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Power – A Quick Introduction

In statistics, power is the probability of rejecting
a false null hypothesis.

Power - Minimal Example

Now, given a sample size of N = 10 and a population correlation ρ = 0.10, what's the probability of correctly rejecting the null hypothesis? This probability is known as power and denoted as (1 - β) in statistics. For the aforementioned example, (1 - β) is only 0.058 (roughly 6%) as shown below.

Gpower Example Single Correlation

If a population correlation ρ = 0.10 and
we sample N = 10 respondents, then
we need to find an absolute sample correlation of |r| > 0.63 for rejecting H0 at α = 0.05.
The probability of finding this is only 0.058.

So even though H0 is false, we've little power to actually reject it. Not rejecting a false H0 is known as a committing a type II error.

Type I and Type II Errors

Any null hypothesis may be true or false and we may or may not reject it. This results in the 4 scenarios outlined below.

Reality: H0 is trueReality: H0 is false
Decision: reject H0Type I error
Probability = α
Correct decision
Probability = (1 - β) = power
Decision: retain H0Correct decision
Probability = (1 - α)
Type II error
Probability = β

As you probably guess, we usually want the power for our tests to be as high as possible. But before taking a look at factors affecting power, let's first try and understand how a power calculation actually works.

Power Calculation Example

A pharmaceutical company wants to demonstrate that their medicine against high blood pressure actually works. They expect the following:

Given these considerations, what's the power for this study? Or -alternatively- what's the probability of rejecting H0 that the mean blood pressure is equal between treated and untreated populations?

Obviously, nobody knows the outcomes for this study until it's finished. However, we do know the most likely outcomes: they're our population estimates. So let's for a moment pretend that we'll find exactly these and enter them into a t-test calculator.

Power For T-Test Excel Example Compute t-test for expected sample sizes, means and SD's in Excel

We expect p = 0.023 so we expect to reject H0.
This is based on a t-distribution with df = 38 degrees of freedom (total sample size N = 40 - 2).
We expect to find t = 2.37 if the population mean difference is 6 mmHg (160 - 154).

Now, this expected (or average) t = 2.37 under the alternative hypothesis Ha is known as a noncentrality parameter or NCP. The NCP tells us how t is distributed under some exact alternative hypothesis and thus allows us to estimate the power for some test. The figure below illustrates how this works.

Central Noncentral T-Distribution For Power

A minor note here is that we'd also reject H0 if t < -2.02 but this probability is almost zero for our first scenario. The exact calculation can be replicated from the SPSS syntax below.

*Enter chosen alpha and expected NCP as raw data.
data list free/alpha ncp.
begin data
0.05 2.37
end data.

*Compute left (lct) and right (rct) critical t-values and power.
compute lct = idf.t(0.5 * alpha,38).
compute rct = idf.t(1 - (0.5 * alpha),38).
compute lprob = ncdf.t(lct,38,ncp).
compute rprob = 1 - ncdf.t(rct,38,ncp).
compute power = lprob + rprob.
execute.

*Show 3 decimal places for all values.
formats all (f8.3).

Power and Effect Size

Like we just saw, estimating power requires specifying

In the previous example, our scientists had an exact alternative hypothesis because they had very specific ideas regarding population means and standard deviations. In most applied studies, however, we're pretty clueless about such population parameters. This raises the question how do we get an exact alternative hypothesis?

For most tests, the alternative hypothesis can be specified as an effect size measure: a single number combining several means, variances and/or frequencies. Like so, we proceed from requiring a bunch of unknown parameters to a single unknown parameter.

What's even better: widely agreed upon rules of thumb are available for effect size measures. An overview is presented in this Googlesheet, partly shown below.

Effect Size Rules Of Thumb

In applied studies, we often use G*Power for estimating power. The screenshot below replicates our power calculation example for the blood pressure medicine study.

Gpower Example Independent Samples T-Test G*Power computes both effect size and power from two means and SD's

Note that estimating power in G*Power only requires

a single estimated effect size measure. Optionally, G*Power computes it for you, given your sample means and SD's.
the alpha level -often 0.05- used for testing the null hypothesis &
one or more sample sizes

Let's now take a look at how these 3 factors relate to power.

Factors Affecting Power

The figure below gives a quick overview how 3 factors relate to power.

Factors Affecting Power In Statistics

Let's now take a closer look at each of them.

Power & Alpha Level

Everything else equal, increasing alpha increases power. For our example calculation, power increases from 0.637 to 0.753 if we test at α = 0.10 instead of 0.05.

Sampling Distributions Power Versus Alpha

A higher alpha level results in smaller (absolute) critical values: we already reject H0 if t > 1.69 instead of t > 2.02. So the light blue area, indicating (1 - β), increases. We basically require a smaller deviation from H0 for statistical significance.

However, increasing alpha comes at a cost: it increases the probability of committing a type I error (rejecting H0 when it's actually true). Therefore, testing at α > 0.05 is generally frowned upon. In short, increasing alpha basically just decreases one problem by increasing another one.

Power & Effect Size

Everything else equal, a larger effect size results in higher power. For our example, power increases from 0.637 to 0.869 if we believe that Cohen’s D = 1.0 rather than 0.8.

Power Versus Effect Size Sampling Distributions

A larger effect size results in a larger noncentrality parameter (NCP). Therefore, the distributions under H0 and HA lie further apart. This increases the light blue area, indicating the power for this test.

Keep in mind, though, that we can estimate but not choose some population effect size. If we overestimate this effect size, we'll overestimate the power for our test accordingly. Therefore, we can't usually increase power by increasing an effect size.

An arguable exception is increasing an effect size by modifying a research design or analysis. For example, (partial) eta squared for a treatment effect in ANOVA may increase by adding a covariate to the analysis.

Power & Sample Size

Everything else equal, larger sample size(s) result in higher power. For our example, increasing the total sample size from N = 40 to N = 80 increases power from 0.637 to 0.912.

Power Versus Sample Size Sampling Distributions

The increase in power stems from our distributions lying further apart. This reflects an increased noncentrality parameter (NCP). But why does the NCP increase with larger sample sizes?

Well, recall that for a t-distribution, the NCP is the expected t-value under HA. Now, t is computed as

$$t = \frac{\overline{X_1} - \overline{X_2}}{SE}$$

where \(SE\) denotes the standard error of the mean difference. In turn, \(SE\) is computed as

$$SE = Sw\sqrt{\frac{1}{n_1} + \frac{1}{n_2}}$$

where \(S_w\) denotes the estimated population SD of the outcome variable. This formula shows that as sample sizes increase, \(SE\) decreases and therefore t (and hence the NCP) increases.

On top of this, degrees of freedom increase (from df = 38 to df = 78 for our example). This results in slightly smaller (absolute) critical t-values but this effect is very modest.

In short, increasing sample size(s) is a sound way to increase the power for some test.

Power & Research Design

Apart from sample size, effect size & α, research design may also affect power. Although there's no exact formulas, some general guidelines are that

3 Main Reasons for Power Calculations

Power calculations in applied research serve 3 main purposes:

Gpower Types Of Power Analyses Different types of power analysis are made simple by G*Power

Software for Power Calculations - G*Power

G*Power is freely downloadable software for running the aforementioned and many other power calculations. Among its features are

Linear Regression Power Sample Size Plot Required sample sizes for multiple linear regression, given desired power,
chosen α and 3 estimated effect sizes

Altogether, we think G*Power is amazing software and we highly recommend using it. The only disadvantage we can think of is that it requires rather unusual effect size measures. Some examples are

This is awkward because the APA and (perhaps therefore) most journal articles typically recommend reporting

These are also the measures we typically obtain from statistical packages such as SPSS or JASP. Fortunately, G*Power converts some measures and/or computes them from descriptive statistics like we saw in this screenshot.

Software for Power Calculations - SPSS

In SPSS, observed power can be obtained from the GLM, UNIANOVA and (deprecated) MANOVA procedures. Keep in mind that GLM - short for General Linear Model- is very general indeed: it can be used for a wide variety of analyses including

Observed Power In SPSS Glm Select Observed power from Analyze - General Linear Model -
Univariate - Options

Other power calculations (required sample sizes or estimating power prior to data collection) were added to SPSS version 27, released in 2020.

Power Analysis In SPSS 27 Power Analysis as found in SPSS version 27 onwards

In my opinion, SPSS power analysis is a pathetic attempt to compete with G*Power. If you don't believe me, just try running a couple of power analyses in both programs simultaneously. If you do believe me, ignore SPSS power analysis and just go for G*Power.

Thanks for reading.

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