Cox PH Cheatsheet

Model Formula

$h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{x}_1+\cdots +{\beta }_P{x}_P}$

$\log h(t,\mathbf{x})=\log h_0(t)+{\beta }_1{x}_1+\cdots +{\beta }_P{x}_P$

Component	Type	Description
$h_0(t)$	Non-parametric	Baseline hazard (distribution ignored)
$e^{{\mathbf{\beta }}^{\prime }\mathbf{x}}$	Parametric	Covariate contribution, parameters estimated

Hazard Ratio

For subjects A and B:

$HR=\frac{h(t,{\mathbf{x}}_B)}{h(t,{\mathbf{x}}_A)}=e^{{\mathbf{\beta }}^{\prime }({\mathbf{x}}_B-{\mathbf{x}}_A)}$

Interpretation

${\beta }_j$	$HR=e^{{\beta }_j}$	Meaning
$>0$	$>1$	Higher risk
$=0$	$=1$	No effect
$<0$	$<1$	Protective

By Covariate Type

Type	$\Delta$ change in $x_j$	HR
Numeric	+1 unit	$e^{{\beta }_j}$
Binary (1 vs 0)	Condition present	$e^{{\beta }_j}$
Numeric	+$\Delta$ units	$e^{{\beta }_j\Delta }$

Dummy Variable Construction

For categorical $X$ with $L$ levels, create $L-1$ dummies, one level as reference:

Stage	Model	Hazard vs Reference
Ref	$h_0(t)$	$1$
Level $k$	$h_0(t)e^{{\beta }_k}$	$e^{{\beta }_k}$

PH Assumption Check

Log-cumulative-hazard plot: Plot $\ln (-\ln \hat{S}(t))$ by group. Parallel curves → PH holds.

library(survival)
 
fit <- survfit(Surv(time, status) ~ group)
plot(fit, fun = "cloglog", col = c("blue", "red"))

Formal test: cox.zph(fit) — significant p-value → PH violated.

Partial Likelihood

No Ties

$L_1(\mathbf{\beta })={\prod }_{j=1}^D\frac{e^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_{(j)}}}{{\sum }_{l\in R_j}{e}^{{\mathbf{\beta }}^{\prime }{\mathbf{x}}_l}}$

• $D$ : number of distinct event times
• $R_j$ : risk set at $t_j$
• ${\mathbf{x}}_{(j)}$ : covariate of subject with event at $t_j$

With Ties ($d_i$ events at $t_i$)

Method	Formula
Breslow	$\prod _{i=1}^D\frac{\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_i)}{[{\sum }_{l\in R_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{x}}_l){]}^{d_i}}$
Efron	$\prod _{i=1}^D\frac{\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_i)}{{\prod }_{j=1}^{d_i}[{\sum }_{l\in R_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{x}}_l)-\frac{j-1}{d_i}{\sum }_{k\in {\mathcal{D}}_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{x}}_k)]}$
Discrete	$\prod _{i=1}^D\frac{\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_i)}{{\sum }_{q\in Q_i}\exp ({\mathbf{\beta }}^{\prime }{\mathbf{Z}}_q^{\ast })}$

where ${\mathbf{Z}}_i={\sum }_{k\in {\mathcal{D}}_i}{\mathbf{x}}_k$

Method	Accuracy	Speed
Breslow	Good for few ties	Fast
Efron	Better	Moderate (R default)
Discrete	Exact	Slow

Model Construction

Numeric Only

$h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{x}_1+\cdots +{\beta }_P{x}_P}$

Categorical (Dummy) Only

$h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{D}_1+\cdots +{\beta }_{L-1}{D}_{L-1}}$

Mixed (Numeric + Categorical)

$h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{D}_1+{\beta }_2{D}_2+{\beta }_3{x}_1+{\beta }_4{x}_2}$

With Interactions

$h(t,\mathbf{x})=h_0(t)e^{{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_1{x}_2}$

For categorical × numeric: each dummy multiplied by numeric variable.

R Quick Reference

library(survival)
 
# Fit Cox PH model
fit <- coxph(Surv(time, status) ~ age + sex + stage, data = dat)
 
# Results
summary(fit)          # coefficients, HR, CI, p-values
exp(coef(fit))        # hazard ratios
exp(confint(fit))     # CI for HR
 
# PH assumption test
cox.zph(fit)          # Schoenfeld residuals test
 
# Prediction (baseline survival)
base_surv <- survfit(fit)
plot(base_surv)

Related

• Cox PH Model
• Hazard Ratio
• Checking PH Assumption
• Partial Likelihood
• Likelihood with Ties