# Follow-up Life Table

This function provides a follow-up life table that displays the survival experience of a cohort.

The table is constructed by the following definitions:

• Interval
• For a Berkson and Gage survival table this is the survival times in intervals.
• For an abridged life table this is ages in groups.
• Deaths
• Number of individuals who die in the interval. [dx]
• Withdrawn
• Number of individuals withdrawn or lost to follow up in the interval. [wx]
• At Risk
• Number of individuals alive at the start of the interval. [nx]
• Adjusted number at risk (half of withdrawals of current interval subtracted). [n'x]
• P(death)
• Probability that an individual who survived the last interval will die in the current interval. [qx]
• P(survival)
• Probability that an individual who survived the last interval will survive the current interval. [px]
• % Survivors (lx)
• Probability of an individual surviving beyond the current interval.
• Proportion of survivors after the current interval.
• Life table survival rate.
• Var(lx%)
• Estimated variance of lx.
• *% CI for lx%
• *% confidence interval for lx%.

- where lx is the product of all px before x.

Technical validation

The Berkson and Gage method is used to construct the basic table (Berkson and Gage, 1950; Armitage and Berry, 1994; Altman, 1991; Lawless, 1982; Kalbfleisch and Prentice, 1980; Le, 1997). The confidence interval for lx is not a simple application of the estimated variance for Ix, instead it uses a maximum likelihood solution from an asymptotic distribution by the transformation of lx suggested by Kalbfleisch and Prentice (1980). This treatment of lx avoids impossible values (i.e. >1 or <0).

Example

Test workbook (Survival worksheet: Year, Died, Withdrawn).

The following data represent the survival of a 374 patients who had one type of surgery for a particular malignancy.

 Years since operation Died in this interval Lost to follow-up 1 90 0 2 76 0 3 51 0 4 25 12 5 20 5 6 7 9 7 4 9 8 1 3 9 3 5 10 2 5

To analyse these data in StatsDirect you must first prepare them in three workbook columns appropriately labelled. Alternatively, open the test workbook using the file open function of the file menu. Then select Follow-up Life Table from the survival analysis section of the analysis menu. Select the column marked "Year" when asked for the times, select "Died" when asked for deaths and "Withdrawn" when asked for withdrawals. Select 374 (total deaths and withdrawals) as the number alive at the start.

For this example:

Follow-up life table

 Interval Deaths Withdrawn At risk Adj. at risk P(death) 0 to 1 90 0 374 374 0.240642 1 to 2 76 0 284 284 0.267606 2 to 3 51 0 208 208 0.245192 3 to 4 25 12 157 151 0.165563 4 to 5 20 5 120 117.5 0.170213 5 to 6 7 9 95 90.5 0.077348 6 to 7 4 9 79 74.5 0.053691 7 to 8 1 3 66 64.5 0.015504 8 to 9 3 5 62 59.5 0.05042 9 to 10 2 5 54 51.5 0.038835 10 up 21 26 47 * *

 Interval P(survival) Survivors (lx%) SD of lx% 95% CI for lx% 0 to 1 0.759358 100 * * to * 1 to 2 0.732394 75.935829 10.57424 71.271289 to 79.951252 2 to 3 0.754808 55.614973 7.87331 50.428392 to 60.482341 3 to 4 0.834437 41.97861 7.003571 36.945565 to 46.922332 4 to 5 0.829787 35.028509 6.747202 30.200182 to 39.889161 5 to 6 0.922652 29.066209 6.651959 24.47156 to 33.805 6 to 7 0.946309 26.817994 6.659494 22.322081 to 31.504059 7 to 8 0.984496 25.378102 6.700832 20.935141 to 30.043836 8 to 9 0.94958 24.984643 6.720449 20.552912 to 29.648834 9 to 10 0.961165 23.724913 6.803396 19.323326 to 28.39237 10 up * 22.803557 6.886886 18.417247 to 27.483099

We conclude with 95% confidence that the true population survival rate 5 years after the surgical operation studied is between 24.5% and 33.8% for people diagnosed as having this cancer.