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Confidence Interval for a Regression Coefficient, Raw or Standardized

Source: R/ci_reg_coef.R
ci_reg_coef.Rd

The general engine behind ci_rc (unstandardized) and ci_src (standardized): computes a confidence interval for a population regression coefficient by the standard t-based approach or the noncentral t approach, in whichever metric the inputs are supplied.

Usage

ci_reg_coef(
  b_j,
  SE_b_j = NULL,
  s_Y = NULL,
  s_X = NULL,
  N,
  p,
  R2_Y_X = NULL,
  R2_j_X_without_j = NULL,
  conf_level = 0.95,
  R2_Y_X_without_j = NULL,
  t_value = NULL,
  alpha_lower = NULL,
  alpha_upper = NULL,
  noncentral = FALSE,
  ...
)

Arguments

b_j

Value of the regression coefficient for the jth predictor variable

SE_b_j

Standard error for the jth predictor variable

s_Y

Standard deviation of Y, the dependent variable

s_X

Standard deviation of \(X_j\), the predictor variable of interest

N

Sample size

p

The number of predictors

R2_Y_X

The squared multiple correlation coefficient predicting Y from the p predictor variables

R2_j_X_without_j

The squared multiple correlation coefficient predicting the jth predictor variable (i.e., the predictor of interest) from the remaining p-1 predictor variables

conf_level

Desired level of confidence for the computed interval (i.e., 1 - the Type I error rate)

R2_Y_X_without_j

The squared multiple correlation coefficient predicting Y from the p-1 predictor variable with the jth predictor of interest excluded

t_value

The t-value evaluating the null hypothesis that the population regression coefficient for the jth predictor equals zero

alpha_lower

The Type I error rate for the lower confidence interval limit

alpha_upper

The Type I error rate for the upper confidence interval limit

noncentral

TRUE or FALSE, specifying whether or not the noncentral approach to confidence intervals should be used

...

Optional additional specifications for nested functions

Value

A 3-row data.frame with columns term, value, prob_less, and prob_greater. The rows are ordered "lower_limit", "reg_coef" (the regression coefficient point estimate), and "upper_limit", so the point estimate sits between its confidence limits. The lower and upper rows give the confidence limits on the regression coefficient. The prob_less and prob_greater columns report the achieved tail probabilities at each limit when the noncentral t method is used (they are NA for the "reg_coef" estimate row).

Details

For standardized variables, do not specify the standard deviation of the variables and input the standardized regression coefficient for b_j.

When b_j is reconstructed from squared multiple correlations (that is, from R2_Y_X, R2_Y_X_without_j, and R2_j_X_without_j rather than a supplied b_j, SE_b_j, or t_value), only the magnitude of the coefficient is identifiable; its sign is not. The positive root is returned and a warning is issued. If the coefficient is negative, negate the point estimate and swap and negate the confidence limits, or supply b_j directly.

Note

Not all of the values need to be specified, only those that contain all of the necessary information in order to compute the confidence interval (options are thus given for the values that need to be specified).

The function ci_rc in DMAR also calculates the confidence interval for the population (unstandardized) regression coefficient. The function ci_src also calculates the confidence interval for the population (standardized) regression coefficient. These two functions perform the same tasks as ci_reg_coef does and are preferred to it because of simpler arguments.

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.)

Smithson, M. (2003). Confidence intervals. Thousand Oaks, CA: Sage Publications.

See also

ss_aipe_reg_coef, conf_limits_nct, ci_rc, ci_src

Other confidence intervals for effect sizes: ci_R2(), ci_c(), ci_c_ancova(), ci_c_ancova_bp(), ci_correlation, ci_cv(), ci_eta_squared(), ci_eta_squared_generalized(), ci_eta_squared_partial(), ci_mahalanobis(), ci_omega_squared(), ci_pvaf(), ci_rc(), ci_rmsea(), ci_sc(), ci_sc_ancova(), ci_sm(), ci_smd(), ci_smd_c(), ci_snr(), ci_src(), ci_srsnr(), contrast_adjusted(), plot_smd()

Author

Ken Kelley kkelley@nd.edu

Examples

ci_reg_coef(b_j = 0.61319, SE_b_j = 0.16098, N = 30, p = 6)
#>  term        value prob_less prob_greater
#>  lower_limit 0.28  0.025     0.975       
#>  reg_coef    0.613 <NA>      <NA>        
#>  upper_limit 0.946 0.975     0.025       
#> 
#> Confidence level: 95%