The SAS System

The GLM Procedure

Number of observations 30

 

 
The SAS System

The GLM Procedure

The X'X Matrix
  Intercept x1 x2 x3 y
Intercept 30 98.36 24.23 139.61 20.58
x1 98.36 332.3352 81.5834 459.4052 61.6502
x2 24.23 81.5834 30.1907 120.395 12.4103
x3 139.61 459.4052 120.395 682.7813 98.4408
y 20.58 61.6502 12.4103 98.4408 20.8074

 

 
The SAS System

The GLM Procedure

X'X Inverse Matrix
  Intercept x1 x2 x3 y
Intercept 1.7145778894 -0.328953365 0.0952905934 -0.146052394 1.8110425306
x1 -0.328953365 0.1062336548 -0.021055571 -0.00050401 -0.531455299
x2 0.0952905934 -0.021055571 0.117063783 -0.025959081 -0.439635791
x3 -0.146052394 -0.00050401 -0.025959081 0.0362447876 0.2089753066
y 1.8110425306 -0.531455299 -0.439635791 0.2089753066 1.1847859196

 

 
The SAS System

The GLM Procedure
Dependent Variable: y

Source DF Sum of Squares Mean Square F Value Pr > F
Model 3 5.50473408 1.83491136 40.27 <.0001
Error 26 1.18478592 0.04556869    
Corrected Total 29 6.68952000      
 
R-Square Coeff Var Root MSE y Mean
0.822889 31.11782 0.213468 0.686000
 
Source DF Type I SS Mean Square F Value Pr > F
x1 1 3.44600764 3.44600764 75.62 <.0001
x2 1 0.85384481 0.85384481 18.74 0.0002
x3 1 1.20488163 1.20488163 26.44 <.0001
 
Source DF Type III SS Mean Square F Value Pr > F
x1 1 2.65871240 2.65871240 58.35 <.0001
x2 1 1.65106256 1.65106256 36.23 <.0001
x3 1 1.20488163 1.20488163 26.44 <.0001
 
Parameter Estimate Standard Error t Value Pr > |t|
Intercept 1.811042531 0.27951935 6.48 <.0001
x1 -0.531455299 0.06957678 -7.64 <.0001
x2 -0.439635791 0.07303727 -6.02 <.0001
x3 0.208975307 0.04064022 5.14 <.0001

 

 
Observations, fitted values, residuals etc

Obs x1 x2 x3 y yhat ehat se ul ll
1 3.05 1.45 5.67 0.34 0.73752 -0.39752 0.06654 0.87429 0.60075
2 4.22 1.35 4.86 0.11 -0.00959 0.11959 0.07871 0.15219 -0.17137
3 3.34 0.26 4.19 0.38 0.79728 -0.41728 0.05438 0.90906 0.68550
4 3.77 0.23 4.42 0.68 0.63001 0.04999 0.06920 0.77225 0.48777
5 3.52 1.10 3.17 0.18 0.11917 0.06083 0.08246 0.28867 -0.05033
6 3.54 0.76 2.76 0.00 0.17234 -0.17234 0.08728 0.35174 -0.00706
7 3.74 1.59 3.81 0.08 -0.07943 0.15943 0.08875 0.10300 -0.26185
8 3.78 0.39 3.23 0.11 0.30567 -0.19567 0.07773 0.46545 0.14590
9 2.92 0.39 5.44 1.53 1.22456 0.30544 0.06779 1.36391 1.08521
10 3.10 0.64 6.16 0.77 1.16945 -0.39945 0.07825 1.33030 1.00860
11 2.86 0.82 5.48 1.17 1.07576 0.09424 0.05914 1.19733 0.95420
12 2.78 0.64 4.62 1.01 1.01770 -0.00770 0.05196 1.12450 0.91089
13 2.22 0.85 4.49 0.89 1.19582 -0.30582 0.08422 1.36893 1.02271
14 2.67 0.90 5.59 1.40 1.16456 0.23544 0.06881 1.30601 1.02311
15 3.12 0.92 5.86 1.05 0.97303 0.07697 0.06192 1.10031 0.84575
16 3.03 0.97 6.60 1.15 1.15352 -0.00352 0.08699 1.33234 0.97471
17 2.45 0.18 4.51 1.49 1.37232 0.11768 0.07591 1.52835 1.21629
18 4.12 0.62 5.31 0.51 0.45853 0.05147 0.08009 0.62315 0.29391
19 4.61 0.51 5.16 0.18 0.21513 -0.03513 0.10992 0.44107 -0.01080
20 3.94 0.45 4.45 0.34 0.44921 -0.10921 0.06837 0.58976 0.30867
21 4.12 1.79 6.17 0.36 0.12388 0.23612 0.09339 0.31585 -0.06809
22 2.93 0.25 3.38 0.89 0.85031 0.03969 0.06612 0.98622 0.71440
23 2.66 0.31 3.51 0.91 0.99459 -0.08459 0.06988 1.13823 0.85094
24 3.17 0.20 3.08 0.92 0.68205 0.23795 0.07238 0.83083 0.53326
25 2.79 0.24 3.98 1.35 1.05449 0.29551 0.06076 1.17938 0.92961
26 2.61 0.20 3.64 1.33 1.09669 0.23331 0.07129 1.24322 0.95015
27 3.74 2.27 6.50 0.23 0.18377 0.04623 0.10893 0.40767 -0.04014
28 3.13 1.48 4.28 0.26 0.39134 -0.13134 0.07107 0.53744 0.24525
29 3.49 0.25 4.71 0.73 0.83063 -0.10063 0.06082 0.95565 0.70561
30 2.94 2.22 4.58 0.23 0.22968 0.00032 0.11784 0.47191 -0.01255

 

 
Output from IML 

                     X     30 rows      4 cols    (numeric)
 
1 3.05 1.45 5.67
1 4.22 1.35 4.86
1 3.34 0.26 4.19
1 3.77 0.23 4.42
1 3.52 1.1 3.17
1 3.54 0.76 2.76
1 3.74 1.59 3.81
1 3.78 0.39 3.23
1 2.92 0.39 5.44
1 3.1 0.64 6.16
1 2.86 0.82 5.48
1 2.78 0.64 4.62
1 2.22 0.85 4.49
1 2.67 0.9 5.59
1 3.12 0.92 5.86
1 3.03 0.97 6.6
1 2.45 0.18 4.51
1 4.12 0.62 5.31
1 4.61 0.51 5.16
1 3.94 0.45 4.45
1 4.12 1.79 6.17
1 2.93 0.25 3.38
1 2.66 0.31 3.51
1 3.17 0.2 3.08
1 2.79 0.24 3.98
1 2.61 0.2 3.64
1 3.74 2.27 6.5
1 3.13 1.48 4.28
1 3.49 0.25 4.71
1 2.94 2.22 4.58
  
                     Y     30 rows      1 col     (numeric)
 
0.34
0.11
0.38
0.68
0.18
0
0.08
0.11
1.53
0.77
1.17
1.01
0.89
1.4
1.05
1.15
1.49
0.51
0.18
0.34
0.36
0.89
0.91
0.92
1.35
1.33
0.23
0.26
0.73
0.23
  
                     A     30 rows      4 cols    (numeric)
 
3.05 1.45 5.67 0.34
4.22 1.35 4.86 0.11
3.34 0.26 4.19 0.38
3.77 0.23 4.42 0.68
3.52 1.1 3.17 0.18
3.54 0.76 2.76 0
3.74 1.59 3.81 0.08
3.78 0.39 3.23 0.11
2.92 0.39 5.44 1.53
3.1 0.64 6.16 0.77
2.86 0.82 5.48 1.17
2.78 0.64 4.62 1.01
2.22 0.85 4.49 0.89
2.67 0.9 5.59 1.4
3.12 0.92 5.86 1.05
3.03 0.97 6.6 1.15
2.45 0.18 4.51 1.49
4.12 0.62 5.31 0.51
4.61 0.51 5.16 0.18
3.94 0.45 4.45 0.34
4.12 1.79 6.17 0.36
2.93 0.25 3.38 0.89
2.66 0.31 3.51 0.91
3.17 0.2 3.08 0.92
2.79 0.24 3.98 1.35
2.61 0.2 3.64 1.33
3.74 2.27 6.5 0.23
3.13 1.48 4.28 0.26
3.49 0.25 4.71 0.73
2.94 2.22 4.58 0.23
  
                    X1     30 rows      1 col     (numeric)
 
3.05
4.22
3.34
3.77
3.52
3.54
3.74
3.78
2.92
3.1
2.86
2.78
2.22
2.67
3.12
3.03
2.45
4.12
4.61
3.94
4.12
2.93
2.66
3.17
2.79
2.61
3.74
3.13
3.49
2.94
  
                    X2     30 rows      1 col     (numeric)
 
1.45
1.35
0.26
0.23
1.1
0.76
1.59
0.39
0.39
0.64
0.82
0.64
0.85
0.9
0.92
0.97
0.18
0.62
0.51
0.45
1.79
0.25
0.31
0.2
0.24
0.2
2.27
1.48
0.25
2.22
  
                    X3     30 rows      1 col     (numeric)
 
5.67
4.86
4.19
4.42
3.17
2.76
3.81
3.23
5.44
6.16
5.48
4.62
4.49
5.59
5.86
6.6
4.51
5.31
5.16
4.45
6.17
3.38
3.51
3.08
3.98
3.64
6.5
4.28
4.71
4.58
  
                     Y     30 rows      1 col     (numeric)
 
0.34
0.11
0.38
0.68
0.18
0
0.08
0.11
1.53
0.77
1.17
1.01
0.89
1.4
1.05
1.15
1.49
0.51
0.18
0.34
0.36
0.89
0.91
0.92
1.35
1.33
0.23
0.26
0.73
0.23
  
                   NOBS      1 row       1 col     (numeric)
 
30
  
                    MU     30 rows      1 col     (numeric)
 
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
  
                     X     30 rows      4 cols    (numeric)
 
1 3.05 1.45 5.67
1 4.22 1.35 4.86
1 3.34 0.26 4.19
1 3.77 0.23 4.42
1 3.52 1.1 3.17
1 3.54 0.76 2.76
1 3.74 1.59 3.81
1 3.78 0.39 3.23
1 2.92 0.39 5.44
1 3.1 0.64 6.16
1 2.86 0.82 5.48
1 2.78 0.64 4.62
1 2.22 0.85 4.49
1 2.67 0.9 5.59
1 3.12 0.92 5.86
1 3.03 0.97 6.6
1 2.45 0.18 4.51
1 4.12 0.62 5.31
1 4.61 0.51 5.16
1 3.94 0.45 4.45
1 4.12 1.79 6.17
1 2.93 0.25 3.38
1 2.66 0.31 3.51
1 3.17 0.2 3.08
1 2.79 0.24 3.98
1 2.61 0.2 3.64
1 3.74 2.27 6.5
1 3.13 1.48 4.28
1 3.49 0.25 4.71
1 2.94 2.22 4.58
  
                    XTX      4 rows      4 cols    (numeric)
 
30 98.36 24.23 139.61
98.36 332.3352 81.5834 459.4052
24.23 81.5834 30.1907 120.395
139.61 459.4052 120.395 682.7813
  
                    XTY      4 rows      1 col     (numeric)
 
20.58
61.6502
12.4103
98.4408
  
                  INVXTX      4 rows      4 cols    (numeric)
 
1.7145779 -0.328953 0.0952906 -0.146052
-0.328953 0.1062337 -0.021056 -0.000504
0.0952906 -0.021056 0.1170638 -0.025959
-0.146052 -0.000504 -0.025959 0.0362448
  
                     B      4 rows      1 col     (numeric)
 
1.8110425
-0.531455
-0.439636
0.2089753
  
                    TSS      1 row       1 col     (numeric)
 
20.8074
  
                    SSR      1 row       1 col     (numeric)
 
19.622614
  
                    SSE      1 row       1 col     (numeric)
 
1.1847859
  
                   NOBS      1 row       1 col     (numeric)
 
30
  
                    RX      1 row       1 col     (numeric)
 
4
  
                    DFE      1 row       1 col     (numeric)
 
26
  
                   SUMY      1 row       1 col     (numeric)
 
20.58
  
                   YBAR      1 row       1 col     (numeric)
 
0.686
  
                    CF      1 row       1 col     (numeric)
 
14.11788
  
                   SSRM      1 row       1 col     (numeric)
 
5.5047341
  
                    MSE      1 row       1 col     (numeric)
 
0.0455687
  
                   COVB      4 rows      4 cols    (numeric)
 
0.0781311 -0.01499 0.0043423 -0.006655
-0.01499 0.0048409 -0.000959 -0.000023
0.0043423 -0.000959 0.0053344 -0.001183
-0.006655 -0.000023 -0.001183 0.0016516
  
                   SEB0      1 row       1 col     (numeric)
 
0.2795193
  
                   SEB1      1 row       1 col     (numeric)
 
0.0695768
  
                   SEB2      1 row       1 col     (numeric)
 
0.0730373
  
                   SEB3      1 row       1 col     (numeric)
 
0.0406402
  
                    T0      1 row       1 col     (numeric)
 
6.4791312
  
                    PR      1 row       1 col     (numeric)
 
7.237E-7
  
                    T1      1 row       1 col     (numeric)
 
-7.6384
  
                    T2      1 row       1 col     (numeric)
 
-6.019335
  
                    T3      1 row       1 col     (numeric)
 
5.1420811
  
                    T4      1 row       1 col     (numeric)
 
33.215715
  
                    PR      1 row       1 col     (numeric)
 
0
  
                     K      4 rows      1 col     (numeric)
 
0
1
0
0
  
                    KB      1 row       1 col     (numeric)
 
-0.531455
  
                   KXXK      1 row       1 col     (numeric)
 
0.1062337
  
                   INVKK      1 row       1 col     (numeric)
 
9.4132128
  
                    SS1      1 row       1 col     (numeric)
 
2.6587124
  
                    MS1      1 row       1 col     (numeric)
 
2.6587124
  
                    F1      1 row       1 col     (numeric)
 
58.345159
  
                    PR      1 row       1 col     (numeric)
 
4.1639E-8
  
                     K      4 rows      1 col     (numeric)
 
0
0
1
0
  
                    KB      1 row       1 col     (numeric)
 
-0.439636
  
                   KXXK      1 row       1 col     (numeric)
 
0.1170638
  
                   INVKK      1 row       1 col     (numeric)
 
8.5423517
  
                    SS2      1 row       1 col     (numeric)
 
1.6510626
  
                    MS2      1 row       1 col     (numeric)
 
1.6510626
  
                    F2      1 row       1 col     (numeric)
 
36.232391
  
                    PR      1 row       1 col     (numeric)
 
2.3423E-6
  
                     K      4 rows      1 col     (numeric)
 
0
0
0
1
  
                    KB      1 row       1 col     (numeric)
 
0.2089753
  
                   KXXK      1 row       1 col     (numeric)
 
0.0362448
  
                   INVKK      1 row       1 col     (numeric)
 
27.590174
  
                    SS3      1 row       1 col     (numeric)
 
1.2048816
  
                    MS3      1 row       1 col     (numeric)
 
1.2048816
  
                    F2      1 row       1 col     (numeric)
 
26.440998
  
                    PR      1 row       1 col     (numeric)
 
0.0000231
  
                     K      4 rows      3 cols    (numeric)
 
0 0 0
1 0 0
0 1 0
0 0 1
  
                    KB      3 rows      1 col     (numeric)
 
-0.531455
-0.439636
0.2089753
  
                   KXXK      3 rows      3 cols    (numeric)
 
0.1062337 -0.021056 -0.000504
-0.021056 0.1170638 -0.025959
-0.000504 -0.025959 0.0362448
  
                   INVKK      3 rows      3 cols    (numeric)
 
9.8455467 2.1413067 1.6705467
2.1413067 10.620937 7.6366567
1.6705467 7.6366567 33.082897
  
                    SS3      1 row       1 col     (numeric)
 
5.5047341
  
                    MS3      1 row       1 col     (numeric)
 
1.8349114
  
                    F3      1 row       1 col     (numeric)
 
40.266933
  
                    PR      1 row       1 col     (numeric)
 
6.454E-10
  
                   YHAT     30 rows      1 col     (numeric)
 
0.737522
-0.009587
0.7972831
0.6300107
0.1191722
0.1723394
-0.079425
0.3056738
1.2245608
1.1694521
1.0757637
1.0176958
1.1958205
1.1645566
0.9730324
1.1535233
1.3723212
0.4585314
0.2151319
0.4492127
0.1238763
0.8503061
0.9945877
0.682046
1.0544914
1.0966872
0.183766
0.3913408
0.8306283
0.2296794
  
                  SEYHAT     30 rows      1 col     (numeric)
 
0.0665365
0.0787054
0.0543806
0.0691986
0.0824599
0.0872781
0.08875
0.0777278
0.0677938
0.0782525
0.0591418
0.0519616
0.0842187
0.0688141
0.0619197
0.086993
0.0759074
0.0800871
0.1099155
0.068373
0.0933924
0.0661197
0.0698814
0.0723817
0.0607562
0.0712891
0.1089287
0.0710741
0.0608219
0.1178412
  
                   X1BAR      1 row       1 col     (numeric)
 
3.2786667
  
                   X2BAR      1 row       1 col     (numeric)
 
0.8076667
  
                   XNEW     30 rows      4 cols    (numeric)
 
1 3.05 1.45 5.67
1 4.22 1.35 4.86
1 3.34 0.26 4.19
1 3.77 0.23 4.42
1 3.52 1.1 3.17
1 3.54 0.76 2.76
1 3.74 1.59 3.81
1 3.78 0.39 3.23
1 2.92 0.39 5.44
1 3.1 0.64 6.16
1 2.86 0.82 5.48
1 2.78 0.64 4.62
1 2.22 0.85 4.49
1 2.67 0.9 5.59
1 3.12 0.92 5.86
1 3.03 0.97 6.6
1 2.45 0.18 4.51
1 4.12 0.62 5.31
1 4.61 0.51 5.16
1 3.94 0.45 4.45
1 4.12 1.79 6.17
1 2.93 0.25 3.38
1 2.66 0.31 3.51
1 3.17 0.2 3.08
1 2.79 0.24 3.98
1 2.61 0.2 3.64
1 3.74 2.27 6.5
1 3.13 1.48 4.28
1 3.49 0.25 4.71
1 2.94 2.22 4.58
  
                   XNEW     30 rows      4 cols    (numeric)
 
1 3.05 0.8076667 5.67
1 4.22 0.8076667 4.86
1 3.34 0.8076667 4.19
1 3.77 0.8076667 4.42
1 3.52 0.8076667 3.17
1 3.54 0.8076667 2.76
1 3.74 0.8076667 3.81
1 3.78 0.8076667 3.23
1 2.92 0.8076667 5.44
1 3.1 0.8076667 6.16
1 2.86 0.8076667 5.48
1 2.78 0.8076667 4.62
1 2.22 0.8076667 4.49
1 2.67 0.8076667 5.59
1 3.12 0.8076667 5.86
1 3.03 0.8076667 6.6
1 2.45 0.8076667 4.51
1 4.12 0.8076667 5.31
1 4.61 0.8076667 5.16
1 3.94 0.8076667 4.45
1 4.12 0.8076667 6.17
1 2.93 0.8076667 3.38
1 2.66 0.8076667 3.51
1 3.17 0.8076667 3.08
1 2.79 0.8076667 3.98
1 2.61 0.8076667 3.64
1 3.74 0.8076667 6.5
1 3.13 0.8076667 4.28
1 3.49 0.8076667 4.71
1 2.94 0.8076667 4.58
  
                   YNEW     30 rows      1 col     (numeric)
 
1.0199147
0.228842
0.5565092
0.3760477
0.2476924
0.1513834
0.2645165
0.1220526
1.0409396
1.0957398
1.0811859
0.9439835
1.2144317
1.2051497
1.0224181
1.2248908
1.0963765
0.3760264
0.084267
0.2919696
0.5557452
0.6051359
0.7757956
0.414894
0.8049248
0.8295351
0.82666
0.6869226
0.5854581
0.8505917
 
  
                   SENEW     30 rows      1 col     (numeric)
 
0.2214892
0.2268009
0.2178583
0.2198903
0.2258799
0.2310058
0.2220617
0.2273427
0.2207817
0.2258366
0.2215397
0.2197556
0.2292368
0.2243998
0.2227596
0.2316591
0.2245896
0.226275
0.2367708
0.22199
0.2329231
0.2243548
0.2259829
0.226334
0.2213099
0.2256503
0.2317523
0.2177675
0.2175053
0.2182904