DMAR (version 1.0.0)

ci_R2: Confidence Interval for the Population Squared Multiple Correlation Coefficient

Description

Constructs a confidence interval for the population squared multiple correlation coefficient \(\rho^2\) by inverting the sampling distribution of the sample \(R^2\). The confidence interval is for the population value \(\rho^2\); the required input is the corresponding sample value, the observed sample squared multiple correlation coefficient \(R^2\) (or, equivalently, the observed F-statistic and degrees of freedom). The right choice of sampling distribution, and so the right interval, depends on whether the predictors are treated as random draws from a joint multivariate normal distribution (the default and the typical case in the behavioral, educational, and social sciences) or as fixed by design (planned dosing levels, factorial covariates, etc.). The function selects the sampling distribution via random_predictors and inverts the corresponding noncentral distribution; the construction is the regression analogue of a noncentral distribution based CI on a standardized effect size (Steiger & Fouladi, 1992; Kelley, 2007).

Usage

ci_R2(
  R2 = NULL,
  df_1 = NULL,
  df_2 = NULL,
  conf_level = 0.95,
  random_predictors = TRUE,
  F_value = NULL,
  N = NULL,
  p = NULL,
  alpha_lower = NULL,
  alpha_upper = NULL,
  tol = 1e-09
)

Value

A 3-row data.frame with columns term, value, prob_less, and prob_greater. The rows are ordered "lower_limit" (lower confidence limit on the population \(\rho^2\)), "R2" (the sample squared multiple correlation coefficient supplied by the user, the point estimate), and "upper_limit" (upper confidence limit on the population \(\rho^2\)), so the point estimate sits between its confidence limits. The prob_less and prob_greater columns report the achieved lower-tail and upper-tail error probabilities at each limit (they are NA for the "R2" estimate row). For random-predictor mode (random_predictors = TRUE) the limits are computed via the Lee (1971) bisection over the multiple-correlation sampling distribution; for fixed-predictor mode they are computed by inversion of the noncentral F distribution.

Arguments

R2

Observed value of the sample squared multiple correlation coefficient

df_1

Numerator degrees of freedom

df_2

Denominator degrees of freedom

conf_level

Confidence interval coverage; 1-Type I error rate

random_predictors

Whether or not the predictor variables are random or fixed (random is default)

F_value

Obtained F-value

N

Sample size

p

Number of predictors

alpha_lower

Type I error for the lower confidence limit

alpha_upper

Type I error for the upper confidence limit

tol

The convergence tolerance passed to uniroot when random_predictors = FALSE and the confidence limits are found by inverting the noncentral F distribution (see ci_nc_F); ignored when random_predictors = TRUE, where the Lee (1971) bisection uses its own fixed tolerance

Author

Ken Kelley [email protected]

Details

Fixed vs.\ random predictors. The two regression models give different sampling distributions for the sample \(R^2\), and so different confidence intervals. Under fixed predictors the design matrix is treated as constant in hypothetical replications of the study, and the omnibus \(F\)-statistic \(F = (R^2 / p) / ((1 - R^2) / (N - p - 1))\) follows a noncentral F with \(p\) and \(N - p - 1\) degrees of freedom and noncentrality \(\lambda = N \rho^2 / (1 - \rho^2)\) (Cohen, 1988); the CI is obtained by inverting that distribution at the supplied confidence level (see ci_nc_F). Under random predictors the design matrix is itself a draw from a joint multivariate normal distribution and the unconditional sampling distribution of the sample \(R^2\) is given by Lee (1971); ci_R2 uses the Lee (1971) bisection (the same construction Algina and Olejnik 2000 implemented in SAS) to invert that distribution. Gatsonis and Sampson (1989) document the comparison and show that treating random predictors as fixed tends to over-state precision (and so under-state the CI width); the discrepancy is modest at moderate to large \(N\) but non-trivial at small \(N\) with moderate-to-large effects. In the behavioral, educational, and social sciences predictor variables are almost always random, so the default is random_predictors = TRUE; pass random_predictors = FALSE for designs in which the predictor variables are fixed by design.

References

Algina, J. & Olejnik, S. (2000). Determining sample size for accurate estimation of the squared multiple correlation coefficient. Multivariate Behavioral Research, 35, 119--137. tools:::Rd_expr_doi("10.1207/s15327906mbr3501_5")

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

Gatsonis, C., & Sampson, A. R. (1989). Multiple correlation: Exact power and sample size calculations. Psychological Bulletin, 106(3), 516--524.

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

Kelley, K. (2008). Sample size planning for the squared multiple correlation coefficient: Accuracy in parameter estimation via narrow confidence intervals. Multivariate Behavioral Research, 43, 524--555. tools:::Rd_expr_doi("10.1080/00273170802490632")

Lee, Y. S. (1971). Some results on the sampling distribution of the multiple correlation coefficient. Journal of the Royal Statistical Society, Series B, 33(1), 117--130.

Maxwell, S. E., Delaney, H. D., & Kelley, K. (2027). Designing experiments and analyzing data: A model comparison perspective (4th ed.). Routledge. (See Chapter 3 on \(R^2\) as a model comparison effect size.)

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

Steiger, J. H., & Fouladi, R. T. (1992). R2: A computer program for interval estimation, power calculations, sample size estimation, and hypothesis testing in multiple regression. Behavior Research Methods, Instruments, & Computers, 24(4), 581--582. tools:::Rd_expr_doi("10.3758/BF03203611")

See Also

ss_aipe_R2, ci_nc_F

Other confidence intervals for effect sizes: 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_reg_coef(), ci_rmsea(), ci_sc(), ci_sc_ancova(), ci_sm(), ci_smd(), ci_smd_c(), ci_snr(), ci_src(), ci_srsnr(), contrast_adjusted(), plot_smd()

Examples

# For random predictor variables.
ci_R2(R2 = .25, N = 100, p = 5, conf_level = .95, random_predictors = TRUE)

ci_R2(F_value = 6.266667, N = 100, p = 5, conf_level = .95, random_predictors = TRUE)

# For fixed predictor variables.
ci_R2(R2 = .25, N = 100, p = 5, conf_level = .95, random_predictors = FALSE)

ci_R2(F_value = 6.266667, N = 100, p = 5, conf_level = .95, random_predictors = FALSE)

# One sided confidence intervals when predictors are random.
ci_R2(R2 = .25, N = 100, p = 5, alpha_lower = .05, alpha_upper = 0,
      conf_level = NULL, random_predictors = TRUE)

ci_R2(R2 = .25, N = 100, p = 5, alpha_lower = 0, alpha_upper = .05,
      conf_level = NULL, random_predictors = TRUE)

# One sided confidence intervals when predictors are fixed.
ci_R2(R2 = .25, N = 100, p = 5, alpha_lower = .05, alpha_upper = 0,
      conf_level = NULL, random_predictors = FALSE)

ci_R2(R2 = .25, N = 100, p = 5, alpha_lower = 0, alpha_upper = .05,
      conf_level = NULL, random_predictors = FALSE)