1. 因子分析，共分散構造分析 Factor Analysis Structural Equations Model
　　　　　主成分分析 Principal Components
第17章　共分散構造分析
Structural Equations Model (SEM) 1

2. 線形構造の図式（ｐ310） Linear Structure
観測変数
Observed V.
潜在変数
Latent V.
誤差項
Error term
重回帰分析 Multiple Linear Regression
(複数の観測変数と誤差で目的の観測変数を表現) x１ y e x2
因子分析 Factor Analysis
(複数の観測変数を
共通の潜在変数で表現)
主成分分析 Principal Components
(複数の観測変数を統合し
集約した潜在変数で表現) y1 e1 x1 f1 h1 e1 y2 x2 e2 h2 e2 f2 y3 x3 e3 2

3. 線形構造の図式（ｐ310） Linear Structure
観測変数
Observed V.
潜在変数
Latent V.
誤差項
Error term
一般線形構造
General Structure δ2 y1 e1 f2 e4 y4 y2 e2 f1 f3 y3 e3 y5 e5
Structural Equation Model (SEM),
Linear Structure Regression with Latent variables(LISREL) δ3 3

4. 線形構造の図式（ｐ310） Linear Structure
誤差項
Error Term
観測変数
Observed　V.
潜在変数
Latent V. y5
一般線形構造 e5 y6 e6 δ2 y7 e7 f2 e1 y1 y8 e8 f1 y2 e2 f3 y9 e9 e3 y3
y10
e10 δ3 y4
y11 e4
e11
y12
e12

5. パッケージを使う Use Additional Package
(1) SEM (Structual Equations Model)
From Package Menu 　メニューから
Select a Mirror Site　CRANミラーサイト指定
Install the package　　プルダウンから選ぶ
Type from Command Line コマンドラインから
install.packages("sem")
Make a Library in the Package Effective
パッケージ内のライブラリーを有効にする
library(sem)

6. パッケージを使う Use Additional Package
(2) Lavaan (latent variable analysis)
From Package Menu 　メニューから
Select a Mirror Site　CRANミラーサイト指定
Type from Command Line コマンドラインから
install.packages(c("lavaan", "psych", "qgraph"))
Make a Library in the Package Effective
library(lavaan)
library(psych)
library(qgraph)

7. 相関係数行列入力（p312） Specify The Correlation Coefficient Matrix
# p312 specify the correlation coefficient matrix (as lower triangular matrix)
coopd<- readMoments(names=c(
"y1","y2","y3","y4","y5","y6","y7","y8","y9","y10","y11","y12"))
1.0

8. 相関係数行列の入力（ｐ313） Specify The Correlation Coefficient Matrix (Alt)
# p312 specify the correlation coefficient matrix (without diagonal values)
coopd <- readMoments (diag=FALSE,names=as.character(paste("y",1:12, sep="")))
.160

9. SEMpackege Description of Equations 方程式の記述
model.coop <- specifyModel()
mother -> y1, b11, NA
mother-> y2, b21, NA
mother-> y3, b31, NA
mother-> y4, b41, NA
interaction-> y5, NA, 1
interaction-> y6, b62, NA
interaction-> y7, b72, NA
interaction-> y8, b82, NA
cooperative -> y9, NA, 1
cooperative -> y10, b103, NA
cooperative -> y11, b113, NA
cooperative -> y12, b123, NA
mother-> interaction, g21, NA
mother-> cooperative, g31, NA
y1 <-> y1, e1, NA
y2 <-> y2, e2, NA
y3 <-> y3, e3, NA
y4 <-> y4, e4, NA
y5 <-> y5, e5, NA
y6 <-> y6, e6, NA
y7 <-> y7, e7, NA
y8 <-> y8, e8, NA
y9 <-> y9, e9, NA
y10 <-> y10, e10, NA
y11 <-> y11, e11, NA
y12 <-> y12, e12, NA
mother<-> mother, NA, 1
interaction<-> interaction, delta2, NA
cooperative <-> cooperative, delta3, NA

10. Description of Relations(係数の記述) 変数　－＞　影響先,推定母数，固定母数 variable -> variable , estimated, fixed
model.coop <- specifyModel()
mother -> y1, b11, NA
mother-> y2, b21, NA
mother-> y3, b31, NA
mother-> y4, b41, NA
interaction-> y5, NA, 1
interaction-> y6, b62, NA
interaction-> y7, b72, NA
interaction-> y8, b82, NA
cooperative -> y9, NA, 1
cooperative -> y10, b103, NA
cooperative -> y11, b113, NA
cooperative -> y12, b123, NA
mother-> interaction, g21, NA
mother-> cooperative, g31, NA

11. Descrition of Variances(分散の記述) 内生変数の分散(変数<->変数),推定母数,固定母数 Variance of Endogenous Variables variable <-> variable, estimated parameter, fixed param.
y1 <-> y1, e1, NA
y2 <-> y2, e2, NA
y3 <-> y3, e3, NA
y4 <-> y4, e4, NA
y5 <-> y5, e5, NA
y6 <-> y6, e6, NA
y7 <-> y7, e7, NA
y8 <-> y8, e8, NA
y9 <-> y9, e9, NA
y10 <-> y10, e10, NA
y11 <-> y11, e11, NA
y12 <-> y12, e12, NA
mother<-> mother, NA, 1
interaction<-> interaction, delta2, NA
cooperative <-> cooperative, delta3, NA

12. SEM Package 推定母数の計算 sem(モデル名,相関係数行列，データ数)
sem.coop <- sem(model.coop, coopd, N=50)
推定結果の表示 Show the result
stdCoef(sem.coop)
summary(sem.coop)

13. 適合度指標
一般に、カイ二乗値、GFI、AGFI、RMSEA、CFI、AIC,CAIC、BIC　などが使われる。（たぶん）まずはRMSEA、それからCFI。で、AICでほかのモデルと比較する。カイ二乗値、GFI系はいろいろよくないらしいが、慣習として (?) 報告だけはしておく。
カイ二乗値：小さくて、有意じゃないとよい。75から200ケースくらいならよいが、それ以上になると常に有意になってしまうのでよろしくない指標。
GFI: Goodness of Fit Index 。0から1までの値で、大きいほどよい。サンプルサイズに依存するのでおすすめしない。
AGFI: Adjusted Goodness of Fit Index。0から1までの値で、大きいほどよい。サンプルサイズに依存するのでおすすめしない。
RMSEA: Root Mean Square Error of Approximation。 0.05以下だとよい。信頼区間の計算を薦める。sqrt([([&chi2/df] - 1)/(N - 1)])
NFI: Normed Fit Index。ヌルモデルとの差。大きいほど、0.95以上だとよい。[&chi2/df(Null Model) - &chi2/df(Proposed Model)]/[&chi2/df(Null Model) - 1]
TFI: Tucker-Lewis Index。NFIを自由度で補正。大きいほどよい。1を超えることもある
CFI: Comparative Fit Index。NFIを自由度で補正 (TFIとは違うやりかた) 。大きいほどよい。0.95以上だとよい。[d(Null Model) - d(Proposed Model)]/d(Null Model)
AIC: Akaike Information Criterion。カイ二乗値に自由度、パラメータ数の補正を加えたもの。小さいほうがよい。相対基準なので、いくつ以下、というのはない。 chi2 + k(k - 1) - 2df
CAIC: Consistent Akaike Information Criterion。AICのサンプルサイズの補正をさらに加えた。X2+(1+log(N))*(((k*(k-1)-2*df))/2)。Nはサンプルサイズ。
BIC: Bayesian Information Criterion。事後の分布との比較。小さいほどよい。chi2 + [k(k - 1)/2 - df]ln(N)

14. SEM Package 標準化解 15 e1 0.81523858 y1 <--> y1
　　　stdCoef(sem.coop, digit=4)
Std. Estimate
1 b y1 <--- mother
2 b y2 <--- mother
3 b y3 <--- mother
4 b y4 <--- mother
y5 <--- interaction
6 b y6 <--- interaction
7 b y7 <--- interaction
8 b y8 <--- interaction
y9 <--- cooperative
10 b y10 <--- cooperative
11 b y11 <--- cooperative
12 b y12 <--- cooperative
13 g interaction <--- mother
14 g cooperative <--- mother
e y1 <--> y1
e y2 <--> y2
e y3 <--> y3
e y4 <--> y4
e y5 <--> y5
e y6 <--> y6
e y7 <--> y7
e y8 <--> y8
e y9 <--> y9
24 e y10 <--> y10
25 e y11 <--> y11
26 e y12 <--> y12
mother <--> mother
28 delta interaction <--> interaction
29 delta cooperative <--> cooperative

15. summary(sem.coop)
Model Chisquare = Df = 52 Pr(>Chisq) =
Chisquare (null model) = Df = 66
Goodness-of-fit index =
Adjusted goodness-of-fit index =
RMSEA index = % CI: ( , )
Bentler-Bonnett NFI =
Tucker-Lewis NNFI =
Bentler CFI =
SRMR =
AIC =
AICc =
BIC =
CAIC =
Normalized Residuals
Min. 1st Qu. Median Mean 3rd Qu. Max.
R-square for Endogenous Variables
y y y y4 interaction y5
y y y8 cooperative y y10
y y12
Parameter Estimates
Estimate Std Error z value Pr(>|z|)
b e-03 y1 <--- mother
b e-03 y2 <--- mother
b e-09 y3 <--- mother
b e-04 y4 <--- mother
b e-11 y6 <--- interaction
b e-09 y7 <--- interaction
b e-18 y8 <--- interaction
b e-03 y10 <--- cooperative
b e-03 y11 <--- cooperative
b e-03 y12 <--- cooperative
g e-06 interaction <--- mother
g e-03 cooperative <--- mother
e e-06 y1 <--> y1
e e-06 y2 <--> y2
e e-03 y3 <--> y3
e e-06 y4 <--> y4
e e-05 y5 <--> y5
e e-05 y6 <--> y6
e e-06 y7 <--> y7
e e-01 y8 <--> y8
e e-05 y9 <--> y9
e e-05 y10 <--> y10
e e-04 y11 <--> y11
e e-05 y12 <--> y12
delta e-03 interaction <--> interaction
delta e-01 cooperative <--> cooperative
Iterations = 48

16. pathDiagram(sem. coop, ignore. double=FALSE, edge
pathDiagram(sem.coop, ignore.double=FALSE, edge.labels="values", digits=3)
> pathDiagram(sem.coop, ignore.double=FALSE, edge.labels="values", digits=3) digraph "sem.coop" { rankdir=LR; size="8,8"; node [fontname="Helvetica" fontsize=14 shape=box]; edge [fontname="Helvetica" fontsize=10]; center=1; "interaction" [shape=ellipse] "cooperative" [shape=ellipse] "mother" [shape=ellipse] "mother" -> "y1" [label="0.43"]; "mother" -> "y2" [label="0.488"]; "mother" -> "y3" [label="0.799"]; "mother" -> "y4" [label="0.521"]; "interaction" -> "y5" [label="1"]; "interaction" -> "y6" [label="0.937"]; "interaction" -> "y7" [label="0.858"]; "interaction" -> "y8" [label="1.138"]; "cooperative" -> "y9" [label="1"]; "cooperative" -> "y10" [label="1.155"]; "cooperative" -> "y11" [label="1.239"]; "cooperative" -> "y12" [label="1.097"]; "mother" -> "interaction" [label="0.597"]; "mother" -> "cooperative" [label="0.39"]; "y1" -> "y1" [label="0.815" dir=both]; "y2" -> "y2" [label="0.762" dir=both]; "y3" -> "y3" [label="0.361" dir=both]; "y4" -> "y4" [label="0.729" dir=both]; "y5" -> "y5" [label="0.298" dir=both]; "y6" -> "y6" [label="0.383" dir=both]; "y7" -> "y7" [label="0.484" dir=both]; "y8" -> "y8" [label="0.091" dir=both]; "y9" -> "y9" [label="0.708" dir=both]; "y10" -> "y10" [label="0.61" dir=both]; "y11" -> "y11" [label="0.552" dir=both]; "y12" -> "y12" [label="0.649" dir=both]; "mother" -> "mother" [label="1" dir=both]; "interaction" -> "interaction" [label="0.345" dir=both]; "cooperative" -> "cooperative" [label="0.14" dir=both]; } >

17. lavaan packege Description of Equations 方程式の記述
model.cooplv <- '
mother =~ y1+y2+y3+y4
interaction =~ y5+y6+y7+y8
cooperative =~ y9+y10+y11+y12
interaction ~ mother
cooperative ~ mother '
{lavaan}パッケージでは以下のような記号の使い方をしています。
=~ 測定方程式
~ 　構造程式（回帰）
~~ 残差の共分散（相関）

18. lavaan Package Estimation
result2 <- sem(model.cooplv,sample.cov=coopd, sample.nobs=50) summary(result2, fit.measures=TRUE, standardized=TRUE)

19. Estimated result from Lavaan
summary(result2) lavaan (0.5-11) converged normally after 45 iterations Number of observations 50 Estimator ML Minimum Function Test Statistic Degrees of freedom 51 P-value (Chi-square) Parameter estimates: Information Expected Standard Errors Standard Estimate Std.err Z-value P(>|z|) Latent variables: mother =~ y y y y
interaction =~ y y y y
cooperative =~ y y y y
Regressions:
interaction ~
mother
cooperative ~
mother
Covariances:
interaction ~~
cooperative

20. Estimated result from Lavaan
diagram(result2, errors=TRUE, lr=FALSE)
Variances: y y y y y y y y y y y y mother interaction cooperative

21. Estimated result from Lavaan