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The hazard function in survival analysis is represented as (https://in.mathworks.com/help/stats/cox-proportional-hazard-regression.html):

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Here the exponential term is termed as hazard ratio as mentioned here:(https://in.mathworks.com/help/stats/cox-proportional-hazard-regression.html):

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However, in some other documentation, the exponential term is denoted as a partial hazard as mentioned here (https://lifelines.readthedocs.io/en/latest/Survival%20Regression.html#model-selection-in-survival-regression):

enter image description here

I have below queries:

  1. Why is it important to subtract the dataset by mean (𝑥 ̅ )? By doing this does the value of the regression coefficient (b) and hence the hazard function (h) change? If not, then is there any documentation/research paper/article where people could have explained this criterion by given datasets?

  2. What is the difference between hazard ratio and partial hazard?

  3. Why is the term 'partial' being used? Or what is the significance of the term 'partial'

Can somebody please clarify my doubts?

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    $\begingroup$ A partial hazard is just the multiplier for the baseline hazard function to calculate the actual hazard function. As you know, the Cox model doesn't estimate the baseline hazard function (it's an unknown function of time). But for the purposes of risk prediction, you can classify subjects as "higher risk" or "lower risk" than each other based on the partial hazard alone. $\endgroup$
    – AdamO
    Commented Jun 21, 2023 at 16:30
  • $\begingroup$ That's not to say that a baseline hazard function can't be estimated, the Breslow step estimator is pretty widely used when going for "full bore prediction". The interepretation is straightforward - it's the hazard as a function of time for a subject with all variables = 0. If you center them in this fashion, then BSE uses "prediction at the means" to calculate the population incidence curve. $\endgroup$
    – AdamO
    Commented Jun 21, 2023 at 21:12

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Subtracting the mean from the covariate values can help in fitting a Cox model, as otherwise the exponentiations can lead to overflow. I recall that the R coxph() function internally mean-centers and standardizes (to unit standard deviation) all continuous covariates for that reason, even though it reports coefficients appropriate to the original scales of the covariates.

In the formula with the mean subtracted, you can factor out the constant terms associated with the mean covariate values into the baseline hazard:

$$ h(t | x) = b_0(t) \exp \left(\sum_{i=1}^n b_i (x_i - \overline{x_i})\right)\\ =\left(\frac{b_0(t)} {\exp \sum_{i=1}^n b_i ( \overline{x_i})}\right)\exp \left(\sum_{i=1}^n b_i (x_i)\right).$$

Thus there's no change in the modeled coefficients, just in the definition of the baseline hazard.

The important "partial" terminology has to do with the "partial likelihood" that a Cox model optimizes to estimate coefficient values. Technically, a likelihood is (proportional to) the probability of observing the data given a set of parameter values. In a Cox model the actual observation times aren't modeled directly, so you don't model the probability of the data per se. The contribution of Cox was to recognize that, if you were willing to make a proportional-hazards assumption, you don't need to model the actual observation times and you can factor out the baseline hazard to start. What's left is then called the "partial likelihood" of the data given the Cox regression coefficients.

The "partial hazard" and "log-partial hazard" terminology isn't uniformly used in books on survival analysis; at least, it didn't show up in a quick search of a few electronic texts that I have on hand, including the classic text by Therneau and Grambsch on Cox models. It might be intended to emphasize the partial-likelihood basis of the coefficient estimates. I wouldn't worry too much about that terminology.

The hazard ratio is simply the ratio of two hazards. It's often represented for an individual having a set of covariate values with respect to the baseline hazard, as in your first examples, but in general you can calculate a hazard ratio between any two sets of covariate values that are included in a Cox model.

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  • $\begingroup$ Thanks a lot@EdM for your valuable comments. $\endgroup$ Commented Jun 22, 2023 at 5:01

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