Abstract Understanding on Cox PH Model

COX PROPORTIONAL HAZARD (Cox PH) MODEL

— — — — — Proposed by D.R. Cox (1972) .

This model is popular in survival analysis. It is used in

the survival time estimation , with

various covariates, through

hazard function.

This model Links these two parameters. So researchers GOAL is to find hat which covariates dominates on patient’s survival time.

Risk of an event happening depends on covariates. For determining an individual, having an event:

a) various covariates affect, and we

b) don’t know the associated parameters — βs.

This is an estimation problem for given data set and β varies along individual. Therefore, this problem uses likelihood function.

Survival Analysis studies the time between the entry of study and the subsequent event(/death). Regression is a statistical term, which represents dependent variable and explanatory variable. Therefore, it tells the relationship between them. If ther is multiple dependent variables are in use, call it multiple regression. Cox Regression Analysis is also a multiple regression that holds an equation for hazard in a function form.

Cox Models

well-recognized statistical technique

survival of patient and several explanatory variables are the variables in use.

Estimates the effect of treatment after adjustment in variables.

Cox Proportional Hazard Models

Assumption We Made Here Intially:- for EACH individual this equation hold

h(t|X) = h(t) exp(X1β1 + · · · + Xpβp)

h(t|X) ::: Hazard function

The probability of an individual who will experience an event within an small interval GIVEN that he/she has survived upto the beginning of an interval. (Risk of dying at time t)

h(t) ::: Baseline hazard function

can be any complicated function on t(time), but greater than 0.

exp(X1β1 + · · · + Xpβp) ::: Exponential part

This section involves covariates and NOT having time variable.

Cox PH Model Explanation :-

  • Doesn’t flow a particular distribution
  • Effect of variables are constant over time
  • And additive in a particular scale

— :: Ratio of hazards of two individuals does not depend on time :: —

If the individual events are independent among them, the overall likelihood can be product of all the individual samples.

Censoring time :: is used instead of Survival time in training. If censored, then NO likelihood.

Hazard ratio is condsidered when comparing two predictors for an event of an individual.

Hazard Ratio

And the point estimate of hazard ratio is given below. Here β^ is the maximum likelihood estimate of β.

Point Estimate of Hazard Ratio

MLE = Maximum Likelihood Estimation :::::::::::::::::::: SE = Standard Error

MLE and SE are used in the Cox model in statistical derivations because for 95% CI(Confidence Interval), of hazard ratio — hr(X*: X) we use the below estimation.

95% CI Point Estimation

β interpretation :-

=0 :: Covariate has no effect on risk of an event

>0 :: Has higher hazard of risk

<0 :: Has lower hazard of risk

We maximize partial likelihood in Cox PH model to estimate parameters.

Partial Likelihood Function

And the log partial likelihood function is given by,

Log Partial Likelihood Function

Time dependent model is a more favorable than the previous with assumptions made.

Typical model for time varying covariates is also proposed. But NOT necessarily, all covariates should be time dependent

Supplementary Terminologies :-

ROC Curve — Receiver Operating Characteristic ::

  • Plots the true positive against the false positive.
  • Shows how the classifier would perform

C-Statistic / Concordance Statistic :: If we want to compare the performance of two classifier models, we need a parameter value; thus, c-index (Concordance Index) came. It is calculated as the area under the ROC curve, and it ranges between 0.5 and 1. It validates the predictive ability of a model.

It is a measure of goodness fit for binary outcomes in a LOGISTIC regression model. In clinical studies, this value expresses the probability of random patient having higher risk than a normal patient.

Interpretation of c-index :-

  • below 0.5 — very poor model
  • 0.5 — similar to a model predicting an random outcome
  • 0.7 — good model
  • 0.8 — strong model
  • 1.0 — model perfectly predicts

Usually ROC curve is applied in many places than c-index, because we prefer a curve better than a value for analysis.

References :-

  1. http://www.statisticshowto.com/c-statistic/
  2. https://www.xlstat.com/en/solutions/features/cox-proportional-hazards-models

Ramraj Chandradevan

Written by

Undergrad student@ University of Moratuwa-Sri Lanka. Former Deep Learning Intern @ Emory University.

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