Statistics Knowledge Base - Post-hoc Statistical Power for Multiple Regression

Friday, March 19, 2010

Post-hoc Statistical Power for Multiple Regression

The power of a statistical test refers to the ability of the test to reject a false null hypothesis, given the structure of the research model. Put another way, power is the ability of a statistical test to detect a significant effect within the confines of a particular research model, presuming that a significant effect exists in the population. Statistical power is very important in quantitative research, as it provides a measure of the adequacy of an investigative model to detect a hypothesized effect -- without sufficient statistical power, there is little value in moving forward with one's research.

PROPERTIES

Some of the most important properties and considerations related to statistical power include:

Statistical power is directly related to a study's Type II error rate (Β), in that Power = 1 - Β. As such, the probability of a Type II error decreases as power increases, and vice-versa.
While no formal standard has been established for what constitutes adequate statistical power, the value of 0.80 proposed by Cohen (1988) has become the de facto minimal power standard for most researchers.
The most common way of increasing statistical power is by increasing the sample size, but power can also be increased by using more reliable metrics, or by incorporating theoretically-justified control variables into the research model.
Statistical power is intimately related with the computation of a-priori sample size.

ONLINE CALCULATOR

To calculate a post-hoc statistical power value for a multiple regression analysis, please click here.

FORMULAE

There are several formulae involved in the computation of post-hoc statistical power for multiple regression. These formulae are detailed below.

F-distribution probability density function:
Where d₁ and d₂ are the numerator and denominator degrees of freedom, and B is the Beta function.

Noncentrality parameter for the F-distribution (lambda):
Where f² is the effect size and n is the sample size.

Noncentral F-distribution probability density function:
Where v₁ and v₂ are the numerator and denominator degrees of freedom, λ is the noncentrality parameter, f is the Fisher F-value, and B is the Beta function.

Normal curve cumulative distribution function:
Where μ is the mean, σ is the standard deviation, and erf is the Error function.

REFERENCES

Cohen, J., Cohen, P., West, S.G., and Aiken, L.S. (2003) "Applied Multiple Regression/Correlation Analysis for the Behavioral Sciences (3rd edition)", Lawrence Earlbaum Associates, Mahwah, NJ

Cohen, J. (1988) "Statistical Power Analysis for the Behavioral Sciences (2nd Edition)", Lawrence Earlbaum Associates, Hillsdale, NJ