^{†}

This paper is an extended version of paper presented at the Ninth International Symposium on Recent Advances in Environmental Health Research, Jackson, MS, USA; 16–19 September 2012.

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

The informational odds ratio (IOR) measures the post-exposure odds divided by the pre-exposure odds (

A useful measure of association is the informational odds ratio (IOR) [

A key advantage of IORs is that the Mantel-Haenszel adjusted ratio estimates are collapsible (

Prior to conducting a study it is important to determine how large a sample is needed to be reasonable confident that estimates are precise and suitable for answering

A 2 × 2 contingency table.

Disease → ↓Exposure | D | Total | |
---|---|---|---|

E | a = 2,352 | b = 1,600 | e = 3,952 |

c = 912 | d = 1,600 | f = 2,512 | |

Total | g = 3,264 | h = 3,200 | i = 6,464 |

The IOR is computed from the above 2 × 2 contingency table as:

There exist two types of error in classical statistical hypothesis testing [_{0}_{A}_{0}_{0}_{0}_{0}_{0}_{0 }

Power and sample size for IORs may be computed by a simple rearrangement of the general formulas for marginal risk ratios [_{1} = proportion of diseased individual who are exposed, p_{0} = proportion of non-diseased individuals who are exposed, r = ratio of non-diseased to diseased individuals, z_{α/2} = 100(1 − α/2) centile of the standard normal distribution, Z_{β} = the standard normal deviate corresponding to β = (1 − power), it follows that Z_{β} = [n·(p_{1} − p_{0})^{2}·r/(r + 1)·ξ·(1 − ξ)]^{1/2} − Z_{α/2} and n = (Z_{α/2} + Z_{β})^{2}·ξ·(1 − ξ)·(r + 1)/(p_{1} − p_{0})^{2}·r, where ξ = (p_{1} + r·p_{0})/(1 + r), and p_{1} = p_{0}·IOR. Power then equals the probability that an observation from the standard normal distribution is less than or equal to Z_{β}. The above formulas assume a log-normal distribution for IOR and the use of a robust variance estimate for the logarithm of IOR based on the delta method [

Assuming an equal number of diseased (n_{1} = 100) and non-diseased (n_{0} = 100) individuals, the plot in _{0} ranging from 0.01 to 0.10. For example, when p_{0} = 0.04, the power to detect an IOR of at least 4.0 equals 80.7% at the α = 0.05 level of statistical significance. In

Power for IOR by proportion of non-diseased individuals who are exposed.

(Alpha = 0.05; No. non-diseased Pts. = 100; non-diseased: diseased ratio = 1.0).

Sample size for IOR by power.

(Alpha = 0.05; proportion of non-diseased who are exposed = 0.10; non-diseased: diseased ratio = 1.0).

An important property of adjusted IORs is their collapsibility and interpretability as an outcome measure of information gained after knowing exposure status (

Based on the mirror relationship of IORs and RRs as marginal measures of association, the formulas used to compute power and sample size for RRs may be simply rearranged and applied to IORs. This is a particularly useful feature in practice given the availability of software for computing the power and sample size of RR estimates.

The power and sample size formulas for IOR are based on asymptotic statistics and only should be used when the sample size is reasonably large and the sampling distribution for log (IOR) is approximately Gaussian. Similar to RRs, IORs are upwardly biased and actual power may be lower than the estimated one, at least for small sample sizes. Furthermore, the methods described for computing power and sample size apply to the simple case of unadjusted IORs and must be modified accordingly for more complex applications.

The author kindly thanks the Center for Health Disparities Research for salary support and the flexibility to explore new statistical methods.

The author declares no conflict of interest.