Power (statistics)

In frequentist statistics, power is a measure of the ability of an experimental design and hypothesis testing setup to detect a particular effect if it is truly present.

is the probability of making a type II error (a false negative) conditional on there being a true effect or association.

If the actual value calculated on the sample is sufficiently unlikely to arise under the null hypothesis, we say we identified a statistically significant effect.

The threshold for significance can be set small to ensure there is little chance of falsely detecting a non-existent effect.

In the case of the comparison of the two crop varieties, it enables us to answer questions like: Suppose we are conducting a hypothesis test.

Statistical power is one minus the type II error probability and is also the sensitivity of the hypothesis testing procedure to detect a true effect.

Tests may have the same size, and hence the same false positive rates, but different ability to detect true effects.

Consideration of their theoretical power proprieties is a key reason for the common use of likelihood ratio tests.

This threshold then implies that the observation must be at least that unlikely (perhaps by suggesting a sufficiently large estimate of difference) to be considered strong enough evidence against the null.

Some statistical tests will inherently produce better power, albeit often at the cost of requiring stronger assumptions.

It can be the expected effect size if it exists, as a scientific hypothesis that the researcher has arrived at and wishes to test.

If the researcher is looking for a larger effect, then it should be easier to find with a given experimental or analytic setup, and so power is higher.

More broadly, the precision with which the data are measured can also be an important factor (such as the statistical reliability), as well as the design of an experiment or observational study.

A smaller sampling error could be obtained by larger sample sizes from a less variability population, from more accurate measurements, or from more efficient experimental designs (for example, with the appropriate use of blocking), and such smaller errors would lead to improved power, albeit usually at a cost in resources.

The effect may exist, but be smaller than what was looked for, meaning the study is in fact underpowered and the sample is thus unable to distinguish it from random chance.

[8] Conclusions about the probability of actual presence of an effect also should consider more things than a single test, especially as real world power is rarely close to 1.

In many contexts, the issue is less about deciding between hypotheses but rather with getting an estimate of the population effect size of sufficient accuracy.

An alternative, albeit related analysis would be required if we wish to be able to measure correlation to an accuracy of +/- 0.1, implying a different (in this case, larger) sample size.

In this setting, the only relevant power pertains to the single quantity that will undergo formal statistical inference.

For instance, in multiple regression analysis, the power for detecting an effect of a given size is related to the variance of the covariate.

For example, if we consider a false positive to be making an erroneous null rejection on any one of these hypotheses, our likelihood of this "family-wise error" will be inflated if appropriate measures are not taken.

Such measures typically involve applying a higher threshold of stringency to reject a hypothesis (such as with the Bonferroni method), and so would reduce power.

[11][12] Falling for the temptation to use the statistical analysis of the collected data to estimate the power will result in uninformative and misleading values.

[11] In fact, a smaller p-value is properly understood to make the null hypothesis relatively less likely to be true.

[citation needed] The following is an example that shows how to compute power for a randomized experiment: Suppose the goal of an experiment is to study the effect of a treatment on some quantity, and so we shall compare research subjects by measuring the quantity before and after the treatment, analyzing the data using a one-sided paired t-test, with a significance level threshold of 0.05.

We can proceed according to our knowledge of statistical theory, though in practice for a standard case like this software will exist to compute more accurate answers.

If n is large, the t-distribution converges to the standard normal distribution (thus no longer involving n) and so through use of the corresponding quantile function

In the trivial case of zero effect size, power is at a minimum (infimum) and equal to the significance level of the test

The success criterion for PPOS is not restricted to statistical significance and is commonly used in clinical trial designs.

Numerous free and/or open source programs are available for performing power and sample size calculations.