前回は主成分分析を説明した。この手法は、
p変量(p次元)の観測値をm個(m次元)の主成分に縮約させる方法であった。
この意味では、因子分析も同じような方法であるのだが、
主成分分析の場合は、
データの散らばり方を捉えてデータ特性を把握する手法である。
今回説明する因子分析は、変数間に(潜在的な)構造を持ち込んで関係を探る手法である
(少し理解しにくいかも)。
この手法は心理学の分野で広く利用されている。
/* Lesson 15-1 */
/* File Name = fact01.sas 10/16/97 */
data food; :
infile 'food.dat'; : ファイルの読み込み
input X01-X10; : 変量リスト、連続的に
label X01='M(-15)' : 各変量に解りやすい名前を付ける
X02='M(16-20)' : M : 男性
X03='M(21-30)' : F : 女性
X04='M(31-40)' : ()内 : 年齢
X05='M(41-)' :
X06='F(-15)' :
X07='F(16-20)' :
X08='F(21-30)' :
X09='F(31-40)' :
X10='F(41-)'; :
:
proc print data=food(obs=10); : データの表示
run; :
proc factor data=food; : 因子分析
var X01-X10; : 解析に使う変量リスト
run; :
SAS システム 1
12:10 Thursday, October 16, 1997
OBS X01 X02 X03 X04 X05 X06 X07 X08 X09 X10
1 7.69 7.31 7.47 7.76 7.87 7.51 7.24 7.70 7.91 7.95
2 6.59 5.56 6.21 6.04 5.81 6.64 6.11 6.53 6.44 6.64
3 4.55 4.18 4.36 4.25 4.53 4.60 3.66 4.04 3.68 4.43
4 6.78 6.11 6.30 5.98 5.56 6.37 6.29 5.43 5.32 5.28
5 6.47 6.24 6.02 5.42 5.88 6.00 5.60 4.60 5.40 5.95
6 6.96 6.81 6.91 6.48 6.23 7.09 7.27 7.13 6.86 7.36
7 6.57 5.70 5.89 5.16 5.30 6.07 5.56 4.50 4.92 5.33
8 7.32 6.95 6.02 4.98 4.88 6.82 6.40 5.53 5.61 5.33
9 6.51 6.15 5.51 4.68 4.16 5.17 4.81 4.70 4.86 3.82
10 6.86 6.05 5.85 6.14 6.75 6.71 5.39 5.42 6.03 6.59
SAS システム 2
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Prior Communality Estimates: ONE
Eigenvalues of the Correlation Matrix: Total = 10 Average = 1
1 2 3 4 5
Eigenvalue 6.8280 1.7619 0.7545 0.2624 0.1216
Difference 5.0661 1.0074 0.4921 0.1408 0.0236
Proportion 0.6828 0.1762 0.0754 0.0262 0.0122
Cumulative 0.6828 0.8590 0.9344 0.9607 0.9728
6 7 8 9 10
Eigenvalue 0.0980 0.0721 0.0441 0.0358 0.0219
Difference 0.0259 0.0280 0.0083 0.0139
Proportion 0.0098 0.0072 0.0044 0.0036 0.0022
Cumulative 0.9826 0.9898 0.9942 0.9978 1.0000
SAS システム 3
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
2 factors will be retained by the MINEIGEN criterion.
SAS システム 4
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Factor Pattern
FACTOR1 FACTOR2
X01 0.74741 -0.59244 M(-15)
X02 0.86579 -0.31836 M(16-20)
X03 0.84491 0.22079 M(21-30)
X04 0.78216 0.47602 M(31-40)
X05 0.68129 0.67325 M(41-)
X06 0.80647 -0.54140 F(-15)
X07 0.89959 -0.33542 F(16-20)
X08 0.90901 -0.04289 F(21-30)
X09 0.90316 0.21817 F(31-40)
X10 0.79262 0.35477 F(41-)
SAS システム 5
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Variance explained by each factor
FACTOR1 FACTOR2
6.827955 1.761873
Final Communality Estimates: Total = 8.589828
X01 X02 X03 X04 X05
0.909618 0.850950 0.762624 0.838371 0.917413
X06 X07 X08 X09 X10
0.943520 0.921775 0.828147 0.863298 0.754112
/* Lesson 15-2 */
/* File Name = fact02.sas 10/16/97 */
data food;
infile 'food.dat';
input X01-X10;
label X01='M(-15)'
X02='M(16-20)'
X03='M(21-30)'
X04='M(31-40)'
X05='M(41-)'
X06='F(-15)'
X07='F(16-20)'
X08='F(21-30)'
X09='F(31-40)'
X10='F(41-)';
proc print data=food(obs=10);
run; :
proc factor data=food nfactor=3 out=fscore; : 因子数3、出力の保存
var X01-X10; :
run; :
proc plot data=fscore; :
plot factor1*factor2/vref=0.0 href=0.0; : 第1因子 x 第2因子、軸
plot factor2*factor3/vref=0.0 href=0.0; : 第2因子 x 第3因子、軸
run; :
SAS システム 8
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
3 factors will be retained by the NFACTOR criterion.
SAS システム 9
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Factor Pattern
FACTOR1 FACTOR2 FACTOR3
X01 0.74741 -0.59244 0.16808 M(-15)
X02 0.86579 -0.31836 0.29190 M(16-20)
X03 0.84491 0.22079 0.38417 M(21-30)
X04 0.78216 0.47602 0.32604 M(31-40)
X05 0.68129 0.67325 0.11067 M(41-)
X06 0.80647 -0.54140 -0.07270 F(-15)
X07 0.89959 -0.33542 -0.14888 F(16-20)
X08 0.90901 -0.04289 -0.25110 F(21-30)
X09 0.90316 0.21817 -0.27989 F(31-40)
X10 0.79262 0.35477 -0.45389 F(41-)
SAS システム 10
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Variance explained by each factor
FACTOR1 FACTOR2 FACTOR3
6.827955 1.761873 0.754451
Final Communality Estimates: Total = 9.344279
X01 X02 X03 X04 X05
0.937870 0.936157 0.910210 0.944673 0.929662
X06 X07 X08 X09 X10
0.948805 0.943939 0.891197 0.941637 0.960129
SAS システム 11
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Scoring Coefficients Estimated by Regression
Squared Multiple Correlations of the Variables with each Factor
FACTOR1 FACTOR2 FACTOR3
1.000000 1.000000 1.000000
SAS システム 12
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Standardized Scoring Coefficients
FACTOR1 FACTOR2 FACTOR3
X01 0.10946 -0.33626 0.22279 M(-15)
X02 0.12680 -0.18069 0.38691 M(16-20)
X03 0.12374 0.12531 0.50920 M(21-30)
X04 0.11455 0.27018 0.43215 M(31-40)
X05 0.09978 0.38212 0.14670 M(41-)
X06 0.11811 -0.30729 -0.09636 F(-15)
X07 0.13175 -0.19038 -0.19733 F(16-20)
X08 0.13313 -0.02434 -0.33282 F(21-30)
X09 0.13227 0.12383 -0.37099 F(31-40)
X10 0.11609 0.20136 -0.60162 F(41-)
SAS システム 13
12:10 Thursday, October 16, 1997
プロット : FACTOR1*FACTOR2. 凡例: A = 1 OBS, B = 2 OBS, ...
5 + |
| |
FACTOR1 | |
| A A |A B A
| A A A AA BBA AAADA BB A AA A A A
0 +----A-------A----AAB--AAA----ACA---BABAAA-B--AAAAA--AA-A--A--A-
| A AA A A A A A A A | ABAAABB BA A A A
| A A A | A A
| | A
| |
-5 + |
--+-----------+-----------+-----------+-----------+-----------+-
-3 -2 -1 0 1 2
FACTOR2
SAS システム 14
12:10 Thursday, October 16, 1997
プロット : FACTOR2*FACTOR3. 凡例: A = 1 OBS, B = 2 OBS, ...
FACTOR2 | |
2.5 + |
| A A A A | AC A A A
| AA ABA ADABA AB| B A AA AA A
0.0 +---BA----AAAA-CBA--A-+-ECB-A------A----A--------A--------------
| A A BAA ABB AB AABAB
| B AA B AAAAA A A
-2.5 + | A
| |
| |
-5.0 + |
--+---------+---------+---------+---------+---------+---------+-
-2 -1 0 1 2 3 4
FACTOR3
/* Lesson 15-3 */
/* File Name = fact03.sas 10/16/97 */
data food;
infile 'food.dat';
input X01-X10;
label X01='M(-15)'
X02='M(16-20)'
X03='M(21-30)'
X04='M(31-40)'
X05='M(41-)'
X06='F(-15)'
X07='F(16-20)'
X08='F(21-30)'
X09='F(31-40)'
X10='F(41-)';
proc print data=food(obs=10);
run;
proc factor data=food nfactor=3 rotate=varimax out=fscore2;
var X01-X10; : 回転の指定
run; :
proc plot data=fscore2;
plot factor1*factor2/vref=0.0 href=0.0;
plot factor2*factor3/vref=0.0 href=0.0;
plot factor3*factor1/vref=0.0 href=0.0;
run;
SAS システム 20
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Orthogonal Transformation Matrix
1 2 3
1 0.65751 0.53576 0.52976
2 -0.73452 0.61238 0.29234
3 0.16779 0.58134 -0.79617
SAS システム 21
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Rotated Factor Pattern
FACTOR1 FACTOR2 FACTOR3
X01 0.95480 0.13534 0.08893 M(-15)
X02 0.85209 0.43859 0.13319 M(16-20)
X03 0.45782 0.81121 0.20628 M(21-30)
X04 0.21933 0.90009 0.29393 M(31-40)
X05 -0.02799 0.84163 0.46962 M(41-)
X06 0.91574 0.05827 0.32684 F(-15)
X07 0.81289 0.19001 0.49704 F(16-20)
X08 0.58706 0.31477 0.66894 F(21-30)
X09 0.38662 0.45477 0.76508 F(31-40)
X10 0.18442 0.37804 0.88499 F(41-)
SAS システム 22
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Variance explained by each factor
FACTOR1 FACTOR2 FACTOR3
3.923686 2.875550 2.545044
Final Communality Estimates: Total = 9.344279
X01 X02 X03 X04 X05
0.937870 0.936157 0.910210 0.944673 0.929662
X06 X07 X08 X09 X10
0.948805 0.943939 0.891197 0.941637 0.960129
SAS システム 23
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Scoring Coefficients Estimated by Regression
Squared Multiple Correlations of the Variables with each Factor
FACTOR1 FACTOR2 FACTOR3
1.000000 1.000000 1.000000
SAS システム 24
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Standardized Scoring Coefficients
FACTOR1 FACTOR2 FACTOR3
X01 0.35634 -0.01776 -0.21769 M(-15)
X02 0.28101 0.18221 -0.29369 M(16-20)
X03 0.07475 0.43906 -0.30323 M(21-30)
X04 -0.05062 0.47805 -0.20440 M(31-40)
X05 -0.19046 0.37274 0.04777 M(41-)
X06 0.28720 -0.18091 0.04945 F(-15)
X07 0.19335 -0.16071 0.17125 F(16-20)
X08 0.04957 -0.13707 0.32839 F(21-30)
X09 -0.06623 -0.06897 0.40164 F(31-40)
X10 -0.17252 -0.16424 0.59935 F(41-)
SAS システム 25
12:10 Thursday, October 16, 1997
プロット : FACTOR1*FACTOR2. 凡例: A = 1 OBS, B = 2 OBS, ...
2 + A | A
| A AA A A | A A
FACTOR1 | A AA |A A A A A
| A B AA AB B C A A A A A
| A A A B | A AA A A A
0 +-----------------A-A---------B-+AA-AA--A-AA--------------------
| A B AA AA| A A A A A
| A AA A|A C AA
| A A AA A AA B | A A A
| A A | AA A
-2 + A A A| A
--+---------+---------+---------+---------+---------+---------+-
-3 -2 -1 0 1 2 3
FACTOR2
SAS システム 26
12:10 Thursday, October 16, 1997
プロット : FACTOR2*FACTOR3. 凡例: A = 1 OBS, B = 2 OBS, ...
FACTOR2 | |
4 + |
| |
| A A |
2 + A | A A
| A AA A | AAA A A A
| A A A A A AAA |ABB BAB A B AA
0 +--------------------------A------B--D--AA+ACB-AAABAB-A---------
| A A AAA A C| BA BA C A A
| A A CA A | A B AB A
-2 + A A AA |
--+---------+---------+---------+---------+---------+---------+-
-4 -3 -2 -1 0 1 2
FACTOR3
SAS システム 27
12:10 Thursday, October 16, 1997
プロット : FACTOR3*FACTOR1. 凡例: A = 1 OBS, B = 2 OBS, ...
FACTOR3 | |
2.5 + |
| A B |A BA A
| A BABA A C ABBAA AAA AB A A A A
0.0 +-------------A-----BA-A-AAA---A--A-A-+-BAAC-AABAA-A--AC-AA-----
| A AA A A A B B| B A ABBB A AA
| A A |A AA A
-2.5 + A A | A
| A |
| |
-5.0 + |
--+-----------+-----------+-----------+-----------+-----------+-
-3 -2 -1 0 1 2
FACTOR1
/* Lesson 15-4 */
/* File Name = fact04.sas 10/16/97 */
data hobby;
infile 'syumi.dat';
input code $ X1-X6;
label X1='M(-29)'
X2='M(30-49)'
X3='M(50-)'
X4='F(-29)'
X5='F(30-49)'
X6='F(50-)';
proc print data=hobby(obs=10);
run;
proc factor data=hobby nfactor=2 out=fscore;
var X1-X6;
run;
proc plot data=fscore; : 回転前
plot factor1*factor2=code/vref=0.0 href=0.0; : コード化した記号
run; :
proc factor data=hobby nfactor=2 rotate=varimax out=fscore2;
var X1-X6;
run;
proc plot data=fscore2; : 回転後
plot factor1*factor2=code/vref=0.0 href=0.0; : コード化した記号
run; :
SAS システム 28
12:10 Thursday, October 16, 1997
OBS CODE X1 X2 X3 X4 X5 X6
1 A 4.00 4.25 3.83 4.50 4.67 4.00
2 B 4.17 3.89 4.00 4.50 4.17 3.75
3 C 3.83 3.44 2.83 3.57 3.17 1.50
4 D 2.83 4.22 3.83 3.71 3.00 2.25
5 E 4.17 4.11 3.83 3.57 4.00 3.75
6 F 2.33 3.56 3.33 2.93 2.83 2.75
7 G 1.83 2.44 2.33 3.71 3.83 3.75
8 H 2.50 1.89 2.00 4.21 3.17 3.75
9 I 2.00 1.44 2.00 4.07 3.33 3.50
10 J 4.00 3.33 3.33 3.00 3.17 2.25
SAS システム 29
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Prior Communality Estimates: ONE
Eigenvalues of the Correlation Matrix: Total = 6 Average = 1
1 2 3
Eigenvalue 2.7435 1.7477 0.7451
Difference 0.9958 1.0027 0.3571
Proportion 0.4573 0.2913 0.1242
Cumulative 0.4573 0.7485 0.8727
4 5 6
Eigenvalue 0.3879 0.2263 0.1495
Difference 0.1616 0.0768
Proportion 0.0647 0.0377 0.0249
Cumulative 0.9374 0.9751 1.0000
SAS システム 30
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
2 factors will be retained by the NFACTOR criterion.
Factor Pattern
FACTOR1 FACTOR2
X1 0.52708 0.63297 M(-29)
X2 0.59628 0.64623 M(30-49)
X3 0.64192 0.47370 M(50-)
X4 0.82757 -0.35514 F(-29)
X5 0.79607 -0.43033 F(30-49)
X6 0.61604 -0.62750 F(50-)
SAS システム 31
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Variance explained by each factor
FACTOR1 FACTOR2
2.743514 1.747721
Final Communality Estimates: Total = 4.491236
X1 X2 X3 X4 X5 X6
0.678467 0.773166 0.636447 0.810993 0.818906 0.773257
SAS システム 32
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Scoring Coefficients Estimated by Regression
Squared Multiple Correlations of the Variables with each Factor
FACTOR1 FACTOR2
1.000000 1.000000
SAS システム 33
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Standardized Scoring Coefficients
FACTOR1 FACTOR2
X1 0.19212 0.36217 M(-29)
X2 0.21734 0.36976 M(30-49)
X3 0.23398 0.27104 M(50-)
X4 0.30164 -0.20320 F(-29)
X5 0.29016 -0.24622 F(30-49)
X6 0.22454 -0.35904 F(50-)
SAS システム 34
12:10 Thursday, October 16, 1997
プロット : FACTOR1*FACTOR2. 使用するプロット文字: CODE の値．
(NOTE: 1 オブザベーションを表示してません．)
2 + A B
| | Z E
FACTOR1 | R |
| |
| 3 Q M | DL O
0 +--------------HG------------S-----2--+--F-------C--------------
| I K P | V N
| | U W
| 1|Y
| T X
-2 + 4 |
--+-----------+-----------+-----------+-----------+-----------+-
-3 -2 -1 0 1 2
FACTOR2
SAS システム 35
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Prior Communality Estimates: ONE
Eigenvalues of the Correlation Matrix: Total = 6 Average = 1
1 2 3
Eigenvalue 2.7435 1.7477 0.7451
Difference 0.9958 1.0027 0.3571
Proportion 0.4573 0.2913 0.1242
Cumulative 0.4573 0.7485 0.8727
4 5 6
Eigenvalue 0.3879 0.2263 0.1495
Difference 0.1616 0.0768
Proportion 0.0647 0.0377 0.0249
Cumulative 0.9374 0.9751 1.0000
SAS システム 36
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
2 factors will be retained by the NFACTOR criterion.
Factor Pattern
FACTOR1 FACTOR2
X1 0.52708 0.63297 M(-29)
X2 0.59628 0.64623 M(30-49)
X3 0.64192 0.47370 M(50-)
X4 0.82757 -0.35514 F(-29)
X5 0.79607 -0.43033 F(30-49)
X6 0.61604 -0.62750 F(50-)
SAS システム 37
12:10 Thursday, October 16, 1997
Initial Factor Method: Principal Components
Variance explained by each factor
FACTOR1 FACTOR2
2.743514 1.747721
Final Communality Estimates: Total = 4.491236
X1 X2 X3 X4 X5 X6
0.678467 0.773166 0.636447 0.810993 0.818906 0.773257
SAS システム 38
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Orthogonal Transformation Matrix
1 2
1 0.77751 0.62886
2 -0.62886 0.77751
SAS システム 39
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Rotated Factor Pattern
FACTOR1 FACTOR2
X1 0.01176 0.82361 M(-29)
X2 0.05723 0.87743 M(30-49)
X3 0.20121 0.77199 M(50-)
X4 0.86678 0.24430 F(-29)
X5 0.88957 0.16603 F(30-49)
X6 0.87359 -0.10049 F(50-)
Variance explained by each factor
FACTOR1 FACTOR2
2.349707 2.141529
SAS システム 40
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Final Communality Estimates: Total = 4.491236
X1 X2 X3 X4 X5 X6
0.678467 0.773166 0.636447 0.810993 0.818906 0.773257
Scoring Coefficients Estimated by Regression
Squared Multiple Correlations of the Variables with each Factor
FACTOR1 FACTOR2
1.000000 1.000000
SAS システム 41
12:10 Thursday, October 16, 1997
Rotation Method: Varimax
Standardized Scoring Coefficients
FACTOR1 FACTOR2
X1 -0.07838 0.40241 M(-29)
X2 -0.06354 0.42417 M(30-49)
X3 0.01147 0.35788 M(50-)
X4 0.36232 0.03170 F(-29)
X5 0.38045 -0.00897 F(30-49)
X6 0.40037 -0.13795 F(50-)
SAS システム 42
12:10 Thursday, October 16, 1997
プロット : FACTOR1*FACTOR2. 使用するプロット文字: CODE の値．
2 + |
| | A
FACTOR1 | I H G 3 | R ZB
| Q | E
| K S |M
0 +---------------------P-2--+------------D-------------
| |F CJ L O
| Y | V N
| 4 1 T | U
| X | W
-2 + |
---+-----------+-----------+-----------+-----------+--
-2 -1 0 1 2
FACTOR2