Introduction to PowerComputing PowerIntroduction to Power Computing PowerPower AnalysisDr. J. Kyle RobertsSouthern Methodist UniversitySimmons School of Education and Human DevelopmentDepartment of Teaching and LearningIntroduction to Power Computing PowerIntroduction to Power•When testing a specific null hypothesis (H0), we have fourpossible outcomes.True StateDecision Made H0= True H0= FalseReject Type I error Correct DecisionFail to Reject Correct Decision Type II error•In the above table, α is referred to as our Type I error rate.•Therefore, if we set α = 0.05, then we have a 5% chance ofmaking a Type I error.Introduction to Power Computing PowerPower Definition cont.•From the table on the previous slide, we call the probability ofmaking a Type II error, β.•Based on these definitions for α and β, we then define poweras the probability of rejecting the H0when it is false, or(1 − β).•Recall that this is specifically what we often want to do.Power is making a correct decision.•Therefore, we maximize power when we minimize β.•Power will have bounds 0 ≤ (1 − β) ≤ 1 with larger valuesmeaning more power. These can also be thought of as apercent.Introduction to Power Computing PowerFactors Affecting Power1. α Level•As the α level increases (say from 0.05 to 0.10), β willdecrease, meaning that power will go up (1 − β).•Conversely, as the α level decreases (say from 0.05 to 0.01), βwill increase, meaning that power will go down (1 − β).2. Sample Size•As sample size increases, the standard error of measurementwill likewise decrease, all other things being equal sinceσ¯X= σX/√n.3. Effect Size•As the effect size increases, so does the power of the analysis.•All other things remaining constant, as the size of the effectincreases, the relative power associated with rejecting the nullalso increases.Introduction to Power Computing PowerPower for the Single Sample t test•Suppose that we have a hypothetical dataset where µ = 100,¯X = 105, SDX= 15, and n = 30.•In this case we can compute t as:t =¯X − µσ/√n=105 − 10015/√30= 1.826•We can then compute the probability of t given our data andlikewise β as:> pt(qt(0.975, 29), 29, 1.826)[1] 0.5770932Introduction to Power Computing PowerGraphical Representation of Heuristic Data> curve(pt(x, 29, ncp = 1.826), from = 0, to = 6)> abline(v = qt(0.975, 29))0 1 2 3 4 5 60.0 0.2 0.4 0.6 0.8 1.0xpt(x, 29, ncp = 1.826)Introduction to Power Computing PowerComputing Power•In order to compute power, we need to know a few thingsabout our data.1. n2. SD3. Effect Size (or difference between means)4. α level•Knowing these four things will then let you compute themissing parameter (1 − β), or power.> power.t.test(delta = 5, sd = 15, n = 30, sig.level = 0.05,+ type = "one.sample")One-sample t test power calculationn = 30delta = 5sd = 15sig.level = 0.05power = 0.4228091alternative = two.sidedIntroduction to Power Computing PowerComputing n based on Power•Suppose that you are conducting a single sample experimentand you want to know how many people you should sample inorder to attain a power of 0.85.•We can compute this by the work backwards principle since nis the only missing piece of data.> power.t.test(delta = 5, sd = 15, power = 0.85, sig.level = 0.05,+ type = "one.sample")One-sample t test power calculationn = 82.74835delta = 5sd = 15sig.level = 0.05power = 0.85alternative = two.sided•This means that we need to sample 83 people to obtain apower of 0.85 given our study parameters.Introduction to Power Computing PowerWays of Improving Power•Reduce α from 0.05 to say 0.10.•Consider methods for reducing the within-group variability(thus increasing the effect size).•Increase the number of individuals in each group.Introduction to Power Computing PowerPower for the One-Way ANOVA•For the ANOVA, we can calculate relative group sample sizegiven our estimated power, α, effect-size, and K.> library(pwr)> pwr.anova.test(f = 0.28, k = 4, sig.level = 0.05,+ power = 0.8)Balanced one-way analysis of variance power calculationk = 4n = 35.75789f = 0.28sig.level = 0.05power = 0.8NOTE: n is number in each groupIntroduction to Power Computing PowerPost-Hoc Power for the One-Way ANOVA•Likewise for the ANOVA, we can calculate post-hoc powerbased on α, effect-size, the average n, and K.> pwr.anova.test(f = 0.28, k = 4, n = 20, sig.level = 0.05)Balanced one-way analysis of variance power calculationk = 4n = 20f = 0.28sig.level = 0.05power = 0.5149793NOTE: n is number in each groupIntroduction to Power Computing PowerOther Types of Tests•For all types of statistical tests, we can compute power or theneeded sample size based on desired power.•Almost all of the power calculations for statistical tests youwill run are covered in Cohen (1988) Statistical PowerAnalysis for the Behavioral