How to increase the power of a test statistics
|
If you're seeing this message, it means we're having trouble loading external resources on our website. Show
If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. The statistical power of a hypothesis test is the probability of detecting an effect, if there is a true effect present to detect. Power can be calculated and reported for a completed experiment to comment on the confidence one might have in the conclusions drawn from the results of the study. It can also be used as a tool to estimate the number of observations or sample size required in order to detect an effect in an experiment. In this tutorial, you will discover the importance of the statistical power of a hypothesis test and now to calculate power analyses and power curves as part of experimental design. After completing this tutorial, you will know:
Kick-start your project with my new book Statistics for Machine Learning, including step-by-step tutorials and the Python source code files for all examples. Let’s get started.  A Gentle Introduction to Statistical Power and Power Analysis in Python Tutorial OverviewThis tutorial is divided into four parts; they are:
Need help with Statistics for Machine Learning?Take my free 7-day email crash course now (with sample code). Click to sign-up and also get a free PDF Ebook version of the course. Download Your FREE Mini-Course Statistical Hypothesis TestingA statistical hypothesis test makes an assumption about the outcome, called the null hypothesis. For example, the null hypothesis for the Pearson’s correlation test is that there is no relationship between two variables. The null hypothesis for the Student’s t test is that there is no difference between the means of two populations. The test is often interpreted using a p-value, which is the probability of observing the result given that the null hypothesis is true, not the reverse, as is often the case with misinterpretations.
In interpreting the p-value of a significance test, you must specify a significance level, often referred to as the Greek lower case letter alpha (a). A common value for the significance level is 5% written as 0.05. The p-value is interested in the context of the chosen significance level. A result of a significance test is claimed to be “statistically significant” if the p-value is less than the significance level. This means that the null hypothesis (that there is no result) is rejected.
Where:
We can see that the p-value is just a probability and that in actuality the result may be different. The test could be wrong. Given the p-value, we could make an error in our interpretation. There are two types of errors; they are:
In this context, we can think of the significance level as the probability of rejecting the null hypothesis if it were true. That is the probability of making a Type I Error or a false positive. What Is Statistical Power?Statistical power, or the power of a hypothesis test is the probability that the test correctly rejects the null hypothesis. That is, the probability of a true positive result. It is only useful when the null hypothesis is rejected.
— Page 60, The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results, 2010. The higher the statistical power for a given experiment, the lower the probability of making a Type II (false negative) error. That is the higher the probability of detecting an effect when there is an effect. In fact, the power is precisely the inverse of the probability of a Type II error. 1 2 Power = 1 - Type II Error Pr(True Positive) = 1 - Pr(False Negative) More intuitively, the statistical power can be thought of as the probability of accepting an alternative hypothesis, when the alternative hypothesis is true. When interpreting statistical power, we seek experiential setups that have high statistical power.
Experimental results with too low statistical power will lead to invalid conclusions about the meaning of the results. Therefore a minimum level of statistical power must be sought. It is common to design experiments with a statistical power of 80% or better, e.g. 0.80. This means a 20% probability of encountering a Type II area. This different to the 5% likelihood of encountering a Type I error for the standard value for the significance level. Power AnalysisStatistical power is one piece in a puzzle that has four related parts; they are:
All four variables are related. For example, a larger sample size can make an effect easier to detect, and the statistical power can be increased in a test by increasing the significance level. A power analysis involves estimating one of these four parameters given values for three other parameters. This is a powerful tool in both the design and in the analysis of experiments that we wish to interpret using statistical hypothesis tests. For example, the statistical power can be estimated given an effect size, sample size and significance level. Alternately, the sample size can be estimated given different desired levels of significance.
— Page 56, The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results, 2010. Perhaps the most common use of a power analysis is in the estimation of the minimum sample size required for an experiment.
— Page 57, The Essential Guide to Effect Sizes: Statistical Power, Meta-Analysis, and the Interpretation of Research Results, 2010. As a practitioner, we can start with sensible defaults for some parameters, such as a significance level of 0.05 and a power level of 0.80. We can then estimate a desirable minimum effect size, specific to the experiment being performed. A power analysis can then be used to estimate the minimum sample size required. In addition, multiple power analyses can be performed to provide a curve of one parameter against another, such as the change in the size of an effect in an experiment given changes to the sample size. More elaborate plots can be created varying three of the parameters. This is a useful tool for experimental design. Student’s t Test Power AnalysisWe can make the idea of statistical power and power analysis concrete with a worked example. In this section, we will look at the Student’s t test, which is a statistical hypothesis test for comparing the means from two samples of Gaussian variables. The assumption, or null hypothesis, of the test is that the sample populations have the same mean, e.g. that there is no difference between the samples or that the samples are drawn from the same underlying population. The test will calculate a p-value that can be interpreted as to whether the samples are the same (fail to reject the null hypothesis), or there is a statistically significant difference between the samples (reject the null hypothesis). A common significance level for interpreting the p-value is 5% or 0.05.
The size of the effect of comparing two groups can be quantified with an effect size measure. A common measure for comparing the difference in the mean from two groups is the Cohen’s d measure. It calculates a standard score that describes the difference in terms of the number of standard deviations that the means are different. A large effect size for Cohen’s d is 0.80 or higher, as is commonly accepted when using the measure.
We can use the default and assume a minimum statistical power of 80% or 0.8.
For a given experiment with these defaults, we may be interested in estimating a suitable sample size. That is, how many observations are required from each sample in order to at least detect an effect of 0.80 with an 80% chance of detecting the effect if it is true (20% of a Type II error) and a 5% chance of detecting an effect if there is no such effect (Type I error). We can solve this using a power analysis. The statsmodels library provides the TTestIndPower class for calculating a power analysis for the Student’s t test with independent samples. Of note is the TTestPower class that can perform the same analysis for the paired Student’s t test. The function solve_power() can be used to calculate one of the four parameters in a power analysis. In our case, we are interested in calculating the sample size. We can use the function by providing the three pieces of information we know (alpha, effect, and power) and setting the size of argument we wish to calculate the answer of (nobs1) to “None“. This tells the function what to calculate. A note on sample size: the function has an argument called ratio that is the ratio of the number of samples in one sample to the other. If both samples are expected to have the same number of observations, then the ratio is 1.0. If, for example, the second sample is expected to have half as many observations, then the ratio would be 0.5. The TTestIndPower instance must be created, then we can call the solve_power() with our arguments to estimate the sample size for the experiment. 1 2 3 # perform power analysis analysis = TTestIndPower() result = analysis.solve_power(effect, power=power, nobs1=None, ratio=1.0, alpha=alpha) The complete example is listed below. 1 2 3 4 5 6 7 8 9 10 # estimate sample size via power analysis from statsmodels.stats.power import TTestIndPower # parameters for power analysis effect = 0.8 alpha = 0.05 power = 0.8 # perform power analysis analysis = TTestIndPower() result = analysis.solve_power(effect, power=power, nobs1=None, ratio=1.0, alpha=alpha) print('Sample Size: %.3f' % result) Running the example calculates and prints the estimated number of samples for the experiment as 25. This would be a suggested minimum number of samples required to see an effect of the desired size. 1 Sample Size: 25.525 We can go one step further and calculate power curves. Power curves are line plots that show how the change in variables, such as effect size and sample size, impact the power of the statistical test. The plot_power() function can be used to create power curves. The dependent variable (x-axis) must be specified by name in the ‘dep_var‘ argument. Arrays of values can then be specified for the sample size (nobs), effect size (effect_size), and significance (alpha) parameters. One or multiple curves will then be plotted showing the impact on statistical power. For example, we can assume a significance of 0.05 (the default for the function) and explore the change in sample size between 5 and 100 with low, medium, and high effect sizes. 1 2 3 # calculate power curves from multiple power analyses analysis = TTestIndPower() analysis.plot_power(dep_var='nobs', nobs=arange(5, 100), effect_size=array([0.2, 0.5, 0.8])) The complete example is listed below. 1 2 3 4 5 6 7 8 9 10 11 # calculate power curves for varying sample and effect size from numpy import array from matplotlib import pyplot from statsmodels.stats.power import TTestIndPower # parameters for power analysis effect_sizes = array([0.2, 0.5, 0.8]) sample_sizes = array(range(5, 100)) # calculate power curves from multiple power analyses analysis = TTestIndPower() analysis.plot_power(dep_var='nobs', nobs=sample_sizes, effect_size=effect_sizes) pyplot.show() Running the example creates the plot showing the impact on statistical power (y-axis) for three different effect sizes (es) as the sample size (x-axis) is increased. We can see that if we are interested in a large effect that a point of diminishing returns in terms of statistical power occurs at around 40-to-50 observations. Power Curves for Student’s t Test Usefully, statsmodels has classes to perform a power analysis with other statistical tests, such as the F-test, Z-test, and the Chi-Squared test. ExtensionsThis section lists some ideas for extending the tutorial that you may wish to explore.
If you explore any of these extensions, I’d love to know. Further ReadingThis section provides more resources on the topic if you are looking to go deeper. Papers
Books
API
Articles
SummaryIn this tutorial, you discovered the statistical power of a hypothesis test and how to calculate power analyses and power curves as part of experimental design. What can be done to increase the statistical power of a test?How do you increase power?. Increase the effect size. ... . Increase sample size. ... . Increase the significance level. ... . Reduce measurement error. ... . Use a one-tailed test instead of a two-tailed test.. What factors increase power in statistics?FACTORS AFFECTING POWER
The 4 primary factors that affect the power of a statistical test are a level, difference between group means, variability among subjects, and sample size.
What makes a test more powerful?The difference between the hypothesized value of a parameter and its true value. This is sometimes called the “magnitude of the effect” in the case when the parameter of interest is the difference between parameter values (say, means) for two treatment groups. The larger the effect, the more powerful the test is.
How can power be increased?Use a larger sample. Using a larger sample provides more information about the population and, thus, increase power. Using a larger sample is often the most practical way to increase power.
|
