DMAR (version 1.0.0)

ss_power_reg_coef: Sample Size for a Targeted Regression Coefficient

Description

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

Usage

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
)

Value

ss

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

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

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

which_predictor

Identifies the predictor of interest when rho_XX and rho_YX are specified

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)

specified_N

Sample size for which power should be evaluated. This is the total sample size.

print_progress

If the progress of the iterative procedure is printed to the screen as the iterations are occurring

Author

Ken Kelley [email protected]

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.

Power is computed from a noncentral t distribution with noncentrality \(\sqrt{N}\,f\), which treats the predictors as fixed (their values held constant across hypothetical replications). This is the standard fixed-predictor power analysis; under random predictors, where the predictor values themselves vary from sample to sample, the sample size required for a given level of power is somewhat larger.

References

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

Kelley, K., & Maxwell, S. E. (2003). Sample size for multiple regression: Obtaining regression coefficients that are accurate, not simply significant. Psychological Methods, 8(3), 305--321. tools:::Rd_expr_doi("10.1037/1082-989X.8.3.305")

Kelley, K., & Maxwell, S. E. (2008). Sample size planning with applications to multiple regression: Power and accuracy for omnibus and targeted effects. In P. Alasuutari, L. Bickman, & J. Brannen (Eds.), The Sage handbook of social research methods (pp. 166--192). Sage.

Maxwell, S. E. (2000). Sample size and multiple regression analysis. Psychological Methods, 5(4), 434--458. tools:::Rd_expr_doi("10.1037/1082-989X.5.4.434")

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 4 on individual comparisons of means and Chapter 6 on trend analysis.)

Maxwell, S. E., Kelley, K., & Rausch, J. R. (2008). Sample size planning for statistical power and accuracy in parameter estimation. Annual Review of Psychology, 59, 537--563. tools:::Rd_expr_doi("10.1146/annurev.psych.59.103006.093735")

Anderson, S. F., Kelley, K., & Maxwell, S. E. (2017). Sample-size planning for more accurate statistical power: A method adjusting sample effect sizes for publication bias and uncertainty. Psychological Science, 28(11), 1547--1562. tools:::Rd_expr_doi("10.1177/0956797617723724")

See Also

ss_aipe_reg_coef, ss_power_R2, ci_nc_F

design_consequences for what a chosen design delivers: power, the Type S (sign) and Type M (exaggeration) errors of the significance filter, and the expected confidence interval width.

Other sample size for power: power_fisher_exact(), ss_aipe_mixed_effects(), ss_power_R2(), ss_power_R2_sensitivity(), ss_power_c(), ss_power_c_ancova(), ss_power_composite_ancova(), ss_power_composite_ancova_2group(), ss_power_composite_anova(), ss_power_composite_factorial_ancova(), ss_power_composite_factorial_ancova_het(), ss_power_composite_factorial_anova(), ss_power_composite_sem(), ss_power_contrast(), ss_power_equivalence_c(), ss_power_factorial_ancova(), ss_power_factorial_anova(), ss_power_indirect_effect(), ss_power_mixed_effects(), ss_power_one_way_anova(), ss_power_pcm(), ss_power_r(), ss_power_rc(), ss_power_reg_coef_sensitivity(), ss_power_rm_anova(), ss_power_sc(), ss_power_sem(), ss_power_smd(), ss_power_split_plot_anova()

Examples

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 %*% solve(RHO.XX.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 %*% solve(RHO.XX.Maxwell.no.1) %*% Rho.YX.Maxwell.no.1

# This procedure arrives at N = 111, whereas Maxwell (2000, p. 445)
# reports N = 113. The two differ because of the noncentrality
# parameterization, not rounding: this function uses the fixed-predictor
# noncentrality sqrt(N) * f (see Details), while the tabled value rests on
# Cohen's (1988) convention. Neither is a random-predictor result; under
# random predictors, where the predictor values vary across replications,
# the sample size needed for the same power is larger still.
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)