## 《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。

[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));

```

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.