# Statistical Power – An Easy Introduction with Examples

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Statistical power is a metric used in research to carry out various tests in a data set. It is a procedural method that incorporates several tools to test data points to come up with credible conclusions.

## Statistical Power – In a Nutshell

• Statistical power is the probability that a test will detect a difference from the null hypothesis if the difference exists.
• A high statistical power implies that a test is more valid in its findings.
• Researchers can use various methods to improve the statistical power in a test by manipulating sample sizes and other variables.

## Definition: Statistical power

Statistical power refers to the probability of a hypothesis test identifying a true effect if one exists. A true effect in statistics implies a non-zero, real association between a set of variables in the target population.1

• High power is a significant likelihood of a test uncovering a true effect
• Low power implies that a statistical test has a slim chance of sensing a true effect.2
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## Importance of statistical power

Statistical power is essential to derive accurate findings from samples selected in a population. Hypothesis testing begins with a null hypothesis and an alternative hypothesis proposed to contradict the null hypothesis.3

Example

A research problem aims to see if having a social circle improves the quality of life.

• Null hypothesis: Interacting with friends and close associates does not make people happier.
• Alternative hypothesis: Regular interaction with friends contributes to individual happiness.

Analysis of study results is prone to two common errors:

Type I error

Disregarding the null hypothesis of zero effect when it is true.

Type II error

Failing to reject the null hypothesis of no effect though it is false.4

A high statistical power such as 80% reduces the chances of a Type II error. Low power tests may completely fail to detect true effects. On the other hand, too much power leads to very sensitive tests.

## Statistical power – Power analysis

Power analysis is a method used to estimate the smallest sample in a study. If you have three of these components, you can estimate the fourth. There are four primary parts of power analysis:

 Significance level (alpha) The highest risk you are willing to consider in disapproving a null hypothesis, usually maintained at 5%. Statistical power The probability that a test will identify the presence of an effect if it exists, starting from 80% and above. Sample size The base number of observations required to note an effect of a certain level with a specified power level. Expected effect size A conventional way of illustrating the strength of the desired outcome of the study.5

You can conduct a power analysis before a study, the recommended significance level is 5%, while the desired power level is set to 80%.

### Sample size

A small sample (up to 30 units) typically has low power, and expanding the sample size increases the power but only to a limited extent.

The designated research design is also affected by power and sample size as follows:

 Within-subjects design Every participant in a study is tested in all conditions, which prevents individual variations from affecting the outcome of various conditions. Between-subjects design Each study member is subjected to a single condition. Since every condition is assigned to a different participant, individual contrasts may affect the outcome.

The within-subjects framework is more capable; therefore, few subjects are needed.

### Significance level

The significance level implies the likelihood of the Type I error and is often maintained at 5%. Your outcomes must have a less than 5% chance of happening to be regarded as statistically significant in a null hypothesis.6

Researchers account for their risk tolerance in making false positives and negatives to regulate the likelihood of making Type I and II errors.

### Effect size

Studies with high power can identify large and medium effects, while low-powered studies can only detect large effects.

Example

In our study of social circles and happiness. The main effect is socialization and the final happiness level from this interaction.

You can begin by analyzing existing research cases similar to this problem. Identify the studies that measure the impact of social circles and highlight happiness as an end goal.

Pick three studies that demonstrate these variables, use their reported effect sizes and calculate an average effect size. You can use this average as your expected effect size.

In fields such as engineering, using low-powered simulations as the norm may result in an exaggerated estimation of true effects.

## Statistical power – Other influencing factors

Besides the main elements of statistical power, researchers should consider other factors in estimating power. They include:

### Variability

Data groups with large variations decrease the sensitivity of a test while a group with smaller variations increases sensitivity. A defined population with demographic markers can reduce the variation of the main variable and boost power.

Example

Income levels are a common measure of the quality of life in many countries. A researcher can narrow down the income levels for people between 25 – 30 to improve the power.

### Measurement error

This is the difference between a true value and the identified value of a measurable object. The two types of measurement error are:

 Error type Definition Example Random errors They occur because of chance since they are unpredictable. Attitudes and moods tend to fluctuate during a study which may affect the quality of measurements. Systematic errors Errors that are predictable as they are technical in nature. Poorly constructed research questions often cause biased outcomes.7

## Increasing the statistical power

There are several ways of increasing statistical power, as discussed below:

### Increasing the effect size

Researchers can adjust the independent variable upwards, for example, expanding the social circle to measure marginal improvement in happiness.

### Increasing the sample size

Additionally, there may be room to increase the sample size using sample size satiates as a basis.

### Increasing the significance level

Though this increases sensitivity, it also creates a higher probability of a Type II error.

### Reducing measurement errors

Using high-quality, well-calibrated machines reduces measurement errors. Researchers may also employ multiple instruments and methods.

### Using a one-tailed test instead of a two-tailed test

One-tailed tests have a higher power in t and z tests. However, one-tailed tests are limited to cases with a good reason to anticipate an effect in a particular direction. A two-tailed test is applicable in testing effects in any direction.8

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## FAQs

#### What is statistical power?

Statistical power is the probability that a statistical test will detect a true effect if one is present, and a high statistical power rejects the null hypothesis.

#### What is a null hypothesis in a statistical test?

The null hypothesis postulates that there is no real difference between populations, and statistical power tests aim to reject the null hypothesis.

#### How can you improve statistical power?

Researchers prefer high statistical power for various reasons. They can improve statistical power by altering variables such as confidence level and adjusting the sample size.

#### What is power analysis?

Power analysis refers to calculations to find the minimum sample for a study. The optimum sample size helps to increase the accuracy of a case study.

## Sources:

1 Brownlee, Jason. “A Gentle Introduction to Statistical Power and Power Analysis in Python.” Machine Learning Mastery. July 13, 2018. https://machinelearningmastery.com/statistical-power-and-power-analysis-in-python/

2 Statistics Teacher. “What is Power?” September 15, 2017. https://www.statisticsteacher.org/2017/09/15/what-is-power/

3 Mintlab. “Informationen zur Nullhypothese und zur Alternativhypothese.” Accessed November 17, 2022. https://support.minitab.com/en-us/minitab/18/help-and-how-to/statistics/basic-statistics/supporting-topics/basics/null-and-alternative-hypotheses/

4 Dr. McLeod, Saul. “What are Type I and Type II Errors?” July 04, 2019. Simply Psychology. https://www.simplypsychology.org/type_I_and_type_II_errors.html

5 Conjointly. “Statistical Power.” Accessed November 17, 2022. https://conjointly.com/kb/statistical-power/

6 Frost, Jim. “Significance level.” Statistics by Jim. Accessed November 17, 2022. https://statisticsbyjim.com/glossary/significance-level/

7 Statistics how to. “Measurement error.”Accessed November 17, 2022. https://www.statisticshowto.com/measurement-error/

8 Statistical Methods and Data Analytics. “What are the differences between one tailed and two tailed tests?” Accessed November 17, 2022. https://stats.oarc.ucla.edu/other/mult-pkg/faq/general/faq-what-are-the-differences-between-one-tailed-and-two-tailed-tests/

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