In statistics, power is the probability of rejecting

a false null hypothesis.

- Power Calculation Example
- Power & Alpha Level
- Power & Effect Size
- Power & Sample Size
- 3 Main Reasons for Power Calculations
- Software for Power Calculations - G*Power

## Power - Minimal Example

- In some country, IQ and salary have a population correlation ρ = 0.10.
- A scientist examines a sample of N = 10 people and finds a sample correlation r = 0.15.
- He tests the (false) null hypothesis H
_{0}that ρ = 0. The significance level for this test, p = 0.68. - Since p > 0.05, his chosen alpha level, he does
**not**reject his (false) null hypothesis that ρ = 0.

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.

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 H_{0} at α = 0.05.

The probability of finding this is only 0.058.

So even though H_{0} is false, we've little power to actually reject it. Not rejecting a false H_{0} 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: H_{0} is true | Reality: H_{0} is false | |
---|---|---|

Decision: reject H_{0} | Type I errorProbability = α | Correct decision Probability = (1 - β) = power |

Decision: retain H_{0} | Correct decision Probability = (1 - α) | Type II errorProbability = 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:

- the average blood pressure in some untreated population is 160 mmHg;
- they expect their medicine to lower this to roughly 154 mmHg;
- the standard deviation should be around 8 mmHg (both populations);
- they plan to use an independent samples t-test at α = 0.05 with N = 20 for either subsample.

Given these considerations,
what's the power for this study?
Or -alternatively- what's the probability of rejecting H_{0} 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.

We expect p = 0.023 so we expect to reject H_{0}.

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 H_{a} 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.

- First off, our H
_{0}is tested using a central t-distribution with df = 38; - If we test at α = 0.05 (2-tailed), we'll reject H
_{0}if t < -2.02 (left critical value) or if t > 2.02 (right critical value); - If our alternative hypothesis H
_{A}is exactly true, t follows a noncentral t-distribution with df = 38 and NCP = 2.37; - Under this noncentral t-distribution, the probability of finding t > 2.02 ≈ 0.637. So this is roughly the probability of rejecting H
_{0}-or the power (1 - β)- for our first scenario.

A minor note here is that we'd also reject H_{0} 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

- an
*exact*null hypothesis and - an
*exact*alternative hypothesis.

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.

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.

G*Power computes both effect size and power from two means and SD'sNote 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.

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.

A higher alpha level results in smaller (absolute) critical values: we already reject H_{0} 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 H_{0} for statistical significance.

However, increasing alpha comes at a cost: it increases the probability of committing a type I error (rejecting H_{0} 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.

A larger effect size results in a larger noncentrality parameter (NCP). Therefore, the distributions under H_{0} and H_{A} 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.

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 H_{A}. 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, \(SE\) *de*creases and therefore t (and hence the NCP) *in*creases.

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

- everything else equal, within-subjects designs tend to have more power than between-subjects designs;
- for ANCOVA, including one or two covariates tends to increase power for demonstrating a treatment effect;
- for multiple regression, power for each separate predictor tends to decrease as more predictors are added to the model;

## 3 Main Reasons for Power Calculations

Power calculations in applied research serve 3 main purposes:

- compute the
**required sample size**prior to data collection. This involves estimating an effect size and choosing α (usually 0.05) and the desired power (1 - B), often 0.80; **estimate power before**collecting data for some planned analyses. This requires specifying the intended sample size, choosing an α and estimating which effect sizes are expected. If the estimated power is low, the planned study may be cancelled or proceed with a larger sample size;**estimate power after**data have been collected and analyzed. This calculation is based on the actual sample size, α used for testing and observed effect size.

## 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

- computing effect sizes from descriptive statistics (mostly sample means and standard deviations);
- computing power, required sample sizes, required effect sizes and more;
- creating plots that visualize how power, effect size and sample size relate for many different statistical procedures. The figure below shows an example for multiple linear regression.

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

- Cohen’s f for ANOVA and
- Cohen’s W for a chi-square test.

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

- (partial) eta-squared for ANOVA and
- the contingency coefficient or (better) Cramér’s V for a chi-square test.

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

- (multiple) linear regression;
- t-tests;
- ANCOVA (analysis of covariance);
- repeated measures ANOVA.

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 as found in SPSS version 27 onwardsIn 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.