Life Tables

Overview Life tables is a statistical procedure which generates duration (time to event) distributions for an entire dataset or separately for each level of a factor. The primary output is a life table in which the rows are researcher-defined time intervals and the columns have to do with counts, probabilities, and cumulative probabilities of the event of interest occuring during the given time interval. See also: Kaplan-Meier survival analysis Cox regression Event history analysis

Contents Key concepts and terms Assumptions Frequently asked questions Bibliography

Key Terms and Concepts

• SPSS. The life tables procedure is selected from the SPSS menu system by Analyze, Survival, Life Tables. This leads to the Life Tables dialog shown below:
  

  
  

• Example. In the discussion which follows, the example used analyzes the thirteen original states as units of analysis, looking at duration defined as days from the adoption of the Constitution until a state actually ratified it. The dataset also contains various factors which may may influence duration One of these factors is a binary variable labeled Rights, with 1 = a state heavily involved in the fight for inclusion of a Bill of Rights in the Constitution, and 0 = other states.

• Variables
  • Time. The duration variable is the time variable, whether in hours, days, years, etc. It must be a quantitative variable. In the illustration above it is days. A time interval must be specified at the researcher's option, here every 50 days. The time intervals determine the rows in the life table.

  • Status. Status is a dichotomous or categorical variable coded in integer format. Often it is a 0, 1 variable where 1 means the event of interest has occurred. In the case of our example, 1 corresponds to ratification and since all states eventually ratified the constitution, all states have a status of 1. After entering the name of the status variable, one must click Define Event to specify the value or range of values of the status variable that indicate(s) that an event occurred(ex., 1 = ratification occurred).

  • Factor. Factors are categorical integer-coded variables which are of research interest. If specified, life tables are generated for each level of the factor. After entering a factor, click "Define Range" to specify the factor's minimum and maximum values.

  • Second-level factor. A second-level factor may be specified in the main Life Tables dialog by entering a categorical variable in the "By Factor" box. If a second-order factor is specified, life tables will be generated for every combination of the first- and second-order factors.

  
  
• Options

  

  The Options button allows the research to select the following:

  • Life tables. This is the main output of the Life Tables procedure, but may be suppressed by unchecking it.

  • Plot. Various plots discussed below in the output section may be requested.

  • Compare Levels of First Factor.. If a first-order factor was specified, the researcher can select None, Overall, or Pairwise as the basis for performing the Wilcoxon (Gehan) test. This test compares the survival of subgroups formed by levels of the first-order factor. If there is a second-order factor, tests are performed for each level of the second-order variable. Selecting pairwise generates both overall and pairwise tests and thus provides the most information to the researcher. See further discussion below.

  
  

• Output
  • Life table. The life table is, of course, the primary output from the Life Tables procedure.
    

    The life table has these columns with corresponding data for each time interval (row).:

    1. Interval start time: Intervals are set by the researcher in the main Life Tables dialog. Intervals go from the start time up to, but not including, the next interval's start time.
    2. Number entering: This is how many surving cases (cases not yet having experience the event) at the start of the interval. This column starts with the number of cases in the category and declines to 1 (the last case to experience the event).
    3. Number leaving: This is the number cases which are censored cases in the given interval. For this example, there are 0 censored cases.
    4. Number exposed to risk: This is the number of surviving cases. However, if there are censored cases, the procedure tries to account for their effect by making the number exposed to risk equal to the number of surviving cases minus half the censored cases.
    5. Number of terminal events. This is how many cases (states in our example) which experience the terminal event (ex., ratification) during the given interval.
    6. Proportion terminating: This is the number of terminal events divided by the number exposed to risk, from previous columns.
    7. Proportion surviving: This is calculated as 1 minus the proportion terminating.
    8. Cumulative proportion surviving with standard error: This is the percentage of cases (ex., states) which are still surviving (ex., states which still have not ratified the Constitution by the end of the given interval.
    9. Probability density with standard error: This is the estimated probability of experiencing the event (ex., ratification) during the given interval.
    10. Hazard rate with standard error: This is the estimated probability of experiencing the event conditional on surviving to the start of the given interval.

  • Median survival times. If a factor is specified, this table shows the median values of the groups formed by the levels of that factor. In this example, the factor is Rights and the table below shows a substantial median difference in time to Constitutional ratification for Bill of Rights states vs. other states with the latter being quicker to ratify (lower duration to event).
    

  • Wilcoxon (Gehan) test. This test is used to compare survival distributions between groups, with the test statistic (D) based on differences in group mean scores for groups formed by the factor variable. This table is output if in the Options dialog the researcher has selected "Overall" or "Pairs" as the type of test. The default is "None" and no output is generated by default. In the example below, the control variable is the first level factor, which was Rights. Because Wilcoxon-Gehan's D is significant, the researcher concludes that there is a significant difference in duration to event (days to Constitutional ratification in the example) between Bill of Rights states and other states. (Of course, for these data, since all thirteen original states are in the dataset, which is not a random sample, significance is irrelevant any any difference is a real difference).
    

    If the control variable has three or more levels and if the researcher has asked for pairwise testing, the comparison table below is generated. That is, the Wilcoxon-Gehan test above is an overall test, but the pairwise tests below show just which level comparisons are contributing to overall significance, if any. Here the factor (control) is size of state, with three levels (small, medium, large). Although large states vs. small states was a major division at the Constitutional Conventions, the example below shows that state size was not a significant factor related to days from adoption to ratification (duration to event).

    

  • Plots
    • Survival function plot. This is a graphical representation of the life table. For this example, it shows Bill of Rights states are associated with longer duration to the event of ratification of the Constitution.
      

    • Log survival function plot. This is a logged version of the survival function plot.

    • Density function plot.
      

    • Hazard function plot.
      

    • One minus survival plot.
      

Assumptions

• Censored cases not different . The Life Table procedure, unlike Kaplan-Meier survival analysis or Cox regression, does not handle censored cases (cases for which the event has not yet occurred). If censored cases are in the dataset, they must not be different in nature from the uncensored cases.

• Probabilities depend on time. The Life Table procedure, unlike Cox regression, does not model multiple causes of time to event. Rather it is assumed that the probabilities for the event of interest depend only on time within any level of the first or second order factors, if specified. If time is not the only cause, Cox regression should be used. If causal factors are not fixed but rather vary over time, then Cox Regression with Time-Dependent Covariates should be used.

Frequently Asked Questions

• What is the command syntax in SPSS for the Life Tables procedure?

  SURVIVAL TABLES=survival varlist
                  [BY varlist (min, max)...][BY varlist (min, max)...]
    /INTERVALS=THRU n BY a [THRU m BY b ...]
    /STATUS=status variable({min, max}) FOR {ALL             } 
                            {value   }      {survival varlist}
   [/STATUS=...]
   [/PLOT  ({ALL     })={ALL            } BY {ALL    }     BY {ALL    }]
            {LOGSURV }  {survival varlis}    {varlist}        {varlist}
            {SURVIVAL}
            {HAZARD  } 
            {DENSITY } 
            {OMS     } 
   [/PRINT={TABLE**}] 
           {NOTABLE}
   [/COMPARE={ALL**           } BY {ALL**  }   BY {ALL**  }] 
             {survival varlist}    {varlist}      {varlist}
   [/CALCULATE=[{EXACT**    }] [PAIRWISE] [COMPARE] ] 
                {CONDITIONAL}
                {APPROXIMATE}
   [/MISSING={GROUPWISE**}  [INCLUDE] ] 
             {LISTWISE   }
   [/WRITE=[{NONE**}] ] 
            {TABLES} 
            {BOTH  }
  
  **Default if the subcommand or keyword is omitted. 
  
  Example
  SURVIVAL TABLES=MOSFREE BY TREATMNT(1,3)
    /STATUS = PRISON (1) FOR MOSFREE
    /INTERVAL=THRU 24 BY 3.
  

Bibliography

• Gehan, E. A. (1975). Statistical methods for survival time studies. In M. J. Staquet, eds. (1975). Cancer Therapy: Prognostic Factors and Criteria. NY: Raven Press.
• Hosmer, D. W. & Lemeshow. S. (1999). Applied survival analysis. NY: John Wiley and Sons.
• Kleinbaum, D. G. 1996. Survival analysis: A self-learning text. NY: Springer-Verlag.
• Lee, E. & Desu, M. (1972). A computer program for comparing k samples with right censored data. Computer Programs in Biomedicine 2, 315-321.