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UNL PSYC 942 - Power Analysis for Correlation

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Power Analysis for Correlation & Multiple RegressionSlide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Power Analysis for Correlation & Multiple Regression•Sample Size & multiple regression•Subject-to-variable ratios•Stability of correlation values•Useful types of power analyses–Simple correlations–Full multiple regression•Considering Stability & Power•Sample size for a studySample Size & Multiple RegressionThe general admonition that “larger samples are better” has considerable merit, but limited utility…• R² will always be 1.00 if k = N-1 (it’s a math thing)• R² will usually be “too large” if the sample size is “too small” (same principle but operating on a lesser scale)• R² will always be larger on the modeling sample than on any replication using the same regression weights• R² & b-values will replicate better or poorer, depending upon the stability of the correlation matrix values• R² & b-values of all predictors may vary with poor stability of any portion of the correlation matrix (any subset ofpredictors)• F- & t-test p-values will vary with the stability & power of the sample size – both modeling and replication samplesSubject-to-Variable RatioHow many participants should we have for a given number of predictors? -- usually refers to the full modelThe subject/variable ratio has been an attempt to ensure that the sample is “large enough” to minimize “parameter inflation” and improve “replicability”. Here are some common admonitions..• 20 participants per predictor• a minimum of 100 participants, plus 10 per predictor• 10 participants per predictor• 200 participants for up to k=10 predictors and 300 if k>10• 1000 participants per predictor• a minimum of 2000 participants, + 1000 for each 5 predictorsAs is often the case, different rules of thumb have grown out of different research traditions, for example…• chemistry, which works with very reliable measures and stable populations, calls for very small s/v ratios • biology, also working largely with “real measurements” (length, weight, behavioral counts) often calls for small s/v ratios• economics, fairly stable measures and very large (cheap) databases often calls for huge s/v ratios• education, often under considerable legal and political scrutiny, (data vary in quality) often calls for fairly large s/v ratios• psychology, with self-report measures of limited quality, but costly data-collection procedures, often “shaves” the s/v ratio a bitProblems with Subject-to-variable ratio #1 neither n, N nor N/k is used to compute R² or b-values• R² & b/-values are computed from the correlation matrix• N is used to compute the significance test of the R² & each b-weight#2 Statistical Power Analyses involves more than N & kWe know from even rudimentary treatments of statistical power analysis that there are four attributes of a statistical test that are inextricably intertwined for the purposes of NHST…• acceptable Type I error rate (chance of a “false alarm”)• acceptable Type II error rate (chance of a “miss”)• size of the effect being tested for• sample sizeWe will “forsake” the subjects-to-variables ratio for more formal power analyses & also consider the stability of parameter estimates (especially when we expect large effect sizes).NHST Power “vs.” Parameter estimate stabilityStability  how much error is there in the sample-based estimate of a parameter (correlation, regression weight, etc.) ?Stability is based on …• “quality” of the sample (sampling process & attrition)• sample sizeStd of r = 1 /  (N-3), so … N=50 r +/- .146 N=100 r +/- .101 N=200 r +/- .07 N=300 r +/- .058 N=500 r +/- .045 N=1000 r +/- .031NHST power  what’s the chances of rejecting a “false null” vs. making a Type II error?Statistical power is based on…• size of the effect involved (“larger effects are easier to find”)• amount of power (probability of rejecting H0: if effect size is as expected or larger)Partial Power Table (taken & extrapolated from Friedman, 1982) r .15 .20 .25 .30 .35 .40 .45 .50 .55 .60 .65 .70power .30 93 53 34 24 18 14 11 9 8 7 6 5 .40 132 74 47 33 24 19 15 12 10 8 7 6 .50 170 95 60 42 30 23 18 14 12 9 8 7 .60 257 143 90 62 45 34 24 20 16 13 11 9 .70 300 167 105 72 52 39 29 23 18 15 12 10 .80 343 191 120 82 59 44 33 26 20 16 13 11 .90 459 255 160 109 78 58 44 34 27 21 17 13“Sufficient power” but “poor stability”How can a sample have “sufficient power” but “poor stability”?Notice it happens for large effect sizes!! e.g., For a population with r = .30 & a sample of 100 …• Poor stability of r estimate  +/- 1 std is .20-.40• Large enough to reject H0: that r = 0  power almost .90The power table only tells us the sample size we need to reject H0: r=0!! It does not tell us the sample size we need to have a good estimate of the population r !!!!!Power Analysis for Simple CorrelationPost hoc I found r (22) = .30, p > .05, what’s the chance I made a Type II error ??N = Power = Chance Type II errorA priori#1 I expect my correlation will be about .25, & want power = .90sample size should be = #2 Expect correlations of .30, .45, and .20 from my three predictors & want power = .80 sample size should be = 24 .30.70160191, based on lowest r = .20Power Analysis for Multiple regression Power analysis for multiple regression is about the same as for simple regression, we decide on values for some parameters and then we consult a table …Remember the F-test of H0: R² = 0 ?? R² / k R² N-k-1 F = -------------------- = ---------- * --------- 1-R² / N - k - 1 1 - R² kWhich corresponds to: significance test = effect size * sample size So, our power analysis will be based not on R² per se, but on the power of the F-test of the H0: R² = 0Using the power tables (post hoc) for multiple regression (single model) requires that we have four values:a = the


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