# Different hazard ratios in Cox model, depending on which dummy is omitted?

I'm using a Cox survival analysis model in Stata. My covariates include a series of mutually-exclusive dummies. As in other regression models, I leave one out as a reference.

If I change the reference dummy, the significance and hazard ratios are changing where I wouldn't expect it.

A fictional example:

6 race/ethnicity categories.

I leave out "white" as the reference.

"Asian" has a hazard ratio of 1.3, and is significant.

I take that to mean the Asian population is 1.3 times more likely to have the event occur.

Now I leave out "Asian" instead of "white", because I want to look at the difference between the Asian population and other groups.

I would expect "white" to have a hazard ratio of 0.7 and be significant, since its relative to Asian. However, it has a vastly different hazard ratio, and/or the difference is no longer significant.

What is the explanation for that? Is there some sort of instability in my model? Is this a mathematical thing my brain is not grasping well?

Example:

To simplify (and make up for formatting difficulties), I've included just the dummies, hazard ratio, and z-score.

Model 1:

variable      haz. ratio   z
category1       .567     -2.41
category2       .906     -0.76
category3       .842     -1.16
category4       .940     -0.49
category5       .654     -3.02  **
category6      2.26e-14  -0.00
category7       .437     -3.14
category8 (I omitted; placeholder)**
category9       .809     -0.46
category10     2.16e-14  -0.00

Model 2:

variable      haz. ratio   z
category1       .868     -0.58
category2      1.38       2.08
category3      1.29       1.50
category4      1.44       2.50
category5 (I omitted; placeholder)**
category6      2.16e-19  -0.00
category7       .669     -1.50
category8      1.53       3.02**
category9      1.24       0.46
category10     7.47e-20  -0.00


I'm trying to figure out why the relationship between categories 5 and 8 is changing, depending on which is omitted.

• Can you add the result of the model(s) and a descriptive of the categorical variable? (you can change the names if the data is sensitive) – Yuval Spiegler Oct 6 '16 at 19:08
• Deleted my misformatted comments; see new question text, above. – ShannonC Oct 10 '16 at 20:32
• can you please also edit in descriptive (N per category) and perhaps also the standard errors and the script uses? I hope that will help shed light on the matter. – Yuval Spiegler Oct 10 '16 at 20:50
• 1 / 0.654 = 1.53. Does that help? – mdewey Oct 10 '16 at 20:57
• Huh! Okay. So, this may go to my interpretation of hazard ratios. I thought of it as a percentage; like, 35% less likely or 53% more likely. Have I been thinking of this wrong? – ShannonC Oct 10 '16 at 21:33