DEMOGRAPHIC

AND

HEALTH

RESEARCH

DHS WORKING PAPERS

2010 No. 70

Hassan Eini-Zinab

September 2010

This document was produced for review by the United States Agency for

International Development.

New Multivariate Methodology for

Assessing the Effect of Child Mortality on 

the Total Fertility Rate and its Components
The DHS Working Papers series is a prepublication series of papers reporting on research in progress that 

is based on Demographic and Health Surveys (DHS) data. This research is carried out with support 

provided by the United States Agency for International Development (USAID) through the MEASURE 

DHS project (#GPO-C-00-08-00008-00). The views expressed are those of the authors and do not 

necessarily reflect the views of USAID or the United States Government.

MEASURE DHS assists countries worldwide in the collection and use of data to monitor and evaluate 

population, health, and nutrition programs. Additional information about the MEASURE DHS project can 

be obtained by contacting MEASURE DHS, ICF Macro, 11785 Beltsville Drive, Suite 300, Calverton, 

MD 20705 (telephone: 301-572-0200; fax: 301-572-0999; e-mail: reports@measuredhs.com; internet: 

www.measuredhs.com).
New Multivariate Methodology for Assessing the Effect of Child Mortality 

on the Total Fertility Rate and Its Components

Hassan Eini-Zinab

ICF Macro

Calverton, Maryland, USA

September 2010

Corresponding author: Hassan Eini-Zinab, Population and Health Studies, Research Program, East-West Center, 

1601 East West Rd. Honolulu, HI 96848, USA; E-mail: eini@hawaii.edu and h_eini_z@yahoo.com
Suggested citation:

Eini-Zinab, Hassan. 2010. New Multivariate Methodology for Assessing the Effect of Child 

Mortality on the Total Fertility Rate and Its Components. DHS Working Papers No. 70.

Calverton, Maryland, USA: ICF Macro.
ABSTRACT

This paper develops and validates new multivariate methodology to assess the effect of 

child mortality on period and cohort fertility. Methodology is based on discrete-time survival 

models that are used to construct multivariate life tables of parity progression. Parity transitions 

are transitions from the woman’s own birth to her first birth (0-1), from first to second birth (1-

2), from second to third birth (2-3), and so on. The basic dimensions of these life tables are age, 

parity, duration in parity, and child mortality. By multivariate is meant that a life table can be 

constructed by values or categories of one socioeconomic predictor variable while holding the 

other socioeconomic predictors constant (socioeconomic predictors may include residence, 

education, wealth index, religion, etc). These multivariate life tables yield various measures of 

the quantum and tempo of fertility and child mortality. The methodology also yields a 

replacement rate, which measures the extent to which parents replace lost children.

The validity of the methodology is tested by applying it to India’s first National Family 

Health Survey, conducted in 1992/93 (NFHS-1). The application is to both cohort and period 

data. The cohort analysis pertains to all women age 45-49 at time of survey. The period analysis 

pertains to the 5-year period preceding the survey, spanning years 1988 to 1992. Tests indicate 

that basic results agree closely with results from the birth history method and other methods, 

insofar as comparisons can be made. The replacement rate in the cohort analysis is 0.32 which 

means about one-third of lost children by cohort of women age 45-49 were replaced by 

additional births. The replacement rate in the period analysis is 0.50.

1

INTRODUCTION

Griffith Feeney (1983, 1986) developed a methodology that uses life tables of parity 

progression to calculate parity progression ratios (PPRs) and a total fertility rate (TFR) from the 

PPRs using either period or cohort birth history data. Feeney’s work inspired the development of 

a multivariate version of his methodology by Retherford et al., based on multivariate life tables 

of parity progression (Retherford et al., 2009, 2010a, 2010b). The term ―multivariate‖ means that 

a life table can be constructed by values or categories of socioeconomic predictors of interest, 

such as residence, education, wealth index, religion, etc. The advantage of this multivariate 

methodology compared with Feeney’s original is that it yields fertility measure by categories or 

values of predictor variables, while holding other covariates constant.

The multivariate methodology approaches the calculation of parity progression ratios 

from the perspective and with the tools of discrete-time event history analysis (Retherford et al., 

2009, 2010a). The event of interest is the transition from parity i to parity i+1 (i=0,1, 2...) as a 

function of the duration t in parity i and a number of socioeconomic predictors of interest. This 

multivariate methodology is referred to as the Pit method, where i denotes parity and t denotes 

duration in parity i. If there are no socioeconomic predictors, the transition probabilities Pit

estimated by the multivariate methodology coincide with the standard ones calculated using 

Feeney’s approach. 

Adding the woman’s age at starting parity i to the Pit method allows construction of 

multi-dimensional (or ―global‖) life tables where the basic dimensions are age, parity, and 

duration in parity (Retherford et al., 2010b). This extended methodology is referred to as the Pait

method, where a denotes a woman’s age, and t denotes duration in parity i. The multivariate life 

tables derived applying the Pait method yield various measures of the quantum and tempo of 

fertility. Measures of the quantum of fertility are parity progression ratios (PPRs), age-specific 

fertility rates (ASFRs), and the total fertility rate (TFR). Measures of the tempo or timing of 

fertility are mean and median ages at childbearing (both all births and by birth order) and mean 

and median closed birth intervals. For ease of exposition, these related measures are collectively 

referred to as ―components of the TFR‖. 
2

This paper builds on the Pait method to incorporate child mortality. Specifically, we 

develop and validate a new multivariate methodology to assess the effect of child mortality on 

fertility. To do so, for each parity i, we take into account of the number of child deaths 

(regardless of child’s age at death) that occurred to women age a at each duration t. This 

methodology is labeled as the Paitmn, where a denotes age, i denotes parity, t denotes duration in 

parity, m denotes child mortality state (number of child deaths, lagged by one year) at the start of 

a one-year age period, and n denotes child mortality state at the end of that period. 

In addition to the quantum and tempo measures of fertility mentioned above, this new 

methodology yields measures of the quantum and tempo of child mortality. Mean number of 

child deaths per woman and child mortality rates by age of woman (not by age of child, who can 

be of any age at death) are the measures of the quantum of child mortality, and mean age of 

women at child death is a measure of the tempo or timing of child mortality. The methodology 

also yields a replacement rate, which is based on the effect of child mortality on the TFR. The 

replacement rate measures the extent to which parents replace lost children (regardless of child’s 

age at death) by having more births. The new methodology is validated using data from India’s 

first National Family Health Survey (NFHS-1), conducted in 1992/93. 
3

THE Paitmn METHOD

The Global Life Table

The new methodology that we develop in this paper builds on the Pit and Pait methods. 

The details of these methods are presented elsewhere (Retherford et al., 2009, 2010a, 2010b)1, 

and we summarize here only their key elements. 

Pit and Pait refer to multivariate models of parity progression for women of reproductive 

age. The event of interest in both cases is the transition from parity i to parity i+1 (i=0,1, 2...), 

and the key outcomes to be estimated are the probabilities of transition from parity i to parity 

i+1.2These probabilities are estimated using discrete-time survival models. Specifically, the 

complementary log-log (CLL) model3is used to model parity progression (i.e. the probability of 

transition) from a woman’s own birth to her first birth (0-1), from first birth to second birth (1-2), 

from second birth to third birth (2-3), and so on4. For each parity transition, duration in parity (t) 

is the basic predictor variable in the CLL model. The set of predictor variables also includes 

additional characteristics of interest such as residence, education, wealth index, religion, etc.

In the Pit method, each CLL model for a particular parity transition yields estimates of the 

transition probability Pit (Retherford et al., 2009, 2010a). These transition probabilities can be 

used to construct a life table for each parity, from which a PPR, mean and median age at first 

birth, and mean and median closed birth intervals are calculated. The PPRs of different parity 

transitions are combined to estimate the TFR. 

Adding the woman’s age to the underlying CLL models in the Pit method allows 

construction of multi-dimensional life tables where the basic dimensions are woman’s age, 

parity, and duration in parity. This extended methodology is labeled as the Pait method, where a

denotes the woman’s age, i denotes parity, and t denotes duration in parity (Retherford et al., 

 

1 The procedure for data construction of this method is similar to the previous methods.

2

It is important to clarify that Pit and Pait represent the names of the methodologies as well as their outcomes (the 

transition probabilities between parities i and i+1 as a function of duration in parity t and woman’s age a).

3

For explanation of why complementary log-log is used instead of other survival models, see Retherford et al., 

2009, 2010a).

4 The 0-1 transition can be disaggregated into separate parity transitions from woman’s birth to her first marriage (BM) and from first marriage to first birth (M-1), if so desired (see Retherford et al., 2009, 2010a). This research uses a 

combined 0-1 transition, since child mortality becomes a determinant of fertility only after a woman has at least one 

birth, so that marriage is largely irrelevant.
4

2010b). A separate model is run for each parity i to estimate the probabilities of transition Pait. 

Collectively, the models for the various parity transitions yield model-predicted estimates of 

probabilities of failure (i.e., first birth or next birth) by age, parity, and duration in parity, and by 

socioeconomic characteristics. The Pait are conditional probabilities of failure between a and 

a+1; i.e., they are conditional on ―survival‖ to age a, parity i, and duration t. 

The Pait for a particular set of values of the socioeconomic characteristics allow 

construction of a ―global life table.‖ The term ―global‖ means that the life table associated to the 

estimated Pait spans all parity transitions, rather than just a single parity transition. From the 

global life table one can calculate not only a TFR but also all of the various components of the 

TFR mentioned above. Because the Pait are multivariate, the global life table is itself 

multivariate, as are all of the measures derived from it.

Child Mortality as an Additional Dimension of the Global Life Table

The Paitmn method that we develop in this paper adds a new dimension (child mortality) to 

the global life tables constructed with the Pait method. In the new methodology, a denotes the 

woman’s age, i denotes parity, t denotes duration in parity, m denotes child mortality state at the 

start of a one-year age interval, and n denotes child mortality state at the end of that interval. 

Child mortality state corresponds to the number of child deaths between the beginning and the 

end of a one-year interval of the woman’s age, irrespective of the child’s age at death. Both m

and n are lagged by one year, to ensure that causation is from child mortality to fertility, not the 

other way around.5In other words, m and n, though denoted at the woman’s age a, actually 

pertain to the number of child deaths between the beginning and end of the (a-1)th age interval. 

Child mortality state is represented by two subscripts in the Paitmn methodology because, as will 

be explained in the next section, women are partitioned into two groups over each one-year age 

interval. One group consists of women who do not experience a child death over the interval (in 

which case n = m), and the other group consists of women who do experience a child death over 

the interval (in which case n = m + 1). In the case of women who experience more than one child 

 

5 A more complete way to control for reverse causation in this research is running separate models for each parity 

transition. Because each parity transition has its own CLL model, the causation is always from child mortality to 

fertility, not the other way around. For example, if a fifth child dies, parity transitions 0-1, 1-2, 3-4, and 4-5 will not 

be affected. But parity transitions 5-6 and higher could be affected after one year of the child’s death.
5

death during the year (a relatively rare event), the additional child deaths are spread forward in 

time, so that no more than one child death occurs in any given year.

To illustrate the mechanisms of the new Paitmn methodology, Figure 2.1 depicts 

transitions of women along the new global life table’s multiple dimensions, for each of the first 

three transitions 1-2, 2-3, and 3-4 starting from any particular age. In the figure, duration in 

parity t starts at 0.6 Mm denotes child mortality state (irrespectively of child’s age at death) for a 

woman age a and duration in parity t. The subscript m refers to the number of dead children at 

the start of the previous age interval (a-1). 

 

6

In a CLL model for a particular parity transition, t starts at 1, which is later translated to zero when computing the 

global life tables.
6

Figure 2.1: Possible child mortality transitions over duration in parity during parity transitions 1-2, 

2-3, and 3-4 

Note: M0, M1, M2, ... denote one-year-lagged child mortality states7. 

 

7

Subscript m in Mm denotes woman's number of previous child deaths at the start of the previous age interval. Child 

deaths pertain to children of any age. The logic of the figure requires that a woman cannot experience more than one 

Panel 2: Possible child mortality transitions within parity transition 2-3:

M1

Panel 3: Possible child mortality transitions within parity transition 3-4:

M0

M0

.

 .

 .

t

0

1

2

9

M0

M0 M1

0

1

2

9

M0 M1

M0 M1

 a a+1 a+2 . . . a+9

M0 M1 M2

M0 M1 M2

 .

 .

 .

 a a+1 a+2 . . . a+9

0 M0 M1

1

2

9

M2

M0 M1 M2

M0 M1 M2 M3

M0 M1 M2

 a a+1 a+2 . . . a+9

 .

 .

 .

.

.

.

.

.

.

.

.

.

Panel 1: Possible child mortality transitions within parity transition 1-2:

M3
7

Child mortality does not come into play in the 0-1 transition, because during this 

transition there are as yet no children born. Thus, Panel 1 of Figure 2.1 shows child mortality 

transitions from one age interval to the next within the 1-2 parity transition. Consider a woman 

who starts in state M0 at age a and duration 0 (i.e., t = 0). At age a+1 and duration 1 (i.e., t = 1), 

there is still only one child mortality state, M0, because child mortality state is lagged by one year 

of age. A woman in the M0 state at age a+1 and duration 1 can continue in that state until age 

a+2 and duration 2 if she does not experience a child death between age a and duration 0 and age 

a+1 and duration 1 (i.e., between exact age a and exact duration 0 up to but not including exact 

age a+1 and duration 1). Alternatively, the woman can experience a child death between age a

and duration 0 and age a+1 and duration 1, in which case she moves from M0 to M1. Once a 

woman attains child mortality state M1, she remains in that state until one year after she has a 

second birth (in which case she is no longer in parity transition 1-2) or reaches ten years of 

duration in parity without having yet had a second birth (which is the situation depicted in the 

figure). The logic of Panels 2 and 3 is similar, the difference being that the number of possible 

child mortality transitions increases as parity increases.

In the next section, we illustrate how we use the CLL model to estimate the transition 

probabilities Paitmn and the associated global life tables that take child mortality into account.

Incorporating Child Mortality into the CLL Models

As in the earlier methodologies, the new probabilities Paitmn are also estimated by a set of 

CLL models, one for each parity transition. The basic predictor variables in these CLL models 

are similar to those in the Pait method. They are woman’s age A at starting parity i (where A=a+t) 

and woman’s duration in parity i (denoted by t or a set of dummy variables T1, T2...). The 

difference is that, for the 1-2 and higher-order parity transitions, the Paitmn model also includes 

the predictor variables D1 (number of previous child deaths at the start of the (a-1)th age interval) 

and D2 (number of previous child deaths at the end of the (a-1)th age interval), representing the 

values of m and n at age a, parity i, and duration t. The variables D1 and D2 are defined in the 

 

child death in any given year. But some actually do. This problem is handled in the following way: If, for example, a 

woman had two child deaths in a year (a rare occurrence), one of the two deaths is randomly moved to the following 

year in the woman's birth history, before the expanded data set is constructed. If a woman had three child deaths in 

the same year, two of them, randomly selected, are moved to the following two years, with one child death assigned 

to each of those two years.
8

same way as the subscripts m and n in Paitmn; i.e., D1 = m and D2 = n, where m and n can take on 

different values at different values of a. The more elaborate notation D1 and D2 is used for 

variable names in the model equations, in order to make clear that the number pair (m, n) is 

variable from one age interval to the next and also to conform to the convention of using uppercase letters to represent variable names. Additional predictor variables are socioeconomic 

characteristics of interest. 

In the CLL models, D1 and D2 are also interacted with t and t

2

. This is because it is 

assumed that the effect of a child death on the probability of having a next birth is nonlinear, and 

that a quadratic specification of duration t adequately captures the nonlinearity. Both D1 and D2

are treated as continuous variables, the ranges of which increase as parity increases. But it is still 

the case that the difference between D1 and D2 is either zero or one. 

The new CLL model for parity transitions higher than 0-1, with place of residence and 

education as socioeconomic predictors of interest, is:

P = 1- exp{-exp[a + b1T1 + b2T2 + ... + b9T9 + A(c1+c2t+c3t

2

) + A2(d1+d2t+d3t

2

) + 

D1(e1+e2t+e3t

2

) + D2(f1+f2t+f3t

2

) + U(g1+ g2t+ g3t

2

) + L(j1+ j2t+ j3t

2

) + 

H(k1+ k2t+ k3t

2

) + mU D1 + nL D1 + oH D1]} (2.1)

where A denotes the woman’s age at the start of the parity transition; U denotes urban/rural 

residence; and L and H denote low and high education (with illiterate as the reference category). 

Only D1 is interacted with the socioeconomic variables, because D1 and socioeconomic 

predictors pertain to the same age interval. Values of Paitmn are calculated from this equation in 

essentially the same way that values of Pit and Pait were calculated in Retherford et al., 2009, 

2010a, 2010b.

At higher parities, where sample sizes get smaller and non-convergence may occur, a 

quadratic specification of duration is used in the CLL model equation: 
9

P = 1- exp{-exp[a + b1t + b2t

2+ A(c1+c2t+c3t2

) + A2(d1+d2t+d3t

2

) + 

D1(e1+e2t+e3t

2

) + D2(f1+f2t+f3t

2

) + U(g1+ g2t+ g3t

2

) + L(j1+ j2t+ j3t

2

) + 

H(k1+ k2t+ k3t

2

) + mU D1 + nL D1 + oH D1]} (2.2)

Since a woman’s age increases by parity, person-year contributions from the lower-order 

parity transitions in an open-ended parity transition are smaller than those from higher-order 

parity transitions. Person-year observations contributed from lower-order parity transitions to the 

open-ended parity transition are also greater than person-years contributed from higher-order 

parity transitions, because not all women continue childbearing to higher-order parities. 

Therefore, age of women at the start of parity is not included in the CLL model for open-ended 

parity transition. However, the global life table itself will include age for open-ended parities

(Retherford et al., 2010b). The CLL model for open-ended parity transition is:

P = 1- exp{-exp[a + b1t + b2t

2+ D1(c1+c2t+c3t2

) + D2(e1+e2t+e3t

2

) + 

U(f1+ f2t+ f3t

2

) + L(g1+ g2t+ g3t

2

) + H(j1+ j2t+ j3t

2

) + kU D1 +

mL D1 + nH D1]} (2.3)

When calculating the global life table, it is assumed in the case of open parity intervals that Paitmn

= Pitmn for ages within the open parity interval, where i denotes the open parity interval (e.g., 6+) 

and Pitmn is calculated from equation (2.3). 

Collectively, the CLL models for the various parity transitions yield model-predicted 

failure (i.e., birth) probabilities Paitmn for specified values (usually representing categories) of the 

socioeconomic characteristics. The Paitmn are conditional probabilities of failure (next birth) 

between age a and a+1; i.e., they are conditional on ―survival‖ to age a, parity i, duration t, and 

child mortality states m at age a and n at age a+1. (Because child mortality variables are lagged 

by one year, the actual ages are a-1 and a.) The Paitmn for a particular set of values of the 

socioeconomic characteristics allow calculation of a global life table (illustrated in the next 

section), from which one can calculate not only TFR and its components, but also child mortality 

measures and a replacement rate indicating the extent to which parents replace children who 

have died.
10

Constructing the Global Life Table from the Birth Probabilities Paitmn 

The predicted estimates of Paitmn from the CLL model are the building blocks for 

constructing the global life table that incorporates child mortality, referred to here as the 

GLTWCM. Because the Paitmn are multivariate, the global life table is itself multivariate, as are all 

of the measures derived from it. 

The two other building blocks of the GLTWCM are the functions Saitmn and faitmn. Saitmn

shows number of ―survivors‖ (women) at age a, parity i, duration in parity t, child mortality state 

m at exact age a, and child mortality state n at exact age a+1. (Recall that, because the child 

mortality variables are lagged by one year, the actual ages are a-1 and a.) When the difference 

between m and n is 0, there is no change in child mortality state over the interval a to a+1 and 

duration t to t+1. When the difference is 1, it indicates that, among the women of parity i who 

attained duration t at age a, child mortality state changes from Mm to Mm+1 over the interval a to 

a+1and duration t to t+1. (A detailed presentation of the formulae to construct the global life 

tables that include child mortality is included in Appendix A.1.)

Because a woman may or may not experience a child death over a one-year interval in the 

global life table, it is necessary to partition the Saitmn women into two groups, according to 

whether or not they experienced a child death. This requires estimation of a partitioning factor, 

denoted by Qaitm, which represents the proportion who experience a child death over the interval. 

The proportion of women who do not experience a child death is then 1–Qaitm (Appendix A.2 

elaborates the procedure to calculate the partitioning factors.) 

Estimating Child Mortality Rates from the Global Life Table

The fact that women are partitioned into two groups at any given age, parity, duration in 

parity, and child mortality state in the GLTWCM allows estimation of the number of changes in 

child mortality state (i.e., the number of child deaths) that occur in the global life table as a 

whole, as women progress from age 10 to age 50. The number of changes in child mortality state 

is the same as the number of child deaths lagged by one year. Over each one-year age interval, 

only one change in child mortality state (i.e., one lagged child death) can occur. So, if one 

calculates the number of women who change their starting child mortality state between ages a
11

and a+1 in the global life table, one can also calculate an age-specific child mortality rate, where 

age refers to woman’s age, and child deaths can occur at any age of the child.

In the global life table, women who change their starting child mortality state are located 

at Saitmn, where n = m + 1. The following formulae calculate the child mortality rate by age of 

woman (lagged by one year) and the overall mean number of child deaths per woman:

Da

= (∑Saitmn)/1,000 (2.4)

where 

Da

is the child mortality rate by woman’s age, with child deaths lagged by one year. The 

summation is over i, t, m, and n, but only for cases where n = m + 1.

If the summation in equation (2.4) also includes age a, the result is overall mean number 

of child deaths per woman:

D

= (∑Saitmn)/1,000 (2.5)

This child mortality measure is different than conventional child mortality measures. It reflects 

fertility as well as conventionally measured child mortality. In the case of conventionally 

measured child mortality, children rather than women are the units of analysis. But women are 

units of analysis in the case of

D

. This measure reflects fertility, since the higher the fertility, the 

higher the level of 

D . 

As mentioned by Retherford et al. (2010b) in the discussion of the Pait method, a 

seemingly illogical property of the global life table is that the number of women at any given 

age, Sa, does not equal the starting radix of 1,000 women, due to the global life table property 

that women can experience more than one event; i.e., next birth; in any given one-year age 

interval. However, this property does not affect the calculation of the lagged child mortality 

rates. The reason is that the global life table does not allow women to experience more than one 

child death in a single year of age. 
12

Estimating the Effect of Child Mortality on Fertility Using the Global Life Table

There are different ways to calculate the effect of child mortality on fertility using the 

Paitmn method. In all cases, the starting point is to construct a child-mortality-effect-free global 

life table, which one can do using the Paitmn approach and setting the partitioning factor equal to 

zero. Setting the partitioning factor to zero means that no one in the global life table experiences 

a change in child mortality state; i.e., there are no child deaths. In other words, when the 

partitioning factor is zero, it is always true that Sa,i,t,m,n = Sa,i,t,0,0. We denote the child-mortalityeffect-free global life table as GLTNCM.

Since all measures derived from the GLTNCM are also child-mortality-effect-free, one can 

estimate the effect of child mortality on fertility by comparing these measures with 

corresponding measures calculated from the standard global life table with child mortality 

(GLTWCM). For example, the difference between TFRWCM (the child-mortality-effect-present 

TFR) and TFRNCM (the child-mortality-effect-free TFR) is attributable to the effect of child 

mortality on TFR:

ΔTFR = TFRWCM – TFRNCM (2.6)

ΔTFR depends on the magnitude of child mortality (i.e., mean number of child deaths per 

woman). The higher the mean number of child deaths per woman, the higher ΔTFR. A more 

accurate way to measure the effect of child mortality on TFR, however, is to use the replacement 

rate.

The replacement rate is defined as the difference between the child-mortality-effectpresent TFR and the child-mortality-effect-free TFR, divided by the overall mean number of 

child deaths per woman, 

D

:

Replacement Rate = 

D

TFR

 (2.7)

The replacement rate indicates the extent to which women replace dead children with additional 

births. 
13

VALIDATION OF THE METHODOLOGY

This section tests the validity of the Paitmn method by comparing the fertility and child 

mortality measures derived from it with estimates derived from the other methods (birth history 

method, Pit, and Pait ), to the extent possible. The methodology is considered valid if the 

estimates derived by the various methods reasonably agree. 

The validation procedure includes the application of the four methods (birth history 

method, Pit , Pait, and Paitmn ) to both cohort and period data from India’s first National Family 

Health Survey (NFHS-1), conducted in 1992/93. The cohort analysis pertains to all women age 

45-49 at the time of the survey. The period analysis pertains to the five-year period preceding the 

survey (1988 to 1992)8. The difference between cohort and period analysis is in how one 

constructs the expanded data set. In cohort analysis the expanded data set includes all personyear observations for women from the age at which the life table starts, e.g., age 10, to their 

current age. In contrast, in period analysis the expanded data set includes only those person-year 

observations located in the period of interest. (Each person-year observation is located in a 

particular calendar year.)

One expects estimates derived by the Paitmn method to agree closely with estimates 

derived by the other methods when cohort data are used. However, when using period data, one 

deals with a synthetic cohort instead of an actual cohort, which makes the comparisons between 

the birth history method and the life table methods more complicated. In period analysis the 

biggest difference in results among the various methods is expected to be between the birth 

history method and the global life table methods (both Pait and Paitmn), because in a global life 

table the distribution of women at higher-order parities depends on the distribution of women at 

lower-order parities. In the actual population the distribution of women at higher-order parities 

does not depend on the distribution of women at lower-order parities, because the synthetic 

cohort splices person-year observations from different cohorts. 

 

8 Calendar years in this analysis do not exactly match actual calendar years. Calendar years in this research are 

calculated by counting by increments of 12 months backward from the first complete survey month. For example, 

1992 refers to the first 12 months preceding the 1992/93 NFHS survey. The second 12 months preceding the survey 

are labeled 1991, and so on. The survey month is not included in the analysis because it is an incomplete month for 

most women.
14

Both quantum and tempo measures of fertility and child mortality are calculated with all 

four methods, insofar as possible. The birth history method directly uses the data set to calculate 

these measures9, while the Pit, Pait, and Paitmn methods rely on discrete-time survival models to 

estimate the measures.

Cohort Analysis

This section applies the Paitmn method to the cohort of women age 45-49 in India’s 

NFHS-1. The fertility and child mortality measures derived by the global life table are then 

compared with the fertility and child mortality measures estimated by the birth history, Pit, and 

Pait methods. The estimates of quantum and tempo measures of fertility and child mortality are 

considered separately. 

Quantum measures of fertility

Table 3.1 compares, to the extent possible, the quantum measures of fertility—TFRs, 

PPRs, and ASFRs—derived by the four methods. 

 

9 To make the comparisons between the birth history method and the life table methods accurate, the data set for 

birth history method is constructed the same way as the data set for the life table methods. For example, all first 

births after age 40 and next births after 10 years of duration in parity are eliminated from both data sets.
15

Table 3.1: Comparison of quantum measures of fertility derived by the birth history, Pit, Pait, and 

Paitmn methods: Cohort analysis, women age 45-49, India's NFHS-1

Birth Pit Pait Paitmn Birth Pait Paitmn

PPR History Method Method Method ASFR History Method Method

p0 0.96 0.96 0.96 0.96 10-14 15 9 9

p1 0.95 0.94 0.94 0.94 15-19 172 156 154

p2 0.92 0.91 0.91 0.91 20-24 294 291 286

p3 0.86 0.85 0.85 0.85 25-29 262 265 260

p4 0.79 0.78 0.79 0.78 30-34 168 170 167

p5 0.73 0.73 0.73 0.72 35-39 81 77 76

p6 0.68 0.67 0.68 0.66 40-44 21 22 22

p7 0.62 0.62 0.62 0.60 45-49 6 5 5

p8 0.56 0.57 0.57 0.54

p9 0.51 0.53 0.48 0.47

p10 0.42 0.45 0.47 0.46

p11 0.37 0.40 0.45 0.44

p12 0.34 0.39 0.42 0.41

p13 0.44 0.46 0.38 0.37

p14 0.54 0.71 0.35 0.33

p15 0.30

TFR 5.03 4.93 4.97 4.89 TFR 5.09 4.97 4.89

CEB 5.03

Notes: In the Pit method, the underlying CLL model for each parity transition includes duration in parity as the sole 

basic predictor variable. In the Pait method, the underlying CLL model for each parity transition includes age and 

duration in parity as the basic predictor variables. In the Paitmn method, the underlying CLL model for each parity 

transition includes age, duration in parity, and child mortality as the basic predictor variables.

The Pit method does not yield ASFRs.

The open-ended parity for both the Pait and Paitmn methods is 9+-10+.

ASFRs are per 1000 women.

TFRPPR and TFRASFR, in birth history method are not equal because TFRPPR takes into account censoring at age 

45-49 while TFRASFR does not.

As the table shows, the TFR calculated with the birth history method equals the mean 

children ever born (CEB) of the cohort of analysis, as it was expected. Taking the birth history as 

the base of the comparison, the table shows that all three life table methods (Pit, Pait, and Paitmn ) 

underestimate the TFR. PPRs at lower-order parities are higher for the birth history method than 

the life table methods, but at higher-order parity transitions the PPR estimated by life table 

methods are higher than the birth history method. The TFRs calculated from different methods, 

however, are at acceptable levels. Experience shows the life table approach usually gives TFRs 

that are as much as 0.5 of a child too low or too high compared with the birth history method 

(Retherford et al., 2009; National Bureau of Statistics of China and East-West Center, 2007). 
16

Part of this underestimation or overestimation could be attributed to the underlying regression 

models of the life table approach. Since the fit of the models to the data set is not perfect, some 

discrepancy may occur. The small discrepancy between the different methods is acceptable, as 

the life table approaches are usually used to study the effect of socioeconomic variables on 

fertility, not to measure the fertility rates in the general population. 

As the table shows, the pattern of ASFRs in the birth history, Pait, and Paitmn methods is 

similar, but the level of fertility at each age group differs by method. The Paitmn method has the 

lowest level of fertility for almost all age groups. The Pait method has lower fertility rates than 

the birth history method and a higher fertility level than the Paitmn method for most age groups. 

However, the differences in ASFRs and TFR of the three methods are minimal.10

The ASFRs at 10-14 and 15-19 are underestimated for both Pait and Paitmn methods (see 

Figure 3.1) 11. This underestimation may be related to the quadratic specification of the effect of 

age on probabilities of failure. It seems that the quadratic specification of age does not 

completely capture the effect of age on fertility. It is worth noting that for both the Pait and Paitmn

methods the effect of age on the probabilities of failure is specified as quadratic by means of A

and A

2

. 

 

10 The TFRs calculated from birth history method are not identical because TFRPPR takes into account the censoring 

at last age groups, i.e., 45-49, but TFRASFR does not take into account censoring at this age group. Censoring occurs 

because not all women are fully exposed to this age group.

11 The Pit method does not yield ASFRs because age is not included in its underlying CLL models.
17

Figure 3.1: Comparison of ASFRs derived by the birth history, Pait , and Paitmn methods: Cohort 

analysis, women age 45-49, India's NFHS-1

Tempo measures of fertility

The tempo measures of fertility include mean and median age at childbearing (both all 

births and by birth order) and mean and median closed birth intervals12. Figure 3.2 compares the 

mean ages at childbearing by birth order and all births for the cohort of women age 45-49 in 

India’s NFHS-1.

 

12 Only mean age at childbearing and mean closed birth intervals are presented in this paper.

0

50

100

150

200

250

300

350

10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49

ASFR (per 1000 women)

Birth History Pait Paitmn
18

Figure 3.2: Comparison of mean age at childbearing derived by the birth history, Pait, and Paitmn

methods: Cohort analysis, women age 45-49, India's NFHS-1

As the figure shows, mean ages at childbearing of all three methods are close to each 

other in parities one to nine, where each parity transition has its own CLL model in the life table 

approaches, i.e., the Pait and Paitmn methods. At higher-order parity transitions mean ages of 

childbearing calculated from life table approaches are different than those of the birth history 

method, due to pooling of all higher-order transitions in a single open-ended parity transition. 

However, the two life table methods have similar mean ages at childbearing at higher-order 

parity transitions. The overall mean ages at childbearing calculated from different methods are 

quite similar.

Figure 3.3 presents the other tempo measures of fertility—that is, mean closed birth 

intervals. Mean closed birth intervals correspond to the interval between the two births. For 

instance, the ―2-3‖ shows the birth interval between the second and the third births. The interval 

between a woman’s own birth and her first birth is not shown here, as it is equivalent to her age 

18

23

28

33

38

43

48 Mean age at childbearing

Birth History Pait Paitmn
19

at first birth and thus is shown in the previous figure. As figure 3.3 shows, all four methods yield 

similar results. At lower parity transitions the means of all methods agree well, while at higherorder transitions the results of different methods do not agree well, due to the pooling of data at 

open-ended parity transition. The Pit method does not use open-ended parity transition, while the 

Pait and Paitmn methods use 9+-10+ as open-ended parity transition. 

Figure 3.3: Comparison of mean closed birth intervals derived by the birth history, Pit, Pait, and 

Paitmn methods: Cohort analysis, women age 45-49, India's NFHS-1 

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

Mean Closed Birth Interval (years)

Birth History Pit Pait Paitmn
20

Measures of child mortality

The child mortality measures considered here are the overall mean number of child 

deaths per woman, child mortality rates by women’s age (not child’s age, which can be any age 

at death), and mean and median age of women at child death (all child deaths, so that if a woman 

experiences two child deaths, two ages for her are included in the calculation of mean and 

median). Only measures derived by the birth history method can be compared with the child 

mortality measures derived by Paitmn method, since the other life table methods do not yield child

mortality measures. Because child mortality is a one-year lagged variable, we apply the birth 

history method also by lagging child mortality by one year. 13

Table 3.2, compares the child mortality measures derived by the birth history method and 

the Paitmn method for cohort of women age 45-49 in India’s NFHS-1. As the table indicates, the 

two methods yield overall mean numbers of child deaths per woman that are close together. The 

overall mean number of child deaths per woman estimated from the Paitmn method is slightly 

lower than the equivalent birth history method value, but it is acceptable. In terms of child 

mortality rates by women’s age, as figure 3.4 shows, the two methods yield quite similar 

patterns. However, the level of the measures is lower for the Paitmn method than for the birth 

history method. As the figure shows, most of the underestimation of child mortality by the Paitmn

method occurs at middle age groups (20-29).

 

13 The procedure to calculate the child mortality (lagged by one year) by age of woman from the original data set is 

as follows: first the original dataset should be expanded by current age of woman (one person-year observation for 

each age). Each person-year observation then gets a value for age that shows age of woman at each person-year 

observation. A counter variable counts number of child mortality experiences at each person-year observation as of 

one-year ago. Using the expanded dataset and child mortality variable, one can calculate child mortality rates by age 

of woman (not child’s age that can be any age at death) just like the calculation of ASFRs. The child mortality rates 

could be reported by single ages of woman or averaged by five-year age groups. The overall mean number of child 

deaths per woman is calculated by either summing the rates by single ages or summing the five-year rates and 

multiplying the sum by five. The overall mean number of child deaths per woman could also be calculated directly 

from the dataset, without expansion, by counting number of child mortality experiences for each woman as of one 

year ago and taking the mean of the variable.
21

Table 3.2: Comparison of child mortality measures derived by the birth history and Paitmn

methods: Cohort analysis, women age 45-49, India's NFHS-1

Age Group Birth History Paitmn Method

10-14 0 0

15-19 16 13

20-24 48 39

25-29 53 48

30-34 42 41

35-39 24 27

40-44 14 14

45-49 8 6

Overall 1.03 0.94

Mean 29.2 29.5

Median 28.7 29.3

TFRWCM 4.89

TFRNCM 4.58

Replacement Rate 0.33

Note: age specific rates are per 1000 women.

“Overall” refers to mean number of child deaths per woman lagged by one year.

Mean and median refer to mean and median age of women at child death.

TFRWCM refers to child-mortality-effect-present TFR. TFRNCM refers to child-mortality-effect-free TFR.

Replacement rate is (TFRWCM - TFRNCM)/mean number of child deaths per woman.

Mean age of women at child death (lagged by one year), calculated from the two methods 

agrees well. The two medians also agree but the median calculated from the birth history method 

is slightly younger than the median estimated by the Paitmn method.

For the Paitmn method, in the table we present also the replacement rate, which cannot be 

calculated by applying the birth history method. The replacement rate is about 0.33, which 

means that in the cohort of study every 100 children who die are replaced by 33 additional births.
22

Figure 3.4: Comparison of child mortality rates (lagged by one year) by mother's age derived by 

the birth history and Paitmn methods: Cohort analysis, women age 45-49, India's NFHS-1

Period Analysis

This section applies the Paitmn method to period data for the five years preceding India’s 

NFHS-1 (1988 to 1992). The fertility and child mortality measures derived by the global life 

table are then compared with fertility and child mortality measures estimated by the birth history, 

Pit, and Pait methods insofar as possible. The comparisons are made, once again, by quantum and 

tempo measures of fertility and child mortality. 

Quantum measures of fertility

Table 3.3 compares quantum measures of fertility derived by all four methods insofar as 

possible.

0

10

20

30

40

50

60

10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49

Child mortality rate (by mother's age)

Birth History Paitmn
23

Table 3.3: Comparison of quantum measures of fertility derived by the Pit, Pait, and Paitmn

methods: 5-year period spanning 1988-92, India's NFHS-1

Pit Pait Paitmn Birth Pait Paitmn

PPR Method Method Method ASFR History Method Method

p0 0.92 0.92 0.92 10-14 0 3 3

p1 0.93 0.91 0.91 15-19 121 99 98

p2 0.78 0.73 0.73 20-24 234 225 222

p3 0.66 0.62 0.61 25-29 172 181 178

p4 0.62 0.58 0.56 30-34 97 90 87

p5 0.58 0.53 0.52 35-39 43 32 30

p6 0.59 0.55 0.53 40-44 15 8 7

p7 0.57 0.52 0.49 45-49 6 2 2

p8 0.50 0.44 0.42

p9 0.50 0.47 0.43

p10 0.46 0.44 0.41

p11 0.46 0.41 0.38

p12 0.30 0.36 0.34

p13 0.41 0.31 0.29

p14 0.67 0.26 0.25

TFR 3.51 3.20 3.13 TFR 3.44 3.20 3.13

Note: Pit method refers to life tables in which the underlying CLL model for each parity transition includes duration in 

parity as the sole basic predictor variable. Pait method refers to life tables in which the underlying CLL model for 

each parity transition includes age and duration in parity as the basic predictor variables. Paitmn method refers to life 

tables in which the underlying CLL model for each parity transition includes age, duration in parity, and child 

mortality as the basic predictor variables.

Pit method does not yield ASFRs.

The open-ended parity for both Pait and Paitmn methods is 9+-10+.

ASFRs are per 1000 women.

As the table shows, the Pit method produces the highest TFR that is closest, although 

slightly high, to the TFRasfr calculated by birth history method14. Once age is introduced into the 

life tables derived by the Pait and Paitmn methods, the TFR drops substantially. This drop could be 

attributed to the fact that the distribution of the actual population and the life table population are 

different and that the actual population has undergone a substantial change in fertility and 

marriage patterns and levels during the previous decade(s). The change in fertility patterns and 

levels is one of the reasons that the two populations have different distributions. As discussed 

earlier, the fact that the global life tables use the synthetic cohort approach makes the distribution 

 

14 The TFRPPR is not calculated for period analysis, as it requires use of Feeney’s method.
24

of women by age, parity, and duration in parity in the global life table different than in the actual 

population. 

The birth history method to calculate the PPRs of period data is to apply Feeney’s period 

parity progression ratios (PPPR) methodology, but it is not applied here.

With regard to the age-specific fertility rates, as the table shows the birth history method 

produces the highest ASFRs and TFRs, while the ASFRs and TFRs of the Pait and Paitmn methods 

are lower, but close to each other. Figure 3.5 shows the ASFRs of all methods. As the figure 

indicates, the ASFRs estimated by the birth history method are almost always higher than the 

ASFRs estimated by the Pait and Paitmn methods. Although the levels of ASFRs of the three 

methods are different, all three methods have a similar pattern. This shows that the Pait and Paitmn

methods underestimate fertility at all age groups.

Figure 3.5: Comparison of ASFRs derived by the birth history, Pait , and Paitmn methods: 5-year 

period spanning 1988-92, India's NFHS-1

0

50

100

150

200

250

10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49

ASFR (per 1000 women)

Birth History Pait Paitmn
25

Tempo measures of fertility

The tempo measures of fertility include mean and median age at childbearing (both all 

births and by birth order) and mean and median closed birth intervals. Figure 3.6 compares mean 

ages at childbearing, derived alternatively by the birth history, the Pait, and the Paitmn methods, 

for the period of interest. The birth history method to calculate mean age at childbearing uses the 

ASFRs. In the case of mean ages at childbearing by birth order, the ASFRs at each birth order 

are used.

Figure 3.6: Comparison of mean age at childbearing derived by the birth history, Pait, and 

Paitmn methods: 5-year period spanning 1988-92, India's NFHS-1

Since the age distribution of the population of the global life table may differ from the 

age distribution of the actual population, the period estimates of mean age at childbearing by 

birth order calculated from the birth history method and the global life table methods may not 

agree well. However, mean ages at childbearing by birth order should agree closely when the 

18

23

28

33

38

43

48

53

Mean age at childbearing (years)

Birth History Pait Paitmn
26

two life table methods (the Pait and Paitmn methods) are compared. As the table shows, the means 

directly calculated from the data set, using the birth history method, are lower than the means 

calculated by the life table methods. But the estimates of the Pait and Paitmn methods are almost 

identical, showing that the two methods yield similar results. These higher means show older 

population for life table populations than for the actual population. The overall age at 

childbearing in the actual population is about one year younger than in the global life table 

populations. This difference could be attributed to the differences in distribution of actual and 

global life table populations. The close proximity of the results of the Pait and Paitmn methods 

shows that the two methods have similar population distributions, and introduction of child 

mortality to the life table does not change the population distribution of the global life table. As 

previously mentioned, the means calculated from the global life table methods at higher-order 

transitions are not comparable to the birth history method, because life table approaches use 

open-ended parity transition and age is not included in the underlying CLL model. 

Figure 3.7 presents the next tempo measure of fertility, mean closed birth intervals. The 

figure shows mean closed birth intervals calculated from the Pit, Pait, and Paitmn methods, for 

transitions 1-2 and higher. Mean closed birth intervals correspond to the mean interval between 

the two births. The mean closed birth intervals are not calculated by the birth history method, as 

one would need to apply Feeney’s approach.

As the table shows, the three methods yield almost similar results. The results of the Pit

method at higher-order births (10th birth and higher) are different than the Pait, and Paitmn results. 

This could be related to the open-ended parity transition in the Pait, and Paitmn methods. The Pit

method does not use open-ended parity transition, while the Pait, and Paitmn methods use 9+-10+ 

as open-ended parity transition. 
27

Figure 3.7: Comparison of mean closed birth intervals derived by the Pit, Pait, and Paitmn

methods: 5-year period spanning 1988-92, India's NFHS-1

0

1

2

3

4

5

6

1-2 2-3 3-4 4-5 5-6 6-7 7-8 8-9 9-10 10-11 11-12 12-13 13-14 14-15

Mean CBI (years)

Pit Pait Paitmn
28

Measures of child mortality

The child mortality measures that could be calculated from a global life table that 

includes child mortality are the overall mean number of child deaths per woman, child mortality 

rates by woman’s age (not child’s age, which can be any age at death), and mean and median age 

of women at the child’s death. Only the birth history method could be compared with the child 

mortality measures derived by the Paitmn method, as the other life table methods do not yield 

child mortality measures. 

Due to lag function, the number of child deaths per woman in the Paitmn method actually 

shows the child mortality measure as of one-year ago. In order to make the measures 

comparable, the birth history method should also lag child mortality by one year. 

Table 3.4 compares the child mortality measures derived by both the birth history method

and the Paitmn method for the five-year period 1988-1992 in India’s NFHS-1. 

Table 3.4: Comparison of child mortality measures derived by the birth history and Paitmn

methods: 5-year period spanning 1988-92, India's NFHS-1

Age Group Birth History Paitmn Method

10-14 0 0

15-19 7 4

20-24 24 15

25-29 27 17

30-34 21 12

35-39 15 7

40-44 10 3

45-49 8 1

Overall 0.56 0.29

Mean 29.0 28.4

Median 27.9 28.0

TFRWCM 3.13

TFRNCM 2.99

Replacement Rate 0.50

Note: age specific rates are per 1000 women.

“Overall” refers to mean number of child deaths per woman lagged by one year.

Mean and median refer to mean and median age of women at child death.

TFRWCM refers to child-mortality-effect-present TFR. TFRNCM refers to child-mortality-effect-free TFR.

Replacement rate is (TFRWCM - TFRNCM)/mean number of child deaths per woman.
29

As Table 3.4 indicates, the Paitmn method underestimates the child mortality measures. 

The underestimation is much worse in the period case in Table 3.4 than in the cohort case in 

Table 3.2. The overall mean number of child deaths per woman estimated by this method is 

about half the number that the birth history method generates. Also the Paitmn method 

underestimates child mortality rates by mother’s age at all ages, and especially at ages 20 and 

above. The main reason for discrepancies in child mortality estimates could be the difference in 

age distributions of the actual period population and the period global life table population. 

Within a particular age group, the distribution of women by parity and duration in parity 

evidently differs substantially between the actual population and the global life table population. 

When child mortality declines dramatically, the period global life table underestimates child 

mortality. The measures estimated by the global life table actually show that these measures will 

be achieved by the hypothetical cohort if the rates stay constant and the synthetic cohort follows 

those rates over the reproductive span.

The mean age of women at time of child death (lagged by one year) is slightly higher for 

the actual population than the global life table population. However, the two populations have 

almost identical median age of women at time of child death15. 

The replacement rate estimated by the Paitmn method is about 0.50 for the period global 

life table population constructed by using data for the five-year period 1988-1992 from India’s 

NFHS-1. The child replacement rate 0.50 shows that, of each two lost children in the period GLT 

population, one is replaced by an additional birth.

 

15 The mean and median ages of woman at child death by the birth history method are also calculated by using the 

age-specific (woman’s age) child mortality rates.
30

Figure 3.8: Comparison of child mortality rates (lagged by one year) by mother's age 

derived by the birth history and Paitmn methods: 5-year period spanning 1988-92, India's 

NFHS-1

As explained earlier, the child mortality measure used in this research is different than 

conventional child mortality measures, as it contains elements of fertility as well as child 

mortality. For example, in Figure 3.8 the age-specific child mortality rate is higher than the agespecific fertility rate at age 45-49. At first it appears that there must be some kind of error. But 

this result could occur inasmuch as there are very few births at age 45-49 but there could be even 

more child deaths at age 45-49, because the child deaths pertain not only to children born to 

mothers age 45-49 but also to all of the woman’s previous births.

0

5

10

15

20

25

30

10-14 15-19 20-24 25-29 30-34 35-39 40-44 45-49

Child mortality by mother's age (per 1000)

Birth History Paitmn 
31

CONCLUSIONS

This paper develops and validates new multivariate methodology to assess the effect of 

child mortality on period and cohort fertility. The methodology is based on discrete-time survival 

models that are used to construct multivariate life tables of parity progression. This new 

methodology is built on previous research on developing multivariate methods of estimating the 

TFR and its components. The methodology is important because it yields comprehensive results 

on the effect of child mortality on fertility. The key measure of the effect is the child replacement 

rate, which shows the extent to which parents replace lost children by having additional births. 

The method is also important due to its applicability to period data and its ability to control for 

reverse causality between child mortality and fertility.

The research tested the validity of the methodology by applying it to India’s first National 

Family Health Survey (NFHS-1), conducted in 1992/93. The application is to both cohort and 

period data. The results of the analysis show that the new estimates of child mortality and 

fertility reasonably agree with the estimates derived from other methods.

Although this methodology overcomes the deficiencies of the previous methods used in 

the literature, it has its own limitation as well. The biggest limitation is the calculation of 

standard errors of estimates in order to conduct significance tests. The jackknife method is 

usually used to estimate the standard error of estimates in this method. The jackknife procedure

requires repeatedly running the whole process of estimating fertility and child mortality measures 

by number of clusters in each sample. One drops one cluster from the sample each time the 

process is repeated until all clusters have dropped once. The process of estimating fertility and 

child mortality measures include expanding datasets, running CLL models for each parity 

transition, constructing the global life tables, and estimating the measures from the global life 

tables. This process should be repeated for each category of predictor variables on each survey. 

In the case of global life tables that include child mortality, calculation of standard errors for a 

study similar to this research with few additional socioeconomic variables requires months of 

super fast computer time.
32

NOTATION AND ABBREVIATIONS

i: woman’s parity (number of children ever born)

t: duration in parity (in years)

T1, T2, ..., Tk-1: dummy variables representing the first k-1of k one-year life table time intervals 

(alternative representation of duration in parity)

A: woman’s starting age (in years) in any given parity transition

a: woman’s age (in years) at a particular duration in parity t, where a = A + t (in the global life 

table, but not in the CLL equations, age a is translated to age a-10, so that a=0 at age 10 

at the beginning of the global life table)

Parity transitions

B-M: transition from a woman’s own birth to her first marriage

M–1: transition from first marriage to first birth

0-1: transition from a woman’s own birth to her first birth

1-2: transition from first to second birth

2-3: transition from second to third birth

3-4: transition from third to fourth birth

4-5: transition from fourth to fifth birth

5-6: transition from fifth to sixth birth

6+ - 7+: transition from sixth or higher-order birth to next higher-order birth

CLL model: complementary log-log model (one type of discrete-time survival model)
33

Pit method: method in which the basic predictor variable in the CLL model for each parity 

transition is duration in parity

Pait method: method in which the basic predictor variables in the CLL model for each parity 

transition are age and duration in parity

Paitmn method: method in which the basic predictor variables in the CLL model for each parity 

transition are age, duration in parity, and two indicators of child mortality

m: child mortality state at the beginning of age a-1 (number of previous child deaths) 

n: child mortality state at the beginning of age a (number of previous child deaths)

n

: average mortality state at the beginning of age a (average number of previous child deaths to 

women at the beginning of age a, as calculated for person-year observations with 

specified characteristics in the expanded data set)

Mm: Alternative notation for m

Mn: Alternative notation for n

D1: Alternative notation for m

D2: Alternative notation for n

D

: Mean number of child deaths per woman, calculated from the entire global life table

Pt: for women of any given parity, Pt is the conditional probability of failure (next birth) between 

durations t and t+1

Pait: for women age a, parity i, and duration t, Pait is the conditional probability of failure (next 

birth) between ages a and a+1 and therefore also between durations t and t+1

Paitmn: for women age a, parity i, duration t, and child mortality states m and n, Paitmn is the 

conditional probability of failure (next birth) between ages a and a+1 and therefore also 

between durations t and t+1
34

a i t m n

P

, , , ,

: for women age a, parity i, duration t, and child mortality states m and 

n , 

a i t m n

P

, , , ,

is the 

conditional probability of failure (next birth) between ages a and a+1 and therefore also 

between durations t and t+1 (

n

is substituted for D2 when computing 

a i t m n

P

, , , ,

from the 

underlying CLL model equation) 

faitmn: number of failures (births) between ages a and a+1 (and therefore also between durations t 

and t+1) in the global life table to women of parity i and child mortality states m and n 

Faitmn: failure rates by age, parity, duration in parity, and mortality states m and n are obtained by 

dividing faitmn by 1,000.

Qa,i,t,m: partitioning factor, the proportion of women at age a, parity i, duration t, and child 

mortality state m for whom child mortality state changes during the transition.

Saitmn: number of women at age a, parity i, duration in parity t, and child mortality states m and n

in the global life table

TFRNCM: child-mortality-effect-free TFR that is calculated from a GLT that incorporates child 

mortality, which is set to zero

TFRWCM: child-mortality-effect-present TFR calculated from a GLT that incorporates child 

mortality

CM+: denotes women with child mortality experience in a GLT that incorporates child mortality

CM-: denotes women with no child mortality experience in a GLT that incorporates child 

mortality
35

REFERENCES

Allison, P. 1982. Discrete-time methods for the analysis of event histories. In S. Leinhardt (ed.), 

Sociological Methodology. San Francisco: Jossey-Bass.

Allison, P. 1995. Survival Analysis Using SAS: A Practical Guide. Cary, N. C.: SAS Institute 

Inc.

Feeney, G. 1983. Population dynamics based on birth intervals and parity progression. 

Population Studies 37: 75-89.

Feeney, G. 1986. Period Parity Progression Measures of Fertility in Japan. NUPRI Research 

Papers Series No. 35. Tokyo: Nihon University Population Research Institute.

Li, X., Y. Hu, J. Chen, R. D. Retherford, and M. K. Choe. 2007. Fertility Estimates for 

Provinces of China, 1975-2000. Beijing: National Bureau of Statistics. Honolulu: EastWest Center.

Retherford, R. D., and M. K. Choe. 1993. Statistical Models for Causal Analysis. New York: 

John Wiley.

Retherford, R. D., N. Ogawa, R. Matsukura, and H. Eini-Zinab. 2009. Multivariate Analysis of 

Parity Progression-based Measures of the Total Fertility Rate and its Components Using 

Individual-level Data. East-West Center Working Paper No. 119. Honolulu: East-West 

Center. http://www.eastwestcenter.org/fileadmin/stored/pdfs/POPwp119.pdf

Retherford, R. D., N. Ogawa, R. Matsukura, and H. Eini-Zinab. 2010a. Multivariate analysis of 

parity progression-based measures of the total fertility rate and its components. 

Demography 47: 97-124.

Retherford, R. D., H. Eini-Zinab, M.K. Choe, N. Ogawa, and R. Matsukura. 2010b. Further 

Development of Methodology for Multivariate Analysis of the Total Fertility Rate and its 

Components Based on Birth-history Data. Paper presented at the Population Association 

of America Annual Meeting, April 15-17, Dallas, Texas.
36

APPENDIX A.1: CONSTRUCTING THE GLOBAL LIFE TABLE FROM THE BIRTH

PROBABILITIES Paitmn

As in the development of the Pait method, the formulae are built up one parity transition 

at a time. The formulae for the 0-1 transition in the new GLT are basically the same as in the 

GLT derived by the Pait method, since there is no previous child mortality in the case of this 

transition. The formulae for the 0-1 (i = 0) transition are: 

S0,0,0,0,0 = 1,000 (A1.1)

Sa,0,t,0,0 = Sa,0,a,0,0 = Sa-1,0,a-1,0,0 (1 - Pa-1,0,a-1,0,0) for a > 0 (A1.2)

fa,i,t,0,0 = Sa,i,t,0,0 Pa,i,t,0,0 (A1.3)

Sa,i,t,0,0 is number of survivors and fa,i,t,0,0 is number of failures (i.e., births) by age, parity, 

and duration in parity, respectively. Because there is no one-year lagged child mortality in the 

case of the 0-1 transition, m and n are both zero for Saitmn and faitmn.

The formulae for higher-order transitions use the same basic logic as that used in the Pait

method, but the formulae are more elaborate because they incorporate non-zero child mortality. 

As discussed in the text, Panel 1 of Figure 2.1 for the 1-2 parity transition shows how women 

progress by age, duration in parity, and child mortality state. Panel 1 of Figure A.1 shows the 

corresponding progression of survivors, starting with Saitm*, where the asterisk indicates that the 

value of n can be either m or m+1. Panel 2 elaborates Panel 1 by partitioning Saitm* into two 

groups, one consisting of women who do not experience a child death during a one-year interval 

and the other consisting of women who do experience a child death during the one-year interval. 
37

Figure A.1: Elaboration of panel 1 of Figure 2.1:

Note: For t > 0, the Sa,i,t,m,* in panel 1 are partitioned into Saitmm and Saitmn (where n = m +1) in panel 2. The 

asterisk in Sa,i,t,m,* indicates that the child mortality state at the end of the interval (equivalently, at the start of the 

next interval) could be either m or m+1

Sa,i,0,0,0

Sa,i,1,0,*

t

0

1

2

3

Sa,i,2,0,* Sa,i,2,1,1

Panel 1: Parity transition 1-2 (Sa,i,t,m*)

 a a+1 a+2 a+3

 Sa,i,3,0,* Sa,i,3,1,1

.

.

.

.

.

.

Sa,i,0,0,0

t

0

1

2

3

Panel 2: Parity transition 1-2 (Sa,i,t,m,n)

 a a+1 a+2 a+3

.

.

.

.

.

.

Sa,i,1,0,0 Sa,i,1,0,1

Sa,i,2,0,0 Sa,i,2,0,1 Sa,i,2,1,1

Sa,i,3,0,0 Sa,i,3,0,1 Sa,i,3,1,1
38

Formulae corresponding to Panel 2 of Figure A.1 for the 1-2 transition are as follows:

If t=0:

,1,0,0,0   ,0, ,0,0 ,0, ,0,0   ,0, ,0,0

( ) a a t a t a t S S P f where the summation is over t, (A1.4) 

Sa,1,0,0,1  0

(A1.5)

Sa,1,0,1,1  0

(A1.6)

If t = 1:

(1 )(1 ) Sa,1,1,0,0  Sa1,1,0,0,0  Pa,1,1,0,0 Qa,1,1,0

 for a > 0 (A1.7)

,1,1,0,1 1,1,0,0,0 1,1,0,0,0 ,1,1,0

(1 ) Sa Sa  Pa Qa

 for a > 0 (A1.8)

Sa,1,1,1,1  0 (A1.9)

If t = 2:

(1 )(1 ) Sa,1,2,0,0  Sa1,1,1,0,0  Pa1,1,1,0,0 Qa,1,2,0

for a > 0 (A1.10)

,1,2,0,1 1,1,1,0,0 1,1,1,0,0 ,1,2,0

(1 ) Sa Sa  Pa Qa

for a > 0 (A1.11)

(1 ) Sa,1,2,1,1  Sa1,1,1,0,1  Pa1,1,1,0,1

for a > 0 (A1.12)

If t > 2:

(1 )(1 ) Sa,1,t,0,0  Sa1,1,t1,0,0  Pa1,1,t1,0,0 Qa,1,t,0

for a > 0 (A1.13)

,1, ,0,1 1,1, 1,0,0 1,1, 1,0,0 ,1, ,0

(1 ) Sa t Sa t  Pa t Qa t

for a > 0 (A1.14)

( )(1 ) ( )(1 ) Sa,1,t,1,1  Sa1,1,t1,0,1  Pa1,1,t1,0,1  Sa1,1,t1,1,1  Pa1,1,t1,1,1

 for a > 0 (A1.15)

aitmn SaitmnPaitmn f  (A1.16)
39

The formulae for higher-order transitions follow the same logic, with the following 

exceptions: (1) When t = 0, one needs to add number of failures (i.e., number of births) from the 

previous parity transition in order to determine how many women start the next parity transition 

in each mortality state. (2) More child mortality states are added as parity increases. 

As an example, the formulae for parity transition 2-3 (i = 2) are presented next. In order 

to clarify the formulae for the 2-3 transition, Panel 2 of Figure 2.1 is elaborated in a Figure A.2 

to show how women are partitioned into two groups at each age, parity, duration in parity, and 

child mortality state. 
40

Figure A.2: Elaboration of panel 2 of Figure 2.1:

Note: Sa,i,t,m,*’s of panel 1 are partitioned to Saitmm and Saitmn (where n = m +1) in panel 2.

The asterisk in Sa,i,t,m,* shows that the ending child mortality state could be equal to m or m+1.

Sa,i,t,0,* Sa,i,t,1,*

t

0

1

2

3

Panel 1: Parity transition 2-3 (Sa,i,t,m*)

 a a+1 a+2 a+3

.

.

.

.

.

.

Sa,i,t,0,* Sa,i,t,1,*

Sa,i,t,0,* Sa,i,t,1,* Sa,i,t,2,2

Sa,i,t,0,* Sa,i,t,1,* Sa,i,t,2,2

Sa,i,t,0,0 Sa,i,t,0,1 Sa,i,t,1,1

t

0

1

2

3

Panel 2: Parity transition 2-3 (Sa,i,t,m,n)

 a a+1 a+2 a+3

.

.

.

 Sa,i,t,0,0 Sa,i,t,0,1 Sa,i,t,1,1 Sa,i,t,1,2

 Sa,i,t,0,0 Sa,i,t,0,1 Sa,i,t,1,1 Sa,i,t,1,2 Sa,i,t,2 ,2 

 Sa,i,t,0,0 Sa,i,t,0,1 Sa,i,t,1,1 Sa,i,t,1,2 Sa,i,t,2 ,2 

.

.

.
41

If t = 0:

( ( ))(1 ) Sa,2,0,0,0   Sa,1,t,0,0Pa,1,t,0,0 Qa,2,0,0

 where the summation is over t (A1.17)

,2,0,0,1 ,1, ,0,0 ,1, ,0,0 ,2,0,0

( ( )) Sa  Sa t Pa t Qa

 where the summation is over t (A1.18)

 ( ) ( ) Sa,2,0,1,1 Sa,1,t,0,1Pa,1,t,0,1 Sa,1,t,1,1Pa,1,t,1,1

 where the summation is over t (A1.19)

Sa,2,0,1,2  0 (A1.20)

Sa,2,0,2,2  0 (A1.21)

If t = 1:

(1 )(1 ) Sa,2,1,0,0  Sa1,2,0,0,0  Pa1,2,0,0,0 Qa,2,1,0

for a > 0 (A1.22)

,2,1,0,1 1,2,0,0,0 1,2,0,0,0 ,2,1,0

(1 )

a a Pa Qa S S    

for a > 0 (A1.23)

( (1 ) (1 ))(1 ) Sa,2,1,1,1  Sa1,2,0,0,1  Pa1,2,0,0,1  Sa1,2,0,1,1  Pa1,2,0,1,1 Qa,2,1,1

 for a > 0 (A1.24)

,2,1,1,2 1,2,0,0,1 1,2,0,0,1 1,2,0,1,1 1,2,0,1,1 ,2,1,1

( (1 ) (1 )) Sa Sa  Pa  Sa  Pa Qa

 for a > 0 (A1.25)

Sa,2,1,2,2  0 (A1.26)

If t = 2:

(1 )(1 ) Sa,2,2,0,0  Sa1,2,1,0,0  Pa1,2,1,0,0 Qa,2,2,0

for a > 0 (A1.27)

,2,2,0,1 1,2,1,0,0 1,2,1,0,0 ,2,2,0

(1 ) Sa Sa  Pa Qa

for a > 0 (A1.28)

( (1 ) (1 ))(1 ) Sa,2,2,1,1  Sa1,2,1,0,1  Pa1,2,1,0,1  Sa1,2,1,1,1  Pa1,2,1,1,1 Qa,2,2,1

 

for a > 0 (A1.29) 

,2,2,1,2 1,2,1,0,1 1,2,1,0,1 1,2,1,1,1 1,2,1,1,1 ,2,2,1

( (1 ) (1 )) Sa Sa  Pa  Sa  Pa Qa

 

for a > 0 (A1.30) 

(1 ) Sa,2,2,2,2  Sa1,2,1,1,2  Pa1,2,1,1,2

 for a > 0 (A1.31)
42

If t > 2: 

(1 )(1 ) Sa,2,t,0,0  Sa1,2,t1,0,0  Pa1,2,t1,0,0 Qa,2,t,0

for a > 0 (A1.32)

,2, ,0,1 1,2, 1,0,0 1,2, 1,0,0 ,2, ,0

(1 ) Sa t Sa t  Pa t Qa t

for a > 0 (A1.33)

( (1 ) (1 ))(1 ) Sa,2,t,1,1  Sa1,2,t1,0,1  Pa1,2,t1,0,1  Sa1,2,t1,1,1  Pa1,2,t1,1,1 Qa,2,t,1

for a > 0 (A1.34) 

,2, ,1,2 1,2, 1,0,1 1,2, 1,0,1 1,2, 1,1,1 1,2, 1,1,1 ,2, ,1

( (1 ) (1 )) Sa t Sa t  Pa t  Sa t  Pa t Qa t

 

for a > 0 (A1.35) 

( (1 ) ( (1 ) Sa,2,t,2,2  Sa1,2,t1,1,2  Pa1,2,t1,1,2  Sa1,2,t1,2,2  Pa1,2,t1,2,2

 for a > 0 (A1.36) 

The rest of the global life table is constructed in a similar fashion. The new global life 

table with child mortality included has all the properties of the original global life table 

(Retherford et al., 2010b). Once the new global life table is constructed, it is straightforward to 

calculate the TFR and its components from it. Formulae are similar to those in the original Pait

method for calculating the original global life table in the previous section. The only difference is 

that summations run over m and n as well as a, i, and t. For example, TFR is calculated from the 

new global life tables as

TFR = (∑faitmn)/1,000 (A1.37)

where the summation is over a, i, t, m, and n.
43

APPENDIX A.2 THE PARTITIONING FACTOR

As mentioned in the text, the main difference between the original global life table and 

the global life table that includes child mortality is that women in some mortality transitions of 

the new global life table are partitioned into two groups, the first of which sustain the starting 

child mortality state and the second of which experience a change in the child morality state. To 

assign women in each group, one needs a partitioning factor. The partitioning factor is defined as 

the proportion of women at age a, parity i, duration t, and child mortality Mm who change their 

child mortality state to Mm+1 by age a+1. During a one-year transition, these women may survive 

having next birth and stay at parity i (in which case duration increases to t+1), or they may 

experience the terminating event as well and change their parity to i+1, in which case their 

duration in parity reverts to t = 0. 

Two approaches are introduced in this research to calculate the partitioning factor: the 

―overall probability‖ approach and the ―logit model‖ approach. The former approach uses the 

overall probability of failure (i.e., next birth) at age a, parity i, duration t, and child mortality 

state m, to estimate the partitioning factor. The latter approach applies logit regression models to 

estimate the probabilities of change in child mortality state by age, parity, duration in parity, and 

child mortality state. The probabilities directly show the proportion of women who change their 

child mortality state by age, parity, duration in parity, and child mortality. The former approach 

is preferred to the latter (see next two sections for a discussion).

A.2.1 “Overall Probability” Approach to Calculation of the Partitioning Factor

Saitm* indicates number of women at age a, parity i, duration in parity t, and child 

mortality state m in the new global life table. The asterisk shows that child mortality state at the 

end of a one-year transition could be either m or m+1. Saitm* is partitioned into two groups, the 

first of which remain in child mortality state m at the end of a one-year transition, and the second 

of which experiences a change in child mortality state to m+1 at the end of the one-year 

transition. 
44

By way of illustration, suppose there are 100 women at age a, duration t, of parity i, with 

child mortality state 2 (Mm = M2). I.e., Saitm* = Sa,i,t,2,* = 100. During age a and duration t some of 

these 100 women will remain in the same child mortality state M2 and some will move to the 

next child mortality state Mm+1 = M3. The number of women in the first group is denoted as 

Sa,i.t.2,2 . The probability of having a birth for this group is Pa,i,t,2,2. The number of women in the 

second group, which changes child mortality state (from M2 to M3) during age a and duration t, is 

denoted as Sa,i,t,2,3 where n = m + 1. The probability of having a next birth for this group is 

Pa,i,t,2,3. Thus Sa,i,t,2,* = Sa,i,t,2,2 + Sa,i,t,2,3. The partitioning factor shows the proportion of Sa,i,t,2,*

who belong to Sa,i,t,2,3.

The main issue is how to divide the Sa,i,t,2,* women into the two groups Sa,i,t,2,2 and Sa,i,t,2,3. 

One possible solution is to use the overall transition probability to calculate the number of 

women in each group. In other words, if one knows how many women move to age a+1 and 

duration t+1 overall, regardless of their subsequent child mortality state, one can calculate the 

number of women in each group. The overall probability of transition is denoted by 

aitmn

P

, where 

n

is the average child mortality state (mean number of previous child deaths as of one year ago) 

at age a-1 and duration t-1. 

n

is always equal or greater than m and equal or less than n. (The 

procedure to estimate 

aitmn

P

is explained shortly.) In this example 

n

is a number between 2 and 

3. Figure A2.1 simplifies this example. 

Figure A2.1: Depicting the “overall probability” approach to the partitioning factor

S a,i,t,2,* = Sa,i,t,2,2 + Sa,i,t,2,3 = 100

Sa+1,i,t+1,2,* =Sa,i,t,2,2 (1-Pa,i,t,2,2)

 Pa,i,t,2,2

Sa+1,i,t+1,3,*=Sa,i,t,2,3(1-Pa,i,t,2,3)

P a,i,t,2,3

S a,i,t,2,* = 100

 

1,i,t 1,n a,i,t,2,* (1 a,i,t,2,n )

t

a S  

 S  P

a i t n

P , , , 2 ,
45

The following equations are generated from the figure:

Sa,i,t,2,* = Sa,i,t,2,2 + Sa,i,t,2,3

(A2.1)

Sa+1,i,t+1,2,* = Sa,i,t,2,2(1-Pa,i,t,2,2) (A2.2)

Sa+1,i,t+1,3,* = Sa,i,t,2,3(1-Pa,i,t,2,3) (A2.3)

It is assumed that

S

a1,i,t1,n

 Sa1,i,t1,2,*  Sa1,i,t1,3,*

 (A2.4)

Equation (A2.4) assumes that the number of women who move to the next age and 

duration in parity, regardless of their child mortality state, equals the sum of the number of 

women who remain in child mortality state m = 2 and those who move to child mortality state 

m+1 = 3. When equations (A2.1) and (A2.4) are solved simultaneously, using elements from 

equations (A2.2) and (A2.3), the numbers of women Sa,i,t,,2,2 and Sa,i,t,2,3 are estimated as

( )

( )

, , ,2.3 , , ,2,2

3 , , ,2,, , ,2,* , , ,2.

, , ,2,2

a i t a i t

a i t a i t a i t n

a i t

P P

S P P

S





 (A2.5)

( )

( )

, , ,2.2 , , ,2,3

2 , , ,2,, , ,2,* , , ,2,

, , ,2,3

a i t a i t

a i t a i t a i t n

a i t

P P

S P P

S





 (A2.6)

Using equation (A2.6), one can estimate the partitioning factor, Qaitm (the proportion of 

women who change mortality state) as 
46

, , ,2.2 , , ,2,3

2 , , ,2,, , ,2,

, , ,2

a i t a i t

a i t a i t n

a i t

P P

P P

Q





 (A2.7)

This equation could be generalized to any child mortality state as

a i t m m a i t m n

a i t m m a i t m n

a i t m

P P

P P

Q

, , , . , , , ,

, , , ,

, , , ,

, , ,





 (A2.8)

where m denotes the starting child mortality state and n denotes the ending child mortality state if 

there is a change in child mortality state (i.e., n = m+1). If there is no change in child mortality 

state, m is used instead of n.

aitmn

P

is estimated by substituting values of age A (where A = a – t + 10, recall that age is 

translated to start from 0 instead of 10 (Retherford et al., 2010b)), duration t, starting child 

mortality state m, and mean child mortality state 

n

at the end of a one-year transition (the values 

of m and 

n

are substituted for D1 and D2) into the CLL model equation for each parity transition. 

Mean child mortality state 

n

at the end of a one-year transition is specific by age a, parity i, 

duration t, and starting child mortality state m. For example, to estimate

n

P25,3,4,2,

, one substitutes 

A = 31, i = 3, t = 5 (recall that duration in CLL models starts from 1 not zero, thus t = 5, instead 

of 4), D1 = 2, and D2 = the computed value of 

n

into equation (2.1), (2.2), or (2.3). Since the 

starting child mortality state in this example is 2, the mean of D2, specified by age, parity, and 

starting child mortality state, is a number between 2 and 3. If no one changes her child mortality 

state, the mean is 2. If everyone changes child mortality state, the mean is 3. If some women 

change their child mortality state and others sustain their starting child mortality state, the mean 

is a number between 2 and 3. If the CLL model includes predictor variables, the mean of 

n

should be specific to the main predictor variable as well. For instance, to estimate the values of 

aitmn

P

by urban/rural residence one should specify the 

n

by residence as well. 
47

Formulae ((A2.7) and (A2.8)) are fine as long as the two sets of probabilities, i.e., Pa,i,t,2,2

and Pa,i,,t,2,3 (or Paitmm and Paitmn , where n = m +1) are not equal. When the probabilities are equal, 

the formulae are undefined and one needs to find another solution to partition women in two 

groups. In this research, this problem is solved with estimating the probabilities of changing 

child mortality state by age, parity, duration in parity, and the starting child mortality state. 

A.2.2 “Logit Model” Approach to Calculation of the Partitioning Factor 

Another solution to partitioning women into two groups is to use a logit model to 

estimate the probabilities of change in child mortality state by age, parity, duration in parity, and 

starting child mortality state. In this method, for any given parity transition, one uses the same 

dataset that is used to estimate the CLL model for that transition. The response variable in the 

logit regression is a dummy variable representing change in child mortality state at each personyear observation, namely

D  D2  D1

. Predictor variables in the logit model include age, 

duration in parity, starting child mortality, and socioeconomic variables, if any. The logit model 

is estimated for each parity transition higher than 0-1, where child mortality is involved. 

The logit model is used instead of a CLL model because the response variable, change in 

child mortality state, can occur at any person-year observation in the data. The CLL model 

should be used only when the event could occur at the last person-year observation.

The probability form of the logit model is:

( ) 1

2 2

1

1

a b t ct d A eA fD gSES D

e

P

      





 (A2.9)

where P∆D is probability of change in child mortality state, a is the intercept, t is duration 

in parity, A is age at starting parity, D1 is child mortality state at the start of age a (not to be 

confused with the intercept) and duration t, SES is a column vector of socioeconomic predictor 

which is included in the CLL model as well, g is a row vector of coefficients, and a, b, c, d, e, f, 

and g are the intercept and coefficients to be estimated.
48

Once the equation is fitted to the dataset, one can estimate the probabilities of change in 

child mortality state by substituting specific values of age, duration, child mortality state, and 

socioeconomic predictors, using the same logic explained for the CLL models. 

The procedure to estimate the probabilities of change in child mortality state by age, 

parity, duration in parity, and child mortality state (Qaitm) is then similar to the estimation of 

Paitmn. Qaitm estimated this way can then be used as a partitioning factor.

The ―logit model‖ approach complements the ―overall probability‖ approach and should 

be used only when the probability of a next birth for women who remain in their initial child 

mortality state and the probability of a next birth for women who experience a child death in the 

interval are equal. In almost all cases, they will not be equal, in which case the overall 

probability approach should be used. The main reason for preferring the overall probability 

approach is that the same CLL models used to construct the global life table are also used to 

estimate the partitioning factor. Because all probabilities needed to construct the global life table 

and to estimate the partitioning factor are derived from similar CLL models, these probabilities 

are internally consistent. This internal consistency constrains the global life table to be internally 

consistent too, inasmuch as the overall probability of transition will always be a number between 

the probabilities of failure for women in the two groups (those who remain in their initial child 

mortality state and those who experience a child death during the one-year interval). 

Another advantage of the ―overall probability‖ approach over the logit model approach is 

that one only needs to estimate a single model for each parity transition in order to construct a 

global life table. By contrast, if the logit model approach is used, one needs to estimate both a 

CLL model and a logit model for each parity transition higher than 1-2 transition. Inasmuch as 

no model fits the data perfectly, the chance of overestimation or underestimation is likely to be 

higher when two models are used than when one model is used. The next section, on validation 

of methodology, compares results from these two approaches, when one or the other is used 

exclusively.
49

A.2.3 Validation of the Partitioning Factor

The two approaches for calculating the partitioning factor are tested here. For reasons 

explained earlier, the ―overall probability‖ approach is preferred over the ―logit model‖ 

approach. In practice, the latter approach substitutes for the former approach when the equation 

for the former approach is undefined. This section tests the sensitivity of the global life tables to 

the choice of partitioning factor.

To test for the sensitivity of the global life table to the choice of partitioning factor, four 

different global life tables are constructed using four methods. The first method uses the ―overall 

probability‖ approach to construct the global life table. When equation (A2.8) is undefined, i.e., 

when the probability of a birth during the interval is the same for the two groups, the partitioning 

factor is set to zero, Qaitm = 0. In this method, when the two groups have equal probabilities of 

next birth, all women are assigned to the first group, i.e., no one changes the child mortality 

state. The second method also uses the ―overall probability‖ approach to construct the global life 

table. When equation (A2.8) is undefined, the partitioning factor is set to one, Qaitm = 1. In this 

method, when the two groups have equal probabilities of next birth, all women are assigned to 

the second group, i.e., everyone changes the child mortality state. The third method uses the 

―logit model‖ approach to calculate the partitioning factor throughout the global life table. 

Finally, the fourth method uses the ―overall probability‖ approach if the probability of a birth 

during the interval differs between the two groups and uses the ―logit model‖ approach if the 

probability of a birth during the interval is the same for the two groups. The global life tables are 

constructed using cohort data (women age 45-49) from India’s NFHS-1. All GLTs use similar 

CLL models that include no socioeconomic predictors other than variables representing the basic 

dimensions of the global life table.

The main purposes of constructing four different global life tables are: 1) to show how 

frequently the issue of similar probabilities of two groups of women in a one-year transition 

occurs; 2) to demonstrate the effect of this issue on final outputs of the GLT; and 3) to show that 

the preferred approach yields better results than other methods. 

Results show that the fourth method is preferred. Table A.1 shows the results of all four 

methods. The table includes TFRs, mean number of child deaths per woman, and child 

replacement rates only.
50

Table A.1: Sensitivity of selected global life table results to the choice of the partitioning 

factor Qaitm; Cohort analysis, women age 45-49, NFHS-1

Method TFR CM RR

1 4.89 0.94 0.33

2 4.89 1.00 0.31

3 5.04 1.27 0.36

4 4.89 0.94 0.33

Notes: Partitioning refers to how, over a one-year interval in the global life table, women are partitioned into two 

groups, the first of which does not experience a child death over the previous interval and the second of which 

experiences a child death over the previous interval. Qaitm denotes the partitioning factor and shows the 

proportion of women assigned to the second group.

Method 1: The "overall probability” approach is used to estimate Qaitm if the probability of a birth during the 

interval differs between the two groups. Qaitm set to zero if the probability of a birth during the interval is the 

same for the two groups.

Method 2: The "overall probability” approach is used to estimate Qaitm if the probability of a birth during the 

interval differs between the two groups. Qaitm set to one if the probability of a birth during the interval is the 

same for the two groups.

Method 3: The “logit model” approach is used to estimate Qaitm.

Method 4: The "overall probability” approach is used to estimate Qaitm if the probability of a birth during the 

interval differs between the two groups. The “logit model” approach is used if the probability of a birth during the 

interval is the same for the two groups.

CM: Mean number of child deaths per woman in the global life table.

RR: Replacement rate.

As the table shows, the three methods that use the ―overall probability‖ approach to 

assign women in the two groups yield similar TFRs, regardless of what partitioning strategy is 

used, when the two groups of women have equal probabilities of birth during the one-year 

interval. This indicates that having equal probabilities is not a frequent event and occurs so rarely 

that the two extreme solutions, Qaitm equal to 1 or 0, yield similar TFRs, and also other fertility 

measures, not shown here. However, since the mean number of child deaths per woman is 

directly affected by the choice of partitioning factor, the three methods yield different child 

replacement rates. When all women are assigned to the first group (method 1), the mean number 

of child deaths per woman is the lowest, since all women who lose a child during one-year 

transitions where both groups have equal failure probabilities are assigned to a group with no 

child deaths. Similarly, when all women are assigned to the second group, the mean number of

child deaths per woman is the highest. Similar results of methods 1 and 4 are due to the fact that 

a changing child mortality state is a rare event (about 4 percent in the overall population), as are 

transitions in the global life table where the two groups have equal failure probabilities. Using 

the ―logit model‖ approach, when equation (A2.8) is undefined, yields more accurate and 

realistic results than assigning all women to one group.
51

Why not use the ―logit model‖ approach only? As mentioned above, the main reason for 

not using the ―logit model‖ approach per se is to avoid additional errors in construction of the 

global life tables. It is obvious that most models do not fit to the dataset perfectly. The 

imperfectly fitted models result in overestimation or underestimation of fertility measures 

derived by the global life table. When only one type of model (CLL models) is used to construct 

the global life tables, they have only one source of error. When the logit model added, the source 

of error is two-fold, which means a bigger discrepancy may occur in the final results. This is 

obvious from the results of the third method in Table A.1. As the table shows, the mean number 

of child deaths per woman in this method is 1.268, which is much higher than the actual number, 

1.027. Although one might get similar child mortality measures with this method, the method 

may erroneously assign women in higher (or lower) child mortality groups, which would result 

in higher (or lower) TFRs (because women with high (low) child mortality experience also have 

high (low) fertility rates) and higher (or lower) child replacement rates.