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Revisiting the relationship between baseline risk and risk under treatment

Hao Wang, Jean-Pierre Boissel and Patrice Nony*

Emerging Themes in Epidemiology 2009, 6:1  doi:10.1186/1742-7622-6-1

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Estimation of treatment effects across a range of baseline rates should not be based on assumptions of either constant relative risks or constant odds ratios

James Scanlan   (2012-01-12 20:11)  James P. Scanlan, Attorney at Law email

Wang et al.[1] note that it is generally accepted that the relationship between baseline risk and risk under treatment is linear (i.e., that the relative risk will be constant) and that absolute benefits will be greatest among high-risk patients, but that there exists an alternative assumption of a curvilinear relationship based on the odds ratio. While observing that there is no theoretical support for either approach, the authors conclude that the assumption of a curvilinear benefit reflected by a constant odds ratio will be more useful for estimating risk reductions.

The authors are correct in questioning the assumption of constant relative risk. The assumption is fundamentally illogical for the simple reason that it is impossible for a factor that causes equal proportionate changes in outcome rates for two groups with different baseline rates of experiencing the outcome to cause equal proportionate changes in the opposite outcome. That is, for example, if Group A has a baseline rate of 5% and Group B has a baseline rate of 10%, a factor that reduces the two rates by equal proportionate amounts, say 20% (from 5% to 4% and from 10% to 8%), would necessarily increase the opposite outcome by two different proportionate amounts (95% increased to 96%, a 1.05% increase; 90% to 92%, a 2.2% increase). And since there is no more reason to expect that two group would undergo equal proportionate changes in one outcome than there is to expect they would undergo equal proportionate changes in the opposite outcome, there is no reason to expect that the two groups would undergo equal proportionate changes in either outcome.

For reasons inherent in the shapes of normal distributions of factors associated with experiencing or avoiding an outcome, it is more reasonable to expect that a treatment that reduces an outcome rate will tend to cause a larger proportionate decrease in that outcome for groups with lower base rates while causing a larger proportionate increase in the opposite outcome for other groups.[2-5] The same aspects of normal distributions that underlie these patterns provide a theoretical basis for expecting a curvilinear relationship between baseline risk and risk under treatment. But the relationship is based, not on the odds ratio, but on the differences between means of the hypothesized underlying risk distributions reflected in the observed risk reduction for a particular baseline rate.[4,5]

A comparison of the estimated absolute risk reductions under that approach compared to the constant risk ratio and constant odds ratio approaches may be found in Table 3 of reference 5.

References:

1. Wang H, Boissel JP, Nony P. Revisiting the relationship between baseline risk and risk under treatment. Emerging Themes in Epidemiology 2009;6:1: http://www.ete-online.com/content/6/1/1

2. Scanlan JP. Race and mortality. Society 2000;37(2):19-35: http://www.jpscanlan.com/images/Race_and_Mortality.pdf

3. Scanlan JP. Divining difference. Chance 1994;7(4):38-9,48: http://jpscanlan.com/images/Divining_Difference.pdf

4. Scanlan JP. Interpreting Differential Effects in Light of Fundamental Statistical Tendencies, presented at 2009 Joint Statistical Meetings of the American Statistical Association, International Biometric Society, Institute for Mathematical Statistics, and Canadian Statistical Society, Washington, DC, Aug. 1-6, 2009: PowerPointPresentation: http://www.jpscanlan.com/images/Scanlan_JSM_2009.ppt; Oral Presentation: http://www.jpscanlan.com/images/JSM_2009_ORAL.pdf

5. Subgroup Effects sub-page of Scanlan��s Rule page of jpscanlan.com: http://www.jpscanlan.com/scanlansrule/subgroupeffects.html

Competing interests

None

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