LifeTables Class

LifeTables Class

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.

Inheritance Hierarchy

SystemObject
Imsl.StatLifeTables

Namespace: Imsl.Stat
Assembly: ImslCS (in ImslCS.dll) Version: 6.5.2.0

Syntax

C++

Copy

[SerializableAttribute]
public class LifeTables

<SerializableAttribute>
Public Class LifeTables

[SerializableAttribute]
public ref class LifeTables

[<SerializableAttribute>]
type LifeTables =  class end

The LifeTables type exposes the following members.

Constructors

	Name	Description
	LifeTables	Constructs a new LifeTables instance. The number of classes, nClasses is equal to the length of the input array nCohort.

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Methods

	Name	Description
	Equals	Determines whether the specified object is equal to the current object. (Inherited from Object.)
	Finalize	Allows an object to try to free resources and perform other cleanup operations before it is reclaimed by garbage collection. (Inherited from Object.)
	GetHashCode	Serves as a hash function for a particular type. (Inherited from Object.)
	GetLifeTable	Compute a cohort table.
	GetPopulationTable	Compute a population table.
	GetType	Gets the Type of the current instance. (Inherited from Object.)
	MemberwiseClone	Creates a shallow copy of the current Object. (Inherited from Object.)
	ToString	Returns a string that represents the current object. (Inherited from Object.)

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Properties

	Name	Description
	PopulationSize	Sets the population size at the beginning of the first age interval in requesting a population table.

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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 , for i = 0, 1, ..., 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 denote the number of individuals dying in age interval i, where age interval i ends at time . For population table, the death rate at the middle of the interval is given by , where is the number of individuals alive at the middle of the interval, and $h_i=t_i-t_{i-1}$ , . The number of individuals alive at the beginning of the interval may be estimated by where is the probability that an individual dying in the interval dies prior to the interval midpoint. For cohort table, is input directly while the death rate in the interval, , is not needed.

The probability that an individual dies during the age interval from $t_{i-1}$ to is given by . It is assumed that all individuals alive at the beginning of the last interval die during the last interval. Thus, = 1.0. The asymptotic variance of 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 occur over a time period equal to . If is measured over a period , where $u_i \neq d_i$ , then nCohort[i] must be adjusted to correspond to by multiplication by , i.e., the value input as nCohort[i] is computed as

Let denote the number of survivors at time from a hypothetical (for population table) or observed (for cohort table) population. Then, =initialPopulation for population table, and = nCohort[0] for cohort table, and 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 while the asymptotic variance of 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 denotes this average expected lifetime, then the variance of 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() = 0.0.

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

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

Reference

Imsl.Stat Namespace

Other Resources

Example