Menu location: Analysis_Survival_Wei-Lachin.
This function gives a two sample distribution free method for the comparison of two multivariate distributions of survival (time-to-event) data that may be censored (incomplete, e.g. alive at end of study or lost to follow up). Multivariate methods such as this should be used only with expert statistical guidance.
Wei and Lachin generalise the Log-rank and Gehan generalised Wilcoxon tests (using a random censorship model) for multivariate survival data with two main groups. (Makuch and Escobar, 1991; Wei and Lachin, 1984).
Data preparation
StatsDirect asks you for a group identifier, this could be a column of 1 and 2 representing the two groups. You then select k pairs of survival time (time-to-event) and censorship columns for k repeat times. Censored data are coded as 0 and uncensored data are coded as 1.
Repeat times may represent separate factors or the observation of the same factor repeated on k occasions. For example, time to develop symptoms could be analysed for k different symptoms in a group of patients treated with drug x and compared with a group of patients not treated with drug x.
Missing data can be code either by entering a missing data symbol * as the time, or by setting censored equal to 0 and time less than the minimum uncensored time in your data set.
For further details please refer to Makuch and Escobar (1991) and Wei and Lachin (1984).
Technical Validation
Wei and Lachin's multivariate tests are calculated for the case to two multivariate distributions, and the intermediate univariate statistics are given. The algorithm used for the method is that given by Makuch and Escobar (1991).
The general univariate statistic for comparing the time to event (of component type k out of m multivariate components) of the two groups is calculated as:
- where n1 is the number of event times per component in group 1; n2 is the number of event times per component in group 2; n is the total number of event times per component; rik is the number at risk at time t(i) in the kth component; Δ is equal to 0 if an observation is censored or 1 otherwise; eik is the expected proportion of events in group i for the kth component; and wj is equal to 1 for the log-rank method or (r1k+r2k)/n for the Gehan-Breslow generalised Wilcoxon method.
The univariate statistic for the kth component of the multivariate survival data is calculated as:
- where σ hatkk is the kth diagonal element of the estimated variance-covariance matrix that is calculated as described by Makuch and Escobar (1991).
An omnibus test that the two multivariate distributions are equal is calculated as:
- where T' is the transpose of the vector of univariate test statistics and Σ-1 is the generalised inverse of the estimated variance-covariance matrix.
A stochastic ordering test statistic is calculated as:
Note that the P value given with the stochastic ordering (linear combination) statistic is two sided, some authors prefer one sided inference (Davis, 1994). If you make a one sided inference then you are considering only ascending or only descending ordering, and you are assuming that observing an order in the opposite direction to that expected would be unimportant to your conclusions.
The test statistics are all asymptotically normally distributed.
Example
From Makuch and Escobar (1991).
Test workbook (Survival worksheet: Treatment Gp, Time m1, Censor m1, Time m2, Censor m2, Time m3, Censor m3, Time m4, Censor m4).
The following data represent the times in days it took in vitro cultures of lymphocytes to reach a level of p24 antigen expression. The cultures where taken from patients infected with HIV-1 who had advanced AIDS or AIDS related complex. The idea was that patients whose cultures took a short time to express p24 antigen had a greater load of HIV-1. The two groups represented patients on two different treatments. The culture was run for 30 days and specimens which remained negative or which became contaminated were called censored (=0). The tests were run over four 30 day periods.
| Treatment Gp | time m1 | censor m1 | time m2 | censor m2 | time m3 | censor m3 | time m4 | censor m4 |
| 1 | 8 | 1 | 0 | 0 | 25 | 0 | 21 | 1 |
| 1 | 6 | 1 | 4 | 1 | 5 | 1 | 5 | 1 |
| 1 | 6 | 1 | 5 | 1 | 28 | 0 | 18 | 1 |
| 1 | 14 | 0 | 35 | 0 | 23 | 1 | 19 | 0 |
| 1 | 7 | 1 | 0 | 0 | 13 | 1 | 0 | 0 |
| 1 | 5 | 1 | 4 | 1 | 27 | 1 | 8 | 1 |
| 1 | 5 | 1 | 21 | 0 | 6 | 1 | 14 | 1 |
| 1 | 6 | 1 | 10 | 1 | 14 | 1 | 18 | 1 |
| 1 | 7 | 1 | 4 | 1 | 15 | 1 | 8 | 1 |
| 1 | 6 | 1 | 5 | 1 | 5 | 1 | 5 | 1 |
| 1 | 4 | 1 | 5 | 1 | 6 | 1 | 3 | 1 |
| 1 | 5 | 1 | 4 | 1 | 7 | 1 | 5 | 1 |
| 1 | 21 | 0 | 5 | 1 | 0 | 0 | 6 | 1 |
| 1 | 13 | 1 | 27 | 0 | 21 | 0 | 8 | 1 |
| 1 | 4 | 1 | 27 | 0 | 7 | 1 | 6 | 1 |
| 1 | 6 | 1 | 3 | 1 | 7 | 1 | 8 | 1 |
| 1 | 6 | 1 | 0 | 0 | 5 | 1 | 5 | 1 |
| 1 | 6 | 1 | 0 | 0 | 4 | 1 | 6 | 1 |
| 1 | 7 | 1 | 9 | 1 | 6 | 1 | 7 | 1 |
| 1 | 8 | 1 | 15 | 1 | 8 | 1 | 0 | 0 |
| 1 | 18 | 0 | 27 | 0 | 18 | 0 | 9 | 1 |
| 1 | 16 | 1 | 14 | 1 | 14 | 1 | 6 | 1 |
| 1 | 15 | 1 | 9 | 1 | 12 | 1 | 12 | 1 |
| 2 | 4 | 1 | 5 | 1 | 4 | 1 | 3 | 1 |
| 2 | 8 | 1 | 22 | 1 | 25 | 0 | 0 | 0 |
| 2 | 6 | 1 | 6 | 1 | 8 | 1 | 5 | 1 |
| 2 | 7 | 1 | 10 | 1 | 10 | 1 | 18 | 1 |
| 2 | 5 | 1 | 14 | 1 | 17 | 0 | 6 | 1 |
| 2 | 3 | 1 | 5 | 1 | 8 | 1 | 6 | 1 |
| 2 | 6 | 1 | 11 | 1 | 6 | 1 | 13 | 1 |
| 2 | 6 | 1 | 0 | 0 | 15 | 1 | 7 | 1 |
| 2 | 6 | 1 | 12 | 1 | 19 | 1 | 8 | 1 |
| 2 | 6 | 1 | 25 | 0 | 0 | 0 | 22 | 0 |
| 2 | 4 | 1 | 7 | 1 | 5 | 1 | 7 | 1 |
| 2 | 5 | 1 | 7 | 1 | 4 | 1 | 6 | 1 |
| 2 | 3 | 1 | 9 | 1 | 7 | 1 | 6 | 1 |
| 2 | 9 | 1 | 17 | 1 | 0 | 0 | 21 | 0 |
| 2 | 6 | 1 | 4 | 1 | 8 | 1 | 14 | 1 |
| 2 | 5 | 1 | 5 | 1 | 7 | 1 | 16 | 0 |
| 2 | 12 | 1 | 18 | 0 | 14 | 1 | 0 | 0 |
| 2 | 9 | 1 | 11 | 1 | 15 | 1 | 18 | 0 |
| 2 | 6 | 1 | 5 | 1 | 9 | 1 | 0 | 0 |
| 2 | 18 | 0 | 8 | 1 | 10 | 1 | 13 | 1 |
| 2 | 4 | 1 | 4 | 1 | 5 | 1 | 10 | 1 |
| 2 | 3 | 1 | 10 | 1 | 0 | 0 | 21 | 0 |
| 2 | 8 | 1 | 7 | 1 | 10 | 1 | 12 | 1 |
| 2 | 3 | 1 | 6 | 1 | 7 | 1 | 9 | 1 |
To analyse these data in StatsDirect you must first prepare them in 9 workbook columns as shown above. Alternatively, open the test workbook using the file open function of the file menu. Then select Wei-Lachin from the Survival Analysis section of the analysis menu. Select the column marked "Treatment GP" when asked for the group identifier. Next, enter the number of repeat times as four. Select "time m1" and "censor m1" for time and censorship for repeat time one. Repeat this selection process for the other three repeat times.
For this example:
Wei-Lachin Analysis
Univariate Generalised Wilcoxon (Gehan)
total cases = 47 by group = 23 24
Observed failures by group = 20 23
repeat time = 1
Wei-Lachin t = -0.527597
Wei-Lachin variance = 0.077575
z = -1.89427
chi-square = 3.588261, P = .0582
Observed failures by group = 14 21
repeat time = 2
Wei-Lachin t = 0.077588
Wei-Lachin variance = 0.056161
z = 0.327397
chi-square = 0.107189, P = .7434
Observed failures by group = 18 19
repeat time = 3
Wei-Lachin t = -0.11483
Wei-Lachin variance = 0.060918
z = -0.465244
chi-square = 0.216452, P = .6418
Observed failures by group = 20 16
repeat time = 4
Wei-Lachin t = 0.335179
Wei-Lachin variance = 0.056281
z = 1.412849
chi-square = 1.996143, P = .1577
Multivariate Generalised Wilcoxon (Gehan)
Covariance matrix:
| 0.077575 | |||
| 0.026009 | 0.056161 | ||
| 0.035568 | 0.020484 | 0.060918 | |
| 0.023525 | 0.016862 | 0.026842 | 0.056281 |
Inverse of covariance matrix:
| 19.204259 | |||
| -5.078483 | 22.22316 | ||
| -8.40436 | -3.176864 | 25.857118 | |
| -2.497583 | -3.020025 | -7.867237 | 23.468861 |
repeat times = 4
chi squared omnibus statistic = 9.242916 P = .0553
stochastic ordering z = -0.30981 one sided P = 0.3784, two sided P = 0.7567
Univariate Log-Rank
total cases = 47 by group = 23 24
Observed failures by group = 20 23
repeat time = 1
Wei-Lachin t = -0.716191
Wei-Lachin variance = 0.153385
z = -1.828676
chi-square = 3.344058, P = .0674
Observed failures by group = 14 21
repeat time = 2
Wei-Lachin t = -0.277786
Wei-Lachin variance = 0.144359
z = -0.731119
chi-square = 0.534536, P = .4647
Observed failures by group = 18 19
repeat time = 3
Wei-Lachin t = -0.372015
Wei-Lachin variance = 0.150764
z = -0.9581
chi-square = 0.917956, P = .338
Observed failures by group = 20 16
repeat time = 4
Wei-Lachin t = 0.619506
Wei-Lachin variance = 0.143437
z = 1.635743
chi-square = 2.675657, P = .1019
Multivariate Log-Rank
Covariance matrix:
| 0.153385 | |||
| 0.049439 | 0.144359 | ||
| 0.052895 | 0.050305 | 0.150764 | |
| 0.039073 | 0.047118 | 0.052531 | 0.143437 |
Inverse of covariance matrix:
| 7.973385 | |||
| -1.779359 | 8.69056 | ||
| -1.892007 | -1.661697 | 8.575636 | |
| -0.894576 | -1.761494 | -2.079402 | 8.555558 |
repeat times = 4
chi squared omnibus statistic = 9.52966, P = .0491
stochastic ordering z = -0.688754, one sided P = 0.2455, two sided P = 0.491
Here the multivariate log-rank test has revealed a statistically significant difference between the treatment groups which was not revealed by any of the individual univariate tests. For more detailed discussion of each result parameter see Wei and Lachin (1984).
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