Sample Size Planning for a Single Regression Coefficient (AIPE)

Computes the necessary sample size for the confidence interval on a targeted regression coefficient \(\beta_j\) (or its standardized counterpart) in a multiple regression with p predictors to be no wider than a user-specified value. This is the accuracy in parameter estimation (AIPE) framework of Kelley and Maxwell (2003), targeted at a specific coefficient rather than at the omnibus \(R^2\). The noncentral = TRUE variant inverts the noncentral t distribution of the standardized \(b_j\) under joint multivariate normality of the predictors; the default central-t variant uses a closed form Wald style approximation. Optionally, supplying assurance returns the larger N that guarantees the realized width with the specified probability rather than just on average.

Usage

ss_aipe_reg_coef(
  rho2_Y_X = NULL,
  rho2_j_X_without_j = NULL,
  p = NULL,
  b_j = NULL,
  width,
  which_width = "Full",
  sigma_Y = 1,
  sigma_X = 1,
  rho_XX = NULL,
  rho_YX = NULL,
  which_predictor = NULL,
  noncentral = FALSE,
  alpha_lower = NULL,
  alpha_upper = NULL,
  conf_level = 0.95,
  assurance = NULL
)

Arguments

rho2_Y_X: Population value of \(\rho^2_{Y \cdot X_1, \ldots, X_p}\), the squared multiple correlation of the outcome Y with all p predictors.
rho2_j_X_without_j: Population value of \(\rho^2_{X_j \cdot X_{-j}}\), the squared multiple correlation when the jth predictor is regressed on the remaining \(p - 1\) predictors. Quantifies the multicollinearity faced by the targeted coefficient.
p: The number of predictor variables.
b_j: The (unstandardized) regression coefficient for the jth predictor, the predictor of interest.
width: Desired (full) width of the two-sided confidence interval on \(\beta_j\).
which_width: Which portion of the confidence interval width refers to. Only "Full" is currently implemented.
sigma_Y: Population standard deviation of Y.
sigma_X: Population standard deviation of the jth predictor.
rho_XX: Population correlation matrix for the p predictor variables. If supplied with rho_YX, rho2_Y_X and rho2_j_X_without_j are derived from the covariance structure.
rho_YX: Length-p vector of population correlations between Y and the p predictors.
which_predictor: Which of the p predictors is the targeted coefficient.
noncentral: If TRUE, plans using the exact noncentral t sampling distribution of the standardized \(b_j\) (Kelley, 2007). If FALSE (the default), uses the central t approximation. The noncentral path requires sigma_Y = sigma_X = 1 (i.e., a standardized solution).
alpha_lower: Type I error rate for the lower confidence limit.
alpha_upper: Type I error rate for the upper confidence limit.
conf_level: Confidence level (i.e., \(1 - \alpha\), where \(\alpha\) is the Type I error rate). Default 0.95. Mutually exclusive with alpha_lower and alpha_upper.
assurance: Optional probability with which the realized confidence interval is to be no wider than width. When NULL (the default), the planning targets the expected width.

Value

A 1-row data.frame with columns term and value. The term value is "necessary_N" and value is the necessary total sample size N given the input specifications.

Details

Calling conventions. The function offers several mutually exclusive ways to supply the population information needed to plan \(N\); the user picks the one most aligned with their available planning values. Specify exactly one of:

Covariance structure path. Supply rho_XX and rho_YX along with which_predictor. The function derives \(\rho^2_{Y \cdot X}\) and \(\rho^2_{X_j \cdot X_{-j}}\) from the covariance structure and also computes the population \(b_j\) for consistency checks.
Squared multiple correlations path. Supply rho2_Y_X, rho2_j_X_without_j, p, and b_j directly when the user already has these planning values from prior research and does not need to specify the full covariance structure.
Standardized solution. For the noncentral = TRUE path, set sigma_Y = sigma_X = 1; the result returns the standardized regression coefficient sample size.

Noncentral vs.\ central planning. The central t closed form is fast and adequate at moderate to large N; the noncentral = TRUE path additionally accounts for the noncentral t sampling distribution of the standardized \(b_j\) and is preferred when planning at small to moderate N or when reporting planning that will be matched against the noncentral CI from ci_reg_coef.

References

Kelley, K. (2007). Confidence intervals for standardized effect sizes: Theory, application, and implementation. Journal of Statistical Software, 20(8), 1–24. doi:10.18637/jss.v020.i08

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., 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

Author

Ken Kelley kkelley@nd.edu

Examples

# 1. Covariance structure path: supply the population correlation
#    matrix and the population YX cross-correlations. Five predictors
#    in an exchangeable structure (all pairwise correlations 0.5,
#    all Y-X correlations 0.3).
rho_YX <- c(.3, .3, .3, .3, .3)
rho_XX <- rbind(c(1, .5, .5, .5, .5), c(.5, 1, .5, .5, .5),
                c(.5, .5, 1, .5, .5), c(.5, .5, .5, 1, .5),
                c(.5, .5, .5, .5, 1))

# Closed-form (central t) planning, targeting the first predictor's
# standardized coefficient.
ss_aipe_reg_coef(width = .10, which_width = "Full",
                 sigma_Y = 1, sigma_X = 1,
                 rho_XX = rho_XX, rho_YX = rho_YX,
                 which_predictor = 1, noncentral = FALSE,
                 conf_level = .95)
#>  term        value
#>  necessary_N 2185 
#> 
#> Confidence level: 95%

# Adding assurance (.85): the realized CI width will be no larger than
# 0.10 in 85 percent of replications. Required N grows accordingly.
ss_aipe_reg_coef(width = .10, which_width = "Full",
                 sigma_Y = 1, sigma_X = 1,
                 rho_XX = rho_XX, rho_YX = rho_YX,
                 which_predictor = 1, noncentral = FALSE,
                 conf_level = .95, assurance = .85)
#>  term        value
#>  necessary_N 2259 
#> 
#> Confidence level: 95%

# Exact noncentral t planning. Required N differs at small to
# moderate samples.
ss_aipe_reg_coef(width = .10, which_width = "Full",
                 sigma_Y = 1, sigma_X = 1,
                 rho_XX = rho_XX, rho_YX = rho_YX,
                 which_predictor = 1, noncentral = TRUE,
                 conf_level = .95)
#>  term        value
#>  necessary_N 2191 
#> 
#> Confidence level: 95%

# 2. Squared multiple correlations path: when the user has planning
#    values for rho^2_Y.X and rho^2_j.X_-j directly (e.g., from a
#    prior power analysis), without specifying the full covariance
#    structure. b_j is required on this path.
ss_aipe_reg_coef(rho2_Y_X = 0.30, rho2_j_X_without_j = 0.20,
                 p = 5, b_j = 0.25,
                 width = .15, which_width = "Full",
                 sigma_Y = 1, sigma_X = 1,
                 noncentral = FALSE, conf_level = .95)
#>  term        value
#>  necessary_N 607  
#> 
#> Confidence level: 95%