DEMO PROGRAM 3 from LAB 4:
/****************************************************
*
Scenario: Researchers record ages of rats in
*
* weeks, total radiation dosage, and count=number of *
* damaged cells (average of 5 determinations) found *
* when rats are sacrificed. The rats are exposed to *
* a constant radiation source in their cages and the *
* total (cumulative) dose is measured by a collar
*
* device. If they did not move around with respect
*
* to the source, dosage would be exactly a linear
*
* function of weeks and we could not invert X’X.
*
*
*
* In this demo, we see that if X1 and X2 (age and
*
* dose) are highly collinear, we have VERY unstable *
* estimates of our parameters. Notice that the
*
* model F test says we cannot omit both X1 and X2,
*
* but our t tests say we CAN omit either one.
*
* They (X1 and X2; Age and Dose) are REDUNDANT in a *
* sense. Look at the plot to see the idea.
*
*
*
* Note that we are NOT saying that age and dose do
*
* not affect count, nor are we saying that age and
*
* dose would be related in some population like they *
* are in this sample. We ARE saying that by the way *
* we did the experiment, we have collected data that *
* does NOT ALLOW us to determine which of these two *
* variables is important! – a very poor design.
*
******************************************************/
Data rats; input weeks radiation;
count = 5 + .3*weeks + .2*radiation + round(4*normal(123));
cards;
18 35
27 45
56 73
40 57
84 100
70 88
10 25
50 68
;
proc print;
proc g3d; scatter weeks*radiation=count;
proc reg; model count = weeks radiation/SS1 SS2;
* get the sequential (SS1) and partial (SS2) sums of squares *; run;
Analysis of Variance
Source
Model
Error
Corrected Total
DF
2
5
7
Sum of
Squares
1124.49930
67.68945
1192.18875
Mean
Square
562.24965
13.53789
Parameter Estimates
F Value
41.53
Pr > F
0.0008
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Type I SS
Type II SS
Intercept
weeks
radiation
1
1
1
11.32486
0.65124
-0.15497
22.30709
1.34848
1.33489
0.51
0.48
-0.12
0.6333
0.6495
0.9121
7546.06125
1124.31685
0.18246
3.48924
3.15753
0.18246
Add in weight of rat (should be related to weeks – right?)
Source
Model
Error
Corrected Total
DF
Sum of
Squares
Mean
Square
3
4
7
1125.66559
66.52316
1192.18875
375.22186
16.63079
F Value
Pr > F
22.56
0.0057
Variable
DF
Parameter
Estimate
Standard
Error
t Value
Pr > |t|
Type I SS
Type II SS
Intercept
weeks
radiation
weight
1
1
1
1
9.80542
0.22179
-0.50886
2.43431
25.38133
2.20540
1.99370
9.19243
0.39
0.10
-0.26
0.26
0.7189
0.9247
0.8111
0.8042
7546.06125
1124.31685
0.18246
1.16629
2.48208
0.16820
1.08339
1.16629
H 0 :  2  3  0
(no radiation or weight effect)
F42 
(0.1825  1.1663) / 2
 0.04
16.6308
(critical value is 6.94, p-value is 0.9607. Cannot reject, i.e. no evidence of radiation or weight
effect).
H0 : 2  0
,
F41 
1.0834
 0.07
16.6308
p-value is 0.8111 (same as t)
Or use t = -0.26
t2=F for calculated test statistics (SINGLE parameter only!)
Also critical value for F is square of t critical value.