So, in this next set of lectures, we'll do a treatise of Cox Regression, multiple Cox regression similar to what we've done in terms of the multiple regression for the other types of regression we cover in this course. We'll kick this off by looking at some examples from the results of multiple Cox regression and compare them to the results from corresponding simple regression models for each corresponding predictor that was included in the multiple Cox model. So, what we're going to do in this lecture is we're going to talk about interpreting the intercept function from multiple Cox regression as well as the slope estimates in a scientific context and compare the results from simple and multiple Cox regression models to assess confounding. In other words unadjusted and adjusted associations. So, as with linear and logistic regression, the slope from multiple Cox regression estimates the adjusted relationship between the time-to-event outcome and each predictor x in the model, adjusted for other predictors in the model. So linear logistic regression don't deal with time-to-event outcomes but the multiple regression in general estimates the relationship between whatever the outcome is and each predictor in the model adjusted for the other predictors in the model. The slopes for Cox regression can be exponentiated to estimate adjusted hazard ratios and then unadjusted and adjusted hazard ratios or slopes can be compared to assess confounding. Additionally, even in the absence of confounding, we can use multiple Cox regression models to better explain a time-to-event outcome by taking into account multiple pieces of information in one model as opposed to looking at each on their own separately via a bunch of simple analysis. So, let's go back to a study we've looked at many times in this course and view it in this context, the trial done at the Mayo Clinic in Rochester Minnesota in the United States for primary Biliary Cirrhosis patients and they were randomized trial and they were randomized to either receive the drug D-Penicillamine also called DPCA or placebo. So, patients were followed from enrollment until death or censoring and the follow up period was up to 12 years from the patient's enrollment. So, we know the story on this, we've looked at it many times in terms of the original intended this study. We've shown over and over again that there was no impact of the drug on reducing mortality and here are some unadjusted Kaplan-Meier curves comparing the survival, the proportion who had not died over the follow-up period between the drug in placebo groups, so you may recall that the unadjusted hazard ratio of mortality for the drug compared to the placebo group was only 1.06 and not statistically significant. So, then again, the unadjusted hazard ratio mortality for the DPCA groups compared to the placebo group was 1.06 with the confidence interval that included one. It's not expected in a study like this that this crude unadjusted hazard ratio will be confounded by other patient characteristics such as that patient age, baseline or biological sex or the bilirubin at the time of randomization and why wouldn't we expect it to be confounded? Well, how are patients assigned to the treatment groups, they were randomized. However, nevertheless, it's possible that taken together, including treatment, we could also use other patient characteristics to add Information about mortality above and beyond treatment and some of these other patient characteristics may be related to each other as well as mortality. So, it might be interesting to look at unadjusted and adjusted associations for things that were not randomized like the age and sex for example of patients with this disease. So, let's look at the results. We're going to look at a table that shows the unadjusted hazard ratio from simple Cox regression models for each of the predictors of interest and then we'll put them side-by-side with our adjusted counterpart. So, let's just look at this. The treatment effect, if you will, in the unadjusted comparison was that hazard ratio of 1.06 percent greater risk of mortality for those who got the drug and placebo is not statistically significant. But of course that would be of little interest after the fact that we saw in estimate that shows some slightly higher incidence in the drug group but we've said this many times, are the age effect, if you will, were associations such that increasing age is generally associated with increased risk of mortality? That's not surprising. Some of the comparisons we make like the age quartile three, 50-57 years, to the reference quartiles less than 42 years are statistically significant and as a whole age the construct is a statistically significant predictor of mortality. Similarly, higher bilirubin is related to higher risk of mortality. It's a measure of disease progression and those with more advanced disease have higher Bilirubin levels and higher risk of mortality and it turns out sex was also a predictor of mortality. Females are less likely to die from the disease compared to males and the result was statistically significant although there was a wide confidence [inaudible] here are mainly because the majority of the sample was actually female because females while they are less likely to die for primary Biliary Cirrhosis compared to males, females are much more likely to get the disease. Let's look at the adjusted comparisons and compare them with unadjusted. So, as expected but the estimate is slightly different but it's qualitatively comparable for the drug comparison, this adjusted hazard ratio 1.10 compares the estimated hazard ratio mortality for those in the drug compared to those placebo adjusted for age, bilirubin and sex and as expected it's very similar but qualitatively and in terms of competence interval to the unadjusted counterpart because of randomization. With regard to age, we still see the same general trend. This compares age groups in terms of their relative mortality after adjusting for treatment assignment, bilirubin levels and sex and the estimated hazard ratios go down slightly, attenuates slightly but the confidence intervals for each comparison overlap and we still see an increase in the hazard of mortality with increasing age for the most part. This is 0.99 effectively one, but let me go up to the third quarter or fourth quartiles there's that increase compared to the reference and it's still an overall statistically significant construct or predictor mortality even after accounting for treatment, bilirubin and sex. Bilirubin effectively, the results do not change at all, indicating that the association we saw initially was not at least in part due to differences in bilirubin levels in the other characteristics like treatment, age or sex. There was no confounding here and similarly, there wasn't much confounding with the sex relationship either. So, all four of these associations still are statistically significant when taken together but there's very little confounding of any of these relationships indicating that each of them was contributing independent information about mortality to the outcome. So, the relationship between mortality and treatment or rather the lack of relationship was not confounded by age, sex, or bilirubin levels at time of randomization. Age, sex and bilirubin were statistically significant predictors of mortality in the unadjusted comparisons and after adjustment for each other and treatment, all three remains statistically significant predictors in the multiple regression model with very similar in magnitude to the unadjusted associations. Let's look at another example here, let's look at predictors of infant mortality among Nepalese newborns in the context of this study of vitamin supplementation for pregnant women to see if that had an influence on mortality in infants. Unfortunately, as what we found with the previous trial, the treatment here did not have any impact. So, let's look at some other characteristics of the children as well, we might want to look at their gestational age and see how that relates to mortality. We've already done that in the simple contexts and just a reminder this is the distribution of gestational ages in the group, the median is on the order of 37 or 38. So the majority by the 36th week cutoff are full term because the 25th percentile is around 36 weeks, so if we use the older standard of 36 weeks is the term cutoff, the majority of these babies were full term but a notable proportion are less than full term. So let's just look at some other sample characteristics, the treatment assignment for the mothers of these children was roughly not quite 33, 33, 33 percent but roughly a third of the mothers in the sample each were assigned to the placebo vitamin A or beta carotene. The majority of the children as we said we're full term if we use 36 weeks as cutoff but little more than a fair 22.5 percent were pre-term. The sample of infants was slightly more females than males, and then the parity categories of the mothers was a little over a fifth, had never had a prior child prior to the one under study here. Another fifth had one prior child. Another two-fifths or 42.5 percent had two to four previous children et cetera. So, these are some of the things we're going to look at on their own in simple regressions and taken together in a multiple Cox regression as predictors of mortality. So let's look at what happens to start, let's focus on gestational age, even though treat this study was intended to look at treatment but even though we've found at least on the whole the treatment of vitamin supplementation was not effective in reducing mortality, we still want to use the other information about these children and mothers to see if there's other factors that could help us triage high-risk pregnancies in the future. So in the unadjusted associations we've done some of these back in the lectures on simple Cox regression, gestational age was a significant predictor and the big picture here as we saw before that generally there's a big drop in the risk of mortality by going from pre-term to full term and then after that there's not much of an additional benefit of having a longer gestational age, again no qualitative or statistically significant differences in mortality between the three treatment groups; babies born to mothers who were treated with either placebo, vitamin A, or beta carotene, no ostensible differences by sex. And then with maternal parody there seems to be a benefit of having had previous children but it tends to disappear as that number of prior children increases, so, there's a law of diminishing returns here in terms of mortality, so those with one prior child have a significantly lower risk of their child dying than those with no previous children but then it starts to move closer back to the null of one even though the estimated mortality is a lot higher in those with a prior children compared to those babies who have no siblings where the mother had no previous children it's not statistically significant. In model two what I put in was just gestational age and parity to see if there was any confounding going on there and I'll let you look at this but these results are almost identical for each of these predictors as to what they were in the unadjusted comparisons. And, when taken together as within the unadjusted both of these are statistically significant predictors of mortality. In the third and final model here we included treatments and sex no changes in gestational age or parody with this last level adjustment as expected no noticeable changes in the treatment estimates because we would not expect this to be confounded by these other characteristics because of randomization, and no ostensible change in the sex association or the lack there of. So, model three though if you were to put this back on the regression scale it's birth by a regression of the following form, the log hazard of depth at any time in the follow-up period given the set of maternal and child characteristics is equal to an intercept evaluated at whatever time you were evaluating this log hazard at, plus we need this first four Xs deal with gestational age categories, there were five categories, so we need four indicators, and these slopes we could actually get the numeric version if we took the natural log of each of the adjusted hazard ratios from that third model in that table, then we had treatment which had three categories so there were two indicators, and then there was sex of the child, biological sex and that was a one for male and zero for female and then finally, there was parity category and there were five categories for that so we need four Xs and four estimated slopes to describe that adjusted associations. Again, these slopes if you wanted to put numbers here you can take the respective natural logs of the corresponding adjusted hazard ratios from that prior table. So, of the four predictors considered, only gestational age and maternal parity were statistically significantly associated with mortality both before and after adjustment for the other factors in the model. There was no indication when we looked at those results across the unadjusted and the two levels of adjustment that the relationship between mortality and gestational age is confounded by maternal parity, that's what we saw on that second model which only include those two as predictor, nor that the relationship between mortality and maternal parity was confounded by gestational age after adjustment, nor was it confounded by treatment status or sex of the child. So across the board we saw no evidence of confounding by any of these predictors when all taken together, the unadjusted and adjusted are counterparts estimated hazard ratios and confidence rules were very close in value. In subsequent sections of this lecture set we'll come back and drill down on some of the details of these models, we will look at how to estimate confidence intervals for individual slopes and turn those into confidence intervals for adjusted hazard ratios, we'll talk about how to compare two competing models when we're trying to test whether a multicategorical predictor is a statistically significant predictor in a multiple Cox regression model, remember we have to test more than one slope at once for that. We'll look at some examples from the literature. We'll talk about how the results from multiple Cox regression models can be converted into estimated survival curves for different groups given different predictor values, and we'll have some other discussions as well about multiple Cox regression. But in summary, multiple Cox regression is a tool that relates the hazard of a binary outcome over occurring over time to multiple predictors x_1 through x_p, and those are really multiple Xs that could encapsulate p or less than p predictors if some of the predictors are multicategorical and require more than one x, but via a linear equation of the form that we've seen before, an intercept plus slopes times respective Xs, the only difference with Cox regression as the intercept is not a singular value with something that takes on values over time but the slopes compare two groups who differ in their respective x values at the same point in time. So, for those comparisons the intercept cancels out even though it changes values over time. So, generally speaking the slope beta hat_i is the estimated adjusted log hazard ratio of the outcome for two groups who differ by one unit in x_i adjusted for all other Xs in the model. The intercept, the log of lambda naught hat over t what that tracks is the natural log of the baseline hazard, the log risk over time in the reference group which could be the group when all Xs are zero. This may not necessarily be a relevant quantity, depending on the predictor set some of our predictors are continuous for example, but it is still necessary to specify the regression equation formally and when we get into showing how these results can be transformed into a predicted survival curves that intercept piece even though we won't observe the values directly, will be key in estimating those curves.
