one unit increment in E, which is equivalent to the log of the hazard ratio: 1 = log (hazard ratio) Exponentiate the coefficient and you get the hazard ratio: hazard ratio = exp ( 1) We observe, however, a key difference between Cox regression and other regression models. I calculated Hazard ratio (HR) using this formula: HR= [Ln(proportion of patients event-free on research arm)] / [Ln(proportion of patients event-free on control arm)] The hazard ratio is the ratio of (chance of an event occurring in the treatment arm)/ (chance of an event occurring in the control arm) (20 ). The HR has also been defined as, the ratio of (risk of outcome in one group)/ (risk of outcome in another group), occurring at a given interval of time ( 21 ). Considering that the hazard ratio is calculated as (0=control, 1=experimental): HR = h. 1(t) h. 0(t) (2) (3) The hazard function is (rate parameter ): h(t) = (4) The hazard function in the control group (=0) is: 5. h. 0(t) = . 0(5) The hazard function in the experimental group (=1) is: h. 1(t) = . The hazard ratio quantifies the difference between the hazard of two groups and it is calculated as the ratio between the ratios of observed events and expected events under the null hypothesis of no difference between the two groups. It is equivalent to the risk ratio when viewed within a time slice, so one can say it is the instantenous risk ratio. Most often what is computed in research papers is the mean hazard ratio across all time slices, which reflects the differences in two survival Hazard ratio is a ratio of two hazard functions HR(t) = 1(t;x 1) 2(t;x 2) (3.1) and we remind the reader that the hazard function is defined as (t;x) = lim +t!0 P(t T<t+ tjT t;X= x) t and that hazard is connected to the survival function via the following formula S(t;x) = e: 1)) = 1 1) = + p) The risk of ASCV mortality increased in a linear manner with higher free thyroxine levels (hazard ratio, 2.41; confidence interval, 1.68-3.47 per 1 ng/dL) and lower thyroid-stimulating hormone one unit. Applying formula (1) eβˆ hazard ratio = = = ˆ ( ) eβ −X X B A eβˆ − *(1 0) eβˆ The hazard ratio in the above SAS output means that the hazard of death in treatment group B is 0.550 times the hazard of death in treatment group A. WHEN THE INCREASE IN THE CATEGORY VARIABLE (X C-X A) DOES NOT EQUAL TO 1 UNIT The p-value for sex is 0.000986, with a hazard ratio HR = exp(coef) = 0.58, indicating a strong relationship between the patients’ sex and decreased risk of death. The hazard ratios of covariates are interpretable as multiplicative effects on the hazard. For example, holding the other covariates constant, being female (sex=2) reduces the hazard by a factor of 0.58, or 42%. We conclude that, being female is associated with good prognostic. Title: Statistical Formulae for Calculating Some 95% Confidence Intervals Author: Allan Hackshaw Created Date: 4/25/2009 1:12:36 PM
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