The authors have declared that no competing interests exist.
Analyzed the data: JO MLB. Wrote the paper: JO MLB.
Sample size calculations are an important part of research to balance the use of resources and to avoid undue harm to participants. Effect sizes are an integral part of these calculations and meaningful values are often unknown to the researcher. General recommendations for effect sizes have been proposed for several commonly used statistical procedures. For the analysis of
Sample size calculations are an integral part of scientifically useful and ethical research
When the researcher is interested in
A problem arises when using the effect size
It is common in randomised controlled trials and case-control studies to fix one of the marginal probabilities in the
The aims of this paper are to demonstrate: (1) the equivalence of effect size measures for
The two-way classification or contingency table is a common method for summarising the relationship between two binary variables, say
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π00 | π01 | π0+ | |
π10 | π11 | π1+ | |
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π+0 | π+1 | 1.0 |
In this formulation,
There are many association measures applicable to
For the random sample
So, Pearson's correlation coefficient for binary random variables
Since
For the analysis of contingency tables, in general (not just the
Simple arithmetic demonstrates the equivalence of
The odds ratio for the association between
Therefore, when the marginal probabilities are fixed, the odds ratio can be computed directly from
It is clear from the above formula that the odds ratio will be greater than one (or less than one) precisely when the joint probability
Although mathematically unattractive, it is clear the odds ratio can then be computed from
When the marginal probabilities are fixed constants,
These bounds are due to all cell probabilities being non-negative and the relationship of
Importantly, odds ratios are not bounded with possible values of
In many practical instances, the marginal probabilities are not equal, making the full range of values for
As an alternative to Cohen's recommendations, increments of
Additionally, the maximal odds ratio, and therefore most anti-conservative, occurs when the marginal probabilities are equal, as expected. Below is the maximum attainable odds ratio for equal margins
It is important to note that when
π1+ | |||||||||
Odds Ratio | 0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 |
1.22 | 8168 | 4688 | 3646 | 3254 | 3188 | 3386 | 3948 | 5282 | 9576 |
1.86 | 724 | 436 | 354 | 330 | 338 | 374 | 454 | 632 | 1188 |
3.00 | 200 | 128 | 110 | 108 | 116 | 134 | 170 | 246 | 480 |
Interestingly, Haddock et al.
This approach can also be applied to the relative risk and risk difference. If
Therefore, recommendations can also be derived for relative risk and are identical to those given for the odds ratio above. This result is expected as the odds ratio converges to the relative risk as the incidence rate approaches
Instead of comparing the risk between two groups as a ratio, it is sometimes useful to compare their differences
where
Alternatively, the
When the allocation ratio is 1:1, this formula simplifies to
This paper was motivated by a reanalysis of passing distances for motor vehicles overtaking a bicyclist
The relevant observed data from Walker
No Helmet | Helmet | Total | |
Safe | 0.491 | 0.462 | 0.953 |
Unsafe | 0.021 | 0.026 | 0.047 |
Total | 0.512 | 0.488 |
We present a demonstration that many contingency table correlation measures are equivalent for the
The use of effect size recommendations should be avoided in situations in which clinical or practical differences are known. However, they can help the researcher balance between overly large or overly small sample size calculations when such information is unknown. In these situations, conservative estimates for odds ratio effect sizes can be derived from only the allocation ratio leading to a general result and, when a 1:1 allocation is chosen for optimal power, odds ratios of
(SAS)
The authors would like to thank Warren May, David Warton and Jakub Stoklosa for their help in the preparation of this manuscript.