## Power Analysis

### Uses

• retrospective: to judge what the present analysis died of
• prospective: to provide a basis for decisions on experimental design of a planned study by suggesting for example the minimum sample size necessary for driving down the error variance sufficiently so a given efect size may be picked up statistically at a given a (i.e., least significant number)

### How this is done

When drawing conclusions from the results of a statistical analysis we can commit one of two errors:

• Alpha error: false positive, reject Ho when you should not reject it
• Beta error: false negative, fail to reject Ho when you actual should reject it

Power analysis provides a quantitative method to estimate the probability of obtaining a significant results in a given situation (i.e., rejecting the null hypothesis when the alternative hypothesis is actually true). Power analysis thus allows you to judge how likely it is that you will detect a treatment effect that really exists. In summary, power measures the probability of replicating your statistical decision given a repeated use of the same experimental design and sampling procedure. Power is affected by:

• Level of Significance (a): changing a is not a useful choice for fending off low power. By changing a you are not making better decisions, you are just substituting one error (a) for the other (b).
• Raw Effect Size (d): Predicting true effect size is not trivial. You may use a combination of approaches, including empirical data from the related literature, pilot data, or a general consideration of the smallest effects that you would regard as biologically meaningful in your scenario. Within an ANOVA setting you can obtain an estimate for d as the square root of your model SS divided by total sample size (i.e., SQRT [SSM / N] )
• Standard Deviation of the Residuals (s): The power of your analysis increases as the error variance decreases. Towards that goal you can reduce treatment variability or control for confounding factors. Obtain an estimate for s in an ANOVA setting as the square root of the error variance (MS error).
• Sample Size (N): Standard deviation of the random error decreases with increasing sample size.