Effect size

Effect size or effect size denotes a ( standardized ) statistical measure that indicates the (relative) size of an effect. An effect (or effects ) occurs when in a ( respective ) statistical test the null hypothesis (= no effect) is rejected. The effect size then provides information about the size of the effect. It can thus also be used to illustrate the practical relevance of significant results. For example, always lead with increasing sample sizes smaller effects to reject the null hypothesis. In the empirical research, however, interested not only whether an effect is present ( rejection of the null hypothesis ) or not ( acceptance of the null hypothesis ), but also how big the effect is.

• 5.3.1 Regression Analysis
• 5.3.2 F-test or ANOVA

Definition

There are different measures of effect size in use. According to Cohen should apply to a measure of effect size:

Example

Comparing the performance of intelligence of children who were taught using a new method, with children who were taught by the conventional method. If a very large number of children was collected per sample, differences of, for example, 0.1 IQ points between the groups may already be significant. Quite obviously mean 0.1 IQ points difference but in spite of a significant test result hardly an improvement.

If only the test and its significant result would be involved, was the conclusion that the new method results in a better performance intelligence, and the old method of teaching would be abolished at a high cost. In contrast, if involved, that the new teaching method has only causes an improvement of 0.1 points, would certainly continue to teach according to the original method.

Use in Research

Effect size indicated in experiments ( especially in medicine, the social sciences and psychology ) the magnitude of the effect of an experimental factor. In regression models, it serves as an indicator of the influence of a variable on the explained variable.

The effect size on the one hand are based on a study to compare differences between groups in a standardized measure can. Often, effect sizes are calculated in meta-analyzes, the results of various studies into a single measure - to be able to compare - the effect size.

Often an effect size but is placed as the minimum effect size prior to conducting an investigation or before the execution of a test. A statistical test is performed, it can almost always be rejected the null hypothesis when only a sufficient number of measurement results are taken into account. The test is performed at a sufficiently large sample size that is almost always significant.

To account for the magnitude of the difference, a minimum effect size is determined before the examination. A significant test result will only be accepted if the effect size is at least such as 0.4 (corresponding to " medium effect size " ) is.

Effect size and statistical significance

In the practical application of statistical tests, a smaller p- value is often associated with a relatively high effect size. While it is indeed the case that, while maintaining the other parameters of a test situation ( sample size chosen significance level, required test strength ), a smaller p- value associated with a larger effect size than a larger one, but this is simply a specific feature of the statistical test ( but or the underlying distributions ), it can not be an interpretation of an error probability as effect size. But it is - for example, when performing a meta-analysis - possible from a reported error probability an associated effect size to determine if the sample size is known. As a statistical test, consisting essentially of, a special ( usefully non-central ) sampling distribution for the used test statistic (eg, t - in the case of a t-test - or F for ANOVA) to check whether the value empirically found the statistics assuming a special to be tested null hypothesis is correct, plausible ( or implausible ) is possible to determine the selected distribution and from this the test underlying effect size from the given probability of error and the information on the sample size of the required parameters (non -centrality parameter ). Similarly, a reported be -preserved level of significance are used to give an estimate of how big the effect size must have been at least, so that could be maintained for a given sample size, the reported level of significance.

In the Fisher'schen test theory of the p- value may represent an effect size, as a small p- value is interpreted as a high probability of applying the research hypothesis. Due to the standardization of the test statistics, however, each effect can be significantly improved by increasing the sample. Under the Neyman - Pearson though, is the fact to take into account that accepting the research hypothesis is always accompanied with a Reject the null hypothesis. A result that is highly significant under the null hypothesis can be much less likely under the research hypothesis, since the test strength greatly reduced. As an effect size of the p- value is thus not suitable because of the effect in the research hypothesis may be too small to be of practical importance.

Measures of the effect size

Bravais -Pearson correlation r

The Bravais -Pearson correlation is one of the most used and oldest measures of effect sizes with regression models. They met in a natural way, the requirements imposed on a Cohen effect size.

Cohen indicates a small effect, a middle and a strong effect.

Alternatively, the determination can be used.

Cohen's d

Cohen's d is the effect size for mean differences between two groups with equal group sizes and the same group variances and helps in assessing the practical relevance of a significant mean difference ( see also t-test):

As estimator was developed by Cohen

Indicated, with the respective mean of the two samples and the estimated variances of the two samples according to the equation, respectively.

Cohen means a small effect, a middle and a strong effect.

Unequal group sizes and group variances

Other authors as Cohen estimate the standard deviation with the help of the pooled variance as

With

Conversion to

Is the membership of the one sample and zero for the other one encoded with such a correlation coefficient may be calculated. It results from Cohen as

In contrast to Cohen, the correlation coefficient is bounded above by one. From a weak effect is called here an r = 0.10, a medium effect from an r = 0.30 and a strong effect from r = 0.50.

Glass's Δ

Glass ( 1976) proposed to use only the standard deviation of the second group

The second group is considered herein as a control group. When comparisons are made with several experimental groups, then it is better to estimate from the control group, so that the effect size does not depend on the estimated variances of the experimental groups.

Assuming equal variances in the two groups, however, the pooled variance of the better estimate.

Hedges g

Hedges struck before 1981 a further modification. with

And

Shows a distorted estimates of effect size. with

And

Yields an unbiased estimator, which is better suited to calculate the confidence intervals of the effect strengths of sampling differences as Cohen's d, which estimates the effect strength in the population. here denotes the Gamma function.

Cohen's f2

Cohen is a measure of the effect size in the ANOVA or the F-test and regression analysis.

Regression analysis

The effect size is calculated

With the coefficients of determination with all the variables of the regression model and without the variable to be tested. If only the combined effect of all variables of interest, the above formula reduces to

Cohen indicates a small effect, a middle and a strong effect.

F-test or ANOVA

The effect size calculated for groups

With an estimator of the total variance of the data set. Cohen indicates a small effect, a middle and a strong effect.

Partial eta -squared

The effect size can also be specified via the partial eta squared. The calculation is as follows:

With the sum of squares of each effect and to be determined as sum of squares of the residual variance. Multiplying the partial Eta- square with 100 it can be used for interpretation of the variance explained. The measure then indicates how much variance of the dependent variable is percentage explained by the independent variable. The SPSS IBM calculated at variance analyzes default partial eta squared. In older versions, this was incorrectly referred to as eta- squared. In a one-factor ANOVA between Eta -squared and partial eta -squared is no difference. Once a factorial ANOVA is calculated, the partial Eta- square must be calculated.

But Eta- square as Effektstärkemaß overestimated the proportion of explained variance. Rasch et al. and Bortz recommend that you use the population effect estimate, which is calculated by Cohen as follows:

Cramer's Phi, Cramer's V, and Cohen's w

A measure of the effect size can be calculated not only on the basis of average or variance differences, but also in terms of probabilities. See, page 4 In this case, from the numbers of cross- table containing the probabilities rather than absolute abundances, calculated and then the square root. The result is Cohen's:

Here, the number of categories of the column variable, the number of categories of the row variable, the observed probability of the cell i, j, and the expected probability of the cell ij Expected cell probabilities are calculated by each corresponding edge probabilities are multiplied together. To calculate see and Cohen and, page 6 As crosstabulation that do not contain absolute frequencies, but probabilities, at the point where normally the case number is to be found, always is 1, instead it can also be calculated, which numerically identical:

Also numerically identical to it, when calculated in terms of cross-tabs that contain the probabilities, the number of rows, number of columns, and the smaller of the two numbers is.

For Cohen's conventionally apply the value 0.1 as small, 0.3 as medium and 0.5 as large. See in this regard, Table 5.12 on page 167 top right. In this table, optimal sample sizes depending on the severity of Cohen as a measure of effect size (" weak," for, " medium " for, " strong " for ) and the degrees of freedom are given.

Small, medium and large effect sizes

The previously specified values ​​for small, medium or large effect sizes depend strongly on the subject. Cohen has chosen the values ​​as part of its analysis and the social scientific usage.

" This is an elaborate way to arrive at the same sample size that HAS BEEN used in past social science studies of large, medium, and small size ( respectively). The method uses a Standardized effect size as the goal. Think about it: for a "medium " effect size, you'll choose the same n Regardless of the accuracy or reliability of your instrument, or the narrowness or diversity of your subjects. Clearly, important considerations are being ignored here. "Medium " is definitely not the message! "

"This is a complicated way to arrive at the same sample sizes that have been used in the past in large, medium and small social science studies. This method has a standardized effect size to the target. Let's think about it: For an effect size "medium" we choose the same sample size regardless of the accuracy or the reliability of the instrument, the similarity or differences of objects. Of course, here important aspects of the investigation are ignored. " Agent" is hardly the message! "

Therefore, they are accepted by many researchers as a guide only.