## Product-Limit Survival Fit

### In this graph, Group B has the worst survival and Group A has the best.  However, there are times some groups' survival rates are close to the survival rates of other groups, such as around 550 days when Group C's survival rate is about as bad as Group B's or around 700 days when Group A's rate is about the same as Group C's.  Although it seems that there are differences in the survival rates over time, "eyeballing" the curves is not sufficient to say that there are fundament differences in the survival rates of the groups that.  We need to use a statistical test (such as the log-rank test, the Wilcoxon test, or Gehan's test) to determine whether there are real differences that could be found in the whole population which would appear again in a replication of this analysis with a second group of data.

Group N Failed N Censored Mean
Std Dev
A 9 31 566.005 Biased 39.0321
B 16 24 398.964 Biased 45.6842
C 12 28 436.703 Biased 31.3025
Combined 37 83 494.276 Biased 24.9866
This section gives some descriptive data about the  individual groups.  For example, Group A had 40 observations, 9 of which were observed to fail (Status = 0) and 31 of which were censored (lost, Status not = 0).  The mean reported is the estimated mean survival time of the individuals in that group. It is not the same as the mean of the actual times of the observed failures in the sample.  "Biased" indicates that the estimated mean is biased by the censored data.  The true mean survival time will be larger.

### Quantiles

Group Median Time Lower95% Upper95% 25% Failures 75% Failures
A 683 594 . 594 .
B 418 197 644 195 644
C 559 351 . 318 .
Comb. 594 466 . 300 .

The word "Quantiles" is a more general form of the word "Percentile".  Recall that the 90th percentile is the number which is greater than 90% of the values in the group.

The median is the 50th percentile.  The "Lower 95%" and "Upper 95%" are the lower and upper limits of a confidence interval for the true median of the population.  For example, in this sample of data the median survival time for Group B was 418 days.  Based on this estimate, the
(unobserveable) median for the whole population is most likely to be between 197 days and 644 days.  For Groups A and C, the upper limit could not be determined.  Along with the median, we also have the first quartile (the 25th percentile) and the third quartile (the 75th percentile).  Thus, 25% of the individuals in Group A are expected to fail before 594 days.

### Tests Between Groups

Test ChiSquare DF Prob>ChiSq
Log-Rank 7.6479 2 0.0218
Wilcoxon 5.4779 2 0.0646

This is the section that tells us whether there are differences in the survival patterns of the various groups.  The null hypotheses of these two tests are the same:

H0: Each group has the same survival function.  There are no differences in the survival rates over time between the various groups.

Large chi-square* values and their correspondingly small p values indicate that the null hypothesis should be rejected and that the survival patterns of the various groups do differ.  The log-rank test puts more emphasis on differences that show up later in time, whereas the Wilcoxon test puts more emphasis on differences that occur at earlier failure times.  The log-rank test is the more powerful test when the groups have proportional hazard functions.
It also has been found to work well when the hazard rate is constant.

In this example, using the 5% level of significance (alpha) the log-rank test indicates that there are differences among the three groups, while the Wilcoxon test does not detect differences.  This is not surprising since the largest differences occured after 300 days.  (The hazards do not seem to be proportional.  Group B has its largest hazard around 200 days, while Group A has its largest hazard around 650 days.)  Also, the hazard rate is not constant.  My conclusion in this case is that these is insufficient evidence to say that there are differences among the groups' survival functions.

*Chi is a Greek letter which is written like an X (it is the first letter in the Greek word for christ, hence the abreviation X-mas for Chritsmas).  English speakers would do well to pronounce it like "kai" -- not like "chee" nor like "chai"  It should rhyme with pie.

### A

Time Survival Failure SurvStdErr N Failed N Censored At Risk
0.000 1.0000 0.0000 0.0000 0 0 40
3.000 0.9750 0.0250 0.0247 1 0 40
9.000 0.9750 0.0250 0.0247 0 1 39
10.000 0.9750 0.0250 0.0247 0 1 38
16.000 0.9750 0.0250 0.0247 0 1 37
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