## CI of Non-Centrality Parameter (ncp)

RMSEA=$\sqrt{\frac{ncp_{\chi^2}}{(N-1)df}}$

Cohen's $\delta$=$\frac{ncp_t}{SD/SE}$

Cohen's $\tilde{f}^2$=$\frac{ncp_F}{N}$

$\eta^2$=$\frac{ncp_F}{ncp_F+N}$

* Steiger (2004, Eq. 48-51) and Smithson (2003, Eq. 5.17-5.19) used different definitions for Cohen's partial $\tilde{f}^2$ ( and partial $\eta^2$). This post adopted Steiger's definition divided by the total sample size $N$, rather than by Smithson's denominator $(df_1 + df_2 + 1)$.

To view density plots

Refereces:

• 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.
• ## 可改进翻译的统计术语歧义

a sample: 一个样本(a case?) > 一组抽样
two sample t test: 两样本t检验(two cases ...?) > 两组样本t检验
sample size: 样本大小(case scale?) > 样本组容量
central tendency: 集中趋势(non-dispersion?) > 中心趋势
dispersion: 离散(discrete?)趋势 > 分散趋势
SD=standard deviation: 标准方差(..variance?) > 标准离差
df=degrees of freedom: 自由度(size of freedom?) > 自由维度(dimensions of freedom)

## Why practitioners discretize their continuous data

Yihui asked this question yesterday. My supervisor Dr. Hau also criticized routine grouping discretization. I encountered two plausible reasons in 2007 classes, one negative, the other at least conditionally positive.

The first is a variant of the old Golden Hammer law -- if the only tool is ANOVA, every continuous predictor need discretization. The second reason is empirical -- ANOVA with discretization steals df(s). Let's demo it with a diagram.
The red are the population points, and the black are samples. Which predicts the population better--the green continuous line, or the discretized blue dashes? R simulation code is given.

## 《结构方程模型及其应用》(侯, 温, & 成,2004)部分章节R代码

John FOX教授的sem包试写了这本教材的几个例子，结果都与LISREL8报告的Minimum Fit Function Chi-Square吻合。不过LISREL其它拟合指标用的是Normal Theory Weighted Least Squares Chi-Square?，所以看上去比Minimum Fit Function Chi-Square报告的结果要好那么一些些。LISREL历史上推的GFI/AGFI曾因为经常将差报好被批评，圈内朋友私下嘲笑这样LISREL就更好卖了，买软件的用户也高兴将差报好，只有读Paper的人上当。

目前只写了chap3_1..到Chap3_2_...，一共5(或6)个例子。尺有所短，寸有所长--sem包不会自动报告所有修正指数，不能做样本量不一样多的多组模型，对复杂的模型要写的代码太多。不过，在已经尝试的几个例子里，有一个是LISREL跑不出来但sem包能跑出结果的。目前sem包还没有达到结构方程众多商业软件的成熟水准，但R庞大的义工武器库已经使sem包至少已经在Missing Data Multiple Imputation、Bootstrapping等等应用上胜出一筹。所以例子中还附带了缺失数据一讲Multiple Impuation的R示范代码。

欢迎各位有兴趣作类似尝试的同学将结果email我， 可以陆续更新到下面的GPL版权代码集合中。

[update] AMos, Mplus 与 SAS/STAT CALIS 缺省报告与使用的都是 Minimum Fit Function Chi-Square，可通过各种软件(Albright & Park, 2009)的结果对比查验。Normal Theory Weighted Least Squares Chi-Square并非总是比Minimum Fit Function Chi-Square报告更“好”的拟合结果（虽然常常如此）。Olsson, Foss, 和 Breivik (2004) 用模拟数据对比了二者，确证Minimum Fit Function Chi-Square计算得到的拟合指标在小样本之外的情形都比Normal Theory Weighted Least Squares Chi-Square的指标更适用。

R的sem包目前在迭代初值的计算上还做得不够好。我遇到的几个缺省初值下迭代不收敛案例，将参数设定合理范围的任意初值（比如TX，TY都设成0-1之间的正实数）之后都收敛了。

--

Albright, J. J., & Park., H. M. (2009). Confirmatory factor analysis using Amos, LISREL, Mplus, and SAS/STAT CALIS. The University Information Technology Services Center for Statistical and Mathematical Computing, Indiana University. Retrieved July 7, 2009, from http://www.indiana.edu/~statmath/stat/all/cfa/cfa.pdf.

Olsson, U. H., Foss, T., & Breivik, E. (2004). Two equivalent discrepancy functions for maximum likelihood estimation: Do their test statistics follow a non-central chi-square distribution under model misspecification? Sociological Methods Research, 32(4), 453-500.

[update]R中的sem包妙处之一是可在线实现结构方程应用界面。下面这个最粗陋的例子对应原书Chap3_1_4_CFA_MB.LS8的结果。
--荣耀属于sem package的作者Rweb的作者、以及服务器运算资源的提供者。

##Input Correlation Matrix
R.DHP<-matrix(0,ncol=17,nrow=17);
R.DHP[col(R.DHP) >= row(R.DHP)] <- c(
1,
.34,1,
.38,.35,1,
.02,.03,.04,1,
.15,.19,.14,.02,1,
.17,.15,.20,.01,.42,1,
.20,.13,.12,.00,.40,.21,1,
.32,.32,.21,.03,.10,.10,.07,1,
.10,.17,.12,.02,.15,.18,.23,.13,1,
.14,.16,.15,.03,.14,.19,.18,.18,.37,1,
.14,.15,.19,.01,.18,.30,.13,.08,.38,.38,1,
.18,.16,.24,.02,.14,.21,.21,.22,.06,.23,.18,1,
.19,.20,.15,.01,.14,.24,.09,.24,.15,.21,.21,.45,1,
.18,.21,.18,.03,.25,.18,.18,.18,.22,.12,.24,.28,.35,1,
.08,.18,.16,.01,.22,.20,.22,.12,.12,.16,.21,.25,.20,.26,1,
.12,.16,.25,.02,.15,.12,.20,.14,.17,.20,.14,.20,.15,.20,.50,1,
.20,.16,.18,.04,.25,.14,.21,.17,.21,.21,.23,.15,.21,.22,.29,.41,1
);
R.DHP<-t(R.DHP);
colnames(R.DHP)<-rownames(R.DHP)<-paste('X',1:17,sep='');
print('Inputted Correlation Matrix');
print(R.DHP);
##
##
##Input Model Specification of Chap3_1_4_CFA_MB.LS8
require(sem);
model.B <- matrix(ncol=3,byrow=TRUE,data=c(
'X1 <-> X1' , 'TD1_1'  , NA  ,
'X2 <-> X2' , 'TD2_2'  , NA  ,
'X3 <-> X3' , 'TD3_3'  , NA  ,
'X5 <-> X5' , 'TD5_5'  , NA  ,
'X6 <-> X6' , 'TD6_6'  , NA  ,
'X7 <-> X7' , 'TD7_7'  , NA  ,
'X8 <-> X8' , 'TD8_8'  , NA  ,
'X9 <-> X9' , 'TD9_9'  , NA  ,
'X10<-> X10', 'TD10_10', NA  ,
'X11<-> X11', 'TD11_11', NA  ,
'X12<-> X12', 'TD12_12', NA  ,
'X13<-> X13', 'TD13_13', NA  ,
'X14<-> X14', 'TD14_14', NA  ,
'X15<-> X15', 'TD15_15', NA  ,
'X16<-> X16', 'TD16_16', NA  ,
'X17<-> X17', 'TD17_17', NA  ,
'xi1<-> xi1', NA       , '1' ,
'xi2<-> xi2', NA       , '1' ,
'xi3<-> xi3', NA       , '1' ,
'xi4<-> xi4', NA       , '1' ,
'xi5<-> xi5', NA       , '1' ,
'xi1<-> xi2', 'PH12'   , NA  ,
'xi1<-> xi3', 'PH13'   , NA  ,
'xi1<-> xi4', 'PH14'   , NA  ,
'xi1<-> xi5', 'PH15'   , NA  ,
'xi2<-> xi3', 'PH23'   , NA  ,
'xi2<-> xi4', 'PH24'   , NA  ,
'xi2<-> xi5', 'PH25'   , NA  ,
'xi3<-> xi4', 'PH34'   , NA  ,
'xi3<-> xi5', 'PH35'   , NA  ,
'xi4<-> xi5', 'PH45'   , NA  ,
'X1 <- xi1' , 'LX1_1'  , NA  ,
'X2 <- xi1' , 'LX2_1'  , NA  ,
'X3 <- xi1' , 'LX3_1'  , NA  ,
'X5 <- xi2' , 'LX5_2'  , NA  ,
'X6 <- xi2' , 'LX6_2'  , NA  ,
'X7 <- xi2' , 'LX7_2'  , NA  ,
'X8 <- xi1' , 'LX8_1'  , NA  ,
'X9 <- xi3' , 'LX9_3'  , NA  ,
'X10<- xi3' , 'LX10_3' , NA  ,
'X11<- xi3' , 'LX11_3' , NA  ,
'X12<- xi4' , 'LX12_4' , NA  ,
'X13<- xi4' , 'LX13_4' , NA  ,
'X14<- xi4' , 'LX14_4' , NA  ,
'X15<- xi5' , 'LX15_5' , NA  ,
'X16<- xi5' , 'LX16_5' , NA  ,
'X17<- xi5' , 'LX17_5' , NA  )
);
class(model.B)<-'mod';
##
##
N=350;##sample size;
##R.DHP[-4,-4] excludes X4
## Result
(summary(sem.B<-sem(model.B, R.DHP[-4,-4], N)));
## Residuals
(round(residuals(sem.B),3));
#####################
boxplot.matrix = function(M,ylim=c(-1,1)) {
M = as.matrix(M);
boxplot(c(M[row(M)>col(M)]),at=1,xlab='',ylab='',ylim=ylim);
points(rep(1,length(c(M[row(M)>col(M)]))),c(M[row(M)>col(M)]),pch='-',col='red');
stem(c(M[row(M)>col(M)]));
boxplot.stats(c(M[row(M)>col(M)]));
}
####################
boxplot(residuals(sem.B));



欢迎在本部的同学明天上午到现场看李崇亮同学演示，地点见会议主页

RWebFriend for WordPress 在线示例(yo2.cn上的示例,  奇想录上的临时示例)

[update2009.07.11]文中MediaWiki与R集成的实验平台已不再开放编辑权限，原例转到另一个平台上（界面是西班牙语，找不到英语界面的平台）。希望能有帮手提供linux服务器自建一个平台。

## Misunderstanding of Eq. 4 in Singer’s (1998) SAS PROC MIXED paper

Singer (1998, p. 327, Eq. 4) gave a big covariance matrix as the following --

...if we combine the variance components for the two random effects together into a single matrix, we would find a highly structured block diagonal matrix. For example, if there were three students in each class, we would have:

If the number of students per class varied, the size of each of these submatrices would also vary, although they would still have this common structure. The variance in MATHACH for any given student is assumed to be $\tau_{00}+\sigma^{2}$ ...

Quiz:

1. Sampling K repetitions with replacement, $Y_{k}$ denotes MATHACH of the k-th random student within a given class, say, the first class. What is the population variance of the Y series? A: $\sigma^{2}$, B: $\tau_{00}+\sigma^{2}$.

2. Sampling $2\times K$ repetitions with replacement, $\left(Y_{1st,k},Y_{2nd,k}\right)$ denotes MATHACH(s) of the k-th pair of random students within a given class, say, the first class. Sometime the 2nd sampled student is just the 1st one by chance. What is the population covariance of the $Y_{1st}$ and $Y_{2nd}$ series? A: 0, B: $\tau_{00}$.

3. Sampling K repetitions with replacement, $Y_{k}$ denotes MATHACH of the k-th random student within one randomly sampled class for each repetition. What is the population variance of the Y series? A: $\sigma^{2}$, B: $\tau_{00}+\sigma^{2}$.

4. Sampling $2\times K$ repetitions with replacement, $\left(Y_{1st,k},Y_{2nd,k}\right)$ denotes MATHACH(s) of the k-th pair of random students within one random class sampled for each pair of students. Sometime the 2nd sampled student is just the 1st one by chance. What is the population covariance of the $Y_{1st}$ and $Y_{2nd}$ series? A: 0, B: $\tau_{00}$.

Correct answers: A A B B.

Your plausible incorrect answers for Quiz 1 and 2 just tell how the matrix caused my misunderstanding when I read the paper the first time. It is difficult to give names for the randomly sampled classes, and the matrix columns and rows.

--
Singer, J. D. (1998). Using SAS PROC MIXED to fit multilevel models, hierarchical models, and individual growth models. Journal of Educational and Behavioral Statistics. 24. 323-355.

## “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 $\tilde{Y}_{\left[1\times1\right]}$ at $x_{\left[1\times p\right]}$ is

$\;\hat{Y}\pm\left(x\left(X^{\tau}X_{\left[N\times p\right]}\right)^{-1}x^{\tau}\right)^{\frac{1}{2}}\hat{\sigma}t_{\frac{\alpha}{2},N-p}$,

while CI of $Y\left(x\right)=\tilde{Y}\left(x\right)+\varepsilon$ is

$\;\hat{Y}\pm\left(1+x\left(X^{\tau}X_{\left[N\times p\right]}\right)^{-1}x^{\tau}\right)^{\frac{1}{2}}\hat{\sigma}t_{\frac{\alpha}{2},N-p}$.

The pivot methods of both are quite similar as following.

$\;\frac{\hat{Y}-\tilde{Y}}{s_{\hat{Y}}}\sim t_{df=N-p}$ ,

so $\tilde{Y}_{critical}=\hat{Y}-s_{\hat{Y}}\times t_{critical}$ .

$\;\frac{\hat{Y}-Y}{s_{\left(\hat{Y}-Y\right)}}\sim t_{df=N-p}$,

so $Y_{critical}=\hat{Y}-s_{\left(\hat{Y}-Y\right)}\times t_{critical}=\hat{Y}-s_{\left(\hat{Y}-\tilde{Y}-\varepsilon\right)}\times t_{critical}$

$R^{2}$ of linear regression is the point estimate of

$\;\eta^{2}\equiv\frac{SS\left(\tilde{Y}_{\left[N\times1\right]}\right)}{SS\left(\tilde{Y}_{\left[N\times1\right]}\right)+N\sigma^{2}}$

for fixed IV(s) model. Or, it is the point estimate of $\rho^{2}$ wherein $\rho$ denotes the correlation of Y and $X\beta$, the linear composition of random IV(s) . The CI of $\rho^{2}$ is wider than that of $\eta^{2}$ with the same $R^{2}$ and confidence level.

[update] It is obvious that CI of $\rho^{2}$ 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 $R^{\prime2}=r^{\prime2}$ can also be constructed. Intuitively, it should be wider than CI of $\rho^{2}$.

$\;\tanh^-\left(r\right)\equiv\frac{1}{2}\log\frac{1+r}{1-r}\;{appr\atop \sim}\; N\left(\tanh^-\left(\rho\right),\frac{1}{N-3}\right)$

Thus,

$\;\tanh^-\left(r^{\prime}\right)-\tanh^-\left(r\right){appr\atop \sim}N\left(0,\frac{2}{N-3}\right)$

CI of $\tanh^-\left(r^{\prime}\right)$ can be constructed as $\tanh^-\left(r\right)\pm\sqrt{\frac{2}{N-3}}z_{\frac{\alpha}{2}}$ . With the reverse transform $\tanh\left(.\right)$, the CI bounds of $R^{\prime2}$ are

$\;\left(\max\left(0,\tanh\left(\tanh^-\left(R\right)-\sqrt{\frac{2}{N-3}}z_{1-\frac{\alpha}{2}}\right)\right)\right)^{2}$

and

$\;\left(\tanh\left(\tanh^{-1}\left(R\right)+\sqrt{\frac{2}{N-3}}z_{1-\frac{\alpha}{2}}\right)\right)^{2}$.

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

$\;\left(N-2-p\right)\left(\tanh^-\left(R\right)\right)^{2}\;{appr\atop \sim}\;\chi_{df=p,ncp=\left(N-2-p\right)\left(\tanh^-\left(\rho\right)\right)^{2}}^{2}$ .

Although it could also be used to construct CI of $\rho^{2}$ , 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 $R^{2}$ in replication once more. Most of them actually refer to CI of $\rho^{2}$ . Authors in social science unfamiliar to $L^AT_EX$ hate to type $\rho$ when they feel convenient to type r or R. Users of experimentally designed fixed IV(s) should have reported CI of $\eta^{2}$ . 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 $\rho^{2}$, even in a looser name "CI of $R^{2}$".

----

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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