In
statistics, Cohen's ''h'', popularized by
Jacob Cohen, is a measure of distance between two proportions or
probabilities
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speakin ...
. Cohen's ''h'' has several related uses:
* It can be used to describe the difference between two proportions as "small", "medium", or "large".
* It can be used to determine if the difference between two proportions is "
meaningful
Semantics (from grc, σημαντικός ''sēmantikós'', "significant") is the study of reference, meaning, or truth. The term can be used to refer to subfields of several distinct disciplines, including philosophy, linguistics and comput ...
".
* It can be used in calculating the
sample size
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...
for a future study.
When measuring differences between proportions, Cohen's ''h'' can be used in conjunction with
hypothesis testing
A statistical hypothesis test is a method of statistical inference used to decide whether the data at hand sufficiently support a particular hypothesis.
Hypothesis testing allows us to make probabilistic statements about population parameters.
...
. A "
statistically significant
In statistical hypothesis testing, a result has statistical significance when it is very unlikely to have occurred given the null hypothesis (simply by chance alone). More precisely, a study's defined significance level, denoted by \alpha, is the p ...
" difference between two proportions is understood to mean that, given the data, it is likely that there is a difference in the population proportions. However, this difference might be too small to be meaningful—the statistically significant result does not tell us the size of the difference. Cohen's ''h'', on the other hand, quantifies the size of the difference, allowing us to decide if the difference is meaningful.
Uses
Researchers have used Cohen's ''h'' as follows.
* Describe the differences in proportions using the
rule of thumb criteria set out by Cohen.
Namely, ''h'' = 0.2 is a "small" difference, ''h'' = 0.5 is a "medium" difference, and ''h'' = 0.8 is a "large" difference.
* Only discuss differences that have ''h'' greater than some threshold value, such as 0.2.
* When the sample size is so large that many differences are likely to be statistically significant, Cohen's ''h'' identifies "meaningful", "
clinically meaningful", or "practically significant" differences.
Calculation
Given a probability or proportion ''p'', between 0 and 1, its
arcsine transformation is
:
Given two proportions,
and
, ''h'' is defined as the difference between their arcsine transformations.
Namely,
:
This is also sometimes called "directional ''h''" because, in addition to showing the magnitude of the difference, it shows which of the two proportions is greater.
Often, researchers mean "nondirectional ''h''", which is just the absolute value of the directional ''h'':
:
In
R, Cohen's ''h'' can be calculated using the
ES.h
function in the
pwr
package or the
cohenH
function in the
rcompanion
package
Interpretation
Cohen
provides the following descriptive interpretations of ''h'' as a
rule of thumb:
* ''h'' = 0.20: "small effect size".
* ''h'' = 0.50: "medium effect size".
* ''h'' = 0.80: "large effect size".
Cohen cautions that:
Nevertheless, many researchers do use these conventions as given.
Sample size calculation
See also
*
Estimation statistics
Estimation statistics, or simply estimation, is a data analysis framework that uses a combination of effect sizes, confidence intervals, precision planning, and meta-analysis to plan experiments, analyze data and interpret results. It complement ...
*
Clinical significance In medicine and psychology, clinical significance is the practical importance of a treatment effect—whether it has a real genuine, palpable, noticeable effect on daily life.
Types of significance Statistical significance
Statistical significance ...
*
Cohen's d
In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, th ...
*
Odds ratio
An odds ratio (OR) is a statistic that quantifies the strength of the association between two events, A and B. The odds ratio is defined as the ratio of the odds of A in the presence of B and the odds of A in the absence of B, or equivalently (du ...
*
Effect size
In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the ...
*
Sample size determination
Sample size determination is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences about a populatio ...
References
{{Experimental design
Effect size
Statistical hypothesis testing
Medical statistics
Clinical research
Clinical trials
Biostatistics
Sampling (statistics)