7.00.02 - Background - Aster Analytics

Teradata Aster® Analytics Foundation User GuideUpdate 2

Product
Aster Analytics
Release Number
7.00.02
Release Date
September 2017
Content Type
Programming Reference
User Guide
Publication ID
B700-1022-700K
Language
English (United States)

In survival analysis, the hazard ratio (HR) is the ratio of the hazard rates corresponding to the conditions described by two levels of an explanatory variable. For example, in a drug study, if the treated population dies at twice the rate as the control population, the hazard ratio is 2, indicating a higher hazard of death from the treatment.

The definition of the Cox proportional hazard model is:

h(t) = h0(t)exp(β1X1 + … + β n X n )

The definition of HR is:

HR = h1(t) / h2(t) =

h0(t)exp(β1X1 + … + β n X n ) / h (t) = h0(t)exp(β1X'1 + … + β n X' n ) =

exp(β1(X1 - X'1) + … + β n (X n - X' n ))

The natural logarithm of HR is:

ln(HR) = β1(X1 - X'1) + … + β n (X n - X' n )

For two groups that differ only in treatment condition, the ratio of the hazard functions is given by e β, where β is the estimated treatment effect derived from the regression model. This hazard ratio (the ratio of the predicted hazard for a member of one group to the predicted hazard for a member of the other group) is given by holding everything else constant (that is, assuming proportionality of the hazard functions).

For a continuous explanatory variable, the same interpretation applies to a unit difference.

Researchers consider probabilities lower than .05 to be significant and provide a 95% confidence interval for the hazard ratio. Statistically significant hazard ratios cannot include unity (one) in their confidence intervals.

Suppose that you have the following Cox proportional hazard model:

h(t) = h0(t)exp(β1X AGE + β2X GENDER + β1X AGE*GENDER + β2X WEIGHT )

You can use the preceding model to calculate hazard ratios such as:

  • The hazard ratio when AGE increases 1 unit
  • The hazard ratio among AGE=20, 40, 60 at the group in which GENDER is female
  • The hazard ratio when WEIGHT increases 1 unit at the group in which GENDER is male and AGE = (20, 40)
  • The hazard ratio between the groups (GENDER=1, AGE=20, WEIGHT=80) and (GENDER=0, AGE=60, WEIGHT=70)
  • The hazard ratio when AGE increases 1 unit and WEIGHT increases 10 units