Absolute Change in Cause-Specific Infant Mortality for Blacks and Whites in the US: 1983–2002

Abstract

This paper examines absolute change in infant mortality from 5 leading causes of death for whites and blacks over a 20 year period. Change in infant mortality varies by cause, race, and birth weight. Absolute decline in mortality from respiratory distress syndrome (RDS) and sudden infant death syndrome (SIDS) in the overall study population has been more rapid for black infants during the period after specific technological innovations were approved and behavioral practices were recommended for these conditions. For low birth weight infants, blacks experienced greater decline in mortality from SIDS and whites experienced greater decline in RDS mortality. Despite remarkable declines in mortality from these causes, relative racial disparities have increased over this time period. For the overall study population, blacks and whites experienced similar rates of mortality decline from congenital anomalies. Mortality decline from this cause among low birth weight infants occurred at a faster pace for whites. Mortality from causes for which no specific innovations were developed increased for blacks but remained relatively constant for whites. An analysis of absolute change complements the relative disparities approach by revealing the dynamics of change, thus providing a more complete understanding of changing racial disparities in infant mortality.

Appendix

Bounding Model-Based Predictions

To allow for uncertainty in the predicted probabilities, we adopt the straightforward strategy of bounding predictions using upper and lower confidence limits of the coefficients (a _jk ). Replacing a _jk in Eq. 2, with $a_{jk} \pm Z_{\alpha /2} {\text{se}}\left( {a_{jk} } \right)$ (the upper and lower endpoints of a _jk ), we can place upper and lower limits on p _jk . This information can also be used to obtain the standard error of p _jk , as the absolute difference between the point estimate and the upper or lower limit divided by $Z_{\alpha /2}$. For the data used here, this simple approach compares favorably to a more complex approach based on asymptotic methods. For example, it can be shown using the delta-method described by Rao (1973, pp. 387–390) that the asymptotic variance of a model-based prediction of the baseline mortality in a single year attributable to cause c is

$$ {\text{var}}\left( {p_{c} } \right) \cong \sum\limits_{i = 1}^{6} {\sum\limits_{j = 1}^{6} {\left[ {\frac{{\partial p_{c} }}{{\partial a_{{i_{{}} }} }}} \right]\left[ {\frac{{\partial p_{c} }}{{\partial a_{j} }}} \right]{\text{cov}}\left( {a_{i} a_{j} } \right)} } , $$

(3)

where cov(a _i ,a _j ) is the variance of a coefficient or the covariance between two coefficients. To obtain the variance in the prediction for a single cause in a single 2-year time period requires 6 variances, 15 covariances, and the evaluation of 6 partial derivatives. Letting j index the 6 causes of death, the partial derivatives are

$$ \frac{{\partial p_{c} }}{{\partial a_{j} }} = p_{c} \left( {1 - p_{c} } \right),\,\,{\text{for}}\,j = c $$

and

$$ \frac{{\partial p_{c} }}{{\partial a_{j} }} = - p_{c} p_{j} ,\,\,{\text{for}}\,\,j \ne c $$

(4)

This calculation is carried out for each cause and each time period for each model.

A simplification is desirable given the computational complexity underlying the formal asymptotic approach. In this case, we can capitalize on specific features of the data. Specifically, for rare causes of death, the estimated logits (a _jk terms) are in the neighborhood of −10.6 to −5.5, with small standard errors and covariance terms that are vanishingly small. These quantities are somewhat larger for more common causes of death, with a _jk never exceeding −3.5. To investigate the sensitivity of the simpler approach under two extreme scenarios, we bounded the probability of death due to congenital anomalies in 1983–1984 for the white low birth weight population (i.e., the population who experienced the highest mortality for this cause at 17.8 deaths per 1,000 births). We applied the same approach to whites born at all weights (i.e., a population experiencing a much lower mortality rate—2.3 deaths per 1,000.). Our test models included several predictors. Results were not sensitive to the choice and number of predictors used based on several alternative model specifications.

The bounds on the rates produced by the simple and formal approaches are expected to agree closely when p _jk is very small, with greater disagreement as p _jk becomes larger. Our sensitivity analysis bears this out. For the high mortality group, the upper and lower bounds on mortality from congenital anomalies for whites in 1983–1984, based on the simple and formal approaches, differed by no more than 0.0001, or 1 death per 10,000 births. For the low mortality group, the different approaches led to a much smaller difference of no more than 0.000004, or 4 deaths per million births. Therefore, we adopt the simpler approach of bounding the probabilities using interval estimates of the logits. Note that the variances of the predicted probabilities are estimated from the bounds, and these quantities are used to place upper and lower bounds on differences in predicted probabilities. Specifically, the differences in probabilities involve comparisons of yearly rates or differences by race. For example, the total change in mortality through time period k compared to the 1983–1984 period (k = 1) associated with cause j is,

$$ \Updelta p_{jk} = \sum\limits_{l = 2}^{k} {\left( {p_{j(l - 1)} - p_{jl} } \right)} = p_{j1} - p_{jk} $$

(5)

$\Updelta p_{jk}$ quantifies the total change in mortality occurring from the base period through the current period. We refer to this as a first difference or as absolute change. This is equivalent to the sum of the increments in the change in IMR through period k. Uncertainty in this estimate is obtained by assuming independence across time periods, yielding the variance of $\Updelta p_{jk}$ as the sum of variances of the respective predicted probabilities

$$ \text{var} \left( {\Updelta p_{jk} } \right) \cong \text{var} \left( {p_{j1} } \right) + {\text{var}}\left( {p_{jk} } \right) . $$

(6)

Assessing race differences in absolute change is a primary concern. The black–white difference in first differences quantifies the racial gap in absolute change in IMR in period k. We refer to this as a second difference, which is computed as

$$ D_{k}^{BW} = \Updelta^{B} p_{jk} - \Updelta^{W} p_{jk} = \left( {p_{j1}^{B} - p_{j1}^{W} } \right) - \left( {p_{jk}^{B} - p_{jk}^{W} } \right) = \sum\limits_{l = 2}^{k} {\left( {d_{j(l - 1)}^{BW} - d_{jl}^{BW} } \right)} , $$

(7)

where $\Updelta^{B} p_{jk}$ and $\Updelta^{W} p_{jk}$ denote the first differences for blacks and whites, respectively. Letting $d_{jk}^{BW} = p_{jk}^{B} - p_{jk}^{W}$ denote the black–white mortality gap in period k, the second difference can be decomposed into the difference between the initial black–white mortality gap and the gap by period k, or as the sum of the increments in the racial gap through period k. The variances of the 2nd differences are computed as the sum of the variances in the first differences for blacks and whites, or

$$ \text{var} \left( {D_{k}^{BW} } \right) \cong \text{var} \left( {\Updelta^{B} p_{jk} } \right) + \text{var} \left( {\Updelta^{W} p_{jk} } \right) $$

(8)

See Tables 3 and 4

Table 3 Predicted rates, black–white differences in rates, and rate ratios by cause of death—all births

Table 4 Predicted rates, black–white differences in rates, and rate ratios by cause of death—low weight births