In linear regression, confidence interval (CI) of population *DV* is narrower than that of predicted *DV*. With the assumption of generalizability, CI of at is

,

while CI of is

.

The pivot methods of both are quite similar as following.

,

so .

,

so

of linear regression is the point estimate of

for fixed *IV*(s) model. Or, it is the point estimate of wherein denotes the correlation of *Y* and , the linear composition of random *IV*(s) . The CI of is wider than that of with the same and confidence level.

[update] It is obvious that CI of relies on the distribution presumption of *IV*(s) and *DV*, as fixed *IV*(s) are just special cases of generally random *IV*(s). Usually, the presumption is that all *IV*(s) and *DV * are from multivariate normal distribution.

In the bivariate normal case with a single random *IV*, through Fisher's *z*-transform of Pearson's *r*, CI of the re-sampled can also be constructed. Intuitively, it should be wider than CI of .

Thus,

CI of can be constructed as . With the reverse transform , the CI bounds of are

and

.

In multiple *p* *IV*(s) case, Fisher's *z*-transform is

.

Although it could also be used to construct CI of , it is inferior to noncentral *F* approximation of *R* (Lee, 1971). The latter is the algorithm adopted by MSDOS software* R2* (Steiger & Fouladi, 1992) and *R-*function ci.R2(...) within package MBESS (Kelley, 2008).

In literature, "CI(s) of R-square" are hardly the literal CI(s) of in replication once more. Most of them actually refer to CI of . Authors in social science unfamiliar to hate to type when they feel convenient to type *r* or *R*. Users of experimentally designed fixed *IV*(s) should have reported CI of . However, if they were too familiar to Steiger's software *R2* to ignore his series papers on CI of effect size, it would be significant chance for them to report a loose CI of , even in a looser name "CI of ".

----

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

Kelley, K. (2008). MBESS: Methods for the Behavioral, Educational, and Social Sciences. R package version 1.0.1. [Computer software]. Available from http://www.indiana.edu/~kenkel

Steiger, J. H., & Fouladi, R. T. (1992). R2: A computer program for interval estimation, power calculation, and hypothesis testing for the squared multiple correlation. *Behavior research methods, instruments and computers, 4*, 581–582.

R Code of Part I:

R Code of Part II:

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