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Confidence Intervals for the Population Correlation and Multiple Correlation

Source: R/ci_correlation.R
ci_correlation.Rd

Two confidence intervals on the correlation scale share this page, named by the convention that lowercase r is the Pearson product-moment correlation between two variables and capital R is the multiple correlation between an outcome and a set of predictors.

ci_r() forms a confidence interval for the population correlation coefficient \(\rho\). The confidence interval is for the population value \(\rho\); the required input is the corresponding sample value, the observed sample correlation coefficient r. This approach assumes that the two variables on which the correlation is based are bivariate normally distributed (e.g., Hays, 1994, Chapter 14).

ci_R() constructs a confidence interval for the population multiple correlation coefficient \(\rho = \sqrt{\rho^2}\) from the sample multiple correlation coefficient (or, equivalently, from the observed F-statistic and degrees of freedom). The interval is obtained by inverting the sampling distribution of the sample \(R^2\) and propagating the limits through the monotone (square root) transform.

The two estimands meet at a single predictor: the multiple correlation from a regression on one predictor is the absolute value of the Pearson correlation between the outcome and that predictor.

Usage

ci_r(r, n, conf_level = 0.95, alpha_lower = NULL, alpha_upper = NULL)

ci_R(
  R = 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,
  ...
)

Arguments

r

Observed value of the sample correlation coefficient (specifically the zero-order Pearson product-moment correlation coefficient), for ci_r()

n

Sample size for ci_r(), which must be at least 4 (see Details)

conf_level

Confidence interval coverage (i.e., 1 - Type I error rate); default is .95

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

R

Observed value of the sample multiple correlation coefficient, for ci_R()

df_1

Numerator degrees of freedom

df_2

Denominator degrees of freedom

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

...

Allows one to potentially include parameter values for inner functions

Value

ci_r() returns a 3-row data.frame with columns term and value. The term values are "lower_limit" (the lower confidence limit on the population correlation \(\rho\)), "r" (the observed sample correlation coefficient), and "upper_limit" (the upper limit on \(\rho\)).

ci_R() returns a 3-row data.frame with columns term, value, prob_less, and prob_greater. The rows are ordered "lower_limit", "R" (the sample multiple correlation coefficient supplied by the user, the point estimate), and "upper_limit", so the point estimate sits between its confidence limits. The lower and upper limits are the confidence limits on the population multiple correlation coefficient \(\rho\) (square roots of the corresponding limits on \(\rho^2\)). 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 "R" estimate row).

Details

The Pearson correlation interval (ci_r). This approach will not generally lead to a symmetric confidence interval. The function first transforms \(r\) into Z', forms a confidence interval for the population value (i.e., \(\zeta\)), and then transforms the confidence limits for \(\zeta\) into the scale of the correlation coefficient. The interval requires a sample size of at least 4. The variance of Z' is \(1/(n - 3)\), which is infinite at \(n = 3\); there the interval would be vacuous, covering \([-1, 1]\) regardless of r, and for smaller n the variance is undefined. The function therefore stops with an error when \(n < 4\).

Fixed vs. random predictors (ci_R). The two regression models give different sampling distributions for the sample \(R^2\), and so different confidence intervals on \(\rho\). Under fixed predictors the design matrix is treated as constant in hypothetical replications of the study, and the omnibus \(F\)-statistic 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 on \(\rho^2\) is obtained by inverting that distribution and then taking the square root (see conf_limits_ncf). 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); the same Lee bisection that ci_R2 uses for the random-predictor CI on \(\rho^2\) is applied here and the limits are mapped to \(\rho\). Gatsonis and Sampson (1989) document the comparison; 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.

Note

The ci_r() confidence interval assumes that the two variables the correlation is based on are bivariate normal. See Hays (1994, Chapter 14) for details.

References

Algina, J. & Olejnik, S. (2000). Determining sample size for accurate estimation of the squared multiple correlation coefficient. Multivariate Behavioral Research, 35, 119–137. 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.

Hays, W. L. (1994). Statistics (5th ed.). Fort Worth, TX: Harcourt Brace College Publishers.

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

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

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. (2004). Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychological Methods, 9(2), 164–182. doi:10.1037/1082-989X.9.2.164

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. doi:10.3758/BF03203611

See also

ci_R2, ss_aipe_r, ss_power_r, var_r, ss_aipe_R2, convert_r_Z, convert_Z_r, conf_limits_nct

Other confidence intervals for effect sizes: ci_R2(), ci_c(), ci_c_ancova(), ci_c_ancova_bp(), 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()

Author

Ken Kelley kkelley@nd.edu

Examples

# Pearson correlation, from Hays. Suppose n = 100 and r = .35.
ci_r(r = .35, n = 100, conf_level = .95)
#>  term        value
#>  lower_limit 0.165
#>  r           0.35 
#>  upper_limit 0.511
#> 
#> Confidence level: 95%

# Here is another way to enter the above example.
ci_r(r = .35, n = 100, conf_level = NULL,
     alpha_lower = .025, alpha_upper = .025)
#>  term        value
#>  lower_limit 0.165
#>  r           0.35 
#>  upper_limit 0.511

# Here are examples of one-sided confidence intervals.
ci_r(r = .35, n = 100, conf_level = NULL, alpha_lower = 0, alpha_upper = .05)
#>  term        value
#>  lower_limit -1   
#>  r           0.35 
#>  upper_limit 0.487
ci_r(r = .35, n = 100, conf_level = NULL, alpha_lower = .05, alpha_upper = 0)
#>  term        value
#>  lower_limit 0.196
#>  r           0.35 
#>  upper_limit 1    

# Multiple correlation from a five-predictor regression.
ci_R(R = .7071, df_1 = 5, df_2 = 50, conf_level = .95,
     random_predictors = TRUE)
#>  term        value prob_less prob_greater
#>  lower_limit 0.489 0.025     0.975       
#>  R           0.707 <NA>      <NA>        
#>  upper_limit 0.799 0.975     0.025       
#> 
#> Confidence level: 95%