7.00.02 - Background - Aster Analytics

Teradata Aster® Analytics Foundation User GuideUpdate 2

Product
Aster Analytics
Release Number
7.00.02
Published
September 2017
Content Type
Programming Reference
User Guide
Publication ID
B700-1022-700K
Language
English (United States)
Last Update
2018-04-17

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