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I am having trouble finding information on how to interpret coxph model hazard ratios with a quadratic term. Some of my variables are continuous count data, whereas others are continuous percentages.

My interpretation:

-for every 1 inch of snow the odds of choosing a snow tire increased 43.8%

-for every 1% increases in tread percent the odds of choosing a snow tire increased 0.08%

-for every 1 additional trip, the odds of choosing a snow tire increased 4.4%

Is this interpretation correct? How do you interpret the quadratic term "time_percent" and "time_percent2" since time_percent2 is the time_percent variable squared.

Output:

                        coef     exp(coef) se(coef)  z    p
snow_inches           0.363450     1.438 4.44e-02  8.19 2.2e-16
tread%                0.007498     1.008 2.49e-03  3.01 2.6e-03
count_trips           0.042730     1.044 1.80e-02  2.38 1.7e-02
time_percent         -0.039268     0.961 6.66e-03 -5.90 3.7e-09
time_percent2         0.000253     1.000 6.94e-05  3.64 2.7e-04
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  • $\begingroup$ What is your response variable and how is it defined? In a coxph model, the response variable is expressed as "time to some event of interest" and the exponentiated estimated coefficients represent hazard ratios (not odds ratios). $\endgroup$ – Isabella Ghement Jan 10 at 16:12
  • $\begingroup$ @IsabellaGhement The response variable is selection for a certain brand of tire. In actuality the model is much more complicated, but, I used this example for simplicity. However, I am using the coxph model to describe choice making as it relates to several predictors (snow_inches, time_percent, count_trips, tread% in this example). $\endgroup$ – KNOX Jan 11 at 17:09
  • $\begingroup$ Why not use an ordinal logistic regression model to handle the categorical outcome variable with more than two categories? Cox regression modelling is suitable when the outcome variable is a (possibly censored) time to event variable. $\endgroup$ – Isabella Ghement Jan 11 at 17:14
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In the case of proportional hazard model using a quadratic term, a unit increase in x doesn't translate in to a static value. It still is a function of x. So, it depends on the base from which unit increase is happening.

if log(T) = b1.x1 + b2.x1.x1 + b3.x2

Then T (at x1 = x1) = exp(b1.x1 + b2.x1.x1 + b3.x2)

While, T (at x1 = x1 + 1) = exp(b1.x1 + b1 + b2.x1.x1 + b2 + 2.b2.x1 + b3x2) Taking their ratios, we get

T(at x1+1)/T(x1) = exp(b1 + b2 + 2.b2.x1) - This is hazard ratio which continues to be a function of x1

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