# ss.power.reg.coef: sample size for a targeted regression coefficient In MBESS: The MBESS R Package

## Description

Determine the necessary sample size for a targeted regression coefficient or determine the degree of power given a specified sample size

## Usage

 ```1 2 3 4 5``` ```ss.power.reg.coef(Rho2.Y_X = NULL, Rho2.Y_X.without.j = NULL, p = NULL, desired.power = 0.85, alpha.level = 0.05, Directional = FALSE, beta.j = NULL, sigma.X = NULL, sigma.Y = NULL, Rho2.j_X.without.j = NULL, RHO.XX = NULL, Rho.YX = NULL, which.predictor = NULL, Cohen.f2 = NULL, Specified.N=NULL, Print.Progress = FALSE) ```

## Arguments

 `Rho2.Y_X` population squared multiple correlation coefficient predicting the dependent variable (i.e., Y) from the `p` predictor variables (i.e., the X variables) `Rho2.Y_X.without.j` population squared multiple correlation coefficient predicting the dependent variable (i.e., Y) from the `p`-1 predictor variables, where the one not used is the predictor of interest `p` number of predictor variables `desired.power` desired degree of statistical power for the test of targeted regression coefficient `alpha.level` Type I error rate `Directional` whether or not a direction or a nondirectional test is to be used (usually `directional=FALSE`) `beta.j` population value of the regression coefficient for the predictor of interest `sigma.X` population standard deviation for the predictor variable of interest `sigma.Y` population standard deviation for the outcome variable `Rho2.j_X.without.j` population squared multiple correlation coefficient predicting the predictor variable of interest from the remaining p-1 predictor variables `RHO.XX` population correlation matrix for the `p` predictor variables `Rho.YX` population vector of correlation coefficient between the `p` predictor variables and the criterion variable `Cohen.f2` Cohen's (1988) definition for an effect size for a targeted regression coefficient: `(Rho2.Y_X-Rho2.Y_X.without.j)/(1-Rho2.Y_X)` `which.predictor` identifies the predictor of interest when `RHO.XX` and `Rho.YX` are specified `Specified.N` sample size for which power should be evaluated `Print.Progress` if the progress of the iterative procedure is printed to the screen as the iterations are occurring

## Details

Determines the necessary sample size given a desired level of statistical power. Alternatively, determines the statistical power for a given a specified sample size. There are a number of ways that the specification regarding the size of the regression coefficient can be entered. The most basic, and often the simplest, is to specify `Rho2.Y_X` and `Rho2.Y_X.without.j`. See the examples section for several options.

## Value

 `Sample.Size` either the necessary sample size or the specified sample size, depending if one is interested in determining the necessary sample size given a desired degree of statistical power or if one is interested in the determining the value of statistical power given a specified sample size, respectively `Actual.Power` Actual power of the situation described `Noncentral.t.Parm` value of the noncentral distribution for the appropriate t-distribution `Effect.Size.NC.t` effect size for the noncentral t-distribution; this is the square root of `Cohen.f2`, because `Cohen.f2` is the effect size using an F-distribution

## Author(s)

Ken Kelley (University of Notre Dame; [email protected])

## References

Cohen, J. (1988). Statistical power analysis for the behavioral sciences (2nd ed.). Hillsdale, NJ: Erlbaum.

Kelley, K. & Maxwell, S. E. (2008). Sample Size Planning with applications to multiple regression: Power and accuracy for omnibus and targeted effects. In P. Alasuuta, J. Brannen, & L. Bickman (Eds.), The Sage handbook of social research methods (pp. 166–192). Newbury Park, CA: Sage.

Maxwell, S. E. (2000). Sample size for multiple regression. Psychological Methods, 4, 434–458.

`ss.aipe.reg.coef`, `ss.power.R2`, `conf.limits.ncf`
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71``` ```Cor.Mat <- rbind( c(1.00, 0.53, 0.58, 0.60, 0.46, 0.66), c(0.53, 1.00, 0.35, 0.07, 0.14, 0.43), c(0.58, 0.35, 1.00, 0.18, 0.29, 0.50), c(0.60, 0.07, 0.18, 1.00, 0.30, 0.26), c(0.46, 0.14, 0.29, 0.30, 1.00, 0.30), c(0.66, 0.43, 0.50, 0.26, 0.30, 1.00)) RHO.XX <- Cor.Mat[2:6,2:6] Rho.YX <- Cor.Mat[1,2:6] # Method 1 # ss.power.reg.coef(Rho2.Y_X=0.7826786, Rho2.Y_X.without.j=0.7363697, p=5, # alpha.level=.05, Directional=FALSE, desired.power=.80) # Method 2 # ss.power.reg.coef(alpha.level=.05, RHO.XX=RHO.XX, Rho.YX=Rho.YX, # which.predictor=5, # Directional=FALSE, desired.power=.80) # Method 3 # Here, beta.j is the standardized regression coefficient. Had beta.j # been the unstandardized regression coefficient, sigma.X and sigma.Y # would have been the standard deviation for the # X variable of interest and Y, respectively. # ss.power.reg.coef(Rho2.Y_X=0.7826786, Rho2.j_X.without.j=0.3652136, # beta.j=0.2700964, # p=5, alpha.level=.05, sigma.X=1, sigma.Y=1, Directional=FALSE, # desired.power=.80) # Method 4 # ss.power.reg.coef(alpha.level=.05, Cohen.f2=0.2130898, p=5, # Directional=FALSE, # desired.power=.80) # Power given a specified N and squared multiple correlation coefficients. # ss.power.reg.coef(Rho2.Y_X=0.7826786, Rho2.Y_X.without.j=0.7363697, # Specified.N=25, # p=5, alpha.level=.05, Directional=FALSE) # Power given a specified N and effect size. # ss.power.reg.coef(alpha.level=.05, Cohen.f2=0.2130898, p=5, Specified.N=25, # Directional=FALSE) # Reproducing Maxwell's (2000, p. 445) Example Cor.Mat.Maxwell <- rbind( c(1.00, 0.35, 0.20, 0.20, 0.20, 0.20), c(0.35, 1.00, 0.40, 0.40, 0.40, 0.40), c(0.20, 0.40, 1.00, 0.45, 0.45, 0.45), c(0.20, 0.40, 0.45, 1.00, 0.45, 0.45), c(0.20, 0.40, 0.45, 0.45, 1.00, 0.45), c(0.20, 0.40, 0.45, 0.45, 0.45, 1.00)) RHO.XX.Maxwell <- Cor.Mat.Maxwell[2:6,2:6] Rho.YX.Maxwell <- Cor.Mat.Maxwell[1,2:6] R2.Maxwell <- Rho.YX.Maxwell RHO.XX.Maxwell.no.1 <- Cor.Mat.Maxwell[3:6,3:6] Rho.YX.Maxwell.no.1 <- Cor.Mat.Maxwell[1,3:6] R2.Maxwell.no.1 <- Rho.YX.Maxwell.no.1 # Note that Maxwell arrives at N=113, whereas this procedure arrives at 111. # This seems to be the case becuase of rounding error in calculations # in Cohen (1988)'s tables. The present procedure is correct and contains no # rounding error # in the application of the method. # ss.power.reg.coef(Rho2.Y_X=R2.Maxwell, # Rho2.Y_X.without.j=R2.Maxwell.no.1, p=5, # alpha.level=.05, Directional=FALSE, desired.power=.80) ```