Measure of effect

Concepts

Risk

Risk: the probability that a subject within a population will develop a given disease, or other health outcome, over a specified follow-up period.

\[ R = {{E}\over{E + N}} \]

E (Event): number of individuals with events
N (Non-event): number of individuals without events

Odds

Odds: the likelihood of a new event occurring rather than not occurring. It is the probability that an event will occur divided by the probability that the event will not occur.

\[ Odds = {E \over N } \]

Hazard

Hazard function \(h(t)\): the event rate at time \(t\) conditional on survival until time \(t\) (namely, \(T≥t\))

\[ h(t) = Pr(t<=T<t_{+1} | T>=t ) \]

T is a discrete random variable indicating the time of occurrence of the event.

Relative risk (RR) and Odds ratio (OR)

2×2 Contingency Table

	Intervention I	Control C
Events E	IE	CE
Non-events N	IN	CN

Relative risk (RR)

RR: relative risk (risk ratio), usually used in cohort studies.

\[ RR = {{R_{Intervention}}\over{R_{ conrol}}}={{IE/(IE+IN)}\over{CE/(CE+CN)}} \]

Odds ratio (OR)

OR: usually used in case control studies.

\[ OR = {{Odds_{Intervention}}\over{Odds_{ conrol}}}={{IE/IN}\over{CE/CN}} = {{IE * CN}\over{CE * IN}} \]

When the event occurs in less than 10% of the unexposed population, the OR provides a reasonable approximation of the RR.

Hazard ratios (HR)

Hazard ratios (relative hazard) are usually estimated from Cox proportional hazards model:

\[ h_i(t) = h_0(t) \times e^{\beta_0 + \beta_1X_{i1} + ... + \beta_nX_{in} } = h_0(t) \times e^{X_i\beta } \]

HR: the ratio of the hazard rates corresponding to the conditions characterised by two distinct levels of a treatment variable of interest.

\[ HR = {{h(t | X_i)}\over{h(t|X_j)}} = {{h_0(t) \times e^{X_i\beta }}\over{h_0(t) \times e^{X_j\beta }}} = e^{(X_i-X_j)\beta} \]