Statistical power

The power or sensitivity of a binary hypothesis test is the probability that the test correctly rejects the null hypothesis (H₀) when the alternative hypothesis (H₁) is true. It can be equivalently thought of as the probability of accepting the alternative hypothesis (H₁) when it is true—that is, the ability of a test to detect an effect, if the effect actually exists. That is,

\mbox{power} = \mathbb P\big( \mbox{reject } H_0 \big| H_1 \mbox{ is true} \big)

The power of a test sometimes, less formally, refers to the probability of rejecting the null when it is not correct, though this is not the formal definition stated above. The power is in general a function of the possible distributions, often determined by a parameter, under the alternative hypothesis. As the power increases, there are decreasing chances of a Type II error (false negative), which are also referred to as the false negative rate (β) since the power is equal to 1−β, again, under the alternative hypothesis. A similar concept is Type I error, also referred to as the “false positive rate” or the level of a test under the null hypothesis.

Power analysis can be used to calculate the minimum sample size required so that one can be reasonably likely to detect an effect of a given size. For example: “how many times do I need to toss a coin to conclude it is rigged?”^[1] Power analysis can also be used to calculate the minimum effect size that is likely to be detected in a study using a given sample size. In addition, the concept of power is used to make comparisons between different statistical testing procedures: for example, between a parametric and a nonparametric test of the same hypothesis.

Background

Statistical tests use data from samples to assess, or make inferences about, a statistical population. In the concrete setting of a two-sample comparison, the goal is to assess whether the mean values of some attribute obtained for individuals in two sub-populations differ. For example, to test the null hypothesis that the mean scores of men and women on a test do not differ, samples of men and women are drawn, the test is administered to them, and the mean score of one group is compared to that of the other group using a statistical test such as the two-sample z-test. The power of the test is the probability that the test will find a statistically significant difference between men and women, as a function of the size of the true difference between those two populations.

Factors influencing power

Statistical power may depend on a number of factors. Some factors may be particular to a specific testing situation, but at a minimum, power nearly always depends on the following three factors:

the statistical significance criterion used in the test
the magnitude of the effect of interest in the population
the sample size used to detect the effect

A significance criterion is a statement of how unlikely a positive result must be, if the null hypothesis of no effect is true, for the null hypothesis to be rejected. The most commonly used criteria are probabilities of 0.05 (5%, 1 in 20), 0.01 (1%, 1 in 100), and 0.001 (0.1%, 1 in 1000). If the criterion is 0.05, the probability of the data implying an effect at least as large as the observed effect when the null hypothesis is true must be less than 0.05, for the null hypothesis of no effect to be rejected. One easy way to increase the power of a test is to carry out a less conservative test by using a larger significance criterion, for example 0.10 instead of 0.05. This increases the chance of rejecting the null hypothesis (i.e. obtaining a statistically significant result) when the null hypothesis is false, that is, reduces the risk of a Type II error (false negative regarding whether an effect exists). But it also increases the risk of obtaining a statistically significant result (i.e. rejecting the null hypothesis) when the null hypothesis is not false; that is, it increases the risk of a Type I error (false positive).

The magnitude of the effect of interest in the population can be quantified in terms of an effect size, where there is greater power to detect larger effects. An effect size can be a direct estimate of the quantity of interest, or it can be a standardized measure that also accounts for the variability in the population. For example, in an analysis comparing outcomes in a treated and control population, the difference of outcome means Y − X would be a direct measure of the effect size, whereas (Y − X)/σ where σ is the common standard deviation of the outcomes in the treated and control groups, would be a standardized effect size. If constructed appropriately, a standardized effect size, along with the sample size, will completely determine the power. An unstandardized (direct) effect size will rarely be sufficient to determine the power, as it does not contain information about the variability in the measurements.

The sample size determines the amount of sampling error inherent in a test result. Other things being equal, effects are harder to detect in smaller samples. Increasing sample size is often the easiest way to boost the statistical power of a test. How increased sample size translates to higher power is a measure of the efficiency of the test—for example, the sample size required for a given power.^[2]

The precision with which the data are measured also influences statistical power. Consequently, power can often be improved by reducing the measurement error in the data. A related concept is to improve the “reliability” of the measure being assessed (as in psychometric reliability).

The design of an experiment or observational study often influences the power. For example, in a two-sample testing situation with a given total sample size n, it is optimal to have equal numbers of observations from the two populations being compared (as long as the variances in the two populations are the same). In regression analysis and Analysis of Variance, there are extensive theories and practical strategies for improving the power based on optimally setting the values of the independent variables in the model.

Interpretation

Although there are no formal standards for power (sometimes referred to as π), most researchers assess the power of their tests using π=0.80 as a standard for adequacy. This convention implies a four-to-one trade off between β-risk and α-risk. (β is the probability of a Type II error; α is the probability of a Type I error, 0.2 and 0.05 are conventional values for β and α). However, there will be times when this 4-to-1 weighting is inappropriate. In medicine, for example, tests are often designed in such a way that no false negatives (Type II errors) will be produced. But this inevitably raises the risk of obtaining a false positive (a Type I error). The rationale is that it is better to tell a healthy patient “we may have found something—let's test further,” than to tell a diseased patient “all is well.”^[3]

Power analysis is appropriate when the concern is with the correct rejection, or not, of a null hypothesis. In many contexts, the issue is less about determining if there is or is not a difference but rather with getting a more refined estimate of the population effect size. For example, if we were expecting a population correlation between intelligence and job performance of around 0.50, a sample size of 20 will give us approximately 80% power (alpha = 0.05, two-tail) to reject the null hypothesis of zero correlation. However, in doing this study we are probably more interested in knowing whether the correlation is 0.30 or 0.60 or 0.50. In this context we would need a much larger sample size in order to reduce the confidence interval of our estimate to a range that is acceptable for our purposes. Techniques similar to those employed in a traditional power analysis can be used to determine the sample size required for the width of a confidence interval to be less than a given value.

Many statistical analyses involve the estimation of several unknown quantities. In simple cases, all but one of these quantities is a nuisance parameter. In this setting, the only relevant power pertains to the single quantity that will undergo formal statistical inference. In some settings, particularly if the goals are more "exploratory", there may be a number of quantities of interest in the analysis. For example, in a multiple regression analysis we may include several covariates of potential interest. In situations such as this where several hypotheses are under consideration, it is common that the powers associated with the different hypotheses differ. For instance, in multiple regression analysis, the power for detecting an effect of a given size is related to the variance of the covariate. Since different covariates will have different variances, their powers will differ as well.

Any statistical analysis involving multiple hypotheses is subject to inflation of the type I error rate if appropriate measures are not taken. Such measures typically involve applying a higher threshold of stringency to reject a hypothesis in order to compensate for the multiple comparisons being made (e.g. as in the Bonferroni method). In this situation, the power analysis should reflect the multiple testing approach to be used. Thus, for example, a given study may be well powered to detect a certain effect size when only one test is to be made, but the same effect size may have much lower power if several tests are to be performed.

It is also important to consider the statistical power of a hypothesis test when interpreting its results. A test's power is the probability of correctly rejecting the null hypothesis when it is false; a test's power is influenced by the choice of significance level for the test, the size of the effect being measured, and the amount of data available. A hypothesis test may fail to reject the null, for example, if a true difference exists between two populations being compared by a t-test but the effect is small and the sample size is too small to distinguish the effect from random chance.^[4] Many clinical trials, for instance, have low statistical power to detect differences in adverse effects of treatments, since such effects may be rare and the number of affected patients small.^[5]

A priori vs. post hoc analysis

Power analysis can either be done before (a priori or prospective power analysis) or after (post hoc or retrospective power analysis) data are collected. A priori power analysis is conducted prior to the research study, and is typically used in estimating sufficient sample sizes to achieve adequate power. Post-hoc power analysis is conducted after a study has been completed, and uses the obtained sample size and effect size to determine what the power was in the study, assuming the effect size in the sample is equal to the effect size in the population. Whereas the utility of prospective power analysis in experimental design is universally accepted, the usefulness of retrospective techniques is controversial.^[6] Falling for the temptation to use the statistical analysis of the collected data to estimate the power will result in uninformative and misleading values. In particular, it has been shown ^[7] that post-hoc power in its simplest form is a one-to-one function of the p-value attained. This has been extended^[7] to show that all post-hoc power analyses suffer from what is called the "power approach paradox" (PAP), in which a study with a null result is thought to show MORE evidence that the null hypothesis is actually true when the p-value is smaller, since the apparent power to detect an actual effect would be higher. In fact, a smaller p-value is properly understood to make the null hypothesis LESS likely to be true.

Application

Funding agencies, ethics boards and research review panels frequently request that a researcher perform a power analysis, for example to determine the minimum number of animal test subjects needed for an experiment to be informative. In frequentist statistics, an underpowered study is unlikely to allow one to choose between hypotheses at the desired significance level. In Bayesian statistics, hypothesis testing of the type used in classical power analysis is not done. In the Bayesian framework, one updates his or her prior beliefs using the data obtained in a given study. In principle, a study that would be deemed underpowered from the perspective of hypothesis testing could still be used in such an updating process. However, power remains a useful measure of how much a given experiment size can be expected to refine one's beliefs. A study with low power is unlikely to lead to a large change in beliefs.

Example

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 compare research subjects by measuring the quantity before and after the treatment, analyzing the data using a paired t-test. Let $A_i$ and $B_i$ denote the pre-treatment and post-treatment measures on subject i respectively. The possible effect of the treatment should be visible in the differences $D_i=B_i-A_i$ , which are assumed to be independently distributed, all with the same expected value and variance.

'The effect of the treatment can be analyzed using a one-sided t-test. The null hypothesis of no effect will be that the mean difference will be zero, i.e. $H_0: \mu_D=0$ . In this case, the alternative hypothesis states a positive effect, corresponding to $H_1: \mu_D>0$ . The test statistic is:

T_n=\frac{\bar{D}_n-0}{\hat{\sigma}_D/\sqrt{n}}.

where n is the sample size, $\bar{D}_n=\frac{1}{n}\sum_{i=1}^n D_i$ , and $\hat{\sigma}_D/\sqrt{n}$ is the standard error. The distribution of the test statistic under the null hypothesis follows a Student t-distribution. Furthermore, assume that the null hypothesis will be rejected at the significance level of $\alpha=0.05$ . Since n is large, one can approximate the t-distribution by a normal distribution and calculate the critical value using the quantile function $\Phi$ of the normal distribution. It turns out that the null hypothesis will be rejected if

T_n > 1.64.

Now suppose that the alternative hypothesis is true and $\mu_D=\theta$ . Then, the power is

\begin{array}{ccl} B(\theta)&=&P(T_n > 1.64|\mu_D=\theta) \\ &=&P(\frac{\bar{D}_n-0}{\hat{\sigma}_D/\sqrt{n} }> 1.64|\mu_D=\theta) \\ &=&P\left(\frac{\bar{D}_n-\theta+\theta}{\hat{\sigma}_D/\sqrt{n}} > 1.64\right|\mu_D=\theta)\\ &=& P\left(\frac{\bar{D}_n-\theta}{\hat{\sigma}_D/\sqrt{n}} > 1.64-\frac{\theta}{\hat{\sigma}_D/\sqrt{n}}\right|\mu_D=\theta)\\ &=& 1- P\left(\frac{\bar{D}_n-\theta}{\hat{\sigma}_D/\sqrt{n}} < 1.64-\frac{\theta}{\hat{\sigma}_D/\sqrt{n}}\right|\mu_D=\theta)\\ \end{array}

For large n, $T_n$ approximately follows a standard normal distribution when the alternative hypothesis is true, the approximate power can be calculated as

B(\theta)\approx 1-\Phi(1.64-\frac{\theta}{\hat{\sigma}_D/\sqrt{n}}).

According to this formula, the power increases with the values of the parameter $\theta$ . For a specific value of $\theta$ a higher power may be obtained by increasing the sample size n.

It is not possible to guarantee a sufficient large power for all values of $\theta$ , as $\theta$ may be very close to 0. The minimum (infimum) value of the power is equal to the size of the test, $\alpha$ , in this example 0.05. However, it is of no importance to distinguish between $\theta=0$ and small positive values. If it is desirable to have enough power, say at least 0.90, to detect values of $\theta >1$ , the required sample size can be calculated approximately:

B(1)\approx 1-\Phi(1.64-\sqrt{n}/\hat{\sigma}_D) >0{.}90\ ,

from which it follows that

\Phi(1.64-\sqrt{n}/\hat{\sigma}_D) <0.10.

Hence

\sqrt{n}/\hat{\sigma}_D > 1.64- z_{0.10}=1.64+1.28\approx 3

n>9\,\hat{\sigma}_D^2,

where $z_{0.10}$ is a standard normal quantile; see Probit for an explanation of the relationship between $\Phi$ and z-values.

Software for power and sample size calculations

Numerous programs are available for performing power and sample size calculations. These include commercial software

and free software

PS
Russ Lenth's power and sample-size page
G*Power (http://www.gpower.hhu.de/)
WebPower Free online statistical power analysis (http://webpower.psychstat.org)
A free online calculator that displays the formulas and assumptions behind the calculations is available at powerandsamplesize.com
R package pwr

Extension

Bayesian power

In Frequentist setting, parameter is assumed to have a specific value which is unlikely to be true. This issue can be addressed by assuming the parameter has a distribution. The resulted power is sometimes referred to as Bayesian power which is commonly used in clinical trial design.

Predictive probability of success

Both Frequentist power and Bayesian power uses statistical significance as success criteria. However statistical significance is often not enough to define success. To address this issue, the power concept can be extended to the concept of predictive probability of success (PPOS). The success criteria for PPOS is not restricted to statistical significance and is commonly used in clinical trial designs.

Notes

↑ http://www.statisticsdonewrong.com/power.html
↑ Everitt 2002, p. 321.
↑ Ellis, Paul D. (2010). The Essential Guide to Effect Sizes: An Introduction to Statistical Power, Meta-Analysis and the Interpretation of Research Results. United Kingdom: Cambridge University Press.
↑ Ellis, Paul (2010). The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results. Cambridge University Press. p. 52. ISBN 978-0521142465.
↑ Tsang, R.; Colley, L.; Lynd, L. D. (2009). "Inadequate statistical power to detect clinically significant differences in adverse event rates in randomized controlled trials". Journal of Clinical Epidemiology 62 (6): 609–616. doi:10.1016/j.jclinepi.2008.08.005. PMID 19013761.
↑ Thomas, L. (1997) Retrospective power analysis. Conservation Biology 11(1):276–280
1 2 Hoenig and Heisey (2001)The Abuse of PowerThe American Statistician 55(1):19-24

References

Everitt, Brian S. (2002). The Cambridge Dictionary of Statistics. Cambridge University Press. ISBN 0-521-81099-X.
Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). ISBN 0-8058-0283-5.
Aberson, C. L. (2010). Applied Power Analysis for the Behavioral Science. ISBN 1-84872-835-2.

External links

Video: Power and Sample Size Primer by NCSS
WebPower - Free online sample size planning for one-way ANOVA, two-way ANOVA, repeated-measure ANOVA, SEM, multilevel modeling
PowerAndSampleSize.com – free, online power and sample size calculators with graphics highlighting sensitivity to input values
PASS – Power analysis and sample size software
Hypothesis Testing and Statistical Power of a Test
G*Power – A free program for Statistical Power Analysis for Mac OS and MS-DOS
Effect Size Calculators Calculate d and r from a variety of statistics.
R/Splus package of power analysis functions along the lines of Cohen (1988)
Examples of all ANOVA and ANCOVA models with up to three treatment factors, including tools to estimate design power
Free A-priori Sample Size Calculator for Multiple Regression from Daniel Soper's Free Statistics Calculators website. Computes the minimum required sample size for a study, given the alpha level, the number of predictors, the anticipated effect size, and the desired statistical power level.
Power calculator from Russ Lenth, University of Iowa

Further Explanations

EffectSizeFAQ.com

Statistics

Descriptive statistics

Continuous data

Location	Mean arithmetic geometric harmonic Median Mode

Dispersion	Range Standard deviation Coefficient of variation Percentile Interquartile range

Shape	Variance Skewness Kurtosis Moments L-moments

Count data

Index of dispersion

Summary tables

Dependence

Statistical graphics

Data collection

Study design	Effect size Standard error Statistical power Sample size determination

Survey methodology	Sampling stratified cluster Opinion poll Questionnaire

Controlled experiments	Design control optimal Controlled trial Randomized Random assignment Replication Blocking Factorial experiment

Uncontrolled studies	Observational study Natural experiment Quasi-experiment

Statistical inference

Statistical theory

Frequentist inference

Confidence interval Testing hypotheses Power

Unbiased estimators	Mean unbiased minimum-variance Median unbiased

Biased estimators	Maximum likelihood Method of moments Minimum distance Density estimation

Parametric tests	Likelihood-ratio Wald Score

Specific tests

Z (normal) Student's t-test F Shapiro–Wilk Kolmogorov–Smirnov

Goodness of fit	Chi-squared G Sample source (Anderson–Darling) Sample normality (Shapiro–Wilk) Skewness / kurtosis normality (Jarque-Bera) Model comparison (Likelihood-ratio) Model quality (Akaike criterion)

Signed-rank	1-sample (Wilcoxon) 2-sample (Mann–Whitney U) 1-way anova (Kruskal–Wallis)

Bayesian inference

Correlation	Pearson product–moment Partial correlation Confounding variable Coefficient of determination

Regression analysis	Errors and residuals Regression model validation Mixed effects models Simultaneous equations models Multivariate adaptive regression splines (MARS)

Linear regression	Simple linear regression Ordinary least squares General linear model Bayesian regression

Non-standard predictors	Nonlinear regression Nonparametric Semiparametric Isotonic Robust Heteroscedasticity Homoscedasticity

Generalized linear model	Exponential families Logistic (Bernoulli) / Binomial / Poisson regressions

Partition of variance	Analysis of variance (ANOVA, anova) Analysis of covariance Multivariate ANOVA Degrees of freedom

Categorical / Multivariate / Time-series / Survival analysis

Categorical

Multivariate

Time-series

General	Decomposition Trend Stationarity Seasonal adjustment Exponential smoothing Cointegration Structural break Granger causality

Specific tests	Dickey–Fuller Johansen Q-statistic (Ljung–Box) Durbin–Watson Breusch–Godfrey

Time domain	Autocorrelation (ACF) partial (PACF) Cross-correlation (XCF) ARMA model ARIMA model (Box–Jenkins) Autoregressive conditional heteroskedasticity (ARCH) Vector autoregression (VAR)

Frequency domain	Spectral density estimation Fourier analysis Wavelet

Survival

Survival function	Kaplan–Meier estimator (product limit) Proportional hazards models Accelerated failure time (AFT) model First hitting time

Hazard function	Nelson–Aalen estimator

Test	Log-rank test

Applications

Biostatistics	Bioinformatics Clinical trials / studies Epidemiology Medical statistics

Engineering statistics	Chemometrics Methods engineering Probabilistic design Process / quality control Reliability System identification

Social statistics	Actuarial science Census Crime statistics Demography Econometrics National accounts Official statistics Population Psychometrics

Spatial statistics	Cartography Environmental statistics Geographic information system Geostatistics Kriging

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