F-test for Cox regression with circular variables

[HP Williams]

Hello. I have a question about interpreting results from a survival analysis with a circular variable (ang) which runs from -180 to +180 deg.
I generated the sin and cos of the angle and ran a regression (this is cox regression with time updated variables)

Cox regression with Breslow method for ties

No. of subjects = 411 Number of obs = 3,297
No. of failures = 62
Time at risk = 337,274
LR chi2(8) = 23.53
Log likelihood = -321.75789 Prob > chi2 = 0.0027

--------------------------------------------------------------------------------------------
_t | Haz. ratio Std. err. z P>|z| [95% conf. interval]
---------------------------+----------------------------------------------------------------
cos_ang | .3659565 .129723 -2.84 0.005 .1826831 .7330954
sin_ang | 1.712673 .583607 1.58 0.114 .8782552 3.339859
male | 1.161865 .3123606 0.56 0.577 .6859862 1.967869
age | 1.001317 .0093995 0.14 0.889 .9830623 1.01991
cad | .8453366 .3206804 -0.44 0.658 .4019059 1.778013
dm | 1.017383 .007039 2.49 0.013 1.00368 1.031273
dose | 1.000743 .0012915 0.58 0.565 .9982152 1.003278
htn | 1.00317 .0084401 0.38 0.707 .9867635 1.019849
--------------------------------------------------------------------------------------------

The cosine term is significant but the sin term is not.

Based on some other reading and forum posts on circular statistics I understand that since sin and cos are interrelated, its best to consider them together rather than individually.

Is the best way to do this to run a F-test?

test cos_ang sin_ang

( 1) cos_ang = 0
( 2) sin_ang = 0

chi2( 2) = 16.12
Prob > chi2 = 0.0003


The F-test is very significant. Does this mean that I can "ignore" that non significant P value for the sine term and conclude that there is indeed a circular variation in HR based on the angle?

Is this case any different than if both sin and cos had significant individual P values in addition to a significant joint test?

My goal is to graph the HR as a function of the angle using the regression coefficients

thanks for your insight!

hpw

[Nick Cox]

Indeed, in https://journals.sagepub.com/doi/pdf...867X0600600408 I follow the late and great Sir Harold Jeffreys in treating sine and cosine as a double act never to be parted. I don't know how they appear in your work but you could choose a different set of polar coordinate axes and results might look different in detail but identical in implication.

The rest of this post is just bizarre personal comment.

Harold Jeffreys holds a strange but distinctive place in my personal pantheon. I am very much aware that he grew up just a short distance from where I work. I never encountered him as a student, but in 1976 I wrote to him about a (non-statistical) paper he wrote in 1918. He replied that when he was awarded a prize in 1939 this work was mentioned and he had already forgotten about it!

Conversely, the Cox here is manifestly not me. The late and great Sir David Cox was a very nice person, although capable of sharp asides. In emailing him about something else a long while back I slipped in a few details about my family tree and we established that we were not knowingly related.