### SAS Textbook Examples Applied Linear Statistical Models by Neter, Kutner, et. al. Chapter 21: Two-Factor Studies--One Case Per Treatment

Inputting the Insurance Premium data, table 21.2a, p. 878. The proc glm was used to produce the ANOVA table. The means statement generated the means of premium for each level of of city and each level of region, table 21.2b, p. 878. The output from proc glm also includes the F-tests of each predictor, p. 879-880.
data insurance;
cards;
140  1  1
100  1  2
210  2  1
180  2  2
220  3  1
200  3  2
;
run;
proc glm data=insurance;
class city region;
means city region;
run;
quit;
The GLM Procedure

Class Level Information

Class         Levels    Values
city               3    1 2 3
region             2    1 2

Number of observations    6

The GLM Procedure

Sum of
Source                      DF         Squares     Mean Square    F Value    Pr > F
Model                        3     10650.00000      3550.00000      71.00    0.0139
Error                        2       100.00000        50.00000
Corrected Total              5     10750.00000
R-Square     Coeff Var      Root MSE    premium Mean
0.990698      4.040610      7.071068        175.0000
Source                      DF       Type I SS     Mean Square    F Value    Pr > F
city                         2     9300.000000     4650.000000      93.00    0.0106
region                       1     1350.000000     1350.000000      27.00    0.0351
Source                      DF     Type III SS     Mean Square    F Value    Pr > F
city                         2     9300.000000     4650.000000      93.00    0.0106
region                       1     1350.000000     1350.000000      27.00    0.0351

The GLM Procedure

city         N             Mean          Std Dev
1            2       120.000000       28.2842712
2            2       195.000000       21.2132034
3            2       210.000000       14.1421356

region       N             Mean          Std Dev
1            3       190.000000       43.5889894
2            3       160.000000       52.9150262
Fig. 21.1, p. 879.
In order to get the lines in the same graph we need to create three variables for region that corresponds to each of the levels of city. The overlay option in the plot statement lets us plot all the lines in the same graph.
symbol v=dot i=join;
legend1 label=none value=(height=1 font=swiss 'Large City' 'Medium City' 'Small City' )
position=(left bottom  inside) mode=share cborder=black;
proc gplot data=insurance;
run;
quit;
Predicting estimates of the treatment means, p. 881.
proc glm data=insurance noprint;
class city region;
model premium = city region ;
output out=temp p=predict;
run;
quit;
proc print data=temp;
run;
Obs    city    region    premium    predict

1       1        1        140        135
2       1        2        100        105
3       2        1        210        210
4       2        2        180        180
5       3        1        220        225
6       3        2        200        195
Creating the dummy variables for city and region, p. 881. Running the regression to get the factor effects alphai and betaj. When looking at the predict values from the regression we see that we get exactly the same values as from the proc glm.
data dummy;
set insurance;
if city=1 then x1=1;
else if city=3 then x1=-1;
else x1=0;
if city=2 then x2=1;
else if city=3 then x2=-1;
else x2=0;
if region=1 then x3=1;
else x3=-1;
run;
proc reg data=dummy;
model premium = x1 x2 x3;
output out=temp p=predict;
run;
quit;
proc print data=temp;
run;
The REG Procedure
Model: MODEL1

Analysis of Variance

Sum of           Mean
Source                   DF        Squares         Square    F Value    Pr > F
Model                     3          10650     3550.00000      71.00    0.0139
Error                     2      100.00000       50.00000
Corrected Total           5          10750

Root MSE              7.07107    R-Square     0.9907
Dependent Mean      175.00000    Adj R-Sq     0.9767
Coeff Var             4.04061
Parameter Estimates

Parameter       Standard
Variable     DF       Estimate          Error    t Value    Pr > |t|
Intercept     1      175.00000        2.88675      60.62      0.0003
x1            1      -55.00000        4.08248     -13.47      0.0055
x2            1       20.00000        4.08248       4.90      0.0392
x3            1       15.00000        2.88675       5.20      0.0351

1       1        1        140        135
2       1        2        100        105
3       2        1        210        210
4       2        2        180        180
5       3        1        220        225
6       3        2        200        195
Tukey test of Additivity for the insurance data, p. 884.
We need to obtain the sums of squares for each predictor and the corrected total sums of squares which is most easily accomplished using the ODS system and then saved as macro variables. Using the macro variables and several sql procedures we can then create the SSAB, SSrem values and the F test in a data step at the end of the program.
ods listing close;
proc glm data=insurance;
class region city;
ods output overallanova=overall modelanova=model;
run;
quit;
ods listing;
ods output close;
data _null_;
set overall;
if source='Corrected Total' then call symput('overall', ss);
run;
data _null_;
set model ;
if hypothesistype=1 and source='city' then call symput('ssa', ss);
if hypothesistype=1 and source='region' then call symput('ssb', ss);
if hypothesistype=1 and source='city' then call symput('dfa', df);
if hypothesistype=1 and source='region' then call symput('dfb', df);
run;
%put here is &overall &ssa &ssb &dfa &dfb; /* the statement will appear in the log file so you can check the calculations */
proc sql;
create table temp1 as
from insurance
group by region;
quit;
proc sql;
create table temp2 as
from temp1
group by city;
quit;
proc sql noprint;
select mean(premium) into :meanp from temp1;
quit;
%put here is  &meanp; /* check value in log file */
proc sql noprint;
select sum( (yi - &meanp)*(yj - &meanp)*premium ) into :total from temp2;
quit;
%put here is &total; /*check value in log file*/
data final;
msa = &ssa/(&dfb+1);
msb = &ssb/(&dfa+1);
ssab = (&total*&total) / ( msa*msb );
ssrem = &overall - &ssa - &ssb - ssab;
f = ssab/( ssrem/((&dfa+1)*(&dfb+1) - (&dfa+1) - (&dfb+1)) );
p_value = 1- cdf('F',f,  1, (&dfa+1)*(&dfb+1) - (&dfa+1) - (&dfb+1) );
run;
proc print data=final;
run;
Obs     msa    msb      ssab      ssrem       f     p_value
1     4650    450    87.0968    12.9032    6.75    0.23391

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