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Sample Size for a Targeted Regression Coefficient

Source: R/ss_power_reg_coef.R
ss_power_reg_coef.Rd

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
)

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

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

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. 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. 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. 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. doi:10.1177/0956797617723724

See also

ss_aipe_reg_coef, ss_power_R2, conf_limits_ncf

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()

Author

Ken Kelley kkelley@nd.edu

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)
#>  term              value
#>  necessary_N       40   
#>  actual_power      0.81 
#>  noncentral_t_parm 2.92 
#>  effect_size       0.462

# 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)
#>  term              value
#>  necessary_N       40   
#>  actual_power      0.81 
#>  noncentral_t_parm 2.92 
#>  effect_size       0.462

# 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)
#>  term              value
#>  necessary_N       40   
#>  actual_power      0.81 
#>  noncentral_t_parm 2.92 
#>  effect_size       0.462

# Method 4
ss_power_reg_coef(alpha_level = .05, cohen_f2 = 0.2130898, p = 5,
                  directional = FALSE, desired_power = .80)
#>  term              value
#>  necessary_N       40   
#>  actual_power      0.81 
#>  noncentral_t_parm 2.92 
#>  effect_size       0.462

# 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)
#>  term              value
#>  specified_N       25   
#>  actual_power      0.591
#>  noncentral_t_parm 2.31 
#>  effect_size       0.462

# 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)
#>  term              value
#>  specified_N       25   
#>  actual_power      0.591
#>  noncentral_t_parm 2.31 
#>  effect_size       0.462

# 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)
#>  term              value
#>  necessary_N       111  
#>  actual_power      0.801
#>  noncentral_t_parm 2.83 
#>  effect_size       0.269