Loglinearand logit models

Solution Preview

Please, find a loglinear model that describes this data well. Interpret associations then treating whether killed as the response, find an equivalent logit model. Interpret the effects.

Solution
/* I first transform the data such that: Seat belt = 1, none = 2; Yes = 1, No = 2; Fatal = 1, Nonfatal = 2 */

data car;
input belt eject injury count;
datalines;
1 1 2 1105
1 1 1 14
1 2 2 411111
1 2 1 483
2 1 2 4624
2 1 1 497
2 2 2 157342
2 2 1 1008
;
run;

/*Log-Linear Model*/
/*Fit saturated model then test significance of the highest interaction using p-values. Note 'belt|eject|injury' explores all possible combinations*/

/*CODE*/
proc genmod data=car;
class belt eject injury;
model count = belt|eject|injury/ dist=poi link=log type3;
run;

/*OUTPUT*/
Analysis Of Maximum Likelihood Parameter Estimates

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

belt*eject 2 2 0 0.0000 0.0000 0.0000 0.0000 . .
injury 1 1 -5.0505 0.0316 -5.1124 -4.9885 25547.5 <.0001
injury 2 0 0.0000 0.0000 0.0000 0.0000 . .
belt*injury 1 1 1 -1.6961 0.0554 -1.8048 -1.5875 936.73 <.0001
belt*injury 1 2 0 0.0000 0.0000 0.0000 0.0000 . .
belt*injury 2 1 0 0.0000 0.0000 0.0000 0.0000 . .
belt*injury 2 2 0 0.0000 0.0000 0.0000 0.0000 . .
eject*injury 1 1 1 2.8200 0.0568 2.7087 2.9314 2464.57 <.0001
eject*injury 1 2 0 0.0000 0.0000 0.0000 0.0000 . .
eject*injury 2 1 0 0.0000 0.0000 0.0000 0.0000 . .
eject*injury 2 2 0 0.0000 0.0000 0.0000 0.0000 . .
belt*eject*injury 1 1 1 1 -0.4420 0.2786 -0.9881 0.1041 2.52 0.1127
belt*eject*injury 1 1 2 0 0.0000 0.0000 0.0000 0.0000 . .
belt*eject*injury 1 2 1 0 0.0000 0.0000 0.0000 0.0000 . .
belt*eject*injury 1 2 2 0 0.0000 0.0000 0.0000 0.0000 . .
belt*eject*injury 2 1 1 0 0.0000 0.0000 0.0000 0.0000 . .
belt*eject*injury 2 1 2 0 0.0000 0.0000 0.0000 0.0000 . .
belt*eject*injury 2 2 1 0 0.0000 0.0000 0.0000 0.0000 . .
belt*eject*injury 2 2 2 0 0.0000 0.0000 0.0000 0.0000 . .
Scale 0 1.0000 0.0000 1.0000 1.0000

NOTE: The scale parameter was held fixed.

LR Statistics For Type 3 Analysis

Chi-
Source DF Square Pr > ChiSq

belt 1 939.10 <.0001
eject 1 8798.41 <.0001
belt*eject 1 1132.90 <.0001
injury 1 26523.5 <.0001
belt*injury 1 512.98 <.0001
eject*injury 1 116.50 <.0001
belt*eject*injury 1 2.85 0.0911 (this is not significant so drop it and refit the model without the three way interaction)

/*Three way interaction is not significant, so drop it and refit the model */
proc genmod data=car;
class belt eject injury;
model count = belt eject injury belt*eject belt*injury eject*injury/ dist=poi link=log type3;
run;

The SAS ...