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© 2006 American Public Health Association
DOI: 10.2105/AJPH.2006.094870
LETTER |
Quanhe Yang is with the Division of Birth Defects and Developmental Disabilities, National Center on Birth Defects and Developmental Disabilities, Centers for Disease Control and Prevention, Atlanta, Ga. Sander Greenland is with the Departments of Epidemiology and Statistics, University of California, Los Angeles. W. Dana Flanders is with the Department of Epidemiology, School of Public Health, Emory University, Atlanta, Ga.
Correspondence: Request for reprints should be sent to Quanhe Yang, PhD, Division of Birth Defects and Developmental Disabilities, National Center on Birth Defects and Developmental Disabilities, Centers for Disease Control and Prevention, 1600 Clifton Rd, Mail Stop E-86, Atlanta, GA 30333 (e-mail: qay0{at}cdc.gov).
We thank Schempf and Becker for pointing out a deficiency in our presentation, although we disagree with their solution. In fact, we were aware of the alternative standards they cite and rejected them as inappropriate. Although researchers are often taught otherwise, the choice of standard distributions is not arbitrary and should not be based on symmetry or other mathematical considerations. Rather, the choice of standard distribution should be based on queries of contextual interest about outcomes under alternative histories for a well-specified target population.1,2 Conversely, the standard distribution chosen for a problem implies the population and histories to which the result applies.
Our equation 3 (Schempf and Becker’s equation 1a) breaks down the change in the crude rates into 2 components. The first component answers the natural query of what we would have seen if the age–parity distribution had not changed after 1980 but the specific rates had changed as observed. The second component answers the query of what we would have seen if the age–parity distribution had changed as observed under the after-1980 assumption but the specific rates had been at their final values (Rij2) throughout. This query preserves additivity of the breakdown, but other contextually sensible alternatives are possible.
One alternative to the first query is to ask what our results would have been if the age–parity distribution had been constant at the 1990 value but the specific rates had changed as observed. This alternative query, about the potential impact of the rate change on a population with the 1990 distribution, is answered by replacing Nij1/N++1 with Nij2/N++2 in the first component of our equation 3. The corresponding estimated change in the population rate is shown in Tables 1
and 2 (column 2; see the original article for the complete tables).
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An alternative to the second query is to ask what our results would have been if the age–parity distribution had changed as observed under the after-1980 assumption but the specific rates had been at their initial values (Rij1) throughout. Replacing Rij2 with Rij1 in the second term of our equation 3 provides the answer. The corresponding estimated change in the population rate is shown in Tables 1
and 2 (column 3). With this substitution, however, the components no longer add to the crude difference between the 1980 and 1990 rates unless we change the weighting in the first component from Nij1/N++1 to Nij2/N++2 (Schempf and Becker’s equation 1b). Forcing additivity leads to artificiality for 1 or both components. We presented the combination that seemed most natural to us and noted that the estimates "depend on the choice of standard, which should reflect the targeted population of interest."4(pxxx) Citing Kitagawa,3 Schempf and Becker recommend decomposition with the simple average distribution for low birthweight and simple average specific rates (their equation 2). The first component answers the query of what change we would have seen if the age–parity distribution had been at its average value throughout but the specific rates had changed as observed, whereas the second answers the query of what change we would have seen if the age–parity distribution had changed as observed after 1980 but the specific rates had been at their average values throughout. At best, these queries each use standards achieved at some unknown interim time, when the standardizing quantities were all at their average values. Given the multidimensional nature of the standardizing quantities, however, there may never have been a single time when all were at their average values, in which case the breakdown is more hypothetical than any considered above. We think that Schempf and Becker’s solution makes both queries unnatural.
The general lesson is an old one: breaking down measures into attributable components sounds deceptively simple but quickly leads to interpretational problems.2 Furthermore, Schempf and Becker’s confidence interval analogy fails because we are summarizing over a high-dimensional region, not an interval, and midpoints need not be typical of points in the region of interest. Even within the analogy, however, one should want the other end of the interval, not its midpoint.
Tables 1 and 2 provide results at the other end of the spectrum from the decomposition we used. Nevertheless, with 1 exception, the overall patterns remain similar in that the impacts of changes in rates exceed the impacts of changes in the age–parity distribution. Compared with the magnitudes in our article, the alternative standards diminish somewhat the magnitude of the impact of changes in rates. The exception is for non-Hispanic Whites in 1980 through 1990, for whom the pattern reverses with the alternative standard. In particular, the impact of changes in the age–parity distribution would be substantially greater had the rates remained at the 1980 levels. In our original analyses we presented the impact associated with changes in age–parity distribution based on the 1990 rates, which we thought were more relevant because of their relative recency. Tables 1 and 2 complete the picture and allow the reader to choose his or her own comparisons.
References
1. Greenland S. Interpretation and estimation of summary ratios under heterogeneity. Stat Med. 1982;1: 217–227.[Medline]
2. Rothman KJ, Greenland S. Modern Epidemiology. 2nd ed. Philadelphia, Penn: Lippincott-Raven; 1998.
3. Kitagawa E. Components of a difference between two rates. J Am Stat Assoc. 1955;50:1168–1194.[CrossRef][Web of Science]
4. Yang Q, Greenland S, Flanders WD. Associations of maternal age- and parity-related factors with trends in low birthweight rates: United States, 1980 through 2000. Am J Public Health. 2006;96:856–861.
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