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Efficient estimation of relative risk, odds ratio and their logarithms for rare events

Luis Mendo

Abstract

Sequential estimators are proposed for the relative risk, odds ratio, log relative risk or log odds ratio of a dichotomous attribute in two populations. The estimators take the same number of observations from each population, and guarantee that the relative mean-square error for the relative risk or odds ratio, or the mean-square error for their logarithmic versions, is less than a given target. The efficiency of the estimators, defined in terms of the Cramér-Rao bound, is high when the considered attribute is rare or moderately rare.

Efficient estimation of relative risk, odds ratio and their logarithms for rare events

Abstract

Sequential estimators are proposed for the relative risk, odds ratio, log relative risk or log odds ratio of a dichotomous attribute in two populations. The estimators take the same number of observations from each population, and guarantee that the relative mean-square error for the relative risk or odds ratio, or the mean-square error for their logarithmic versions, is less than a given target. The efficiency of the estimators, defined in terms of the Cramér-Rao bound, is high when the considered attribute is rare or moderately rare.

Paper Structure

This paper contains 18 sections, 10 theorems, 91 equations, 9 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Estimation procedure
  3. Average number of input pairs
  4. For relative risk and log relative risk
  5. For odds ratio and log odds ratio
  6. Estimation efficiency
  7. Conclusions
  8. Proofs
  9. Proof of Proposition \ref{['prop: inner RR LRR']}
  10. Proof of Proposition \ref{['prop: inner OR LOR']}
  11. Proof of Theorem \ref{['theo: MSE']}
  12. Proof of Proposition \ref{['prop: outer RR LRR']}
  13. Proof of Proposition \ref{['prop: E max cota RR LRR']}
  14. Proof of Proposition \ref{['prop: outer OR LOR']}
  15. Proof of Theorem \ref{['theo: effic RR LRR OR LOR']}
  16. ...and 3 more sections

Key Result

Proposition 1

Algorithm algo: RR LRR, inner terminates with probability $1$. Its output has $\Pr[Y=1] = p$, $\Pr[Y=0] = 1-p$, with $p$ given by eq: p RR LRR. In addition, $\blacktriangleleft$$\blacktriangleleft$

Figures (9)

  • Figure 1: Sampling efficiency factor for RR estimation
  • Figure 2: Estimation efficiency for RR
  • Figure 3: Estimation efficiency for LRR
  • Figure 4: Estimation efficiency for OR
  • Figure 5: Estimation efficiency for LOR
  • ...and 4 more figures

Theorems & Definitions (10)

  • Proposition 1
  • Proposition 2
  • Theorem 1
  • Proposition 3
  • Proposition 4
  • Proposition 5
  • Theorem 2
  • Proposition 6
  • Theorem 3
  • Theorem 4