1. Chapter 11Prepared by Samantha Gaies, M.A.1 –Power is based on the Alternative Hypothesis Distribution (AHD) –Usually, the Null Hypothesis Distribution (NHD) is specified exactly, but –the alternative hypothesis is just stated as the complement to the null and is there- fore not specific The Four Major Factors of Power Analysis: –The significance criterion, α –The sample size, N –The population effect size, d –Power, or 1 – β (the probability of rejecting H 0 when it is not true) Chapter 11: Introduction to Power Analysis

2. Chapter 11Prepared by Samantha Gaies, M.A.2 Power for Testing the Mean of a Single Population –Power Determination Specifying an exact H 1 (i.e., µ 1 ) will help to determine d: where µ 0 is the mean of the reference population (i.e., the value of the null hypothesis) If d needs to be estimated, the guidelines are: –d =.2 small effect size –d =.5 medium effect size –d =.8 large effect size

3. Chapter 11Prepared by Samantha Gaies, M.A.3 Power for the Mean of a Single Population (cont.) Once d is determined (or estimated, or guessed at), δ for the one-sample case can be found from this formula: Given α and the number of tails, we can then use a table to look up power. Sample Size Determination –You need to specify d; the value of δ needed for your desired level of power and α (and # of tails) is looked up in a table. –.80 is commonly considered a reasonable level for power. –The size of the sample needed for the desired level of power is found from the following formula:

4. Chapter 11Prepared by Samantha Gaies, M.A.4 Power for the Test of the Proportion of a Single Population –Power determination: Specifying an exact H 1 (i.e., π 1 ) and a null hypothesis proportion ( π 1 ) will allow you to calculate d: –which in turn allows you to find δ, in order to look up power: –If you decide on a desired level of power, you can look up δ, and then calculate the needed sample size:

5. Chapter 11Prepared by Samantha Gaies, M.A.5 The Power of a Test of Pearson’s r –Power Determination: Decide on the value for the population correlation (ρ 1 ) that you expect, and calculate δ, in order to look up power. –Sample Size Determination: If you can decide on a desired level of power, and an expected value for (ρ 1 ), the needed sample size is given by the following formula:

6. Chapter 11Prepared by Samantha Gaies, M.A.6 Power for Testing the Difference between Independent Means –Power Determination: the effect size for the difference of population means is given by: where θ = value of μ 1 – μ 2 specified by your alternative hypothesis (H 1 ), and σ = the population standard deviation (a single value because it is assumed that σ 1 = σ 2 ). –Then, δ is given by the following formula: where n is the size of each of the two groups to be tested. –If you are stuck with unequal ns, use the harmonic mean of your two ns in the above formula (see next slide).

7. Chapter 11Prepared by Samantha Gaies, M.A.7 Power for Testing Two Independent Means (cont.) Sample Size Determination: the n for each of two equal-sized groups needed to attain the power level corresponding to δ is given by the following formula: If you need to estimate d, the following guidelines from Cohen (1988) can be used: –d =.2 small effect size –d =.5 medium effect size –d =.8 large effect size If you need to calculate the harmonic mean of two numbers (e.g., sample sizes), the following formula can be used:

8. Chapter 11Prepared by Samantha Gaies, M.A.8 Power for Testing the Difference between Two Matched Populations Power Determination: where n is the size of each group, d is the same as the effect size for independent means, and ρ is the correlation between the two sets of scores in the population (measures the degree of matching). –As ρ gets closer to +1.0, the term involving ρ gets larger, so δ gets larger, which means power increases. A ρ of.5 increases power as much as doubling the sample sizes. –Sample Size Determination:

9. Chapter 11Prepared by Samantha Gaies, M.A.9 Estimating d for a Power Analysis Involving Independent Means –You can use the sample effect size (g) from a previous similar study to estimate d; g can be found from the t value and n for the previous study: –If the ns in the previous study are not equal, use the following formula, which has the harmonic mean built in: –If you are stuck with two samples of size n, and you want to know the minimum effect size for adequate power, find δ for the power you want and use this formula:

10. Chapter 11Prepared by Samantha Gaies, M.A.10 Interpretation of t Values Significant t obtained with large Ns: implies a smaller effect size than if the same t value were obtained with small Ns Negative (i.e., nonsignificant) results with small Ns: less conclusive and less trustworthy than negative results with large Ns Comparing t values from different studies: –p values tell us nothing automatically about effect sizes. However, with a series of similar tests based on the same Ns, smaller p values will be associated with larger estimates of effect size.

11. Chapter 11Prepared by Samantha Gaies, M.A.11 Four Ways to Manipulate Power –Change sample sizes –Change alpha and/or number of tails –Change the effect size –Add or improve matching between two sets of scores Try this two independent-group example… Let’s say d = 1.47; n 1 = n 2 = 10 How much power does your two- group t test have? For a.05, two-tailed test, power =.91