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We know from even rudimentary treatments of statistical power

analysis that there are four attributes of a statistical test that drive

Power Analysis for Correlational Studies the issue of selecting the sample size needed a particular

analysis...

? acceptable Type I error rate (chance of a "false alarm")

? Remember that both "power" and "stability" are important! ? acceptable Type II error rate (chance of a "miss")

? Useful types of power analyses ? size of the effect being tested for (.1=small, .3=med, .5=large)

- simple correlations

- correlation differences between populations (groups, etc.) ? sample size for that analysis

- differences between correlated correlations We also know that power is not the only basis for determining "N"

- multiple correlation models

- differences between nested multiple correlation models The stability/variability of each r in the correlation matrix is related

- semi-partial and partial correlations to N

- differences between non-nested multiple correlation models

- differences between multiple regression models for different groups Std of r = 1 / (N-3), so ...

- Differences between multiple regression models for different criteria N=50 r +/- .146 N=100 r +/- .101 N=200 r +/- .07

N=300 r +/- .058 N=500 r +/- .045 N=1000 r +/- .031

Power Analysis for Simple Correlation

On the following page is a copy of the power analysis table

from the first portion of the course. Some practice...

Post hoc

I found r (22) = .30, p < .05, what's the chance I made a

Type II error ??

N = 24 Power = .30 Chance Type II error .70

A priori

#1 I expect my correlation will be about .25, & want power = .90

sample size should be = 160

#2 Expect correlations of .30, .45, and .20 from my three

predictors & want power = .80

sample size should be = 191, based on lowest r = .20

Power analysis for correlation differences between populations Power Analysis for Comparing "Correlated Correlations"

? the Bad News

? thias pisaraticvuelrayrwr-eravkaltueestth--arneqtouitreesst rfoour gahclyom2xpathraebNletor-vteaslut efor It takes much more power to test the H0: about correlations

differences than to test the H0: about each r = .00

? the Good News ? Most discussions of power analysis don't include this model

? the test is commonly used, well-understood and tables have

been constructed for our enjoyment (from Cohen, 1988) ? Some sources suggest using the tables designed for comparing

Important! Decide if you are comparing r or |r| values correlations across populations (Fisher's Z-test)

r1 - r2 .10 .20 .30 .40 .50 .60 .70 .80

? Other sources suggest using twice the sample size one would

Power

use if looking for r = the expected r-difference (works out to

.25 333 86 40 24 16 12 10 8

about the same thing as above suggestion)

.50 771 195 88 51 34 24 19 15 ? Each of these depends upon having a good estimate of both

.70 1237 333 140 89 52 37 28 22 correlations, so that the estimate of the correlation

.80 1573 395 177 101 66 47 35 28 difference is reasonably accurate

.90 2602 653 292 165 107 75 56 44 ? It can be informative to consider the necessary sample sizes for

all values for = .05 Values are "S" which is total sample size differences in the estimates of each correlation

Here's an example ...

Suppose you want to compare the correlations of GREQ and

GREA with graduate school performance.

Based on a review of the literature, you expect that...

? GREQ and grad performance will correlate about .4

? GREA and grad performance will correlate about .6

? so you would use the value of r-r = .20 ...

? and get the estimated necessary sample size of N = 395

To consider how important are the estimates of r...

? if the correlations were .35 and .65, then with r-r = .30, N= 177

? if the correlations were .45 and .55, the with r-r=.10, N= 1573

Power Analysis for comparing nested multiple Using the power tables (post hoc) for R? (comparing nested

models) requires that we have four values:

regression models (R?)... a = the p-value we want to use (usually .05)

The good news is that this process is almost the same as was

w = # predictors different between the two models)

the power analysis for R?. Now we need the power of ... u = # predictors associated with the smaller model

v = df associated with F-test error term (N - u - w - 1)

R?L - R?S / kL - ks R?L - R?S N - kl - 1 f? = (effect size estimate) = (R?L - R?S) / (1 - R?L)

F = -------------------------- = --------------- * ------------ = f? * ( u + v + 1) , where

1 - R?L / N - kl - 1 1 - R?L kL - ks

Post Hoc E.g., N = 65, R?L (k=5) = .35, R?S (k=3) = .15

Which, once again, corresponds to: a = .05 w = 2 u = 3 v = 65 - 2 - 3 - 1 = 59

significance test = effect size * sample size f? = .35 - .15 / 1 - .35 = .3077 = .3077 * (3 + 59 + 1) = 19.4

the notation we'll use is ... R2Y-A,B - R2Y-A Go to table -- a = .05 & u = 3 = 20

-- testing the contribution of the "B" set of variables power about .97 v = 60 .97

a priori power analyses for nested model comparisons are probably most easily

done using the "what if " approach

I expect that my 4-predictor model will account for about 12% of the variance in

the criterion and that including an additional 3 variables will increase the R? to

about .18 -- what sample size should I use ???

a = .05 w = 3 u = 4 f? = (RL? - RS?) / (1 - R?) = (.18 - .12) / (1 - .18) = .073

"what if.." N = 28 N = 68 N = 128 N = 208 ()

v = (N - u - w - 1) = 20 60 120 200 ()

= f? * ( u + v + 1) = 1.83 4.75 9.13 15.0

Using the table...

power = < .15 about .37 about .64 about .89

If we were looking for power of .80, we might then try N = 158

so v = 150, = 11.3 power = about .77 (I'd go with N = 180 or so)

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