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  • Birth-Weight-Specific Infant and Neonatal Mortality: Effects of Heterogeneity in the Birth Cohort
Abstract

Birth-weight-specific infant mortality is examined using a novel statistical procedure, parametric mixtures of logistic regressions. The results indicate that birth cohorts are composed of two or more subpopulations that are heterogeneous with respect to infant mortality. One subpopulation appears to account for the “normal” process of fetal development, while the other, which accounts for the majority of births at both low and high birth weights, may represent fetuses that were “disturbed” during development. Surprisingly, estimates of neonatal and infant mortality indicate that the “disturbed” subpopulation has lower birth-weight-specific mortality, although overall crude mortality rates are higher for this subpopulation. It is hypothesized that this is due to high rates of fetal loss among the “disturbed” subpopulation, resulting in a highly selected group at birth. The heterogeneity identified in the birth cohort could be responsible for recent decelerations in the decline in infant mortality, and might be the cause of unexplained ethnic differences in birth-weight-specific infant mortality. The novel statistical methodology developed here has broad application within human biology. In particular, it could be used in any context where parametric mixture modeling is applied, such as complex segregation analysis.

KEY WORDS

BIRTH WEIGHT, INFANT MORTALITY, FINITE MIXTURES OF LOGISTIC REGRESSIONS

Analyses of birth outcomes are generally carried out assuming that birth cohorts are homogeneous after controlling for birth weight and/or gestational age and sometimes the infant’s sex. However, the fact that birth weight and gestational age distributions are significantly skewed has often been interpreted as evidence that birth cohorts are composed of several heterogeneous subpopulations (Brimblecombe et al. 1968; Fryer et al. 1984; Karn and Penrose 1951; Wilcox and Russell 1983). This view is supported statistically by recent applications of parametric (Fryer et al. 1984; Gage and Therriault 1998) and partially parametric (Umbach and Wilcox 1996; Wilcox and Russell 1983) mixture models to human birth-weight distributions. The parametric methods have also been applied to gestational age with similar results (Gage 2000). This paper examines the hypothesis that [End Page 165] the subpopulations identified by the fully parametric mixture models of birth weight are in fact heterogeneous with respect to the risk of infant and neonatal mortality. The specific aims are to (1) estimate birth-weight-and-sex-specific infant mortality for each of the subpopulations identified by the mixture models, (2) ascertain the statistically significant variation between the sexes in mortality patterns, and (3) examine the trends across the first year of life by decomposing sub-population and sex-specific mortality into neonatal and postneonatal components. The present paper builds upon the univariate mixture models described in Gage and Therriault (1998). In particular, it estimates birth-weight-and-subpopulation-specific mortality (with covariates) for the 1988 New York State European American birth cohort (the largest birth cohort examined by Gage and Therriault [1998]).

Background

The fully parametric two-component Gaussian mixture model of birth-weight/gestational age used in Gage and Therriault (1998) can be expressed as: inline graphic

where N[ ] are Gaussian distributions with means μp and μs, and standard deviations σp and σs, 1/ρ is the proportion of births in the component subscripted p, and x is birth weight or gestational age (Fryer et al. 1984; Gage and Therriault 1998; Gage 2000). Cross-population comparisons using these methods on birth weight and gestational age by sex and ethnicity identify very similar underlying component structures (Fryer et al. 1984; Gage and Therriault 1998; Gage 2000). In all applications, the parsimonious model includes at least two Gaussian components. The primary component (subscripted p) accounts for 80% to 95% of births and has a higher/longer mean birth-weight/gestational age, and a relatively small/short standard deviation. The secondary component (subscripted s) represents the remaining births and is characterized by a lower/shorter mean birth-weight/gestational age and a very large/long standard deviation in birth-weight/gestational age. Consequently, the secondary component accounts for most of the individuals in both the lower and upper tails of the birth-weight or gestational age distributions (see Figure 1 for an example). The question remains, does the mixture model identify truly heterogeneous subpopulations, or is the two-component Gaussian mixture model simply a better empirical fit compared to a single-component model? If the two components of the univariate birth-weight/gestational age mixture models display statistically different patterns of mortality, then this would support the view that the components identified represent heterogeneous subpopulations. Here, subpopulation-and-birth-weight-specific mortality is examined to determine if mortality is heterogeneously distributed across the two components of the mixture model. [End Page 166]

Figure 1. Two-component mixture model of the human birth-weight distribution (from ). The solid line represents the total probability density function. The long-dashed line represents the primary component, and the short-dashed line represents the secondary component.
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Figure 1.

Two-component mixture model of the human birth-weight distribution (from Gage and Therriault 1998). The solid line represents the total probability density function. The long-dashed line represents the primary component, and the short-dashed line represents the secondary component.

Methods of statistically controlling for heterogeneity take several forms. The traditional approach is to measure the phenomenon directly and introduce it as a covariate, for example, using age as a covariate in an analysis of covariance. In many cases, however, the phenomenon is not directly measurable, but is “hidden.” Vaupel et al. (1979) showed how uncontrolled heterogeneity could bias the interpretation of trends in mortality and proposed using a gamma distribution to define the unobservable distribution of heterogeneity that they referred to as frailty. Heckman and Singer (1982), however, showed that choice of alternative parametric models in place of the gamma could significantly influence interpretation of the results. They suggested a finite mixtures, continuous hazard regression model in which the mixture is nonparametric, but the hazard is parametrically specified. Unfortunately, these models are not robust either, in this case to various parametric specifications of the hazard (Trussell and Richards 1985). Nevertheless, the methodology of the “finite mixtures” has been extended to logistic regression, where a time- (age-) dependent parametric hazard need not be specified (McLachlan and Peel 2000). The difficulty with this final methodology is that identification depends upon sample size within binomial experiments, which may be impractical when there are several covariates (Wang 1994). The problem with the models of Vaupel et al. and Heckman and Singer is the need to choose an appropriate parametric model a priori without reference to any empirical data to base this choice upon. The problem with the finite mixtures of logistic regression model is that binomial experiments are required to identify the nonparametric specification of the mixture. [End Page 167]

Here, I propose a slightly different approach: parametric mixtures of logistic regressions. In many cases heterogeneity can be inferred from the marginal distribution or density of an observable covariate, as is the case for birth weight presented above. Thus, a parametric mixture model can be specified and empirically evaluated against the marginal distribution of the covariate using standard procedures (McLachlan and Peel 2000). This mixture model can then be incorporated into a parametric mixture of logistic regressions. The advantage of this approach over the models of Vaupel et al. and Heckman and Singer is that the parametric specification of the model can be empirically evaluated. In addition, the method may not suffer from the identification problems of finite mixtures of logistic regressions as discussed below. This basic methodology has broader applications than studies of infant mortality, birth weight, and gestational age. It could be used anywhere mixture modeling is relevant (Pearson et. al. 1992; McLachlan and Peel 2000). Of particular interest to readers of this journal is its potential for studying “genotypes” identified by commingling analysis or complex segregation analysis.

Data and Methods

The difficulty in estimating subpopulation-specific mortality for the components of a mixture model is that subpopulation membership is only probabilistically known. Consequently, logistic regression, the standard method of studying infant mortality, is not immediately applicable. The solution proposed here is to assume that the mortality of the total population at birth weight x is the weighted sum of the mortality of the p and s subpopulations at birth weight x, where the weights are the posterior probabilities of subpopulation membership, that is: inline graphic

where h(x) are birth-weight-specific total mortality rates, the q(x) are posterior probabilities that a birth at birth-weight x belongs to subpopulation p, the (exp (∑ bx))/(1 + exp (∑ bx)) are fitted subpopulation-specific probabilities of dying for the p and s subpopulations, ∑ bx is defined in the most general case examined here as b0 + b1x + b2x2, and the bi are p- and s-specific covariates of the logistic probabilities of dying. This parameterization of birth-weight-specific mortality (a second-degree polynomial) assumes that there may be separate U-shaped birth-weight-specific mortality curves for each subpopulation. Previous research has shown that a second-degree polynomial provides the parsimonious result under the assumption that the birth cohort is homogeneous, that is, using standard logistic regression (Fryer et al. 1984).

The model is fitted in two stages. First, the mixture model is fitted by maximum likelihood as described in Gage and Therriault (1998). The posterior probability that each birth belongs to subpopulation p is estimated as: [End Page 168] inline graphic

based on the stage 1 estimates of the mixture parameters. Finally, the logistic regression terms are fitted by minimizing the sum across all births of: inline graphic

where y is coded 1 if the infant died and 0 otherwise. The result is a maximum likelihood estimate of subpopulation and birth-weight-specific mortality, conditional upon the estimated probabilities of group membership.

This differs significantly from most applications of finite mixtures of logistic regressions (Wang 1994; McLachlan and Peel 2000). These models are generally used to adjust for over-dispersion in binomial response data under the assumption that the over-dispersion is due to heterogeneity. In general, the number of components in the mixture, and the probabilities of group membership [q(x)] are treated as unknown parameters to be fitted to adjust for over-dispersion. In the Bernoulli case, where each trial is considered an independent experiment, this is not possible because a mixture of Bernoulli distributions is Bernoulli. On the other hand, over-dispersion can be identified in the binomial case, that is, where a number of trials are carried out at various experimental levels. The models employed in this paper differ from these applications in that the evidence for heterogeneity is derived from a prior mixture model analysis fitted to the marginal distribution of birth weight and not from over-dispersion in the dependent variable of the logistic regression. The approaches ask slightly different questions. If the q(x) are estimated from over-dispersion, the fitted q(x) adjust for overall heterogeneity in the data. If the q(x) are estimated from a prior Gaussian mixture model analysis, the issue is whether the Gaussian components divide the data into significantly heterogeneous components. The two approaches are identical only if the Gaussian mixture model captures all of the heterogeneity in the birth cohort.

Mixtures of logistic regressions with q(x) estimated from over-dispersion are considered identified in the binomial case if the design matrixes on the q(x) and the logistic probabilities of the model are of full rank and the number of components, g < (n + 1)/2, where n is the number of trials in the experimental level with the maximum number of trials (Wang 1994). In fact, the applications to birth weight presented here meet these criteria in practice, because birth weight is only [End Page 169] reported to the nearest gram (the experimental level or dose) and because the sample sizes are very large. The data could be combined into wider bins if necessary. On the other hand, in the models employed here the q(x) are estimated by a prior parametric mixture model analysis based on the marginal distribution of birth weight. Consequently, it is not clear that the concerns with identification of finite mixtures of logistic regression where the q(x) are estimated from over-dispersion (Wang 1994) apply to parametric mixtures of logistic regressions. Our method is likely to be useful even when the data are considered Bernoulli trials rather than binomial experiments, since evidence for heterogeneity is not derived from the binomial assumption. Formal statistical studies of our methods will be necessary to define the necessary conditions for identification. The examples presented here, however, meet the sufficient conditions for identification in the more complex case.

The parsimonious model is found using standard maximum likelihood procedures. First, separate hierarchical analyses are conducted for each sex to determine the parsimonious parameterization of subpopulation-birth-weight-and-sex-specific mortality. This consists of comparing the full model with seven “nested” models. These include the six possible combinations (among the two subpopulations) of the three parameterizations of birth-weight-and-subpopulation-specific mortality, that is, second-degree polynomial, linear, and constant. The seventh “nested” model is the homogeneous case, that is, a single (three-parameter) second-degree polynomial applied to both subpopulations identified by the mixture model analysis. The second hierarchical analysis identifies the significant sex differences in the subpopulation-and-birth-weight-specific mortality patterns. For this purpose the ∑ bx in Eq. (2) are parameterized as a second-degree polynomial using interaction terms to distinguish females from males. In the most general case this takes the form b0 + b1x + b2x2 + b3z + b4xz + b5x2z, for each subpopulation, where z is sex (coded 0 for males and 1 for females) and b3, b4, and b5 represent the sex difference in the constant, linear, and squared terms of the polynomial, respectively. In this case there are a large number of possible “nested” models, 48 in all. The most important are the two “nested” models that constrain b3 = b4 = b5 = 0.0 for each subpopulation in turn. These two comparisons test the overall hypotheses that subpopulation-and-birth-weight-specific mortality varies between the sexes. The standard likelihood ratio criterion is used for all hierarchical analyses of “nested” models (Titterington et al. 1985; McLachlan and Peel 2000). All hypothesis testing is conducted at the p < 0.05 level.

Standard errors of the parameter estimates are calculated for the parsimonious model using “bootstrap” methods (Staudte and Sheather 1990). This consists of resampling with replacement from the original data set a sample of identical size to the original sample and then refitting the sampled data by stage 1 and stage 2 analyses. Since the q(x) obtained with stage 1 are estimates of the underlying population parameters, the entire two-stage procedure is used to analyze each bootstrap. Thus, the errors due to conditioning stage 2 on stage 1 estimates [End Page 170] of q(x), as opposed to known q(x), are incorporated into the standard errors. The estimates presented here are based on 100 samples (bootstraps). Standard errors are computed as a standard deviation from the 100 bootstrap estimates of each parameter.

To provide a visual assessment of the goodness of fit, comparisons of the total (across both components) fitted birth-weight-specific probabilities of dying are compared to estimates of the “observed” birth-weight-specific mortality. The “observed” birth-weight-specific mortality estimates are based on 500-gram-wide birth-weight categories. Confidence intervals of 95% for these estimates were calculated using the standard binomial variance. It should be noted, however, that the (necessary) use of categories to obtain “observed” birth-weight-specific mortality estimates tends to smooth the dynamics of the true underlying birth-weight-specific mortality curve. The degree of smoothing is smaller with narrower categories, while 95% confidence intervals are greater. The 500-gram width appears to provide reasonable comparisons. Nevertheless, the “observed” estimates are only a crude assessment of the dynamics of birth-weight-specific mortality and hence of the true goodness of fit.

Subpopulation-specific crude mortality rates (that is, across all birth weights) can be estimated from the fitted models. The first (additive) term of Eq. (2) represents the density of deaths to the primary subpopulation, while the second additive term represents the density of deaths to the secondary subpopulation. The crude death rates for each subpopulation are therefore simply the integrals of these terms divided by the number exposed to risk. There are several ways to operationalize these estimates. Here, estimates of the subpopulation-specific crude death rates are obtained as: inline graphic inline graphic

where the CDR are the crude death rates for the primary (p) and secondary (s) subpopulations, and the sums are taken across all observed births.

The two-stage fitting procedure is a novel statistical method. Consequently, there are a number of statistical issues concerning its application that must be considered. Formal statistical examination of this method is underway. Preliminary studies, however, indicate that the statistical methodology is sound. Simulation studies of the two-stage methodology suggest that the conditional maximum likelihood estimates obtained are asymptotically consistent, and normally distributed.2 [End Page 171] Only a subset of these results is included here, in particular simulations that test the results for “estimation bias.” These consist of 100 trials, for each of which a random cohort is simulated based on the parsimonious birth-weight and mortality models, and then analyzed using the stage 1 and stage 2 methods presented above. Specifically, each trial consists of generating a cohort of 50,000 births randomly from the optimum birth-weight mixture model. The probability of dying is generated for each simulated birth based on birth-weight and subpopulation membership using the parsimonious logistic regression model. Deaths to the cohort are assigned by generating a random number in the range 0.0 to 1.0 from a uniform distribution and comparing this random number to the probability of dying. When the probability of dying exceeds the random number, then the birth is coded as an infant death. These simulated data are then analyzed using the stage 1 and stage 2 methodologies presented above. This entire process is repeated 100 times. A series of t tests is then conducted to determine if the original parameters could come from the simulated populations of 100 estimates of each parameter. If the null hypotheses can not be rejected, the method is considered to provide unbiased estimates with samples larger then 50,000. The populations analyzed here are both slightly larger than 50,000.

A second set of simulations is conducted to determine if the application of the likelihood ratio criterion provides accurate type I errors (that is, that the likelihood ratio criterion is chi square), and to estimate the level of type II errors. The null hypothesis of the stage 2 hierarchical analysis assumes that mortality is homogeneous among the subpopulations. Consequently, a type I error is the probability of rejecting the homogeneous hypothesis when it is true. If the stage 2 methods work properly, this should be p < 0.05, since this is the level at which the hierarchical analyses are conducted. A type II error is the probability of accepting the null hypothesis of homogeneity when it is in fact false. In both simulation studies, the parsimonious stage 2 model identified by the original analysis is compared to the homogeneous stage 2 model. Again, the simulations consist of 100 trials, in which data sets with known properties are generated and then analyzed using the stage 2 methodology. Types I and II errors are estimated directly from the 100 hypothesis tests generated by 100 trials.

The data sets for the two simulations differ, however. To test for a type I error, 100 data sets are needed for which mortality is homogeneous across the two subpopulations. Consequently, the observed birth cohort is used, and a probability of dying is estimated for each individual based on the individual’s birth weight and the birth-weight-specific mortality rates obtained from the homogeneous [End Page 172] stage 2 model. One hundred cohorts are assembled by generating 100 random numbers from a uniform distribution for each observed birth (based on birth weight) and comparing these to the individuals probability of dying. The result is 100 data sets for which mortality is homogeneously distributed with respect to subpopulation, but retains the basic birth-weight-specific mortality pattern. To estimate for a type II error, 100 data sets are needed for which mortality is heterogeneously distributed across the two subpopulations as specified by the parsimonious stage 2 model. These data sets are generated using the same algorithm employed to generate data for testing estimation bias presented above.

The data consist of all European American, non-Hispanic singleton births of each sex born in New York State in 1988. Births from interethnic unions and multiple births are excluded. These data include 70,428 male births, associated with 449 infant (296 neonatal) deaths, and 66,476 female births, associated with 353 infant (230 neonatal) deaths. These are the same data concerning European Americans examined by Gage and Therriault (1998) in their study of sex and ethnic variation in birth weight distributions using the fully parametric mixture model approach. The parameter estimates for the stage 1 fit are taken directly from Gage and Therriault (1998). These are presented in Table 1.

Results

Hierarchical analyses to determine the parsimonious subpopulation-and-birth-weight-specific infant mortality model are presented for each sex in Table 2. In both cases, the parsimonious model is heterogeneous with separate U-shaped birth-weight-specific mortality schedules for each subpopulation (model 8, Table 2). All simpler models, including the homogeneous logistic regression model in which birth-weight-specific mortality is constrained to be identical for both components of the mixture model (model 1, Table 2), are rejected at p < 0.05. The surprising aspect of these results, however, is that the infant mortality curve for the

Table 1. Estimated Parameters and Standard Errors for the Optimal Mixture Model of Birth Weight for the European Male and Female Birth Cohorts, New York State, 1988
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Table 1.

Estimated Parameters and Standard Errors for the Optimal Mixture Model of Birth Weight for the European Male and Female Birth Cohorts, New York State, 1988a

[End Page 173]

Table 2. Maximum Likelihood Values for Single-Sex Models of Infant Mortality
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Table 2.

Maximum Likelihood Values for Single-Sex Models of Infant Mortality

secondary subpopulation is lower at every birth weight compared to the birth-weight-specific infant mortality of the primary subpopulation (Figure 2a)! Further, total birth-weight-specific mortality is no longer precisely U-shaped (Figure 2b), when combined across both subpopulations. Shifts in the proportions of the subpopulations at various birth weights produce a shoulder in the total mortality curve at about 2000 grams and another at about 5000 grams. The fact that model 1 can be rejected in favor of model 8 indicates that at least some of these previously unrecognized dynamics are supported statistically (Table 2). Comparisons of the fitted mortality curve with crude empirical estimates of birth-weight-specific mortality (Figure 2c and d) clearly indicate the shoulder at about 2000 grams in the observed data. Evidence concerning the dynamics at birth weights above 5000 grams, on the other hand, is problematic due to the large confidence intervals about the observed estimates of mortality.3

Despite consistently lower birth-weight-specific mortality rates, the secondary subpopulation is, nevertheless, the subpopulation at overall greatest risk of infant death. This is due to differences in the distribution of birth weight between the two subpopulations (Figure 1). To explore this issue further, crude (non-birth-weight-specific) death rates are estimated for each subpopulation. The crude mortality rate of the primary subpopulation is CDRp = 3.46/1000 for males and CDRp = 3.37/1000 for females. The corresponding overall rates for the secondary subpopulation are CRDs = 40.62/1000 for males and CDRs = 28.53/1000 for females. About 48% of male deaths and 39% of female deaths originate from [End Page 174]

Figure 2. Birth-weight-specific infant mortality for European American males (fine lines) and females (heavy lines). a, The fitted subpopulation-and-birthweight-specific mortality curves for the primary subpopulation (short dashed lines) and the secondary subpopulation (long dashed lines). b, Comparison of the total birth-weight-specific mortality curve for males and females. c, Total fitted birth-weight-specific mortality of males compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth-weight categories). d, Total fitted birth-weight-specific mortality of females compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth-weight categories).
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Figure 2.

Birth-weight-specific infant mortality for European American males (fine lines) and females (heavy lines). a, The fitted subpopulation-and-birthweight-specific mortality curves for the primary subpopulation (short dashed lines) and the secondary subpopulation (long dashed lines). b, Comparison of the total birth-weight-specific mortality curve for males and females. c, Total fitted birth-weight-specific mortality of males compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth-weight categories). d, Total fitted birth-weight-specific mortality of females compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth-weight categories).

the secondary subpopulation, although less then 10% of births occur in this sub-population.

The second set of hierarchical analyses indicate significant differences in male and female infant mortality patterns for both subpopulations. Comparison of the model in which the primary mortality curve is held constant between the sexes (full model = −3370.459; reduced model = −3378.853; p = 0.0008, degrees of freedom [df] = 3), and a similar comparison holding secondary mortality constant between the sexes (full model = −3370.459; reduced model = −3374.49; p = 0.0447, df = 3) can both be rejected at the p < 0.05 level. Further analysis of “nested” models indicates that the “best fit” is achieved when b4 = b5 = 0.0 for the primary component and b4 = 0.0 for the secondary component. However, a number of models of the same complexity fit almost as well and cannot be rejected in favor [End Page 175] of the “best fit.” Thus, the parameter values are not biologically interpretable, and the nature of the differences between the sexes is best evaluated from graphical representations of the full model (Figure 2). The parameter values for the full models are presented in Table 3.

Comparisons suggest that females in the primary subpopulation are slightly more sensitive to deviations of birth weight away from the optimum than males. Female primary mortality is higher at both lower and higher birth weights compared to males, although mortality in the optimum range is lower among females (Figure 2a). The optimum birth-weight-specific mortality for females in the primary subpopulation occurs at 3695 grams, 288 grams heavier than the mean primary birth weight of females (3407 grams). The optimum primary birth-weight-specific mortality for males occurs at 3,864, 319 grams heavier than the mean primary birth weight of males (3545). Overall, however, the sex differences in primary mortality are small. If the female curve is shifted 179 grams to the right, the difference in birth weight between the male and female optimum, then the only obvious differences are that the female curve is slightly more U-shaped with a small female mortality advantage near the optimum. On the other hand, female secondary mortality is lower than male secondary mortality at every birth weight. The optimum female secondary birth weight occurs at 4605 grams, 1614 grams greater than the female mean secondary birth weight, while the optimum for males is 4483 grams, 1433 grams greater than the male mean secondary birth weight. Clearly, directional selection is stronger in the secondary subpopulation than the primary subpopulation. Overall, the female curve is flatter than the male curve. As a result of the sex differences in both the primary and secondary sub-populations,

Table 3. Parameters of the Logistic Regressions by Sex and Subpopulation for Infant and Neonatal Mortality
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Table 3.

Parameters of the Logistic Regressions by Sex and Subpopulation for Infant and Neonatal Mortality

[End Page 176]

the dynamics of total female birth-weight-specific infant mortality are exaggerated over those of males (Figure 2b). Based on the crude death rates, 94% of the sex differential in infant mortality originates in the secondary component.

To reveal more about the distribution of deaths across the first year of life, the analyses presented above for infant mortality are also applied to neonatal mortality. Once again the likelihood ratio tests indicate that the birth-weight-specific mortality of both subpopulations is U-shaped and well described by a second-degree polynomial (Figure 3, Table 4). Again, the secondary subpopulation has lower birth-weight-specific mortality curves at all birth weights compared to the primary subpopulation. Comparisons between the sexes, however, indicate fewer differences in neonatal mortality than were found for overall infant mortality

Figure 3. Birth-weight-specific neonatal mortality for European American males (fine lines) and females (heavy lines). a, Fitted subpopulation-and-birth-weight-specific mortality curves for the primary subpopulation (short dashed lines) and the secondary subpopulation (long dashed lines). b, Comparison of the total birth-weight-specific mortality curve for males and females. c, Total fitted birth-weight-specific mortality of males compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth weight categories). d, Total fitted birth-weight-specific mortality of females compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth weight categories).
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Figure 3.

Birth-weight-specific neonatal mortality for European American males (fine lines) and females (heavy lines). a, Fitted subpopulation-and-birth-weight-specific mortality curves for the primary subpopulation (short dashed lines) and the secondary subpopulation (long dashed lines). b, Comparison of the total birth-weight-specific mortality curve for males and females. c, Total fitted birth-weight-specific mortality of males compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth weight categories). d, Total fitted birth-weight-specific mortality of females compared to a crude estimate of the “observed” birth-weight-specific mortality (the points are binomial means, and the vertical bars represent 95% confidence intervals based on the binomial variance for 500-gram-wide birth weight categories).

[End Page 177]

Table 4. Maximum Likelihood Values for Single Sex Models of Neonatal Mortality
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Table 4.

Maximum Likelihood Values for Single Sex Models of Neonatal Mortality

Among primary births, neonatal mortality does not appear to differ between the sexes at p < 0.05 (full model = −1910.104; reduced model = −1912.201; p = 0.2413; df = 3), although significant sex differences in mortality remain in the secondary component (full model = −1910.104; reduced model = −1916.871, p = 0.0036; df = 3). Female secondary mortality is again lower at every birth weight compared to males (Figure 3). These results suggest that the sex differences in mortality during the neonatal period are largely due to sex differences in the secondary subpopulation, while sex differences in the primary subpopulation emerge only after the first month of life.

Decomposition of the overall subpopulation-specific infant death rates into neonatal and postneonatal rates suggests that during the neonatal period the risk from the secondary subpopulation predominates, whereas during the postneonatal period the risk is more evenly divided between the primary and secondary sub-population. In particular, among males in the secondary subpopulation the neonatal death rate is 34.50/1000, which declines to 0.556/1000/month (6.12/1000 over the total 11-month period) during the postneonatal period, while males in the primary subpopulation have a neonatal death rate of 1.51/1000, which declines to 0.177/1000/month (1.95/1000 over the total 11-month period) during the post-neonatal period. Thus, in males there is a 62-fold decline in secondary mortality compared to an 8.5-fold decline in primary mortality. The relative risk of secondary mortality falls from 22.8 during the neonatal period to 3.2 during the postneonatal period. Similarly, among females in the secondary subpopulation the neonatal death rate is 25.81/1000, which declines to 0.305/1000/month (3.35/1000 over the total 11-month period) during the postneonatal period, while females in the primary subpopulation have a neonatal death rate of 1.76/1000, which declines to 0.146/1000/month (1.61/1000 over the total 11-month period) in the postneonatal period. The neonatal primary mortality rate is higher in females than in males because the mean primary birth weight is lower in females, [End Page 178] and birth-weight-specific risk of mortality is the same for both sexes. In females there is an 84-fold decline in secondary mortality and only a 12-fold decline in primary mortality. The relative risk of secondary mortality falls from 14.7 during the neonatal period to 2.1 during the postneonatal period.

The simulation studies of the statistical methods indicate that the methods are unbiased, and provide accurate type I errors.4 The results of the analyses of estimation bias are presented in Table 5 for both males and females. For each sex, simple t tests on all five of the birth-weight distribution (stage 1) parameters, and all six logistic regression (stage 2) parameters fail to reject the null hypothesis that the original parameter estimate (used to generate the simulation data) differs from the mean of 100 analyses conducted on simulated data. This indicates that the novel two-stage methods employed here provide unbiased results. Second, the

Table 5. Analyses of Estimation Bias for Stage 1 and Stage 2 Analyses Infant Mortality
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Table 5.

Analyses of Estimation Bias for Stage 1 and Stage 2 Analyses Infant Mortality

[End Page 179]

simulation study of type I errors estimates these errors at 5/100 for males, and 3/100 for females. These results agree closely with the p < 0.05 level, the level at which all analyses have been conducted. On the other hand, the simulation estimates of type II errors are relatively large. The methods fail to reject the null hypotheses of homogeneity of mortality when it is false, 31/100 trials among males and 0.0/100 among females. This suggests that the two-stage method may have considerably lower power for males compared to females. This is due to the relatively lower heterogeneity in mortality between the subpopulations of males compared to females, as reflected in the likelihood values in Table 2 and in the sub-population differences shown in Figure 2.

Discussion

The skew in birth weight and gestational age distributions is often interpreted as heterogeneity in the birth cohort (Brimblecombe et al. 1968; Fryer et al. 1984; Karn and Penrose 1951; Wilcox and Russell 1983). This view is supported by the results presented above. It is clear that Gaussian mixture models successfully subdivide birth cohorts into two subpopulations that are heterogeneous with respect to mortality. The secondary subpopulation has lower birth-weight-specific mortality at every birth weight, but higher infant mortality overall. This latter finding occurs because the secondary subpopulation accounts for the majority of births in both the lower and upper tails of the birth-weight distribution where mortality is generally higher. Additionally, the heterogeneity in the birth cohort creates complex and counterintuitive dynamics in the birth-weight-specific mortality curve. In particular, total birth-weight-specific mortality appears to have a shoulder at about 2000 grams, which is evident in the empirical data but appears to have been overlooked in previous descriptions of birth-weight-specific mortality. Finally, when compared to primary mortality, mortality in the secondary sub-population tends to be concentrated in the neonatal period. In general, the patterns tend to be stronger for females then males. These results are similar to those of other simple classification systems for infant deaths. For instance, secondary mortality has many of the characteristics of “endogenous” deaths, while primary mortality displays many of the characteristics of “exogenous” deaths as defined by Bourgeois-Pichat (1952).

Gage and Therriault (1998) hypothesized that the primary and secondary components represent subpopulations of the birth cohort that experienced “normal” and “disturbed” fetal development, respectively. In general, most disturbances in fetal growth would be expected to lower birth weight, either by lowering fetal growth rates or initiating premature delivery. For example, pre-eclampsia with proteinuria is associated with low birth weights (Chamberlain 1989; McFadyen 1989). However, a few disturbances are associated with heavy birth weights. For example, maternal diabetes appears to increase fetal growth rates (Brunskill et al. 1991; Meshari, De Silva, and Rahman 1990). Taken together these two types of disturbance in fetal development could account for the generally low [End Page 180] mean but large standard deviation in birth weight characteristic of the secondary subpopulation of a birth cohort. If so, why does the “disturbed” subpopulation, which might be expected to be already compromised, have lower birth-weight-specific mortality than the “normal” subpopulation?

One hypothesis is that members of the secondary subpopulation who are subjected to some shock (disturbance) during fetal development may be at birth a highly selected and constitutionally more robust subpopulation than their birth-weight-specific peers in the primary subpopulation. In particular, the secondary subpopulation may be at greater risk of fetal loss so that the members of the secondary subpopulation that survive until birth are highly selected and more robust. If so, then the evidence presented above indicates that this process of selection continues throughout the neonatal period and into the postneonatal period, particularly in males. The relative risk of membership in the secondary versus the primary subpopulation declines rapidly from the neonatal to the postneonatal periods. Further, the selection process appears to be accelerated in females, that is, more of the selection process occurs fetally (or at a younger age). This would explain the lower birth-weight-specific mortality of female secondary births compared to male secondary births, as well as the rapid decline in the relative risk of secondary infant mortality. Most of the sex differential in infant mortality can be attributed to this process. There is some evidence that sex differences in the primary subpopulation emerge postneonatally. The timing of this phenomenon suggests an environmental origin.

Little concrete information exists concerning the true level of fetal loss in humans. The hypothesis presented above poses several questions. First, is fetal loss higher for female conceptuses than for male fetuses? There is no conclusive evidence suggesting that the sex of a conceptus influences the rates of fetal loss (Kallen 1988), although the sex ratio at birth, which favors males, has been attributed to higher female fetal losses. Further, the existence of sex differentials in fetal loss is consistent with the general observation that the growth and development of females is more canalized than among males (Frisancho 1981), that is, that the development of females is more closely correlated with chronological age than the development of males. Hence, females generally display lower variances in physical traits compared to males. The sex differences in the variance in birth weight of the primary subpopulation (Table 1) is a good example. In general, the sex differences in mortality described above are consistent with the concept that females are more canalized than males. In any event, due to differential canalization a disturbance might result in a higher likelihood of fetal loss if the fetus is female.

Second, is the fetal loss rate high enough to drive the system described above? There is little consensus concerning the overall rates of fetal loss. Estimates of miscarriage rates calculated on known pregnancies (observable fetal loss) suggest rates of 5% to 15%, depending upon the age of the mother. Overall, loss rates may be considerably greater (Kallen 1988). If all of the observable fetal loss is assumed to come from the secondary subpopulation, which accounts for [End Page 181] 5% to 15% of births depending upon the population (Gage and Therriault 1998), then the fetal survivorship in the secondary subpopulation might be as low as 0.5 or even lower. These levels of mortality selection should be sufficient to create the mortality differentials presented above.

Heterogeneity can cause counterintuitive trends in mortality (Vaupel et al. 1979). For example, if the interpretation of infant mortality presented above is correct, improvements in maternal health that reduce fetal losses in the secondary subpopulation could cause infant mortality rates to increase. This might explain why the decline in infant mortality has slowed or even stopped over the last few years. Heterogeneity in birth cohorts might also explain the “pediatric paradox,” that is, why African American births have lower mortality at low birth weights than European American births. In general, because African Americans receive poorer health care than European Americans, it is logical to assume that African American infant mortality should be higher than European American infant mortality. However, higher African American infant mortality only occurs at normal birth weights not at low birth weights (Wilcox and Russell 1990). Gage and Therriault (1998) have shown that African Americans tend to have a larger secondary (disturbed) subpopulation than European Americans. This alone could create the “pediatric paradox,” since a larger proportion of births would occur in the sub-population with lower birth-weight-specific infant mortality, particularly at low birth weights. Additional analyses will be necessary to determine if this is in fact the explanation.

Conclusions

Estimates of infant and neonatal mortality for the two components of the birth-weight distribution identified using two-component Gaussian mixture models indicate that:

  1. 1. The two components are heterogeneous with respect to the risk of mortality, and hence can be considered to be separate subpopulations.

  2. 2. Subpopulation-and-birth-weight-specific mortality is U-shaped in both subpopulations.

  3. 3. Compared to the primary subpopulation, the secondary subpopulation displays lower birth-weight-specific mortality at every birth weight, but is nevertheless at higher risk of overall infant mortality because it accounts for a higher proportion of low- and high-birth-weight infants, where mortality is higher.

  4. 4. There is a significant shoulder in the birth-weight-specific mortality curve at about 2000 grams that has not been previously described in the infant mortality literature. This shoulder is a direct result of the heterogeneity in the birth cohort.

  5. 5. The lower birth-weight-specific mortality attributed to the secondary subpopulation might result from higher fetal losses in the secondary [End Page 182] subpopulation. This is consistent with earlier interpretations of mixture model results suggesting that the primary and secondary subpopulations are groups undergoing “normal” and “disturbed” fetal development, respectively.

  6. 6. Heterogeneity in the birth cohort might be responsible for the deceleration in the decline of infant mortality over the past few years, as well as unexplained differences in the dose response to birth weight among ethnic groups.

Received 14 May 2001; revision received 28 November 2001.

Acknowledgments

This work was supported by the National Institute of Child Health and Human Development through grant HD 37405.

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Footnotes

1. Department of Anthropology and Department of Epidemiology, University at Albany, Albany, NY 12222, and Department of Genetics, Southwest Foundation for Biomedical Research, San Antonio, TX 78245.

2. Extensions of the simulations to test the results presented in this paper for bias provide an indication that the two-stage procedure is asymptotically consistent and normally distributed. These additional simulations are not reported in the paper and are based on the results for females. The simulations are identical to those for bias except that the sample sizes are varied (12,500, 25,000, 50,000, 100,000, and 200,000). The results indicate that any bias disappears, the variance of the parameter values declines (consistency), and the distributions of the parameter estimates become increasingly Gaussian (normally distributed) as sample size increases. These preliminary results suggest that samples larger then 25,000 may be necessary for reliable results. Results for females were employed for these preliminary studies because they are expected to be conservative due to lower infant mortality among females.

3. Among males, there are 267 births at birth weights greater than 5000 grams and only two (both neonatal) deaths (a death rate of 7.5/1000, 95% confidence interval, 0.0 to 17.8/1000). Among females there are 118 births at birth weights greater than 5000 grams and only one (neonatal) death (a death rate of 8.5/1000, 95% confidence interval, 0.0 to 25.0/1000). The dynamics of total birth-weight-specific infant mortality above 5000 grams is questionable. The decline predicted by the model can not be ruled out, although a continued increase in mortality with birth weight cannot be excluded either. In both sexes, the death rate for all births 5000 grams and greater falls at about the same level as the peak of the predicted shoulder occurring at approximately 5000 grams.

4. Standard qq plots of the likelihood ratio criterion with a chi-square distribution suggest that the likelihood ratio criterion is chi-square.

Additional Information

ISSN
1534-6617
Print ISSN
0018-7143
Launched on MUSE
2002-04-01
Open Access
No
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