sample size for a targeted regression coefficient

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Description

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

Usage

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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; KKelley@ND.Edu)

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.

See Also

ss.aipe.reg.coef, ss.power.R2, conf.limits.ncf

Examples

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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)

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