Kaplan-Meier Survival Analysis

Overview Kaplan-Meier survival analysis (KMSA) is a method of generating tables and plots of survival or hazard functions for event history data (time to event data). Time to event data might include, for instance, time to a report of symptomatic relief following a treatment, or time to making a contribution following receipt of a fund-raising appeal. However, unlike parametric event history analysis models or the semi-parametric Cox regression approach, KMSA is not designed to assess the effects of covariates on either function. That is, KMSA is a descriptive procedure for time-to-event variables for use when time is considered the only salient variable. If covariates other than time are thought to be important in determining duration to outcome, results reported by KMSA will represent misleading averages obscuring important differences in groups formed by the covariates (ex., men vs. women). Since this is often the case, methods which incorporate covariates, such as event history models and Cox regression, may be preferred. Even when covariates may be found to be important, however, KMSA may still be fruitful in exploratory stages of research.

Contents Key concepts and terms Statistical output Assumptions Frequently asked questions Bibliography

Key Concepts and Terms

• Survival functions, hazard functions, and other terms basic to event history analysis are discussed in the sections on event history analysis and Cox regression.

• Censored cases (technically, right-censored cases) are those for which the event of interest has not yet happened. KMSA can estimate survival functions even in the presence of censored cases.
  

• Time variable. There must be a continuous variable representing time. The initial date must be clearly defined (ex., date of first treatment in a study comparing real and placebo treatments). Note the time variable is a duration variable, not a date variable. It is number of time periods since the initial date. If the user's dataset contains two date variables (start date, stop date), they must be transformed and subtracted to get a time as a duration.
  

• Status variable. There must be a categorical or continuous variable representing the event of interest. That is, it is the numeric or string variable which contains the code for the terminal event. In SPSS, one may enter a single value, a numeric range of values, or a list of values. The status variable must clearly define a terminal event (ex., the date when the subject first reports symptomatic relief).

• Stratification variable. This is a categorical variable representing a grouping effect. In a medical setting, a stratification variable might be type of flu. If flu_type is set as the stratification variable, one will get survival plots for each type of flu. One may then compare survival rates between types of flu.

• Factor variable. This is a categorical variable representing a causal effect (ex., treatments). In a medical setting, a factor variable might be type of treatment (ex., type of shot). If treatment is set as the factor variable, output will include survival plot curves for each treatment within each strata (ex., each flu type). One may then compare survival rates between treatment groups, for each flu type. One may think of the factor variable as the main grouping variable and the stratification variable as the subgrouping variable. Both are optional.
  • Compare factor. The "Compare Factor" button under Analyze, Survival, Kaplan-Meier in SPSS, supports various statistics to test for equality of survival function between different levels of the factor variable (ex., different treatments in the example). For each test, a finding of significance means the the survival function is not equal between factor levels (between treatments). Usually the substantive inference from all three tests is the same. In the illustration below, days duration from adoption of the U. S. Constitution until its ratification by a state was estimated from a binary predictor, Rights, where the "1" level corresponded to being a state heavily identified with the fight for a Bill of Rights and the "0" level referred to other states. All tests were significant, indicating that survival functions were not equal between Bill of Rights states and other states.
    
    • Log rank. Tests equality of survival functions by weighting all time points the same.

    • Breslow. Tests equality of survival functions by weighting all time points by the number of cases at risk at each time point. This test is also called the generalized Wilcoxon test or the Gehan test. The Breslow test has diminished power (more chance of Type II errors - incorrect findings of non-significance) when the percentage of censored cases is large.

    • Tarone-Ware. Tests equality of survival functions by weighting all time points by the square root of the number of cases at risk at each time point. This is considered a compromise between the log rank and Breslow methods.

    In SPSS, various types of comparisons are available for test selected:

    • Pooled over strata.. A single test is computed for all factor levels, testing for equality of survival function across all levels of the factor variable.

    • Pairwise over strata. A separate test is computed for each pair of factor levels. Pairwise tests are generally used only after a pooled test shows non-equality of survival functions and the researcher is trying to assess just which pairs were non-equal. This two-step testing procedure is recommended since the pairwise strategy may involve numerous paired comparisions, 5% of which will be significant just by chance, as is the problem in all multiple comparison tests.

    • For each stratum. A separate test is computed for each group formed by the stratification variable.

    • Pairwise for each stratum. A separate test is computed for each pair of factor levels of the factor variable, for each stratum of the stratification variable.

    • Linear trend for factor levels. Tests for a linear trend across levels of the factor variable (not available if pairwise options are chosen).

  
  
• Output
  In SPSS, the Options button of the Kaplan-Meier Survival Analysis procedure supports the selection of various statistics and plots: If there are factor variables, statistics and plots are computed for each group. The Options button is illustrated below.
  

  Statistics:

  • Survival table. The survival table shows time until event (ex., effect of the treatment). Rows are arranged in ascending order of the time variable. If there are factors, the table is sectioned by levels of the factor variable (ex., by treatment). Columns include time, status (if the terminal event has taken effect), cumulative survival to event (this will descend toward 0 over time, as "survival" means not yet experiencing the event), standard error of the cumulative survival, number of cumulative events, and number remaining in the risk set (that is, remaining and possible to still experience the event). Note: Since a row is produced for each case, the survival table can become very long.
    

    The illustration above is a survival table from a study of causes of duration till ratification of the U. S. Constitution, using as a factor the binary variable Rights, where 0 = a state not heavily identified with the fight for a Bill of Rights, and 1 = a state that was so identified. For each of the two levels of Rights, the Survival Table shows the number of days from adoption of the Constitution until ratification by a state.

  • Mean and median survival. If this is checked, mean and median survival time is also computed for each level of the factor variable, with standard error and 95% confidence intervals. The means and medians show "typical" effects of levels of the factor variable (ex., different treatments). If confidence intervals do not overlap between levels (treatments), differences in effect on time to event can be inferred.
    

    For the same constitutional ratification case, the figure above shows a substantial mean difference in days duration until ratification between states the were or were not heavily identifed as centers of Bill of Rights support (the Rights variable).

  • Quartiles. This option generates a percentiles table showing average time to event at 25%, 50%, and 75% markers of the cumulative survival curve. Rows represent levels of the factor (treatment) variable. The percentiles table enables the researcher to see differences between factor effects at various times along the survival curve. For instance, the 50% column is the survival time estimate for each treatment when 50% of the subjects have experienced terminal status (ex., have experienced an effect of treatment).
    

    For the illustration above, the Percentiles table shows that when 50% of states had ratified the Constitution, Bill of Rights states were predicted to take 282 more days to ratification but other states only 92 more days.

  Plots:

  • Survival plot. Selection of "Survival" invokes plotting of the cumulative survival function on a linear scale. The x axis is time to effect ; the y axis is cumulative survival . The curve will slope down, with fewer surviving in the risk pool as time goes on, because as subjects experience terminal events they fall out of the risk pool. The cumulative survival percent at any given time on the x axis can be interpreted as the probability of survival to that time (that is, probability of not experience the terminal event). Log survival (plotting the survival function on a logarithmic scale) and "0ne minus survival" plots are also available.
    

    Using the same example, the survival plot above shows that non-Bill of Rights states ratified the Constitution more quickly than Bill of Rights states.

  • Hazard plot. Selection of "Hazard" invokes plotting of the cumulative hazard function on a linear scale. The hazard plot below uses the same Constitutional ratification data as the survival plot above, with similar interpretation.
    

  Created variables. The Save button of the Kaplan-Meier dialog in SPSS supports saving the following variables to one's dataset:
  • Survival. Cumulative survival probability estimate. The default variable name is the prefix sur_ with a sequential number appended to it.

  • Standard error of survival The standard error of the cumulative survival estimate. The default variable name is the prefix se_ with a sequential number appended to it.

  • Hazard. Cumulative hazard function estimate. The default variable name is the prefix haz_ with a sequential number appended to it.

  • Cumulative events. Cumulative frequency of events when cases are sorted by their survival times and status codes. The default variable name is the prefix cum_ with a sequential number appended to it.

Assumptions

• Events dependent only on time. Because KMSA models survival based only on time dependence, without covariate effects, it is assumed that event probabilities depend only on time. All subjects are assumed to behave similarly and computed survival functions are assumed to describe all subjects. This further implies that censored and uncensored cases behave the same. If the censored cases (those for whom the event has not yet happened) are different from the uncensored cases, results may be biased. This is a strong assumption which may be violated in the real world, which is why event history and Cox regression models which do take covariates into account are now the more common approach for time-to-event data.

Frequently Asked Questions

• How do you obtain KMSA output in SPSS?
  Assuming you have SPSS with the "Advanced Models" module installed, select Analyze, Survival, Kaplan-Meier from the menu. Specify the time variable. Specify the status variable and click Define Event to specify coding for the terminal event. Select options such as the strata or factor variables, if any. Press OK to run the analysis.

• How do you obtain KMSA output in Stata?
  First, one inputs one's data with the input command. For instance, if the data are three columns consisting of id, time, and event (coded 0, 1), then the command would be input id time event, followed by the data list, followed by the end statement. Next one declares the time and event variables for time-to-event data, using the stset command: stset time, failure(event). To get the Kaplan-Meier survivor function with standard error and confidence limits, sts list suffices as the command since Stata knows to use the input dataset as declared in the stset command. To get the Nelson-Aalen estimate of the cumulative survival function instead, add the na parameter: st list, na. To get descriptive statistics such as number of subjects, first entry and final exit times, failures, and more, simply use the stdes command with no parameters. To get the number of missing values and for how many subjects the vartiable is constant or varying, ust the stvary command with no parameters. The sts graph command generates graphs of the Kaplan-Meier survival functions. To get time at risk, incidence rates, and quartile survival times, use the stsum command with no parameters. To do factor (group) comparisons, add the by(factor) parameter to stlist, stsum, or stgraph. For instance, for gender as a factor, a command could be stgraph, by(gender) or stlist, by(gender) compare.

• How are Kaplan-Meier survival tables related to "Life Tables"?
  Life tables are a separate SPSS option under Analyze, Survival, Life Tables. A life table also shows survival times but groups time periods into equal intervals in a form of actuarial chart. Life tables are almost always more concise than Kaplan-Meier survival tables because the width of time intervals in the Kaplan-Meier procedure is based on when events actually occur, not on fixed equal intervals. The smaller the interval widths selected in the life tables procedure, the more the life table will resemble the Kaplan-Meier survival table. Usually, however, the selected intervals in a life table are relatively broad, making it unsuitable for estimating the underlying survival or hazard functions.

Bibliography

• Kaplan, E. L. & Meier, P. (1958). Nonparametric estimation from incomplete data. Journal of the American Statistical Association 53: 457-481.