Effect Size Formula

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Effect Size

Effect size is a statistical concept that measures the strength of the relationship between two variables on a numeric scale.  For instance, if we have data on the height of men and women and we notice that, on average, men are taller than women, the difference between the height of men and the height of women is known as the effect size.  The greater the effect size, the greater the height difference between men and women will be.  Statistic effect size helps us in determining if the difference is real or if it is due to a change of factors.  In hypothesis testing, effect size, power, sample size, and critical significance level are related to each other.  In Meta-analysis, effect size is concerned with different studies and then combines all the studies into single analysis.  In statistics analysis, the effect size is usually measured in three ways: (1) standardized mean difference, (2) odd ratio, (3) correlation coefficient.

Types of effect size

Pearson r correlation: Pearson r correlation was developed by Karl Pearson, and it is most widely used in statistics.  This parameter of effect size is denoted by r.  The value of the effect size of Pearson r correlation varies between -1 to +1.  According to Cohen (1988, 1992), the effect size is low if the value of r varies around 0.1,  medium if r varies around 0.3, and large if r varies more than 0.5.  The Pearson correlation is computed using the following formula:

Where
r = correlation coefficient
N = number of pairs of scores
∑xy = sum of the products of paired scores
∑x = sum of x scores
∑y = sum of y scores
∑x2= sum of squared x scores
∑y2= sum of squared y scores

Standardized means difference: When a research study is based on the population mean and standard deviation, then the following method is used to know the effect size:

The effect size of the population can be known by dividing the two population mean differences by their standard deviation.

Cohen’s d effect size: Cohen’s d is known as the difference of two population means and it is divided by the standard deviation from the data. Mathematically Cohen’s effect size is denoted by:

Glass’s Δ method of effect size: This method is similar to the Cohen’s method, but in this method standard deviation is used for the second group.  Mathematically this formula can be written as:

Hedges’ g method of effect size: This method is the modified method of Cohen’s d method. Hedges’ g method of effect size can be written mathematically as follows:

Cohen’s f2 method of effect size: Cohen’s f2 method measures the effect size when we use methods like ANOVA, multiple regression, etc.  The Cohen’s f2 measure effect size for multiple regressions is defined as the following:

Where R2 is the squared multiple correlation.

Cramer’s φ or Cramer’s V method of effect size: Chi-square is the best statistic to measure the effect size for nominal data.  In nominal data, when a variable has two categories, then Cramer’s phi is the best statistic use.  When these categories are more than two, then Cramer’s V statistics will give the best result for nominal data.

Odd ratio: The odds ratio is the odds of success in the treatment group relative to the odds of success in the control group.  This method is used in cases when data is binary.  For example, it is used if we have the following table:

What does effect size tell you?

Statistical significance is the least interesting thing about the results. You should describe the results in terms of measures of magnitude – not just, does a treatment affect people, but how much does it affect them.

What is effect size?

Effect size is a quantitative measure of the magnitude of the experimental effect. The larger the effect size the stronger the relationship between two variables.

You can look at the effect size when comparing any two groups to see how substantially different they are. Typically, research studies will comprise an experimental group and a control group. The experimental group may be an intervention or treatment which is expected to effect a specific outcome.

For example, we might want to know the effect of a therapy on treating depression. The effect size value will show us if the therapy as had a small, medium or large effect on depression.

How to calculate and interpret effect sizes

Effect sizes either measure the sizes of associations between variables or the sizes of differences between group means.

Cohen’s d

Cohen’s d is an appropriate effect size for the comparison between two means. It can be used, for example, to accompany the reporting of t-test and ANOVA results. It is also widely used in meta-analysis.

To calculate the standardized mean difference between two groups, subtract the mean of one group from the other (M1 – M2) and divide the result by the standard deviation (SD) of the population from which the groups were sampled.

d of 1 indicates the two groups differ by 1 standard deviation, a d of 2 indicates they differ by 2 standard deviations, and so on. Standard deviations are equivalent to z-scores (1 standard deviation = 1 z-score).

Cohen suggested that d = 0.2 be considered a ‘small’ effect size, 0.5 represents a ‘medium’ effect size and 0.8 a ‘large’ effect size. This means that if the difference between two groups’ means is less than 0.2 standard deviations, the difference is negligible, even if it is statistically significant.

Pearson r correlation

This parameter of effect size summarises the strength of the bivariate relationship. The value of the effect size of Pearson r correlation varies between -1 (a perfect negative correlation) to +1 (a perfect positive correlation).

According to Cohen (1988, 1992), the effect size is low if the value of r varies around 0.1, medium if r varies around 0.3, and large if r varies more than 0.5.

Why report effect sizes?

The p-value is not enough

A lower p-value is sometimes interpreted as meaning there is a stronger relationship between two variables. However, statistical significance means that it is unlikely that the null hypothesis is true (less than 5%).

Therefore, a significant p-value tells us that an intervention works, whereas an effect size tells us how much it works.

It can be argued that emphasizing the size of effect promotes a more scientific approach, as unlike significance tests, effect size is independent of sample size.

To compare the results of studies done in different settings

Unlike a p-value, effect sizes can be used to quantitatively compare the results of studies done in a different setting. It is widely used in meta-analysis.

What is Effect Size?

Effect size is one of the concepts in statistics which calculates the power of a relationship amongst the two variables given on the numeric scale and there are three ways to measure the effect size which are the 1) Odd Ratio, 2) the standardized mean difference and 3) correlation coefficient.

For example, suppose in a class of students with boys and girls if the average height of all the boys is greater than the average height of all the girls, then with the help of effect size, we can make out that whether the difference in the height is moderate, high or not as much. It is also applicable for various statistical applications like correlation.

It is measured to find out the strength of the relation of two variables. It is standardized when it is calculated to be able to compare the two variables. The effect size is calculated by dividing the difference between the mean of two variables with the standard deviation.

Effect Size Formula

The formula is given below

Effect Size = (µ1-µ2)/α

Examples

Let’s see some simple to advanced examples to understand it better.

Example #1

Let us try to understand the concept with the help of an example. Suppose a class has 12 boys and 12 girls. And the mean height of the boys in the class is 120 cm, and the mean height of the girls of that class is 115 cm. Then we can say in a normalized way that the difference is 5 cm. But this does not quantify the effect as this number of 5cm difference is not standardized. Let us say the standard deviation for the two populations in this example is 4; then, we can calculate the effect size with the help of the formula.

Use the following data for the calculation.

=(120-115)/4

In order to get a sense of the effect of the difference between the two variables, we need to divide the difference between the two means of the two sets of the variables with their standard deviation number

From the calculation, we can see that the effect size is 1.3. With the help of this value, we can find out the shape of the distribution and also figure out how much percentage of the population falls under that percentage.

Example #2

Let us try to understand the concept with the help of another example. Suppose a class has 10 boys and 10 girls. And the mean GPA of the boys in the class is 2.64, and the mean GPA of the girls in that class is 3.64. Then we can say in a normalized way that the difference is 1. But this does not quantify the effect as this number of 1 difference is not standardized. Let us say the standard deviation for the two populations in this example is 2. Then we can calculate the effect size with the help of the equation.

Use the following data for the calculation of effect size.

=2.64-3.64/2

Example #3

Let us try to understand the concept with the help of another example. Suppose a class has 10 boys and 10 girls. And the mean weight of boys in the class is 60kg, and the mean weight of girls in a class is 55 kg. Then we can say in a normalized way that the difference is 5kg. But this does not quantify the effect as this number of 5 kg difference is not standardized. Let us say the standard deviation for the two populations in this example is 3. Then we can calculate the effect size with the help of the formula.

Below is given data for calculation of effect size.

=(60-55)/2

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