Power and sample size
1. Power and sample size
Power is an important aspect to consider.2. Power defined
Power is the probability of correctly rejecting the null hypothesis when the null is false, or not making a type II error of not rejecting the null when it should be. For example, the probability that there is a difference in enjoyment of the Cheese and Pepperoni pizza, given we found a significant difference. Ideally, an experiment has a small probability of a type II error and large power.3. Power benefits
Power can determine whether a test will be useful and likely to reject the null hypothesis, provided the null should be rejected. We can use the expected power to determine the sample size needed before collecting data. With too few subjects, the analysis will not have the support to reject the hypothesis regardless of if it should be rejected, wasting time and funding on an experiment unable to answer the hypothesis. We can also determine the power of a test after it has been run, aiding in the trust given to the test results.4. Sample size
A smaller sample produces a more narrow distribution, requiring a larger test value to reject the null, and greater power and effect size. To determine the sample size needed for a useful experiment, the three values needed are the estimated effect size, estimated power, point-eight is commonly large power, and the decision criteria alpha, commonly point-zero-five. The pwr package is used to determine sample size, inputting the effect size as d, power as power, alpha as sig-dot-level, type of t-test as type, and tails as alternative. For a two-tailed, one-sample t-test with a large effect size of point-eight, power of point-eight, and alpha of point-zero-five, 15 subjects are needed to test one sample of pizza enjoyment. Always round up to a whole number of subjects.5. Effect size
Effect size is an indicator of the difference from the null hypothesis that is detected, or the expected size of the effect. It is commonly calculated by taking the difference between the mean of the control group and the experimental group. Prior to an analysis, effect sizes can be determined with background information from similar studies, or calculated using preliminary data. After an analysis, effect size can be calculated with the full data set.6. Power analysis of test
A power analysis of a test aids in determining if the results of the experiment can be trusted. A test with higher power denotes a higher probability of correctly rejecting the null hypothesis. Three aspects are needed to run a power analysis: sample size, effect size, and alpha. Suppose our one-sample t-test of pizza enjoyment with twenty subjects had a significance level, or alpha, of point-zero-four-five and effect size of point-eight-one. The t-test power is point-nine-two-two, meaning we can trust our results.7. Pizza distributions
Power gives the probability that the alternative hypothesis is true if a significant difference is found between groups. In our Cheese and Pepperoni pizza study, the distributions may end up similar and have no significant difference, indicating the groups derive from the same population distribution or may be significantly different, indicating they derive from different distributions.8. Pizza hypotheses
Provided the topping distributions are similar, the group difference would be small and fall under the blue null distribution. If the distributions are significantly different, the group difference will fall beyond the vertical line, representing the rejection value, into the pink alternative distribution.9. Pizza power
If a significant difference of the distributions is found, the result lies beyond the rejection value, into the region contributing to power or the probability of not incorrectly accepting the null, or making a Type II error.10. Let's practice!
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