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.
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,
...
)
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).
Observed value of the sample correlation coefficient
(specifically the zero-order Pearson product-moment correlation
coefficient), for ci_r()
Sample size for ci_r(), which must be at least 4 (see
Details)
Confidence interval coverage (i.e., 1 - Type I error rate); default is .95
The Type I error rate for the lower confidence interval limit
The Type I error rate for the upper confidence interval limit
Observed value of the sample multiple correlation coefficient,
for ci_R()
Numerator degrees of freedom
Denominator degrees of freedom
Whether or not the predictor variables are random or fixed (random is default)
Obtained F-value
Sample size
Number of predictors
Allows one to potentially include parameter values for inner functions
Ken Kelley [email protected]
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 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); 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.
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.
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. 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")
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. tools:::Rd_expr_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. tools:::Rd_expr_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. tools:::Rd_expr_doi("10.3758/BF03203611")
ci_R2, ss_aipe_r,
ss_power_r, var_r, ss_aipe_R2,
convert_r_Z, convert_Z_r,
ci_nc_t
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()
# Pearson correlation, from Hays. Suppose n = 100 and r = .35.
ci_r(r = .35, n = 100, conf_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)
# Here are examples of one-sided confidence intervals.
ci_r(r = .35, n = 100, conf_level = NULL, alpha_lower = 0, alpha_upper = .05)
ci_r(r = .35, n = 100, conf_level = NULL, alpha_lower = .05, alpha_upper = 0)
# Multiple correlation from a five-predictor regression.
ci_R(R = .7071, df_1 = 5, df_2 = 50, conf_level = .95,
random_predictors = TRUE)
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