Power Analysis For One-Way Anova, Independent Samples, SAS PDF

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Power Analysis For One-Way Anova, Independent Samples, SAS...

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Power Analysis For One-Way ANOVA, Independent Samples, SAS Background Although Proc Power has a “onewayanova” option, I prefer to approach this analysis as a multiple regression. If you use the “onewayanova” option, you must specify the effect size in terms of group means and standard deviations. Psychologists are accustomed to reporting the effect size in terms of Cohen’s f. f2 . Cohen considered an f of .10 1+ f 2 to be a small effect, .25 a medium effect, and .40 a large effect. These benchmarks correspond to η2 values of .0099, .0588, and .1379. The one-way ANOVA is no different from a multiple regression analysis where the predictors are k-1 dummy variables, where k is the number of groups in the analysis. The R2 from such a multiple regression analysis is identical to η2 in ANOVA. Solving for Sample Size Suppose that we wish to determine how many subjects we need to have 80% power for an ANOVA with three groups. Assume that we are using the usual .05 criterion of statistical significance. Here is the SAS code and output: Cohen’s f can be converted into η2 : η 2 =

proc power; multreg model = fixed nfullpred = 2 nredpred= 0 RsqFull = 0.0099, 0.0588, .1379 RsqRed = 0 ntotal = . power = 0.8; run;

Notice that I specified “fixed.” Power depends, in part, on whether the predictor variables are fixed or random. I nearly all cases psychologists assume that the predictors in ANOVA are fixed. Since we have three groups, we have 3-1 = 2 predictors (the treatment df from the ANOVA), . We are testing our model versus a model (the null hypothesis) that has no parameters other than the intercept (the overall mean), so nredpred= 0. As values for η2 (RsqFull), I have entered those corresponding to Cohen’s benchmarks for small, medium, and large. The value of RsqRed is that of the null hypothesis, 0. Since I want to solve for sample size, I specify “.” for ntotal. The POWER Procedure Type III F Test in Multiple Regression Fixed Scenario Elements Method Model Number of Predictors in Full Model Number of Predictors in Reduced Model R-square of Reduced Model

Exact Fixed X 2 0 0

Nominal Power Alpha

0.8 0.05

Computed N Total

Index

R-square Full

Actual Power

N Total

1 2 3

0.0099 0.0588 0.1379

0.800 0.802 0.805

967 158 64

Let’s compare our solution with that provided by G*Power. F tests - ANOVA: Fixed effects, omnibus, one-way Analysis: A priori: Compute required sample size Input:

Output:

Effect size f

= 0.1

α err prob

= 0.05

Power (1-β err prob)

= 0.8

Number of groups

= 3

Noncentrality parameter λ

= 9.6900000

Critical F

= 3.0050418

Numerator df

= 2

Denominator df

= 966

Total sample size

= 969

Actual power

= 0.8011010

SAS told us we need 967 subjects. That is 967/3 = 322.3 per group. Rounding up to 323 per group and multiplying by 3 give us the 969 reported by G*Power. Input:

Output:

Effect size f

= 0.25

α err prob

= 0.05

Power (1-β err prob)

= 0.8

Number of groups

= 3

Noncentrality parameter λ

= 9.9375000

Critical F

= 3.0540042

Numerator df

= 2

Denominator df

= 156

Total sample size

= 159

Actual power

= 0.8048873

SAS told us 158 subjects, = 52.67 per group; 3(53) = 159, as reported by G*Power. Input:

Effect size f

= 0.4

α err prob

= 0.05

Power (1-β err prob)

= 0.8

Number of groups

= 3

Output:

Noncentrality parameter λ

= 10.5600000

Critical F

= 3.1428085

Numerator df

= 2

Denominator df

= 63

Total sample size

= 66

Actual power

= 0.8180744

SAS told us 64 subjects, = 21.3 per group; 3(22) = 66, as reported by G*Power. Suppose we intend to have five groups. Changing one line of SAS code, we obtain:

nfullpredictors = 4,

Computed N Total

Input:

Output:

Input:

Output:

Input:

Output:

Index

R-square Full

Actual Power

N Total

1 2 3

0.0099 0.0588 0.1379

0.800 0.800 0.803

1199 196 80

Effect size f

= 0.1

α err prob

= 0.05

Power (1-β err prob)

= 0.8

Number of groups

= 5

Noncentrality parameter λ

= 12.0000000

Critical F

= 2.3793764

Numerator df

= 4

Denominator df

= 1195

Total sample size

= 1200

Actual power

= 0.8006464

Effect size f

= 0.25

α err prob

= 0.05

Power (1-β err prob)

= 0.8

Number of groups

= 5

Noncentrality parameter λ

= 12.5000000

Critical F

= 2.4179625

Numerator df

= 4

Denominator df

= 195

Total sample size

= 200

Actual power

= 0.8097710

Effect size f

= 0.4

α err prob

= 0.05

Power (1-β err prob)

= 0.8

Number of groups

= 5

Noncentrality parameter λ

= 12.8000000

Critical F

= 2.4936960

Numerator df

= 4

Denominator df

= 75

Total sample size

= 80

Actual power

= 0.8030845

As you can see, the SAS analysis is equivalent to that provided by G*Power. Solving for Power Suppose that we have four groups with 100 subjects in each group. What is power? proc power; multreg model = fixed nfullpredictors = 3 nredpred= 0 RsqFull = 0.0099, 0.0588, .1379 RsqRed = 0 ntotal = 400 power = .; run; Computed Power

Index

R-square Full

Power

1 2 3

0.0099 0.0588 0.1379

0.355 0.993 >.999

F tests - ANOVA: Fixed effects, omnibus, one-way Analysis: Post hoc: Compute achieved power Input:

Output:

Input:

Output:

Input:

Effect size f

= 0.1

α err prob

= 0.05

Total sample size

= 400

Number of groups

= 4

Noncentrality parameter λ

= 4.0000000

Critical F

= 2.6274408

Numerator df

= 3

Denominator df

= 396

Power (1-β err prob)

= 0.3552592

Effect size f

= 0.25

α err prob

= 0.05

Total sample size

= 400

Number of groups

= 4

Noncentrality parameter λ

= 25.0000000

Critical F

= 2.6274408

Numerator df

= 3

Denominator df

= 396

Power (1-β err prob)

= 0.9928175

Effect size f

= 0.4

α err prob

= 0.05

Output:

Total sample size

= 400

Number of groups

= 4

Noncentrality parameter λ

= 64.0000000

Critical F

= 2.6274408

Numerator df

= 3

Denominator df

= 396

Power (1-β err prob)

= 1.0000000

Return to Wuensch’s SAS Lessons Page Karl L. Wuensch, Dept. of Psychology, East Carolina Univ., Greenville, NC 27858 USA February, 2010...