We will illustrate the use of piece-wise exponential survival models using data from an analysis of infant and child mortality in Colombia done by Somoza (1980). The data were collected in a 1976 survey conducted as part of the World Fertility Survey. The sample consisted of women between the ages of 15 and 49. The questionnaire included a maternity history, recording for each child ever born to each respondent the sex, date of birth, survival status as of the interview and (if applicable) age at death.
As if often the case with survival data, most of the work goes into preparing the data for analysis. In the present case we started from tables in Somoza’s article showing living children classified by current age, and dead children classified by age at death. Both tabulations reported age using the groups shown in Table 7.1, using fine categories early in life, when the risk if high but declines rapidly, and wider categories at later ages. With these two bits of information we were able to tabulate deaths and calculate exposure time by age groups, assuming that children who died or were censored in an interval lived on the average half the length of the interval.
Table 7.1. Infant and Child Deaths and Exposure Time by
Age of Child and Birth Cohort, Colombia 1976.
| Exact | Birth Cohort | |||||||
| Age | 1941–59 | 1960–67 | 1968-76 | |||||
| deaths | exposure | deaths | exposure | deaths | exposure | |||
| 0–1 m | 168 | 278.4 | 197 | 403.2 | 195 | 495.3 | ||
| 1–3 m | 48 | 538.8 | 48 | 786.0 | 55 | 956.7 | ||
| 3–6 m | 63 | 794.4 | 62 | 1165.3 | 58 | 1381.4 | ||
| 6–12 m | 89 | 1550.8 | 81 | 2294.8 | 85 | 2604.5 | ||
| 1–2 y | 102 | 3006.0 | 97 | 4500.5 | 87 | 4618.5 | ||
| 2–5 y | 81 | 8743.5 | 103 | 13201.5 | 70 | 9814.5 | ||
| 5–10 y | 40 | 14270.0 | 39 | 19525.0 | 10 | 5802.5 | ||
Table 7.1 shows the results of these calculations in terms of the number of deaths and the total number of person-years of exposure to risk between birth and age ten, by categories of age of child, for three groups of children (or cohorts) born in 1941–59, 1960–67 and 1968–76. The purpose of our analysis will be to assess the magnitude of the expected decline in infant and child mortality across these cohorts, and to study whether mortality has declined uniformly at all ages or more rapidly in certain age groups.
Let \( y_{ij} \) denote the number of deaths for cohort \( i \) (with \( i=1,2,3 \)) in age group \( j \) (for \( j=1,2,\ldots,7 \)). In view of the results of the previous section, we treat the \( y_{ij} \) as realizations of Poisson random variables with means \( \mu_{ij} \) satisfying
\[ \mu_{ij} = \lambda_{ij} t_{ij}, \]where \( \lambda_{ij} \) is the hazard rate and \( t_{ij} \) is the total exposure time for group \( i \) at age \( j \). In words, the expected number of deaths is the product of the death rate by exposure time.
A word of caution about units of measurement: the hazard rates must be interpreted in the same units of time that we have used to measure exposure. In our example we measure time in years and therefore the \( \lambda_{ij} \) represent rates per person-year of exposure. If we had measured time in months the \( \lambda_{ij} \) would represent rates per person-month of exposure, and would be exactly one twelfth the size of the rates per person-year.
To model the rates we use a log link, so that the linear predictor becomes
\[ \eta_{ij} = \log \mu_{ij} = \log \lambda_{ij} + \log t_{ij}, \]the sum of two parts, \( \log t_{ij} \), an offset or known part of the linear predictor, and \( \log \lambda_{ij} \), the log of the hazard rates of interest.
Finally, we introduce a log-linear model for the hazard rates, of the usual form
\[ \log \lambda_{ij} = \boldsymbol{x}_{ij}'\boldsymbol{\beta}, \]where \( \boldsymbol{x}_{ij} \) is a vector of covariates. In case you are wondering what happened to the baseline hazard, we have folded it into the vector of parameters \( \boldsymbol{\beta} \). The vector of covariates \( \boldsymbol{x}_{ij} \) may include a constant, a set of dummy variables representing the age groups (i.e. the shape of the hazard by age), a set of dummy variables representing the birth cohorts (i.e. the change in the hazard over time) and even a set of cross-product dummies representing combinations of ages and birth cohorts (i.e. interaction effects).
Table 7.2. Deviances for Various Models Fitted to
Infant and Child Mortality Data From Colombia
| Model | Name | \(\log \lambda_{ij}\) | Deviance | d.f. |
| \(\phi\) | Null | \(\eta\) | 4239.8 | 20 |
| \(A\) | Age | \(\eta+\alpha_i\) | 72.7 | 14 |
| \(C\) | Cohort | \(\eta+\beta_j\) | 4190.7 | 18 |
| \(A+C\) | Additive | \(\eta+\alpha_i+\beta_j\) | 6.2 | 12 |
| \(AC\) | Saturated | \(\eta+\alpha_i+\beta_j+(\alpha\beta)_{ij}\) | 0 | 0 |
Table 7.2 shows the deviance for the five possible models of interest, including the null model, the two one-factor models, the two-factor additive model, and the two-factor model with an interaction, which is saturated for these data.
The null model assumes that the hazard is a constant from birth to age ten and that this constant is the same for all cohorts. It therefore corresponds to an exponential survival model with no covariates. This model obviously does not fit the data, the deviance of 4239.8 on 20 d.f. is simply astronomical. The m.l.e. of \( \eta \) is \( -3.996 \) with a standard error of 0.0237. Exponentiating we obtain an estimated hazard rate of 0.0184. Thus, we expect about 18 deaths per thousand person-years of exposure. You may want to verify that 0.0184 is simply the ratio of the total number of deaths to the total exposure time. Multiplying 0.0184 by the amount of exposure in each cell of the table we obtain the expected number of deaths. The deviance quoted above is simply twice the sum of observed times the log of observed over expected deaths.
The age model allows the hazard to change from one age group to another, but assumes that the risk at any given age is the same for all cohorts. It is therefore equivalent to a piece-wise exponential survival model with no covariates. The reduction in deviance from the null model is 4167.1 on 6 d.f., and is extremely significant. The risk of death varies substantially with age over the first few months of life. In other words the hazard is clearly not constant. Note that with a deviance of 72.7 on 14 d.f., this model does not fit the data. Thus, the assumption that all cohorts are subject to the same risks does not seem tenable.
Table 7.3 shows parameter estimates for the one-factor models \( A \) and \( C \) and for the additive model \( A+C \) in a format reminiscent of multiple classification analysis. Although the \( A \) model does not fit the data, it is instructive to comment briefly on the estimates. The constant, shown in parentheses, corresponds to a rate of \( \exp\{-0.7427\}=0.4758 \), or nearly half a death per person-year of exposure, in the first month of life. The estimate for ages 1–3 months corresponds to a multiplicative effect of \( \exp\{-1.973\}=0.1391 \), amounting to an 86 percent reduction in the hazard after surviving the first month of life. This downward trend continues up to ages 5–10 years, where the multiplicative effect is \( \exp\{-5.355\}=0.0047 \), indicating that the hazard at these ages is only half-a-percent what it was in the first month of life. You may wish to verify that the m.l.e.’s of the age effects can be calculated directly from the total number of deaths and the total exposure time in each age group. Can you calculate the deviance by hand?
Let us now consider the model involving only birth cohort, which assumes that the hazard is constant from birth to age ten, but varies from one birth cohort to another. This model is equivalent to three exponential survival models, one for each birth cohort. As we would expect, it is hopelessly inadequate, with a deviance in the thousands, because it fails to take into account the substantial age effects that we have just discussed. It may of of interest, however, to note the parameter estimates in Table 7.3. As a first approximation, the overall mortality rate for the older cohort was \( \exp\{-3.899\} = 0.0203 \) or around 20 deaths per thousand person-years of exposure. The multiplicative effect for the cohort born in 1960–67 is \( \exp\{-0.3020\}=0.7393 \), indicating a 26 percent reduction in overall mortality. However, the multiplicative effect for the youngest cohort is \( \exp\{0.0742\}=1.077 \), suggesting an eight percent increase in overall mortality. Can you think of an explanation for this apparent anomaly? We will consider the answer after we discuss the next model.
Table 7.3. Parameter Estimates for Age, Cohort and Age\( + \)Cohort
Models
of Infant and Child Mortality in Colombia
| Factor | Category | Gross Effect | Net Effect |
| Baseline | \(-\)0.4485 | ||
| Age | 0–1 m | (\(-\)0.7427) | – |
| 1–3 m | \(-\)1.973 | \(-\)1.973 | |
| 3–6 m | \(-\)2.162 | \(-\)2.163 | |
| 6–12 m | \(-\)2.488 | \(-\)2.492 | |
| 1–2 y | \(-\)3.004 | \(-\)3.014 | |
| 2–5 y | \(-\)4.086 | \(-\)4.115 | |
| 5–10 y | \(-\)5.355 | \(-\)5.436 | |
| Cohort | 1941–59 | (\(-\)3.899) | – |
| 1960–67 | \(-\)0.3020 | \(-\)0.3243 | |
| 1968–76 | 0.0742 | \(-\)0.4784 |
Consider now the additive model with effects of both age and cohort, where the hazard rate is allowed to vary with age and may differ from one cohort to another, but the age (or cohort) effect is assumed to be the same for each cohort (or age). This model is equivalent to a proportional hazards model, where we assume a common shape of the hazard by age, and let cohort affect the hazard proportionately at all ages. Comparing the proportional hazards model with the age model we note a reduction in deviance of 66.5 on two d.f., which is highly significant. Thus, we have strong evidence of cohort effects net of age. On the other hand, the attained deviance of 6.2 on 12 d.f. is clearly not significant, indicating that the proportional hazards model provides an adequate description of the patterns of mortality by age and cohort in Colombia. In other words, the assumption of proportionality of hazards is quite reasonable, implying that the decline in mortality in Colombia has been the same at all ages.
Let us examine the parameter estimates on the right-most column of Table 7.3. The constant is the baseline hazard at ages 0–1 months for the earliest cohort, those born in 1941–59. The age parameters representing the baseline hazard are practically unchanged from the model with age only, and trace the dramatic decline in mortality from birth to age ten, with half the reduction concentrated in the first year of life. The cohort affects adjusted for age provide a more reasonable picture of the decline in mortality over time. The multiplicative effects for the cohorts born in 1960–67 and 1068–76 are \( \exp\{-0.3243\}= 0.7233 \) and \( \exp\{-0.4784\} = 0.6120 \), corresponding to mortality declines of 28 and 38 percent at every age, compared to the cohort born in 1941–59. This is a remarkable decline in infant and child mortality, which appears to have been the same at all ages. In other words, neonatal, post-neonatal, infant and toddler mortality have all declined by approximately 38 percent across these cohorts.
The fact that the gross effect for the youngest cohort was positive but the net effect is substantially negative can be explained as follows. Because the survey took place in 1976, children born between 1968 and 76 have been exposed mostly to mortality at younger ages, where the rates are substantially higher than at older ages. For example a child born in 1975 would have been exposed only to mortality in the first year of life. The gross effect ignores this fact and thus overestimates the mortality of this group at ages zero to ten. The net effect adjusts correctly for the increased risk at younger ages, essentially comparing the mortality of this cohort to the mortality of earlier cohorts when they had the same ages, and can therefore unmask the actual decline.
A final caveat on interpretation: the data are based on retrospective reports of mothers who were between the ages of 15 and 49 at the time of the interview. These women provide a representative sample of both mothers and births for recent periods, but a somewhat biased sample for older periods. The sample excludes mothers who have died before the interview, but also women who were older at the time of birth of the child. For example births from 1976, 1966 and 1956 come from mothers who were under 50, under 40 and under 30 at the time of birth of the child. A more careful analysis of the data would include age of mother at birth of the child as an additional control variable.
So far we have focused attention on the hazard or mortality rate, but of course, once the hazard has been calculated it becomes an easy task to calculate cumulative hazards and therefore survival probabilities. Table 7.4 shows the results of just such an exercise, using the parameter estimates for the proportional hazards model in Table 7.3.
Table 7.4. Calculation of Survival Probabilities for Three Cohorts
Based on the Proportional Hazards Model
| Age | Width | Baseline | Survival for Cohort | |||||
| Group | Log-haz | Hazard | Cum.Haz | \(<\)1960 | 1960–67 | 1968–76 | ||
| (1) | (2) | (3) | (4) | (5) | (6) | (7) | (8) | |
| 0–1 m | 1/12 | \(-\)0.4485 | 0.6386 | 0.0532 | 0.9482 | 0.9623 | 0.9676 | |
| 1–3 m | 2/12 | \(-\)2.4215 | 0.0888 | 0.0680 | 0.9342 | 0.9520 | 0.9587 | |
| 3–6 m | 3/12 | \(-\)2.6115 | 0.0734 | 0.0864 | 0.9173 | 0.9395 | 0.9479 | |
| 6–12 m | 1/2 | \(-\)2.9405 | 0.0528 | 0.1128 | 0.8933 | 0.9217 | 0.9325 | |
| 1–2 y | 1 | \(-\)3.4625 | 0.0314 | 0.1441 | 0.8658 | 0.9010 | 0.9145 | |
| 2–5 y | 3 | \(-\)4.5635 | 0.0104 | 0.1754 | 0.8391 | 0.8809 | 0.8970 | |
| 5–10 y | 5 | \(-\)5.8845 | 0.0028 | 0.1893 | 0.8275 | 0.8721 | 0.8893 | |
Consider first the baseline group, namely the cohort of children born before 1960. To obtain the log-hazard for each age group we must add the constant and the age effect, for example the log-hazard for ages 1–3 months is \( -0.4485-1.973=-2.4215 \). This gives the numbers in column (3) of Table 7.3. Next we exponentiate to obtain the hazard rates in column (4), for example the rate for ages 1–3 months is \( \exp\{-2.4215\}=0.0888 \). Next we calculate the cumulative hazard, multiply the hazard by the width of the interval and summing across intervals. In this step it is crucial to express the width of the interval in the same units used to calculate exposure, in this case years. Thus, the cumulative hazard at then end of ages 1–3 months is \( 0.6386\times1/12 + 0.0888\times2/12 = 0.0680 \). Finally, we change sign and exponentiate to calculate the survival function. For example the baseline survival function at 3 months is \( \exp\{-0.0680\} = 0.9342 \).
To calculate the survival functions shown in columns (7) and (8) for the other two cohorts we could multiply the baseline hazards by \( \exp\{-0.3242\} \) and \( \exp\{-0.4874\} \) to obtain the hazards for cohorts 1960–67 and 1968–76, respectively, and then repeat the steps described above to obtain the survival functions. This approach would be necessary if we had time-varying effects, but in the present case we can take advantage of a simplification that obtains for proportional hazard models. Namely, the survival functions for the two younger cohorts can be calculated as the baseline survival function raised to the relative risks \( \exp\{-0.3242\} \) and \( \exp\{-0.4874\} \), respectively. For example the probability of surviving to age three months was calculated as 0.9342 for the baseline group, and turns out to be \( 0.9342^{\exp\{-0.3242\}}=0.9520 \) for the cohort born in 1960–67, and \( 0.9342^{\exp\{-0.4874\}}=0.9587 \) for the cohort born in 1968–76.
Note that the probability of dying in the first year of life has declined from 106.7 per thousand for children born before 1960 to 78.3 per thousand for children born in 1960–67 and finally to 67.5 per thousand for the most recent cohort. Results presented in terms of probabilities are often more accessible to a wider audience than results presented in terms of hazard rates. (Unfortunately, demographers are used to calling the probability of dying in the first year of life the ‘infant mortality rate’. This is incorrect because the quantity quoted is a probability, not a rate. In our example the rate varies substantially within the first year of life. If the probability of dying in the first year of life is \( q \), say, then the average rate is approximately \( -\log(1-q) \), which is not too different from \( q \) for small \( q \).)
By focusing on events and exposure, we have been able to combine infant and child mortality in the same analysis and use all available information. An alternative approach could focus on infant mortality (deaths in the first year of life), and solve the censoring problem by looking only at children born at least one year before the survey, for whom the survival status at age one is know. One could then analyze the probability of surviving to age one using ordinary logit models. A complementary analysis could then look at survival from age one to five, say, working with children born at least five years before the survey who survived to age one, and then analyzing whether or not they further survive to age five, using again a logit model. While simple, this approach does not make full use of the information, relying on cases with complete (uncensored) data. Cox and Oakes (1980) show that this so-called reduced sample approach can lead to inconsistencies. Another disadvantage of this approach is that it focuses on survival to key ages, but cannot examine the shape of the hazard in the intervening period.
