### SAS Textbook Examples Applied Linear Statistical Models by Neter, Kutner, et. al. Chapter 6: Multiple Regression I

Inputting the data shown on page 241.
data ch6fig05;
input x1 x2 y;
label x1='targtpop'
x2='dispoinc';
cards;
68.5  16.7  174.4
45.2  16.8  164.4
91.3  18.2  244.2
47.8  16.3  154.6
46.9  17.3  181.6
66.1  18.2  207.5
49.5  15.9  152.8
52.0  17.2  163.2
48.9  16.6  145.4
38.4  16.0  137.2
87.9  18.3  241.9
72.8  17.1  191.1
88.4  17.4  232.0
42.9  15.8  145.3
52.5  17.8  161.1
85.7  18.4  209.7
41.3  16.5  146.4
51.7  16.3  144.0
89.6  18.1  232.6
82.7  19.1  224.1
52.3  16.0  166.5
;
run;
Creating the x1x2 variable to be used in Fig. 6.7
data ch6fig05a;
set ch6fig05;
x1x2 = x1*x2;
run;
Fig. 6.4a, p. 237.
Scatterplot matrix.
Note: Invoking a macro for the scatter matrix.
%include "c:\neter\scatter.sas";
%scatter(data = ch6fig05a, var = y x1 x2);
Fig. 6.4b, p. 237.
Correlation matrix.
proc corr data = ch6fig05a;
run;
The CORR Procedure

4  Variables:    x1       x2       y        x1x2

Simple Statistics

Variable          N         Mean      Std Dev          Sum      Minimum      Maximum   Label

x1               21     62.01905     18.62033         1302     38.40000     91.30000   targtpop
x2               21     17.14286      0.97035    360.00000     15.80000     19.10000   dispoinc
y                21    181.90476     36.19130         3820    137.20000    244.20000
x1x2             21         1077    373.86333        22609    614.40000         1662

Pearson Correlation Coefficients, N = 21
Prob > |r| under H0: Rho=0

x1            x2             y          x1x2

x1             1.00000       0.78130       0.94455       0.99442
targtpop                      <.0001        <.0001        <.0001

x2             0.78130       1.00000       0.83580       0.83951
dispoinc        <.0001                      <.0001        <.0001

y              0.94455       0.83580       1.00000       0.95558
<.0001        <.0001                      <.0001

x1x2           0.99442       0.83951       0.95558       1.00000
<.0001        <.0001        <.0001
Fig. 6.5a and b, p. 241.
Note that output statement is used to create outfig05 with fitted and residual values.
proc reg data = ch6fig05a;
var x1x2;
model y = x1 x2/ i;
output out=outfig05 p = fitted r = residual;
run;
quit;
The REG Procedure
Model: MODEL1
Dependent Variable: y

X'X Inverse, Parameter Estimates, and SSE

Variable       Label             Intercept                x1                x2                 y

Intercept      Intercept      29.728923483      0.0721834719      -1.992553186      -68.85707315
x1             targtpop       0.0721834719      0.0003701761      -0.005549917      1.4545595828
x2             dispoinc       -1.992553186      -0.005549917      0.1363106368      9.3655003765
y                             -68.85707315      1.4545595828      9.3655003765      2180.9274114

Analysis of Variance

Sum of           Mean
Source                   DF        Squares         Square    F Value    Pr > F

Model                     2          24015          12008      99.10    <.0001
Error                    18     2180.92741      121.16263
Corrected Total          20          26196

Root MSE             11.00739    R-Square     0.9167
Dependent Mean      181.90476    Adj R-Sq     0.9075
Coeff Var             6.05118

Parameter Estimates

Parameter       Standard
Variable     Label        DF       Estimate          Error    t Value    Pr > |t|

Intercept    Intercept     1      -68.85707       60.01695      -1.15      0.2663
x1           targtpop      1        1.45456        0.21178       6.87      <.0001
x2           dispoinc      1        9.36550        4.06396       2.30      0.0333
Show all variables, including fitted and residual as shown in Fig. 6.5b, p. 24
proc print data = outfig05;
var y x1 x2 fitted residual;
run;
Obs      y       x1      x2      fitted    residual

1    174.4    68.5    16.7    187.184    -12.7841
2    164.4    45.2    16.8    154.229     10.1706
3    244.2    91.3    18.2    234.396      9.8037
4    154.6    47.8    16.3    153.329      1.2715
5    181.6    46.9    17.3    161.385     20.2151
6    207.5    66.1    18.2    197.741      9.7586
7    152.8    49.5    15.9    152.055      0.7449
8    163.2    52.0    17.2    167.867     -4.6666
9    145.4    48.9    16.6    157.738    -12.3382
10    137.2    38.4    16.0    136.846      0.3540
11    241.9    87.9    18.3    230.387     11.5126
12    191.1    72.8    17.1    197.185     -6.0849
13    232.0    88.4    17.4    222.686      9.3143
14    145.3    42.9    15.8    141.518      3.7816
15    161.1    52.5    17.8    174.213    -13.1132
16    209.7    85.7    18.4    228.124    -18.4239
17    146.4    41.3    16.5    145.747      0.6530
18    144.0    51.7    16.3    159.001    -15.0013
19    232.6    89.6    18.1    230.987      1.6130
20    224.1    82.7    19.1    230.316     -6.2161
21    166.5    52.3    16.0    157.064      9.4356
Note: To recreate the 3-D plots in Fig. 6.6 use interactive data analysis in SAS, visit our web page http://www.ats.ucla.edu/stat/sas/teach/reg_int/reg_int_cont.htm .
Fig. 6.7, p. 246, showing 4 different diagnostic plots.
proc gplot data = outfig05;
plot residual*fitted;
run;
proc gplot data = outfig05;
plot residual*x1;
run;
proc gplot data = outfig05;
plot residual*x2;
run;
proc gplot data = outfig05;
plot residual*x1x2;
run;
Fig 6.8a-Fig 6.8d, page 247 could have been obtained all in one proc gplot command as shown below.
proc gplot data = outfig05;
plot residual*fitted;
plot residual*x1;
plot residual*x2;
plot residual*x1x2;
run;
Fig 6.8a, page 247.
data outfig08;
set outfig05;
absresid = abs(residual);
run;

proc gplot data=outfig08;
plot absresid*fitted;
run;
Fig. 6.8b, p. 247, normal probability plot.
Note: The labels on the X-axis differs from the book.
proc univariate data = outfig05 noprint ;
qqplot residual / normal;
run;
Estimation of Mean Response and Prediction Limits for New Observations, p. 249-251. Adding an extra line of data in order to predict.
data ch6fig05h;
input x1 x2 y;
cards;
68.5  16.7  174.4
45.2  16.8  164.4
91.3  18.2  244.2
47.8  16.3  154.6
46.9  17.3  181.6
66.1  18.2  207.5
49.5  15.9  152.8
52.0  17.2  163.2
48.9  16.6  145.4
38.4  16.0  137.2
87.9  18.3  241.9
72.8  17.1  191.1
88.4  17.4  232.0
42.9  15.8  145.3
52.5  17.8  161.1
85.7  18.4  209.7
41.3  16.5  146.4
51.7  16.3  144.0
89.6  18.1  232.6
82.7  19.1  224.1
52.3  16.0  166.5
65.4  17.6    .
53.1  17.7    .
;
run;
Getting the predicted value and the CI's for E[Yh] and Yh(new), p. 249-251. Upper and Lower CLMean is for E[Yh] and Upper and Lower CL is for Yh(new).
proc reg data = ch6fig05h  ;
model y = x1 x2 / r cli clm;
ods output OutputStatistics=temp;
run;
quit;
The REG Procedure
Model: MODEL1
Dependent Variable: y

Analysis of Variance

Sum of           Mean
Source                   DF        Squares         Square    F Value    Pr > F

Model                     2          24015          12008      99.10    <.0001
Error                    18     2180.92741      121.16263
Corrected Total          20          26196

Root MSE             11.00739    R-Square     0.9167
Dependent Mean      181.90476    Adj R-Sq     0.9075
Coeff Var             6.05118

Parameter Estimates

Parameter       Standard
Variable     DF       Estimate          Error    t Value    Pr > |t|

Intercept     1      -68.85707       60.01695      -1.15      0.2663
x1            1        1.45456        0.21178       6.87      <.0001
x2            1        9.36550        4.06396       2.30      0.0333

The REG Procedure
Model: MODEL1
Dependent Variable: y

Output Statistics

Dep Var Predicted    Std Error
Obs         y     Value Mean Predict     95% CL Mean        95% CL Predict    Residual

1  174.4000  187.1841       3.8409  179.1146  195.2536  162.6910  211.6772  -12.7841
2  164.4000  154.2294       3.5558  146.7591  161.6998  129.9271  178.5317   10.1706
3  244.2000  234.3963       4.5882  224.7569  244.0358  209.3421  259.4506    9.8037
4  154.6000  153.3285       3.2331  146.5361  160.1210  129.2260  177.4311    1.2715
5  181.6000  161.3849       4.4300  152.0778  170.6921  136.4566  186.3132   20.2151
6  207.5000  197.7414       4.3786  188.5424  206.9404  172.8533  222.6295    9.7586
7  152.8000  152.0551       4.1696  143.2952  160.8150  127.3259  176.7843    0.7449
8  163.2000  167.8666       3.3310  160.8684  174.8649  143.7053  192.0280   -4.6666
9  145.4000  157.7382       2.9628  151.5136  163.9628  133.7895  181.6869  -12.3382
10  137.2000  136.8460       4.0074  128.4268  145.2653  112.2354  161.4566    0.3540
11  241.9000  230.3874       4.2012  221.5610  239.2137  205.6346  255.1402   11.5126
12  191.1000  197.1849       3.4109  190.0188  204.3510  172.9744  221.3954   -6.0849
13  232.0000  222.6857       5.3808  211.3810  233.9904  196.9448  248.4266    9.3143
14  145.3000  141.5184       4.1735  132.7502  150.2866  116.7863  166.2506    3.7816
15  161.1000  174.2132       5.0377  163.6294  184.7971  148.7807  199.6458  -13.1132
16  209.7000  228.1239       4.1214  219.4652  236.7826  203.4304  252.8174  -18.4239
17  146.4000  145.7470       3.7331  137.9041  153.5899  121.3276  170.1664    0.6530
18  144.0000  159.0013       3.2529  152.1672  165.8354  134.8870  183.1157  -15.0013
19  232.6000  230.9870       4.4176  221.7059  240.2681  206.0684  255.9056    1.6130
20  224.1000  230.3161       5.8120  218.1054  242.5267  204.1647  256.4675   -6.2161
21  166.5000  157.0644       4.0792  148.4944  165.6344  132.4018  181.7270    9.4356
22         .  191.1039       2.7668  185.2911  196.9168  167.2589  214.9490         .
23         .  174.1494       4.5986  164.4881  183.8107  149.0867  199.2121         .

Output Statistics

Std Error     Student                         Cook's
Obs  Residual    Residual      -2-1 0 1 2              D

1    10.316      -1.239    |    **|      |       0.071
2    10.417       0.976    |      |*     |       0.037
3    10.006       0.980    |      |*     |       0.067
4    10.522       0.121    |      |      |       0.000
5    10.077       2.006    |      |****  |       0.259
6    10.099       0.966    |      |*     |       0.059
7    10.187      0.0731    |      |      |       0.000
8    10.491      -0.445    |      |      |       0.007
9    10.601      -1.164    |    **|      |       0.035
10    10.252      0.0345    |      |      |       0.000
11    10.174       1.132    |      |**    |       0.073
12    10.466      -0.581    |     *|      |       0.012
13     9.603       0.970    |      |*     |       0.098
14    10.186       0.371    |      |      |       0.008
15     9.787      -1.340    |    **|      |       0.159
16    10.207      -1.805    |   ***|      |       0.177

The REG Procedure
Model: MODEL1
Dependent Variable: y

Output Statistics

Std Error     Student                         Cook's
Obs  Residual    Residual      -2-1 0 1 2              D

17    10.355      0.0631    |      |      |       0.000
18    10.516      -1.427    |    **|      |       0.065
19    10.082       0.160    |      |      |       0.002
20     9.348      -0.665    |     *|      |       0.057
21    10.224       0.923    |      |*     |       0.045
22         .           .                           .
23         .           .                           .

Sum of Residuals                           0
Sum of Squared Residuals          2180.92741
Predicted Residual SS (PRESS)     3002.92331
We use Where Observation >= 22 to show just the last two observation
proc print data = temp;
where Observation >= 22;
run;
                                                              StdErr
Predicted       Mean      Lower      Upper
Obs  Model   Dependent  Observation     DepVar    Value      Predict     CLMean     CLMean

22  MODEL1      y             22            .   191.1039     2.7668   185.2911   196.9168
23  MODEL1      y             23            .   174.1494     4.5986   164.4881   183.8107

StdErr     Student
Obs    LowerCL      UpperCL     Residual    Residual    Residual    Picture      CooksD

22   167.2589     214.9490            .          .           .                    .
23   149.0867     199.2121            .          .           .                    .

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