# Follow-up Life Table

Menu location: **Analysis_Survival_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**Withdrawn**- Number of individuals withdrawn or lost to follow up in the interval.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]

**Adj. at risk**- 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 l_{x} is the product of all p_{x} 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__

From Armitage and Berry (1994, p. 473).

Test workbook (Survival worksheet: Year, Died, Withdrawn).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