Simple Linear Regression: y = alpha + beta*x + error

[안재형]

cars.csv

proc import datafile="F:\publish\data analysis using R\data\cars.csv" out=cars dbms=csv replace;
run;

proc mixed data=cars method=reml;
  model dist = speed / s;
run;

Cov Parm     Estimate

Residual       236.53

                              Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|

Intercept    -17.5791      6.7584      48      -2.60      0.0123
speed          3.9324      0.4155      48       9.46      <.0001

        Type 3 Tests of Fixed Effects

               Num    Den
Effect         DF      DF    F Value    Pr > F

speed          1      48      89.57    <.0001

proc mixed data=cars method=ml;
  model dist = speed / s;
run;

Cov Parm     Estimate

Residual       227.07

                              Standard
Effect       Estimate       Error      DF    t Value    Pr > |t|

Intercept    -17.5791      6.6219      48      -2.65      0.0107
speed          3.9324      0.4071      48       9.66      <.0001

        Type 3 Tests of Fixed Effects

              Num     Den
Effect         DF      DF    F Value    Pr > F

speed           1      48      93.30    <.0001

proc genmod data=cars;
  model dist = speed / dist=normal link=identity;
run;

                                     Standard   Wald 95% Confidence      Wald
Parameter  DF   Estimate     Error               Limits             Chi-Square    Pr > ChiSq

Intercept     1    -17.5791      6.6219   -30.5578    -4.6004          7.05           0.0079
speed         1      3.9324      0.4071      3.1345      4.7303         93.30          <.0001
Scale         1     15.0689     1.5069     12.3868    18.3316

proc nlmixed data=cars;
  parms alpha=10 beta=10 s2e=100;
  model dist ~ normal(alpha+beta*speed, s2e);
run;

/* df=50? It should be 48 */

                              Standard
Parameter  Estimate    Error     DF   t Value   Pr > |t|    Alpha    Lower     Upper    Gradient

alpha       -17.5791     6.6219     50     -2.65     0.0106     0.05    -30.8796    -4.2786   -6.95E-6
beta            3.9324     0.4071     50      9.66     <.0001     0.05      3.1147      4.7501    -0.0001
s2e             227.07    45.4146    50     5.00     <.0001      0.05     135.85       318.29    5.603E-7