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.


                        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
  • 1
    $\begingroup$ 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). $\endgroup$ Jan 10 '19 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 '19 at 17:09
  • 3
    $\begingroup$ 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. $\endgroup$ Jan 11 '19 at 17:14
  • 2
    $\begingroup$ I'll echo Isabella on this. Your outcome is not appropriate for a cox model. $\endgroup$ Mar 1 '20 at 5:22

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 $\log(T) = \beta_1 x_1 + \beta_2 x_1^2 + \beta_3 x_2$ then $$T= \exp(\beta_1 x_1 + \beta_2 x_1^2 + \beta_3 x_2)$$

while the same prediction for $x_1$ being 1 unit higher is $$T = \exp(\beta_1 + \beta_1 x_1 + \beta_2 x_1^2 + 2 \beta_2 x_1 + \beta_2 + \beta_3 x_2)$$.

Taking their ratios, we get $\exp(\beta_1 + \beta_2 + 2 \beta_2 x_1)$.

This is the hazard ratio which continues to be a function of $x_1$.


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