Hello,
It's straight forward to see how the HR's are calculated using the coefficients displayed in the HR table, but, I am trying to figure out how the coefficients are determined (especially the one highlighted in blue below) when using restricted cubic splines in calculating the HR's. A mathematical notation/equation using the ML estimates below when I=0.3 would be great. Or, if someone can point me in the direction of how the SAS output correlates with the math behind it, that would be great!
1. Variable I has a range (0.05 - 0.729)
2. Treatments A and B (2 groups)
3. using 3 equally spaced knots
/*Code*/
proc phreg data = dataset outest=RegParms;
effect I_RCS = spline(I / basis=tpf(noint) NATURALCUBIC details knotmethod=equal(3));
class Treatment(ref='B') Gender(ref='Female');
model T*Status(0, 2, 3) = I_RCS Treatment I_RCS*Treatment Gender Age Score1 Score2 / rl;
hazardratio Treatment / at(I=0.3 0.6) e;
run;
/*SAS Output*/
|
Knots for Spline Effect
I_RCS
|
|
Knot Number
|
Frailty
|
|
1
|
0.22297
|
|
2
|
0.39189
|
|
3
|
0.56081
|
|
Basis Details for Spline Effect
I_RCS
|
|
Column
|
Power
|
Break Knot
|
|
1
|
1
|
|
|
2
|
3
|
0.22297
|
|
Analysis of Maximum Likelihood Estimates
|
|
Parameter
|
|
|
DF
|
Parameter
Estimate
|
Standard
Error
|
Chi-Square
|
Pr > ChiSq
|
Hazard
Ratio
|
95% Hazard Ratio Confidence
Limits
|
Label
|
|
I_RCS
|
1
|
|
1
|
-1.34409
|
1.86171
|
0.5212
|
0.4703
|
.
|
.
|
.
|
I_RCS 1
|
|
I_RCS
|
2
|
|
1
|
16.14923
|
7.64104
|
4.4668
|
0.0346
|
.
|
.
|
.
|
I_RCS 2
|
|
Treatment
|
A
|
|
1
|
-1.80983
|
0.93087
|
3.7801
|
0.0519
|
.
|
.
|
.
|
Treatment Group A
|
|
I_RCS*Treatment
|
1
|
A
|
1
|
3.47685
|
3.29380
|
1.1142
|
0.2912
|
.
|
.
|
.
|
I_RCS 1 * Treatment Group A
|
|
I_RCS*Treatment
|
2
|
A
|
1
|
-15.76844
|
14.59523
|
1.1672
|
0.2800
|
.
|
.
|
.
|
I_RCS 2 * Treatment Group A
|
|
Gender
|
Male
|
|
1
|
0.12607
|
0.23512
|
0.2875
|
0.5918
|
1.134
|
0.716
|
1.798
|
Gender Male
|
|
Age
|
|
|
1
|
0.01625
|
0.01025
|
2.5147
|
0.1128
|
1.016
|
0.996
|
1.037
|
Age
|
|
Score1
|
|
|
1
|
0.01209
|
0.00804
|
2.2624
|
0.1325
|
1.012
|
0.996
|
1.028
|
Score1
|
|
Score2
|
|
|
1
|
-0.0004364
|
0.00613
|
0.0051
|
0.9432
|
1.000
|
0.988
|
1.012
|
Score2
|
|
Hazard Ratios for Treatment Group
|
|
Description
|
I_RCS1 Coefficient
|
I_RCS2 Coefficient
|
TrtA Coefficient
|
I_RCS1TrtA Coefficient
|
I_RCS2TrtA Coefficient
|
GENDERMale Coefficient
|
Age Coefficient
|
Score1 Coefficient
|
Score2 Coefficient
|
Point Estimate
|
95% Wald Confidence Limits
|
|
Trt A vs B At I=0.3
|
0
|
0
|
1.000000
|
0.300000
|
0.001353
|
0
|
0
|
0
|
0
|
0.455
|
0.280
|
0.739
|
|
Trt A vs B At I=0.6
|
0
|
0
|
1.000000
|
0.600000
|
0.105460
|
0
|
0
|
0
|
0
|
0.250
|
0.057
|
1.088
|
