Editing Survival analysis (section)

===Example: Acute myelogenous leukemia survival data===

This example uses the [[Acute myeloid leukemia|Acute Myelogenous Leukemia]] survival data set "aml" from the "survival" package in R. The data set is from Miller (1997)<ref name="Miller1997">{{Citation
|last1= Miller
|first1= Rupert G.
|title= Survival analysis
|year=1997
|publisher= John Wiley & Sons
|isbn=0-471-25218-2
}}
</ref> and the question is whether the standard course of chemotherapy should be extended ('maintained') for additional cycles.

The aml data set sorted by survival time is shown in the box.
{| class="wikitable mw-collapsible"
|+Aml data set sorted by survival time
!observation
!time
(weeks)
!status
!x
|-
|12
|5
|1
|Nonmaintained
|-
|13
|5
|1
|Nonmaintained
|-
|14
|8
|1
|Nonmaintained
|-
|15
|8
|1
|Nonmaintained
|-
|1
|9
|1
|Maintained
|-
|16
|12
|1
|Nonmaintained
|-
|2
|13
|1
|Maintained
|-
|3
|13
|0
|Maintained
|-
|17
|16
|0
|Nonmaintained
|-
|4
|18
|1
|Maintained
|-
|5
|23
|1
|Maintained
|-
|18
|23
|1
|Nonmaintained
|-
|19
|27
|1
|Nonmaintained
|-
|6
|28
|0
|Maintained
|-
|20
|30
|1
|Nonmaintained
|-
|7
|31
|1
|Maintained
|-
|21
|33
|1
|Nonmaintained
|-
|8
|34
|1
|Maintained
|-
|22
|43
|1
|Nonmaintained
|-
|9
|45
|0
|Maintained
|-
|23
|45
|1
|Nonmaintained
|-
|10
|48
|1
|Maintained
|-
|11
|161
|0
|Maintained
|}

* Time is indicated by the variable "time", which is the survival or censoring time
* Event (recurrence of aml cancer) is indicated by the variable "status". 0{{nbsp}}= no event (censored), 1{{nbsp}}= event (recurrence)
* Treatment group: the variable "x" indicates if maintenance chemotherapy was given

The last observation (11), at 161 weeks, is censored. Censoring indicates that the patient did not have an event (no recurrence of aml cancer). Another subject, observation 3, was censored at 13 weeks (indicated by status=0). This subject was in the study for only 13 weeks, and the aml cancer did not recur during those 13 weeks. It is possible that this patient was enrolled near the end of the study, so that they could be observed for only 13 weeks. It is also possible that the patient was enrolled early in the study, but was lost to follow up or withdrew from the study. The table shows that other subjects were censored at 16, 28, and 45 weeks (observations 17, 6, and{{nbsp}}9 with status=0). The remaining subjects all experienced events (recurrence of aml cancer) while in the study. The question of interest is whether recurrence occurs later in maintained patients than in non-maintained patients.

====Kaplan–Meier plot for the aml data====

The [[survival function]] ''S''(''t''), is the probability that a subject survives longer than time ''t''. ''S''(''t'') is theoretically a smooth curve, but it is usually estimated using the [[Kaplan–Meier estimator|Kaplan–Meier]] (KM) curve. The graph shows the KM plot for the aml data and can be interpreted as follows:

*The ''x'' axis is time, from zero (when observation began) to the last observed time point.
*The ''y'' axis is the proportion of subjects surviving. At time zero, 100% of the subjects are alive without an event.
*The solid line (similar to a staircase) shows the progression of event occurrences.
*A vertical drop indicates an event. In the aml table shown above, two subjects had events at five weeks, two had events at eight weeks, one had an event at nine weeks, and so on. These events at five weeks, eight weeks and so on are indicated by the vertical drops in the KM plot at those time points.
*At the far right end of the KM plot there is a tick mark at 161 weeks. The vertical tick mark indicates that a patient was censored at this time. In the aml data table five subjects were censored, at 13, 16, 28, 45 and 161 weeks. There are five tick marks in the KM plot, corresponding to these censored observations.

====Life table for the aml data====

A [[life table]] summarizes survival data in terms of the number of events and the proportion surviving at each event time point. The life table for the aml data, created using the R{{nbsp}}software, is shown.

{| class="wikitable mw-collapsible"
|+ Life Table for the aml Data
|-
! time !! n.risk !! n.event !! survival !! std.err !! lower 95% CI !! upper 95% CI
|-
| 5 || 23 || 2 || 0.913 || 0.0588 || 0.8049 || 1
|-
| 8 || 21 || 2 || 0.8261 || 0.079 || 0.6848 || 0.996
|-
| 9 || 19 || 1 || 0.7826 || 0.086 || 0.631 || 0.971
|-
| 12 || 18 || 1 || 0.7391 || 0.0916 || 0.5798 || 0.942
|-
| 13 || 17 || 1 || 0.6957 || 0.0959 || 0.5309 || 0.912
|-
| 18 || 14 || 1 || 0.646 || 0.1011 || 0.4753 || 0.878
|-
| 23 || 13 || 2 || 0.5466 || 0.1073 || 0.3721 || 0.803
|-
| 27 || 11 || 1 || 0.4969 || 0.1084 || 0.324 || 0.762
|-
| 30 || 9 || 1 || 0.4417 || 0.1095 || 0.2717 || 0.718
|-
| 31 || 8 || 1 || 0.3865 || 0.1089 || 0.2225 || 0.671
|-
| 33 || 7 || 1 || 0.3313 || 0.1064 || 0.1765 || 0.622
|-
| 34 || 6 || 1 || 0.2761 || 0.102 || 0.1338 || 0.569
|-
| 43 || 5 || 1 || 0.2208 || 0.0954 || 0.0947 || 0.515
|-
| 45 || 4 || 1 || 0.1656 || 0.086 || 0.0598 || 0.458
|-
| 48 || 2 || 1 || 0.0828 || 0.0727 || 0.0148 || 0.462
|}

The life table summarizes the events and the proportion surviving at each event time point. The columns in the life table have the following interpretation:

*time gives the time points at which events occur.
*n.risk is the number of subjects at risk immediately before the time point, t. Being "at risk" means that the subject has not had an event before time t, and is not censored before or at time t.
*n.event is the number of subjects who have events at time t.
*survival is the proportion surviving, as determined using the Kaplan–Meier product-limit estimate.
*std.err is the standard error of the estimated survival. The standard error of the Kaplan–Meier product-limit estimate it is calculated using Greenwood's formula, and depends on the number at risk (n.risk in the table), the number of deaths (n.event in the table) and the proportion surviving (survival in the table).
*lower 95% CI and upper 95% CI are the lower and upper 95% confidence bounds for the proportion surviving.

====Log-rank test: Testing for differences in survival in the aml data====

The [[log-rank test]] compares the survival times of two or more groups. This example uses a log-rank test for a difference in survival in the maintained versus non-maintained treatment groups in the aml data. The graph shows KM plots for the aml data broken out by treatment group, which is indicated by the variable "x" in the data.

[[File:Kaplan-Meier by treatment in AML.svg|thumb|320px|Kaplan–Meier graph by treatment group in aml]]

The null hypothesis for a log-rank test is that the groups have the same survival. The expected number of subjects surviving at each time point in each is adjusted for the number of subjects at risk in the groups at each event time. The log-rank test determines if the observed number of events in each group is significantly different from the expected number. The formal test is based on a chi-squared statistic. When the log-rank statistic is large, it is evidence for a difference in the survival times between the groups. The log-rank statistic approximately has a [[Chi-squared distribution]] with one degree of freedom, and the [[p-value]] is calculated using the [[Chi-squared test]].

For the example data, the log-rank test for difference in survival gives a p-value of p=0.0653, indicating that the treatment groups do not differ significantly in survival, assuming an alpha level of 0.05. The sample size of 23 subjects is modest, so there is little [[Power of a test|power]] to detect differences between the treatment groups. The chi-squared test is based on asymptotic approximation, so the p-value should be regarded with caution for small [[Sample size determination|sample sizes]].