So in this section we're going to show how to estimate hazard ratios for groups comparing groups who differ in more than one predictor value using the results of multiple Cox regression models. So after viewing this section you will be able to estimate a hazard ratio comparing the hazard of a time to event outcome for two groups who differ in more than one predictor value. So let's first look at the example with the primary biliary cirrhosis trial. We're going to use the results from model two here which was the model that included four predictors in it and treatment whether a person was randomized to the DPCA or placebo group, their age put into quartiles to classified into one of four quartiles. Their bilirubin levels which was in milligrams per deciliter and the sex which was females versus males where males for the reference group. If we were to write this down on the regression scale and you can take this as a given but you could have gotten this by taking for each predictor to get its respective slope taking the natural log of the adjusted hazard ratios presented in that column for model two. So for example, the adjusted hazard ratio for treatment compared to placebo after adjusting for the other predictors, was 1.10 the natural log of that is 0.10 and similarly you could take the natural log of the adjusted hazard ratios for each of the other predictors to get these slopes. So let's estimate the hazard ratio mortality for 60 year old males in the drug group on DPCA with Bilirubin equal to one Compared to forty 40 females on the placebo arm with Bilirubin equal to point 0.5 milligrams per deciliter. So if we write out the log hazard for each of these groups in terms of the regression results, we would do it simply by substituting the appropriate x values for each group in the equation and seeing what fell out. So for the first group, they are in the DPCA group, so their value for treatment are x1 is one, and in the fourth age quartile so their indicator is the x4 is equal to one and the other two are zeros. Their bilirubin is one so there x5 value is one and they are male in the reference group for sex so their value for sex or x6 is zero. So at any given time we would start to get the log hazard for this group, we would start with our intercept evaluated at that time and then add 0.1 times one for the treatment component, 0.87 times one for the age component and add 0.15 times one for the bilirubin component and what we'd end up with to get the log hazard at a specific time we would have to find out what the intercept evaluated that time was, take that value and add 1.12 to it which is the sum of 0.1 plus 0.87 plus 0.15. If we were to do the same thing for the second group of 40 year-old females and substitute in their appropriate x values, so there are the DPCA group and the reference group so they get nothing added for treatment, they have bilirubin levels of 0.5 so we take the slope for Bilirubin of 0.15 and multiply it by 0.5 then the reference group for age and then we add in the difference for females to males since this group is female, the slope for sex and negative 0.52 times one. This is what we'd add to whatever the starting point was for any specific time where we're estimating the log hazard. So we take the intercept evaluated at specific time and add negative 0.445 to get that, to get the log hazard at any time for this group. And negative 0.445 is the sum of these two pieces here, they come from the slopes indexes. So if we were to take the difference in this at a specific time as long as we're comparing the two groups at the same time in the follow-up period, the difference in log hazards will be 1.2 minus negative 0.445. And that's equal, the difference is equal on the difference in log, hazards to the log, hazard ratio is 1.565. So if the log hazard ratio estimate is 1.565, to estimate the hazard ratio we'd exponentiate that. e to the 1.565 is equal to 4.78 approximately. So the first group is older, so on the treatment has higher bilirubin and is male. The cumulative impact of those two things compared to the differences for the other group was younger and female on the placebo group with lower bilirubin is the estimated hazard ratio of 4.78. It's not easy to do but using the computer you could get a 95 percent confidence interval for this, and really was the difficulty of this is getting the standard error because this is a function of the respective combination of slopes and x's that come in this comparison. The standard error would be the function of that. But the resulting confidence interval just want you to know that they can be computed for these things where we're comparing more than two groups is 2.3 to 10.0. So I just want to show you something when it comes to taking differences on the log scale. We've done this before but it's worth showing again in the context of multiple Cox regression. If we were to look at this difference in log hazards piece wise, and keep each individual component represented, the interceptive of course would cancel between these two groups, the difference that comes from being in the different treatment groups would come from the difference in treatments times the slope for treatment. It's binary. So the first group was a one because they were in the DPCA group, the second group is a zero because they were in the placebo so this is the part of the overall difference in log hazards that comes from the treatment difference. The part that comes from age is 0.87 we add to the first group because the're in the oldest age quartile minus zero for the second group because they're in the reference quartile. The difference that comes from bilirubin differences comes from the slope of bilirubin 0.15 times the difference in bilirubin values one minus 0.5 and then the difference that comes from sex is the zero for the first group because they are males, minus the portion the negative 0.52 that comes from being female for the second group. So if we write this out piece-wise and then we exponentiate this sum. So we go through this, we exponentiate the sum. We keep that difference we saw before in some form and look at each individual component piece, we get e to the 0.10 times e to the 0.87 times e to the 0.15 times five times e to the 0.52 because the negative negative 0.52 becomes a plus. If we rewrite this out what the exponentiated version, this is the adjusted hazard ratio for treatment from that table. This is the adjusted hazard ratio for age quartile four versus the reference. This is the adjusted hazard ratio for bilirubin per one unit difference and this gets raised to the difference in values for the two groups we're comparing the 0.5 power and then this is one over the adjusted hazard ratio for sex because that was given in the direction of females to males but the groups were comparing on the opposite direction. If you multiply these out you get the same thing that you did before when we added this all up and got the overall difference in exponentiated with that. So you can do this piece wise by multiplying the appropriate portions on the hazard ratio scale if you wish instead of taking it back to the log scale doing the math and then re- exponentiating. So let's look at that approach for predictors of infant mortality. So let's refer to model three here where we had gestational age, treatment, sex and maternal parody, let's use those results to estimate the estimated hazard ratio mortality for preterm male children born to mothers but to randomize to receive vitamin A who had five previous children compared to female children born at 40 weeks who were born to mothers randomized to receive beta carotene who had no previous children. So you could go back to this and write the entire regression out on the log scale like we did in the previous example. Do the computations for these two groups on the log scale, take the difference in the estimated log hazards and then exponentiate that. But you could just start on the adjusted hazard ratio scale as well as we had shown at the end of the previous example when we exponentiated the pieces of the difference separately. So what we're comparing here on the numerators preterm male children whose mothers were randomized the vitamin A group and mothers had five previous children, to a group of children who are 40 weeks gestation, who are female whose mothers are randomized to beta carotene and had no previous children. So if we look at this in terms of the pieces they're the adjusted hazard ratios but first part comes from comparing preterm children to 40 weeks gestation children, the second part compares male children to female children, the third piece is the part that has to do with vitamin A beta carotene and then we multiply that by the fourth piece which comes from the comparison of parody five previous children compared to no previous children. And so if we do this based on the adjusted hazard ratios in that table as presented and we don't have to take things back to the log scale as we showed before, the comparison that comes from gestational age is preterm at 40 weeks gestation, well the hazard ratio in the opposite direction for those 40 weeks gestation to preterm 0.32. So we're comparing this in the opposite direction preterm to 40 weeks so we take the reciprocal or one over that hazard ratio is given for male children to female children the estimated adjusted hazard ratio was simply one, so we'll take that. For the vitamin A group compared to the beta-carotene while the hazard ratio vitamin A to the placebo was 1.04 and the hazard ratio beta-carotene to the placebo is 0.99 and we take the ratio of those two ratios we'll get the adjusted hazard ratio for vitamin A compared to beta-carotene and then the five previous children and no previous children part is a hazard ratio of 0.79 to one. If we multiply these together, this is the same result as if we did the computations on the log scale where the differences are additive and exponentiate the overall difference. When we exponentiate something that's additive in the log scale it becomes multiplicative on the exponentiated scale. So if you don't believe me you can go back and do this on the log scale by writing out the Cox regression equation, by taking the natural logs of the adjusted hazard ratio given in that results table doing the math, getting the overall difference and exponentiate it and you'll get the same result and adjusted hazard ratio comparison for these two groups of 2.59. If you were to put a confidence interval and we will get I had to go the computer to do this and get the standard error and it's involved to some degree but the 95 percent confidence interval is 1.77 to 3.98. So in summary the results from multiple Cox regression can also be used to estimate hazard ratios, comparing the hazard of the outcome for groups who differ by more than one predictor. These can be computed on the regression scale, they are linear combination of slopes and then transformed or using the adjusted hazard ratios and multiplying and dividing these the adjusted hazard ratios given by those exponentiated slopes. Confidence intervals can be computed for each of the above qualities but it is complicated and I'm not saying you couldn't handle it, but it's just more involved in using the computer and so I'm not going to focus on that here but I do want you to know that these things can be estimated. Confidence intervals for differences, for hazard ratios, comparing groups who differ by more than one, x value can be computed as well.
