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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). $\endgroup$ Jan 10, 2019 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, 2019 at 17:09
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    $\begingroup$ 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, 2019 at 17:14
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    $\begingroup$ I'll echo Isabella on this. Your outcome is not appropriate for a cox model. $\endgroup$ Mar 1, 2020 at 5:22
  • $\begingroup$ The Cox model is not limited to analyzing time. It is in the class of ordinal semi parametric models so is an ordinal model with log-log link function, just as the proportional odds model has a logit link. A model can be selected based on which link function fits the data better. And censoring need not be present for the Cox PH model to work well. As case study is here. $\endgroup$ Dec 24, 2023 at 8:53

1 Answer 1

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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 $\log(T) = \beta_1 x_1 + \beta_2 x_1^2 + \beta_3 x_2$ 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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  • $\begingroup$ One common approach is to the the inter-quartile-range hazard ratio for a predictor. This is discussed here. $\endgroup$ Dec 24, 2023 at 8:54

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