proc phreg - mathematical notation for calculating HR from ML when using restricted cubic splines

kc — Wed, 14 Sep 2022 22:48:50 GMT

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_RCSTreatment**	1	A	1	3.47685	3.29380	1.1142	0.2912	.	.	.	I_RCS 1 * Treatment Group A
*I_RCSTreatment**	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

Re: proc phreg - mathematical notation for calculating HR from ML when using restricted cubic spline

JosvanderVelden — Thu, 15 Sep 2022 10:37:15 GMT

Have you studied these pages:
https://documentation.sas.com/doc/en/pgmsascdc/9.4_3.5/statug/statug_phreg_overview.htm
Most examples have some details/explanations on the output for each example.

Re: proc phreg - mathematical notation for calculating HR from ML when using restricted cubic spline

kc — Thu, 15 Sep 2022 12:46:27 GMT

Yes. Unfortunately, the examples don't cover restricted cubic splines and how the coefficients in this specific scenario are calculated.

Re: proc phreg - mathematical notation for calculating HR from ML when using restricted cubic spline

StatDave — Thu, 15 Sep 2022 21:59:47 GMT

The last section of this note might help to answer your question. This note might also be useful.

topic proc phreg - mathematical notation for calculating HR from ML when using restricted cubic splines in Statistical Procedures

proc phreg - mathematical notation for calculating HR from ML when using restricted cubic splines

Re: proc phreg - mathematical notation for calculating HR from ML when using restricted cubic spline

Re: proc phreg - mathematical notation for calculating HR from ML when using restricted cubic spline

Re: proc phreg - mathematical notation for calculating HR from ML when using restricted cubic spline