Citation: Olivier J, Bell ML (2013) Effect Sizes for 2×2 Contingency Tables. PLoS ONE 8(3):
e58777.
https://doi.org/10.1371/journal.pone.0058777
Editor: Fabio Rapallo,
University of East Piedmont, Italy
Received: December 19, 2012; Accepted: February 6, 2013; Published: March 7, 2013
Copyright: © 2013 Olivier and Bell. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The authors have no support or funding to report.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Sample size calculations are an integral part of scientifically useful and ethical research [1]. A study which is too small may not answer the research question, wasting resources and potentially putting participants at risk for no purpose [2]. Studies which are too large can also waste resources and expose participants to the potential harms of research needlessly, as well as delaying results and their translation into practice. The computation of sample size a priori is usually dependent upon predetermined values for power and level of significance, an estimate of the expected variability in the sample and an effect size of practical or clinical importance. By convention, the choice of power and level of significance is usually at least 
and no more than 
respectively. When a practically important effect size is unknown, there are several recommendations in the literature to guide the researcher. In his seminal paper, Cohen [3] gives operationally defined small, medium and large effect sizes for various, common significance tests. The use of effect size recommendations should not replace differences of clinical or practical importance [4] and may not be appropriate for all disciplines. In basic science research, for example, large effect sizes by Cohen's criteria are common and, therefore, require small sample sizes. On the other hand, clinical and epidemiological research often deals with small effect sizes and often requires large, population-based studies. While there are some approaches to estimating a minimum important effect [5], there are instances where this information is simply not known. Thus, effect size recommendations assist with the balance between overly small and overly large sample sizes.
When the researcher is interested in 
contingency tables, a common measure of effect size is 
which, in this instance, is equivalent to Pearson's correlation coefficient [6]. Cohen [3] recommends effect sizes of 
and 
for small, medium and large effect sizes respectively and are identical to his recommendations for the correlation coefficient. Although Cohen [3] denotes this statistic as 
, much of the literature uses 
[6]–[10] and the remainder of this manuscript follows this convention. To support his recommended effect sizes for correlation coefficients, Cohen [11] chose equivalent values for the difference in two means through the connection with point biserial correlation. Additionally, 
is applicable to logistic regression since it can be converted to an odds ratio (
) when the row (or column) marginal probabilities of the 
table are fixed. For example, when the marginal probabilities are uniform (i.e., 
for row and column probabilities), Cohen's recommended effect sizes are equivalent to odds ratios of 
and 
. It will be demonstrated that the connection between the odds ratio and 
is largely dependent on the marginal probabilities and these 
values should not be used in general.
A problem arises when using the effect size 
for 
tables as the full range of correlation coefficients are only possible under very restrictive circumstances and are not justified in general [12]. On the other hand, odds ratios are valid effect size measures that are not constrained by the marginal probabilities. Ferguson [10] recommends small, medium, and large odds ratio effect sizes of 
and 
, but urges caution in their use as they are not “anchored” to Pearson's correlation coefficient. Although many have pointed out problems with 
as an association measure and advocate the use of odds ratios as an alternative, effect size recommendations for odds ratios do not exist in general.
It is common in randomised controlled trials and case-control studies to fix one of the marginal probabilities in the 
table as it directly relates to the ratio of participant allocation. For instance, a marginal probability of 
corresponds to a 1:1 case-control ratio while a 2:1 ratio is a marginal probability of 
(or equivalently 
for 1:2).
The aims of this paper are to demonstrate: (1) the equivalence of effect size measures for 
contingency tables, in particular the relationship between 
and the odds ratio; (2) that recommended odds ratio effect sizes can be derived from Cohen's work using the maximum value of 
as a guideline for fixed marginal probabilities; (3) the shortcomings of 
and the strength of the odds ratio as an effect size measure; and (4) that conservative odds ratio effect size recommendations can be derived without relying on fixed margins. We provide an example that investigates the association between helmet wearing by bicyclists and overtaking distance by automobiles.
Equivalence of Effect Size Measures for 2×2 Contingency Tables
Contingency tables
The two-way classification or contingency table is a common method for summarising the relationship between two binary variables, say 
and 
. Table 1 gives the joint probability distribution of 
and 
when their individual outcomes are from the set 
.
In this formulation, 
, for 
, is the joint probability of 
and 
, 
is the marginal probability of 
, and 
is the marginal probability of 
. Under an assumption of independence between 
and 
, the product of the marginal probabilities equals the cell probabilities, i.e., 
. Alternatively, the 
table could be represented by the frequency of observations so that 
where 
. Similarly, the marginal frequencies are 
and 
. Note that 
is assumed to be the population proportion as the focus of this paper is the use of effect sizes as a planning tool and not statistical inference per se. In a case-control study, for example, 
may indicate the presence or absence of disease while 
is an indication of exposure. Thus, 
would represent the joint probability of being diseased and exposed.
Effect size 
and Equivalences for 
Tables
There are many association measures applicable to 
tables which, with the exception of the odds ratio and relative risk, are equivalent or similar to 
. The equivalence of some of these association measures is outlined below.
For the random sample 
, Pearson's correlation coefficient is
where 
and 
are the sample means of the 
and 
respectively. Although used primarily as a measure of linear association, Pearson's correlation coefficient can be applied to binary variables and is often given the notation 
. For the 
table case, we get
So, Pearson's correlation coefficient for binary random variables 
and 
is
Since 
under the hypothesis of independence, 
can be interpreted as measuring the departure from independence between 
and 
. Note that Cramér's 
is equivalent to this equation for the 
table case [11] as well as the square root of Goodman and Kruskal's 
[13].
For the analysis of contingency tables, in general (not just the 
table case) the effect size formula for 
total cells is
where 
and 
are cell probabilities under the null and alternative hypotheses respectively. Note that 
is related to the usual chi-square statistic 
by 
and is sometimes called the contingency coefficient. Using this formula, Cohen [3] recommends 
and 
for small, medium and large effect sizes. Making note that 
is the probability of each cell (
) and 
is the cell probability under an independence assumption (so that 
), we can then write the effect size formula for the 
table as follows
Simple arithmetic demonstrates the equivalence of 
with 
. The 
function is used to give the appropriate sign since the chi-square statistic is inherently non-directional.
The relationship of 
to the odds ratio
The odds ratio for the association between 
and 
is 
. When the marginal probabilities are held constant and the cell probability 
is known, the remaining cell probabilities can be written as
Therefore, when the marginal probabilities are fixed, the odds ratio can be computed directly from 
, which can then be expressed as
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 
is greater (or less) than expected under an assumption of independence, i.e., 
. Additionally, the formula for 
can be rearranged to solve for 
, i.e.,
Although mathematically unattractive, it is clear the odds ratio can then be computed from 
, 
, and 
. Note that when 
(i.e., no correlation), we get 
(i.e., 
and 
are independent) and the odds ratio is 
. When 
, the term 
is then a measure of the departure from independence.
Maximum 
and Modified Effect Sizes
When the marginal probabilities are fixed constants, 
is an increasing linear function of 
. Further, 
is bounded by
These bounds are due to all cell probabilities being non-negative and the relationship of 
with the other cell probabilities given above. As a result, 
is bounded as well and attains its maximum when 
. Using the upper bound of the above inequality, it can be shown that
where 
to ensure 
. It is clear from the formula for 
that the full range of correlation coefficients, i.e., 
, is attainable only when the marginal probabilities are equal, i.e., 
or 
. This has an intuitive appeal as perfect correlation for two binary variables is only possible when two cell probabilities are zero. For example, when all observations are in either the 
or 
cells, 
. However, it would appear highly unlikely both marginal probabilities will be equal in practice. For example, in a 1:1 case-control study with mortality as the primary outcome, half of all patients would need to die for perfect correlation to be possible. On the other hand, if 
of all patients die, the maximum correlation possible is 
which is near a medium recommended effect size. So, in this situation, all estimates of 
, computed from observed proportions, are bounded by
Importantly, odds ratios are not bounded with possible values of 
as 
varies on the interval 
. In fact, as 
approaches 
, the 
increases without bound. Figure 1 demonstrates this relationship. Importantly, this indicates 
has serious limitations as a measure of association and that these limitations are not applicable to the odds ratio.
Effect Sizes Relative to 
In many practical instances, the marginal probabilities are not equal, making the full range of values for 
impossible with the potential of making Cohen's recommended effect sizes unusable for 
tables. Although not equivalent to perfect correlation, 
can be interpreted as the maximum possible correlation given the marginal probabilities. In fact, 
/
has been proposed as an association measure with the interpretation as the proportion of observed correlation relative to the maximum attainable with fixed marginal probabilities [7], although the researcher is cautioned when the marginal probabilities diverge [6]. Note that 
is not equivalent to Cohen's similarity/agreement measure 
. However, 
suffers from the same boundary problems as 
and the two are equivalent when scaled to their maximum values, i.e., 
/
/
, making the two measures similar [6].
Recommended effect sizes in terms of the odds ratio
As an alternative to Cohen's recommendations, increments of 
can be related to the odds ratio, say 
, where 
. Note that values of 
or 
coincide with Cohen's usual recommendations when 
. The relationship between 
and the odds ratio can be simplified by choosing marginal probabilities for commonly used participant allocations. As an example, Figures 2 and 3 demonstrate the relationship between 
and odds ratios for 
, 
and 
for 1:1 and 1:2 allocations respectively. Note that the minimal odds ratios, and therefore most conservative when used to compute sample size, occur when 
tends to 
. Although the odds ratio does not exist when 
, the limit exists and is
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 
for increments 
of 
,
It is important to note that when 
, as is often true for case-control studies where cases are harder to identify or enrol than controls, the minimal odds ratio will be smallest for evenly allocated studies, i.e., 
. Further, it is generally recommended to use 1:1 allocation as it is the most statistically efficient ratio, i.e., maximum power for a fixed overall sample size. So, odds ratios of 
and 
can be used as small, medium and large effect sizes without assumptions regarding marginal probabilities. Sample sizes computed using these odds ratios for 1:1 allocation are given in Table 2 for 
power and 
level of significance. A SAS macro that will compute sample sizes from given marginal probabilities for small, medium and large odds ratios has been provided as a supplementary file.
Interestingly, Haddock et al. [12] as a rule of thumb consider odds ratios greater than 
large effect sizes, although there is no clear justification given. In a situation where an allocation ratio other than 1:1 is used, recommended odds ratios can be computed directly using the above formula. These results are also applicable for other values of 
through its complement 
. This is equivalent to swapping the columns (or rows) and the researcher should be aware the recommended odds ratio effect sizes are now the reciprocals of those above, i.e., 
and 
for small, medium and large respectively.
This approach can also be applied to the relative risk and risk difference. If 
is taken as the grouping variable and 
as the outcome, the relative risk is 
. Simple substitution of 
and the marginal probabilities 
and 
results in a relative risk identical to 
for 
, i.e.,
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 [14]. Again taking 
as the grouping variable and 
as the outcome, the risk difference can be written as
where 
to ensure 
as above. It is clear from the numerator in this representation that 
is a measure of the departure from independence, i.e., 
. Simple substitution of 
into 
yields
where the subscript 
is used to distinguish between risk difference formulae. This formula can be simplified somewhat for 1:1 allocations, i.e., 
; however, a general result independent of the marginal probabilities is clearly not possible in this instance as 
and therefore 
.
Alternatively, the 
formula can be solved for 
and compared to previously given odds ratio recommendations. In terms of 
and 
, we get
When the allocation ratio is 1:1, this formula simplifies to 
which has a form identical to Yule's 
[15]. So, Ferguson's [10] odds ratio recommendations of 
and 
therefore correspond to proportions of maximum correlation of 
and 
. This suggests Ferguson's recommendations have the potential to be anti-conservative from a sample size viewpoint.
Example
This paper was motivated by a reanalysis of passing distances for motor vehicles overtaking a bicyclist [16]. One of the primary results of this study was a significant association between helmet wearing and less overtaking distance, supporting a theory of risk perception for motor vehicle drivers directed towards bicyclists. Prior to collecting data, Walker [16] reported computing a sample size of 
overtaking manoeuvres based on a 
fixed effects factorial ANOVA for a small effect size 
, 
level of significance and 
power. The factors for this study were helmet wearing (2 levels) and bicycle position relative to the kerb (5 levels). It has been noted, however, that passing distances are often recommended and sometimes legislated to one metre or more [17]. So, passing manoeuvres of at least a metre are considered safe and less than a metre unsafe, with the implication that large differences in passing distance are unimportant beyond one metre in terms of bicycle safety. When compared with helmet wearing, safe/unsafe passing distances can be analysed using a 
table. Since Walker's study was powered at an unusually high level with subsequent increased probability of a type I error, bootstrap standard errors were estimated for more reasonable values for power of 
, 
and 
. Operationally defined small, medium and large effect sizes were also used since a meaningful difference in overtaking distance is unknown.
The relevant observed data from Walker [16] is given in Table 3. The observed marginal proportions here are 
for helmet wearing and 
for unsafe passing manoeuvres. Using the marginal probabilities, the maximum attainable effect size is 
and the estimated correlation is 
. A consequence is the effect size for the association between helmet wearing and safe passing distance is, at best, much less than a small effect size by Cohen's index. The corresponding small, medium and large odds ratio effect sizes using increments of 
are 
and 
for 
and 
. Note that these values are not much greater than the minimal recommended odds ratios mentioned in the previous section, further suggesting the association between safe/unsafe passing distance and helmet wearing is, at best, a small effect size. In fact, the unadjusted odds ratio is 
and non-significant by the chi-square test (
). Conversely, sample sizes for a future study can be computed from the observed probabilities using G*Power for logistic regression with a single binomially distributed predictor for 
and 
power [18] resulting in 
and 
observations for small, medium and large odds ratios. To put these sample size computations into perspective, a future study would need to extend the sampling period by a factor greater than seven to detect a significant association between helmet wearing and safe/unsafe overtaking distance given a small effect size and identical marginal probabilities.
Discussion
We present a demonstration that many contingency table correlation measures are equivalent for the 
case and their use is limited due to constraints created by fixed marginal probabilities. The odds ratio, which is a function of these measures for fixed marginal probabilities, is not problematic, is regularly used in statistical analyses and has a direct application to logistic regression. Recommended odds ratios have been proposed from Cohen's small, medium and large effect sizes for 
relative to the maximum attainable correlation 
. Further, minimal odds ratios can be computed with only knowledge of participant allocation.
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 
and 
correspond to small, medium and large effect sizes.
Acknowledgments
The authors would like to thank Warren May, David Warton and Jakub Stoklosa for their help in the preparation of this manuscript.
Author Contributions
Analyzed the data: JO MLB. Wrote the paper: JO MLB.
References
- 1.
Lewis J (1999) Statistical principles for clinical trials (ICH E9): an introductory note on an inter-national guideline. Statistics in Medicine 18: 1903–1942.
- 2.
Halpern S, Karlawish J, Berlin J (2002) The continuing unethical conduct of underpowered clinical trials. JAMA: The Journal of the American Medical Association 288: 358–362.
- 3.
Cohen J (1992) A power primer. Psychological Bulletin 112: 155–159.
- 4.
Lenth R (2001) Some practical guidelines for effective sample size determination. The American Statistician 55: 187–193.
- 5.
King M (2011) A point of minimal important difference (MID): a critique of terminology and methods. Expert Review of Pharmacoeconomics & Outcomes Research 11: 171–184.
- 6.
Davenport Jr E, El-Sanhurry N (1991) Phi/phimax: review and synthesis. Educational and Psy-chological Measurement 51: 821–828.
- 7.
Ferguson G (1941) The factorial interpretation of test difficulty. Psychometrika 6: 323–329.
- 8.
Guilford J (1965) The minimal phi coefficient and the maximal phi. Educational and Psychological Measurement 25: 3–8.
- 9.
Breaugh J (2003) Effect size estimation: Factors to consider and mistakes to avoid. Journal of Management 29: 79–97.
- 10.
Ferguson C (2009) An effect size primer: A guide for clinicians and researchers. Professional Psychology: Research and Practice 40: 532–538.
- 11.
Cohen J (1988) Statistical Power Analysis for the Behavioral Sciences. Hillsdale, New Jersey: Lawrence Erlbaum Associates.
- 12.
Haddock C, Rindskopf D, Shadish W (1998) Using odds ratios as effect sizes for meta-analysis of dichotomous data: A primer on methods and issues. Psychological Methods 3: 339–353.
- 13.
Agresti A (2002) Categorical Data Analysis. New York: Wiley-interscience.
- 14.
Greenberg R, Daniels S, Flanders W, Eley J, Boring J (1996) Medical Epidemiology. Appleton & Lange.
- 15.
Liebetrau A (1983) Measures of Association, volume 32. Sage Publications, Incorporated.
- 16.
Walker I (2007) Drivers overtaking bicyclists: Objective data on the effects of riding position, helmet use, vehicle type and apparent gender. Accident Analysis & Prevention 39: 417–425.
- 17.
Olivier J (2013) Bicycle helmet wearing is not associated with close overtaking: A re-analysis of Walker 2007. Submitted.
- 18.
Faul F, Erdfelder E, Buchner A, Lang A (2009) Statistical power analyses using G*Power 3.1: Tests for correlation and regression analyses. Behavior research methods 41: 1149–1160.
