interval is given between 2 and 12 months in which the relapse has
occurred, and so this variables is somewhat more loose than the
usual variable time to event. An appropriate statistic for the
current variable would be the mean time to relapse inferenced from
a generalized linear model with an interval censored link function,
rather than the proportional hazard method of Cox.
6 Interval Censored Analysis in Generalized Linear Models
For analysis the module Generalized
Linear Models is required. It consists of two submodules:
Generalized Linear Models and Generalized Estimation Models. The
first submodule covers many statistical models like gamma
regression (Chap. 30), Tweedie regression (Chap.
31), Poisson regression (Chaps.
21 and 47), and the analysis of paired
outcomes with predictors (Chap. 3). The second is for analyzing
binary outcomes (Chap. 42). For the censored data analysis
the Generalized Linear Models submodule of the Generalized Linear
Models module is required.
Command:
-
Analyze….click Generalized Linear Models….click once again Generalized Linear Models….Type of Model….mark Interval censored survival….click Response…. Dependent Variable: enter Result….Scale Weight Variable: enter “time to first check”….click Predictors….Factors: enter “treatment”….click Model….click once again Model: enter once again “treatment”….click Save….mark Predicted value of mean of response….click OK.
Parameter estimates
|
Parameter
|
B
|
Std. Error
|
95 % Wald confidence interval
|
Hypothesis test
|
|||
|---|---|---|---|---|---|---|---|
|
Lower
|
Upper
|
Wald Chi-Square
|
df
|
Sig.
|
|||
|
(Intercept)
|
.467
|
.0735
|
.323
|
.611
|
40.431
|
1
|
.000
|
|
[treatment = 0]
|
−.728
|
.1230
|
−.969
|
−.487
|
35.006
|
1
|
.000
|
|
[treatment = 1]
|
0a
|
||||||
|
(Scale)
|
1b
|
||||||
The generalized linear model shows,
that, after censoring the intervals, the treatment 0 is, compared
to treat 1, a very significant better maintainer of remission. When
we return to the data, we will observe as a novel variable, the
mean predicted probabilities of persistent remission for each
patient. This is shown underneath for the first ten patients. For
the patients on treatment 1 it equals 79,7 %, for the patients
on treatment 0 it is only 53,7 %. And so, treatment 1
performs, indeed, a lot better than does treatment 0
(mths = months).
|
Time to first check (mths)
|
Result (0 = remission 1 = relapse)
|
Treatment (0 or 1)
|
Mean Predicted
|
|
11
|
0
|
1
|
,797
|
|
12
|
1
|
0
|
,537
|
|
9
|
1
|
0
|
,537
|
|
12
|
0
|
1
|
,797
|
|
12
|
0
|
0
|
,537
|
|
12
|
0
|
1
|
,797
|
|
5
|
1
|
1
|
,797
|
|
12
|
0
|
1
|
,797
|
|
12
|
0
|
1
|
,797
|
|
12
|
0
|
0
|
,537
|
7 Conclusion
This chapter assesses, whether an
appropriate statistic for the variable “time to first check” in
survival studies is the mean time to relapse, as inferenced from a
generalized linear model with an interval censored link function.
The current example shows that, in addition, more sensitivity of
testing is obtained with p-values of 0,054 versus 0,0001. Also,
predicted probabilities of persistent remission or risk of relapse
for different treatment modalities are given. This method is an
important tool for analyzing such data.
8 Note
More background, theoretical and
mathematical information of survival analyses is given in
Statistics applied to clinical studies 5th edition, Chaps. 17, 31,
and 64, Springer Heidelberg Germany, 2012, from the same
authors.