“Confidence interval of R-square”, but, which one?

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:



Confidence Region and Not-reject Region

Either Confidence Interval (CI) or Null Hypothesis Significance Test (NHST) has the same business, to advise whether some sample is or is not disliked by some hypothesized parameter .

NHST.com manages a database. For each Miss , NHST spies out all she dislikes. Mr X logs in NHST.com and inputs a girl name and his credit card number, to bet his luck whispering-- Does she dislike me?

CI.com manages a database too. For each Mr X, CI only needs his credit card with his name X on it, then serves him a full list of available girls.

NHST.com has been historically monopolizing the market. Nevertheless, somebody prefer visiting CI.com and find that the two may share database in most cases.

Not-reject Region of is defined as .

Confidence Region of x is defined as .

So,

答:有同学认为不应该浪费时间教三遍p值和置信区间

如果确实大部分同学认真跟着我学三遍后还不能明白区间估计的假设检验,我承认是我教学上的失败。然而我不介意讲第四遍第五遍(实际上,在结构方程部分,、方程结构和S的关系我至少重复了五遍。但是五遍都能听懂,一定胜过三遍还没听懂?)假如有同学有兴趣,欢迎贡献一个问卷调查有多少人终于弄懂区间估计和假设检验,还没有弄懂的同学中有多少同学仍然有足够的兴趣企图花时间去弄懂。做在线问卷只需要动机,不需要写代码的能力。我很希望有更多同学去实践在线问卷这项重要的技能。

到底教什么是重要的,我的判断没有改变。我仍确信选讲p值、区间估计符合我对大家学术倾向的最初预期。倾向学术还是倾向职场,对每位同学无所谓对错。要错就是我最初的预期错。现在确实有同学认为,不搞懂p值、power、区间估计照样可以安心作学术,照样可以面对海量的报告p值的文献,照样可以在自己的学术作品中每篇都报告p值。对这一类同学,我以为这是把学术当作普通谋生行业。我要编量表宁可划这类同学为职场倾向。但如果有同学对p、CI这类貌似非应用的学术问题感兴趣(当然有),我认为太有必要在北大的研究生课程里占用足够的正课时间。这是我的公开立场。

同样,我也相信大家对于什么东西是重要的自有度量。但这并不意味着我的课程需要符合多数同学目前的偏好。我相信许多同学考入北大,是为了有一个机会让北大改变自己的偏好(或品味?),而不是相反。

至于上课提点考试题目的噱头,建议对此有意见的同学把它正确地理解为我的一种调侃方式,而不是我对考题的允诺或背书。同时建议对此调侃方式不能接受的同学,错误地把它理解为我对试题的某种程度的提示,我尽量弄假成真促成喜剧。

最后,我很清楚以上这些颇为偏激的意见显然不适合作为一个comment跟在任何一个同学的学习笔记之后。因为每一位在学习笔记中花费时间陈述课程意见的同学,他们是在为课程作自己的一份义工。义工身后,多的是搭便车的沉默群众。甚至我的comment本身,更多时是基于我对原贴的片面误读而不是全面的解读,因为原贴全篇超过60%的篇幅在正面肯定我的课程教学。显然,我的这篇答复意见已经完全不针对原贴和发表原贴的那位同学,所以我决定把这个回复贴在自己的教学笔记,并欢迎所有同学匿名或者不匿名评论。