Prevalence odds ratio from a prevalence ratio and the unexposed prevalence

Converts a prevalence ratio into the prevalence odds ratio that logistic regression reports for the same cross-sectional data, given the prevalence among the unexposed. The cross-product a d / (b c) of the two-by-two table gives the same value. The factor (1 minus P_0) / (1 minus P_1) exceeds 1 whenever P_1 exceeds P_0 and falls below 1 whenever P_1 is lower, so the POR lies further from 1 than the PR whenever the two prevalences differ.

Signature

P_1 = PR * P_0; POR = PR * (1 - P_0) / (1 - P_1)
Inputs
InputsDefinitionUnit
PRCrude prevalence ratio of the exposed to the unexposed group, below 1 / P_0ratio
P_0Proportion of unexposed people who have the condition, above zero and below 1proportion
Output
P_1Proportion of exposed people who have the condition, implied by the PR and P_0, below 1proportion
POROdds of the condition among the exposed divided by the odds among the unexposedratio, zero or above

Function

Prevalence ratio function for cross-sectional comparisons and attributable cost inputs

Maps the prevalence of a condition among exposed people and among unexposed people, measured at the same time, to their ratio, and carries that ratio into the prevalence odds ratio, a population attributable fraction of prevalent cases and subgroup prevalences for cost-of-illness and budget impact work. Applying the attributable fraction to aggregate annual costs is the top-down attributable burden HE-FM-COI-002, and the attack rate ratio HE-FM-ATR-003 has the same ratio form for new cases in an outbreak. Survey-weighted prevalence and the link between prevalence, incidence and duration are HE-FM-XSD-001 and HE-FM-XSD-002. Notation follows the Prevalence Ratio article.

Try this function

Implementations

  • Excel

    Prevalence odds ratio from a prevalence ratio in Excel

    With the PR in a cell named PrevRatio and the unexposed prevalence in PrevUnexposed, Excel returns the POR.

    =PrevRatio*(1-PrevUnexposed)/(1-PrevRatio*PrevUnexposed)

Assumptions

  • Crude ratios from one table in the odds ratio conversion

    The identity holds exactly for crude ratios from the same two-by-two table. Adjusted estimates from a log-binomial model and a logistic model fitted to the same data need not satisfy it, because controlling for confounding is not equivalent for the two measures.

  • Implied exposed prevalence below one in the odds ratio conversion

    The product of the PR and P_0 must be below 1, otherwise the odds among the exposed are undefined. A PR from one population combined with an unexposed prevalence from another can break this condition.

Worked examples

  • Prevalence odds ratio when the condition is common

    With the article's PR of 2.00 and unexposed prevalence of 0.15, the exposed prevalence is 0.30 and the POR is about 2.4286, about 21% above the PR. The cross-product of the survey table, 214,200 divided by 88,200, gives the same value.

    PR = 2.00; P_0 = 0.15; P_1 = 0.30; POR = 2.4286

  • Prevalence odds ratio when both prevalences are low

    With a PR of 2.00 and an unexposed prevalence of 0.01, the exposed prevalence is 0.02 and the POR is about 2.0204, so the two measures nearly agree, as the article shows for prevalences of 2% and 1%.

    PR = 2.00; P_0 = 0.01; P_1 = 0.02; POR = 2.0204

  • Prevalence odds ratio at a prevalence ratio of 1

    When the prevalences are equal the conversion factor is 1, so a PR of 1 gives a POR of 1 at any unexposed prevalence, a limiting case that checks the implementation.

    PR = 1; P_0 = 0.15; P_1 = 0.15; POR = 1.0000

Common errors

  • Reporting a prevalence odds ratio as a prevalence ratio

    With the survey prevalences of 0.30 and 0.15, the POR of 2.43 is about 21% above the PR of 2.00. Read as a relative prevalence, or as a relative risk beside true relative risks when priorities are set, it exaggerates the association, as Barros and Hirakata warn for frequent outcomes in cross-sectional studies.

  • Falling back to logistic regression when a prevalence ratio model fails

    Log-binomial models that estimate an adjusted PR may fail to converge, most often with a continuous covariate, several categorical covariates or a high prevalence. Switching silently to logistic regression changes the reported measure from a PR to a POR. Poisson regression with a robust sandwich variance estimates the PR directly.

Sources

  • Tamhane and colleagues on the prevalence odds ratio identity

    Tamhane AR, Westfall AO, Burkholder GA, Cutter GR. Prevalence odds ratio versus prevalence ratio: choice comes with consequences. Statistics in Medicine. 2016;35(30):5730-5735. Table 3: POR = ad/bc and POR = [1 minus c/(c+d)] / [1 minus a/(a+b)] x PR; Discussion: the two are closer when the outcome is rare.

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  • Zocchetti and colleagues on prevalence rate ratios and odds ratios

    Zocchetti C, Consonni D, Bertazzi PA. Relationship between prevalence rate ratios and odds ratios in cross-sectional studies. International Journal of Epidemiology. 1997;26(1):220-223. Abstract: the POR is always further from the null value than the prevalence rate ratio, and the discrepancy depends much more on the prevalence of the condition than on that of the exposure.

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  • Barros and Hirakata on models that estimate the prevalence ratio directly

    Barros AJD, Hirakata VN. Alternatives for logistic regression in cross-sectional studies: an empirical comparison of models that directly estimate the prevalence ratio. BMC Medical Research Methodology. 2003;3:21. Background (the odds ratio can strongly overestimate the PR for frequent outcomes; controlling for confounding is not equivalent) and Methods (log-binomial convergence, Poisson regression with robust variance).

    View source →

Canonical Identity

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