IRREDUCIBLE MINIMUM ISSUES
(Jan. 5, 2009; rev. Apr. 24, 2011)
On occasion, in the context of interpreting apparent departures from standard patterns of changes in relative differences in experiencing an outcome that has become very rare, I have addressed the concept of the “irreducible minimum.” In Race and Mortality,[1] I addressed it in the following terms:
Very likely, however, we will one day see a reduction in the racial disparity in infant mortality. But that result will have to be appraised with great caution. In the case of infant mortality and other relatively rare and declining conditions, we are likely to approach a certain irreducible minimum mortality rate that will persist regardless of socioeconomic status and access to the best medical resources. When this occurs, the gap between observed black and white mortality rates probably will diminish regardless of any true change in the relative health of black and white infants. Yet, if one separated out preventable mortality – which is society’s true concern – one would likely find the racial gap continuing to grow.
That is, suppose that given the state of medical science, even among the most affluent groups infant deaths cannot reasonably be reduced below 3 deaths per thousand live births. Suppose also that when the white rate reached 5 the black rate reached 11. Researchers might well see progress in the fact that the black-white infant mortality ratio had been reduced to 2.2 from the 2.4 ratio of 1997. Yet, the ratio of black to white preventable infant mortality would be 4.0 – i.e., 8(11 minus 3) over 2 (5 minus 3). This would be higher than the black-white ratio of preventable infant mortality in 1997. Assuming the same irreducible minimum, the 14.2 black rate and 6.0 white rate that year would translate into a ratio of preventable deaths of only 3.7 – i.e. 11.2 (14.2 minus 3) over 3 (6 minus 3).
Others have also addressed the concept of an adverse outcome rate of the advantaged group that it may be difficult or impossible to further reduce, using the term “minimum achievable level.”[2] Discussions of “ceiling effects” or “floor effects” involve related issues.
I have differed from others in the interpretation of these patterns in the following respect. Victora et al. [2] interpreted the pattern of increasing relative differences during a period of overall decline in an outcome to reflect a meaningful worsening of the disparity; they also interpreted the reversal of the pattern following the approaching of an irreducible minimum to reflect a meaningful reduction in the disparity. By contrast, I have regarded the former pattern of increasing relative differences, being in accordance with the usual direction of changes in the circumstances, as not reflecting a meaningful change in disparity. [1,3,4] On the other hand, since a decrease in the relative difference during periods of overall declines in an outcome is contrary to the usual (statistically driven) direction of changes in the circumstances, I have suggested that such decrease could be cautiously interpreted as reflecting a meaningful decline in disparity. But, as indicated in the quoted material, that interpretation would not hold where the decrease in the relative difference occurred as a result of the approaching of an absolute minimum. I think that my interpretation is correct.
I have also touched upon the irreducible minimum issue with respect to the approach to measuring disparities by deriving a difference between hypothesized means on the basis of two rates, though previously merely noting it as an additional complication, as in reference 5 (as well as in all the 2008 presentations addressing that approach). But I am here giving the matter greater attention and have augmented the Solutions Database to address the issue.
Initially I note, however, that the description of the issue in the quoted language from Race and Mortality (as well as the treatment at pages 13-14 of reference 3) was arguably incorrect in the following respect. Consider a situation where the adverse outcome rates of the advantaged and disadvantaged groups are 15% and 30% in Year 1 and 10% and 22% in Year 2 and the irreducible minimum rate is 5%. According to the reasoning in the quotation from Race and Mortality, the avoidable outcome rates would be 10% and 25% in Year 1 and 5% and 17% in Year 2. But such reasoning overlooks that the 5 percentage points reflecting the irreducible minimum rate should be excluded from the denominator in the calculation of rates of the avoidable adverse outcomes. Thus, properly adjusting for the irreducible minimum, the avoidable adverse outcome rates would be 10.5% (10/95) and 26.3% (25/95) in Year 1 and 5.3% (5/95) and 17.9% (17/95) in Year 2. This correction to the calculation does not affect the points made with regard to irreducible minimums in various places. But precision requires that the avoidable rates be calculated as just described. It is that approach that underlies the irreducible minimum adjustment in the Solutions Database.
The implications of an irreducible minimum adjustment to the approach in the Solutions Database may be illustrated as follows. Observers relying on relative differences in adverse outcome rates have often noted that disparities in mortality and many other adverse outcomes appear to be greater among the young than among the old. But I have pointed out in various places both that nrelative differences in adverse outcome rates would be expected among the young simply because adverse outcomes tend to be rarer among the young and that relative differences in favorable outcome rates tend to be greater among the old than the young.
There remains the issue of whether, properly analyzed, health disparities would be found to be greater among the young than the old. There is some reason to expect that health disparities would be greater among the old than the young simply because, whatever the factors underlying the observed disparities, the old have been subjected to them for a longer period of time.[6,7]. There are points to be made to the contrary as well. But when observers note that a ceiling effect may be reducing the disparity in later years, they are making what I have suggested is the mistake of regarding changes in overall patterns that are functions of effects of the roles of irreducible minimums as meaningful changes, overlooking that society’s actual concern involves disparities in avoidable outcomes.
In any case, the adjustment to the Solutions Database attempts to address this issue. Table A (which can be accessed at Table A) presents the mortality rates of black and white men and women by various age groups, along with the relative differences in mortality and survival for each age-gender group and the estimated effect size of the difference between hypothesized mean derived by the standard approach in the solutions database. It shows that, as one ought to expect, relative differences in mortality are greater among the younger age groups than the older age groups, but relative differences in survival are greater among the older age groups than the younger age groups.
But the EES column in the table also show that, using an approach that ought not to be affected by the overall prevalence of an outcome, the disparities are indeed larger among the young than the old. But there are two considerations that call that interpretation into question.
The first, which is what this page was initially created to address in January 2009, involves the implications of the existence an irreducible minimum. Table A shows that the pattern of decreasing EES would not necessarily hold if one adjusted for an irreducible minimum. As an example, under the standard approach, the EES for men is .325 standard deviations for the 45-49 age group but only .21 standard deviations for the 60-64 age group. But if we assume that the 10% of the white male deaths in the younger age group are unavoidable (which would translate into an irreducible minimum death rate of 0.04% for that age group) while 50% of the white male deaths in the 60-64 year age group are unavoidable (which would translate into an irreducible minimum death rate of 0.93%% for that age group), the EES would be .34 standard deviations for men in the younger group and .38 standard deviations for men in the older group. The second is addressed in an April 2011 Addendum.
Regarding the irreducible minimum issue, the obvious difficulty is the estimation of the proportion of deaths that is unavoidable. It does seem very likely that only a small proportion of deaths in the younger age group are unavoidable and the proportions increase with each age group, probably reaching quite high levels in the oldest age groups. It also seems likely that irreducible minimums, being the minimum achievable level given the current state of medical knowledge and other factors associated with health outcomes, change over time. Thus, the analysis of changes in racial differences in infant mortality in Section A.7 of the main Scanlan’s Rule page probably should be redone, not only with adjustments for irreducible minimums, but with different adjustments for the two points in time.
Nevertheless, there is no evident scientific basis for any of the adjustment figure underlying Table A or for such figures as might be used in an analysis of changing disparities in infant mortality. Notwithstanding such fact, it does seem clear that the role of the absolute minimum could be a potentially important issue with regard to the utility of the measurement approach employed in the Solutions Database or any other approach to measuring disparities.
In any event, the Solutions Database has now been modified to enable one to calculate estimated effects sizes both with and without adjustment for irreducible minimums. And based on what adjustments the user might deem reasonable, he or she can compare the results.
Finally, I add that the potential role of irreducible minimums and the difficulty of estimating the irreducible minimum may affect the utility the approach implemented by the Solutions Database and may do so substantially. Whatever the implications of such fact, it is not an argument for use of the standard measures of differences between rates in the manner they are commonly.
Addendum – Cohort Considerations
(April 24, 2011)
In May 2010, I created Cohort Considerations sub-page to MHD to address the tendency for EES figures based on the proportion of persons alive at the beginning of the measurement period tend systematically to be smaller than the figures that would be based on total number of persons in a cohort that that had died by the end of a particular period. Thus, even if the factors influencing the differential mortality never change, calculation of EES figures based on persons living at the beginning of each measurement time frame will suggest that the strengths of such facts is decreasing. See also the Life Table Information document.
References:
1. Scanlan JP. Race and mortality. Society 2000;37(2):19-35 (reprinted in Current 2000 (Feb)): http://www.jpscanlan.com/images/Race_and_Mortality.pdf
2. Victora CG, Vaughan JP, Barros FC, et al. Explaining trends in inequities: evidence from Brazilian child health studies. Lancet 2000;356:1093-1098
3. Scanlan JP. The Misinterpretation of Health Inequalities in the United Kingdom, presented at the British Society for Populations Studies Conference 2006, Southampton, England, Sept. 18-20, 2006: http://www.jpscanlan.com/images/BSPS_2006_Complete_Paper.pdf
4. Scanlan JP. “Inverse equity hypothesis” overlooks important statistical tendencies. Journal Review Dec. 2, 2008 (responding to Victora CG, Vaughan JP, Barros FC, et al. Explaining trends in inequities: evidence from Brazilian child health studies. Lancet 2000;356:1093-1098), which appears at http://journalreview.org/v2/articles/view/11009159.html
5. Scanlan JP. Comparing the size of inequalities in dichotomous measures in light of the standard correlations between such measures and the prevalence of an outcome. Journal Review Jan. 14, 2008 (responding to Boström G, Rosén M. Measuring social inequalities in health – politics or science? Scan J Public Health 2003;31:211-215):
http://journalreview.org/v2/articles/view/12850975.html
6. Geronimus A, Hicken M, Keene D, and Bound J. Weathering and Age Patterns of Allostatic Load Scores Among Blacks and Whites in the United States. Am J Pub Health 2006;96:826-833
7. Scanlan JP. Illustrating whether the relationship between race and allostatic load scores increases with age. Journal Review July 24, 2008 (responding to Geronimus A, Hicken M, Keene D, and Bound J. Weathering and age patterns of allostatic load scores among blacks and whites in the United States. Am J Pub Health 2006;96:826-833): http://journalreview.org/v2/articles/view/16380565.html
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