# Question about Hazard Ratio Interpretation

#### afarrey11

##### New Member
Hi, all. I've done all the coding already for this, I just don't know how to interpret it. For an assignment, we were given this data taken from the Stanford Heart Study from 1978 or so:
I used equal signs because I thought dashes might be confusing. They haven no mathematical function in the following 'tables'

The variables in question are:
Age in years in reference to age 48
Year Waiting time in program for acceptance into the program
Surgery 0=no previous surgery; 1=previous surgery
Transplant 0=no transplant perform; 1=transplant performed

A series of Cox regression/hazard ratio tables was run, and this was the output

When run singly:
Variable = Coef = HR
Age = 0.03069 = 1.031
Year = -0.19077 = 0.826
Surgery = -0.73911 = 0.478
Transplant = 0.12567 = 1.134

When run together:
Variable = Coef = HR
Age = 0.02715 = 1.028
Year = -.014611 = 0.864
Surgery = -.063582 = 0.530
Transplant = -0.01189 = 0.988

When run together with interaction:
Variable = Coef = HR
Age = 0.02988 = 1.030
Year = -0.25211 = 0.777
Surgery = -0.66270 = 0.515
Transplant = -0.62253 = 0.537
Yr x Transplant = 0.19697 = 1.218

The question I've been asked to answer is: "Explain why transplantation is a risk for death when taken alone, is protective when used in conjunction with the other variables, and why the risk is "absorbed" by the interaction of year and transplantation. How does this cohere with the observation that survival is clearly extended by transplantation (see graph)?" (Graph shows transplantation definitively lengthens survival time).

So transplantation itself is dangerous and was more so when first being performed; surgical technique and improved immunosuppressant medications have since come into use so the change for organ rejection was higher. When the other variables are taken into account, I'd assume this risk is reduced when the other factors are considered. But I don't know why. Does anyone have any idea?

Thanks.

#### EdGr

##### Member
One minor suggestion: For confounding to be present the pattern of associations should be supportive of it. Have you looked to see if age, year, or surgery are associated with having a transplant done? The interaction being involved in confounding is something I would have to really think about!

#### afarrey11

##### New Member
One minor suggestion: For confounding to be present the pattern of associations should be supportive of it. Have you looked to see if age, year, or surgery are associated with having a transplant done? The interaction being involved in confounding is something I would have to really think about!
There was quite a lot of speculation in the literature at the time about whether hardier individuals were more likely to have surgery done, etc. and there wasn't a whole lot of agreement on the subject. Most researchers were just pleased with how successful the transplants were at extending life.

I've done a simple logistic regression because I know exactly how to do that but I'm not sure if the output is of any use.

proc logistic data=SHTD descending;
model transplant = age year surgery / covb;
run;
Model Fit Statistics

Criterion Intercept Only Intercept and Covariates
AIC 232.648 237.445
SC 235.789 250.012
-2 Log L 230.648 229.445

Testing Global Null Hypothesis: BETA=0
Test Chi-Square DF Pr > ChiSq
Likelihood Ratio 1.2026 3 0.7524
Score 1.1920 3 0.7549
Wald 1.1830 3 0.7571

Analysis of Maximum Likelihood Estimates
Parameter DF Estimate Standard Error Wald Chi-Square Pr > ChiSq
Intercept 1 -0.6371 0.3453 3.4039 0.0650
Age 1 0.0117 0.0172 0.4576 0.4988
Year 1 0.0700 0.0897 0.6086 0.4353
Surgery 1 0.1685 0.4167 0.1636 0.6859

Odds Ratio Estimates
Effect Point Estimate 95% Wald Confidence Limits
Age 1.012 0.978 1.047
Year 1.073 0.900 1.279
Surgery 1.184 0.523 2.678

Association of Predicted Probabilities and
Observed Responses
Percent Concordant 52.2 Somers' D 0.069
Percent Discordant 45.3 Gamma 0.071
Percent Tied 2.5 Tau-a 0.034
Pairs 7038 c 0.535

Estimated Covariance Matrix
Parameter Intercept Age Year Surgery
Intercept 0.119229 -0.00031 -0.02663 -0.01033
Age -0.00031 0.000298 0.000289 -0.0002
Year -0.02663 0.000289 0.00805 -0.00583
Surgery -0.01033 -0.0002 -0.00583 0.173631