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Effect Size
In statistics, an effect size is a measure of the magnitude of a treatment effect. It is an indicator of how important an obtained effect is. Unlike statistical significance tests, effect size does not depend on the sample size of an underlying study. It is helpful to report the effect size, not just the statistical significance, when assessing the effectiveness of a specific intervention in medical studies as well as studies in other sciences. It has been also widely used in meta-analysis, which combines and compares estimates from different but relevant studies.
In medical studies, such as comparison of a new treatment with other traditional ones, the following question is often asked: How well does the new treatment work? In answering this question, the researchers are actually trying to quantify the difference between the effect of the new treatment and those of the traditional ones. Similar things happen in social studies and studies in educational and behavioral sciences. Effect size is a simple way of answering the question, and it has many advantages over the use of tests of statistical significance alone. Effect size measures directly the size of the difference rather than confounding this with the sample size of the study. It is easy to calculate and to interpret, and it can be applied to any measured outcome of medical, social, and educational sciences to quantify the effectiveness of a particular intervention in comparison with others.
Effect Size for Two Independent Groups with Continuous Outcomes
Let us consider comparing the outcome of two groups, the experimental group (the one for which a new treatment is going to be applied) and the control group (the one for which a traditional treatment is going to be applied). The outcome of the study is a kind of continuous measurement. The effect size in such a case is defined as the standardized difference in means between the two groups. In other words,
It is very natural to take the difference of two group means when comparing the two groups of measurements. The standard deviation in the denominator, which is a measure of the spread of a set of values, is to standardize this difference. The same value of difference may represent totally different meanings when the standard deviations are different. It could be explained as a huge difference if the standard deviation is small, such that the two groups of values are completely separated; whereas if the corresponding standard deviation is large, the two sets of values might be well overlapped and the same value of difference might mean just nothing. The difference in means is standardized when it is divided by the standard deviation. In practice, however, the standard deviation is not known. It can be estimated either from the control group or from a pooled value of both groups.
The above-defined effect size is exactly equivalent to the z score of a standard normal distribution. For example, an effect size of 1 means that the average of the experimental group is 1 standard deviation higher than that of the control group. With the assistance of a graph of standard normal distribution curve, one can observe that the average of the experimental group, which is 1 standard deviation higher than that of the control group, is indeed the 84th percentile of the control group. In other words, 84% of the measurements of the control group are below the average of the experimental group. The value of 84% is calculated from the standard normal distribution as the probability that the standard normal random variable is less than or equal to 1. In case the effect size takes different values, the underlying effect size will replace the value 1 in the calculation. Another percentage rather than 84% will be obtained correspondingly. This provides an idea of how the two groups overlap with each other.
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