Computes population (current) or cohort life tables based upon the observed population sizes at the middle (for population table) or the beginning (for cohort table) of some user specified age intervals.

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.0.0

Syntax

C#
[SerializableAttribute]
public class LifeTables
Visual Basic (Declaration)
<SerializableAttribute> _
Public Class LifeTables
Visual C++
[SerializableAttribute]
public ref class LifeTables

Remarks

The number of deaths in each of these intervals must also be observed.

The probability of dying prior to the middle of the interval, given that death occurs somewhere in the interval, may also be specified. Often, however, this probability is taken to be 0.5. For a discussion of the probability models underlying the life table here, see the references.

Let t_i, for i = 0, 1, ..., t_n denote the time grid defining the n age intervals, and note that the length of the age intervals may vary. Following Gross and Clark (1975, page 24), let d_i denote the number of individuals dying in age interval i, where age interval i ends at time t_i. For population table, the death rate at the middle of the interval is given by r_i=d_i/(M_i h_i), where M_i is the number of individuals alive at the middle of the interval, and h_i=t_i-t_{i-1}, t_0=0 . The number of individuals alive at the beginning of the interval may be estimated by P_i=M_i+(1-a_i)d_i where a_i is the probability that an individual dying in the interval dies prior to the interval midpoint. For cohort table, P_i is input directly while the death rate in the interval, r_i, is not needed.

The probability that an individual dies during the age interval from t_{i-1} to t_i is given by q_i=d_i/P_i. It is assumed that all individuals alive at the beginning of the last interval die during the last interval. Thus, q_n = 1.0. The asymptotic variance of q_i can be estimated by

\sigma_i^2=q_i(1-q_i)/P_i

For a population table, the number of individuals alive in the middle of the time interval (input in nCohort[i]) must be adjusted to correspond to the number of deaths observed in the interval. The algorithm assumes that the number of deaths observed in interval h_i occur over a time period equal to h_i. If d_i is measured over a period u_i, where u_i \neq d_i , then nCohort[i] must be adjusted to correspond to d_i by multiplication by u_i/h_i , i.e., the value M_i input as nCohort[i] is computed as

M_i^*=M_iu_i/h_i

Let S_i denote the number of survivors at time t_i from a hypothetical (for population table) or observed (for cohort table) population. Then, S_0 =initialPopulation for population table, and S_0= nCohort[0] for cohort table, and S_i is given by S_i=S_{i-1}-\delta_{i-1} where \delta_i=S_iq_i is the number of individuals who die in the ith interval. The proportion of survivors in the interval is given by V_i=S_i/S_0 while the asymptotic variance of V_i can be estimated as follows:

\textup{var}(V_i)=V_i^2\sum_{j=1}^{i-1}\frac{ \sigma_j^2}{\left ( 1-q_j \right )^2}

The expected lifetime at the beginning of the interval is calculated as the total lifetime remaining for all survivors alive at the beginning of the interval divided by the number of survivors at the beginning of the interval. If e_i denotes this average expected lifetime, then the variance of e_i can be estimated as (see Chiang 1968):

\textup{var}(e_i)=\frac{\sum_{j=i}^{n-1}P_j^2 \sigma_j^2\left [ e_{j+1}+h_{j+1}(1-a_j) \right ]^2}{P_j^2}

where var(e_n) = 0.0.

Finally, the total number of time units lived by all survivors in the time interval can be estimated as:

U_i=h_i[S_i-\delta_i(1-a_i)]

Inheritance Hierarchy

System..::.Object
Imsl.Stat..::.LifeTables

See Also