In this section, I guarantee there won't be any surprises. We'll talk about accounting for the uncertainty in the estimates, namely the estimated slopes from simple Cox regression. So, what you'll be able to do after this section, is create 95 percent confidence intervals for the slopes from simple Cox regression models, and convert these to 95 percent confidence intervals for hazard ratios, and also estimate and interpret p-values for testing the null hypothesis that the true slope or the true log hazard ratio, comparing to groups zero, and hence the corresponding hazard ratio is one at the population level. So, the previous sections we showed the results from several simple Cox regression models. So, for example, in the analysis of death by treatment group in the primary biliary cirrhosis drug trial, the resulting Cox regression equation included an estimated slope of 0.057 times the predictor X1, where X1 was equal to one for persons randomized to the DPCA group, the drug group, and zero for persons randomized to the placebo group. This was estimated from individual level data using the computer package. So, again, just like in any other regression, there must be a consistent algorithm to estimate the parameters of a regression model when it's time to event data, such that the results will be consistent, regardless of that computer package used to fit the regression. So, for Cox regression, this approach which was invented by Sir David Cox is called partial maximum likelihood. The estimates for the slope, beta one hat, more slopes will have multi-categorical predictors is or are the values that make our observed time to event data outcomes in our sample most likely among all possible choices for the slope estimate. The slope is estimated after the baseline hazard function, lambda hat not of T, is estimated via another process, and this process also generates estimates of uncertainty for this estimated function as over time. So, standard error is for the estimated function at all points along the follow-up period. We won't look at these in this section because there's no practical way to use these resulting standard errors to create type-specific confidence intervals by hand. That wouldn't be a good use of our time, but these will factor in to the uncertainty balance on survival curves estimated by the results from Cox regression, and we'll take a quick look at that idea in the next lecture section. In general, in fact, always this estimation, this partial likelihood, partial maximum likelihood estimation must be done with a computer. So, again, the values chosen for any given slope are just estimates based on the single sample. So, in our primary biliary cirrhosis DPCA trial, we had 309 patients with primary biliary cirrhosis who were randomized to one of two groups. And if we randomly selected a different group of patients to participate, and or randomized them with a different method, we would get different samples in our treatment and control groups just by chance, and the values of the slope beta one hat would vary from study to study across different sample configurations. So, again, as such, all regression coefficients from Cox models have an associated standard error that can be used to make statements about the true relationship between the log hazard of y equals one and x1, for example, the true population level slope based on our estimates from a single sample. The method of partial maximum likelihood also yield standard error estimates for the slope estimate, and that's what we'll focus on in this section. The standard errors for the slopes allow for the computation of 95 percent confidence intervals, and P values for the slope. Just like all other regression coefficients, we've seen that random sampling behavior of regression slopes is approximately normal because these log hazard ratios are estimated in large samples generally, and adjustments or exact methods can be used and done by a computer in smaller samples. But we'll focus on how we can do these by hand if we didn't have a computer, but had some summary results. But in general, I just want you to get the idea it's business as usual for getting 95 percent confidence intervals, and doing hypothesis test. The only caveat here is the confidence intervals for our results are done on the log scale, the slope scale, and then exponentiate it to put them on the ratio scale. So, again, just it's always good to review this concept, even though we drilled it in the first term. Every time we see it, the random sampling behavior of our estimated slope, if we were to replicate our study over and over again, and do a histogram of our estimated slopes, it would be approximately normal, and on average would equal the true underlying quantity, the true underlying slope. So, because this distribution is approximately normal, we'll get one estimate under this curve, it describes the distribution of all estimates, most 95 percent of the estimates we would get, would be within plus or minus two standard errors of this unknown truth. So, if we start with our sample-based estimate, and go plus or minus two standard errors in either direction, we have good opportunity to include the truth in that interval. We started from the other perspective instead declared a possibility for the truth and null that says there's no association between the time you add an outcome, and our predictor in other words the slope or log hazard ratio of zero, and hence this is equivalent to saying that the true hazard ratio is equal to one. What we would do, is we know that under this truth, the sampling behavior of our estimates of this truth should be normally distributed around the truth. So, we would expect our estimate to be relatively close to this truth if this is the truth that generated the data. So, we measure how far our estimated slope is from what we'd expect it to be zero under the null, in terms of standard errors. If it's relatively far, we'd reject the null. If it's closer, we would fail to reject the null. So, let's go back to our analysis of death by treatment group in the primary biliary cirrhosis data. The resulting Cox regression model yield the slope estimate of 0.057, and the corresponding standard error estimate of 0.18. So, let's start with that to get confidence intervals for the slope, and then the exponentiated slope or hazard ratio. So, the estimated slope was 0.057. We'll do the same thing as usual, take the estimate plus or minus two estimated standard errors to get a confidence interval for the slope that span negative and positive values. So, it would include the null value for slopes of zero, and so we know right off the bat this result is not statistically significant. We want to actually present this on the hazard ratio scale though both in terms of the estimate and the confidence interval. To do this, to get the estimate, we know we just exponentiate our estimated slope of 0.057 giving us that estimated hazard ratio 1.06. Then the estimated confidence interval for the true population hazard ratio would be obtained by exponentiating the endpoints of the confidence interval for the slope i.e those confidence interval for the log hazard ratio. If we do this, we get a confidence interval that goes from 0.74 to 1.5 and certainly includes the null value for ratios of one. So we know the result is not statistically significant, but if we wanted to get an exact p-Value for this, we can do this by specifying our null hypothesis, and then measuring all the computations are done on the slope scale. We can measure the distance of our estimated slope from what we'd expect the slope to be under the null hypothesis which is zero in units of standard error. So we have a slope of 0.057, which after standardized by the uncertainty unit is 0.32 standard errors above zero. We know that the sampling distribution under the null should be approximately normal around zero. We have a result that's 0.32 standard errors above zero, we want to get the p-Value which is the proportion of results as far or farther from zero in either direction. We know that would be a relatively large percentage because that's relatively close to zero. We looked it up and using the computer the p-Value is approximately 0.75. So not only greater than 0.05 but greater than 0.05 by a large amount. So we wanted to summarize these findings we could say that between 1974 and 1983, total 312 patients, 36 male and 276 female with primary biliary cirrhosis were enrolled in he randomized clinical trial conducted at the Mayo Clinic in Rochester, Minnesota. These patients were randomized to receive D-penicillamine aka DPCA or a placebo. The patients were followed for up to 12 years until death or censoring. A total of 125 patients die during the follow-up period, 65 in the DPCA group and 60 in the placebo group. The relative hazard of death in the follow-up period for the DPCA group compared to the placebo arm is 1.06 with 95 percent confidence interval 0.74 to 1.5. So, the results showed no discernible benefit of DPCA and extending the life of subjects with PBC and this drug was therefore not adopted as standard treatment. Let's look at another example from the same dataset this is where we had a continuous predictor and looked at the relationship between the log hazard of mortality in these patients with PBC and their baseline bilirubin level. We showed under treating, by treating, when treating bilirubin's continuous we showed a positive association between the log hazard of mortality and increase in bilirubin level. So, here the estimated slope or log hazard ratio mortality is 0.15 with an estimated standard error of 0.013. So, if we wanted again to give 95 percent confidence interval for the slope. Then the resulting confidence interval the 95 percent confidence interval for the true slope or log hazard ratio associated with bilirubin is found by taking the estimated slope of 0.15 adding or subtracting two standard errors gives a resulting confidence interval that includes only positive values and does not include the null for log hazard ratios of zero. So we know the result is statistically significant to put this on the hazard ratio scale we know to get the estimated hazard ratio we exponentiate that slope of 0.15 and the estimated hazard ratio is 1.16. To get the estimated confidence interval on the hazard ratio scale, we would exponentiate the end points of our confidence interval for the slope. We get a confidence interval that goes from 1.13 to 1.19 and it does not include the null value for ratios of one. So we know this result is statistically significant, we want to get a p-Value for testing the null with the population slope is zero or the underlying hazard ratio is one. We assume the null is true, we do our calculations on the slope scale and calculate the distance of our estimated slope from zero in units of standard error. So our slope was 0.15, standard error is 0.013, we have a result that's 11.5 standard errors above what we'd expect under the null hypothesis. So if we translate this to a p-Value, we know already it's more than two standard errors away from zero, the p-Values less than 0.05. We already knew that as well based on the confidence intervals but in this example the p-Value is very small way less than 0.001. So if we were to summarize this, we could say that between 1974 and 1983, again, a total of 312 patients with primary biliary cirrhosis were enrolled in a randomized clinical trial conducted at the Mayo Clinic. These patients were randomized to receive the D-penicillamine or placebo. Patients were followed for up to 12 years until death or censoring, baseline bilirubin levels were assessed on each enrollee at the time of randomization. The mean bilirubin for the sample was 3.3 milligrams per deciliter, standard deviation of 4.5 and a range of 0.3 to 28. Higher bilirubin levels at baseline were associated with a higher risk of death in the study follow-up period. Each additional milligram per deciliter baseline bilirubin was associated with a 16 percent increase in the risk of death. With a 95 percent confidence interval on that percent increase of 13 percent to 19 percent. So, in summary, it's business as usual the construction of confidence intervals and the computation of p-Values for Cox regression results is business as usual. For example, to get the confidence interval for the slope, the true population level so we take the estimated slope and add or subtract two estimated standard errors for large samples. That's the only thing I would ask you to do by hand was when we had an ample large enough sample to do it. In smaller samples, the 95 percent confidence interval and p-Values are based on the exact computations but that detail will be handled by a computer. The interpretation of the confidence intervals and p-Values are the same regardless of sample size and confidence intervals for the slopes or confidence intervals for log hazard ratios comparing two groups. The risk of the outcome for two groups who differ by one unit next one the predictor. Of course those endpoints of the confidence intervals on the slope scale can be exponentiated to get a confidence interval for a hazard ratio. In the next section, we'll finish treatise of simple Cox regression, and we'll show briefly that the results from these models can be transformed back to the survival curves scale. We can estimate survival curves with confidence limit is based on the results from Cox regression models, and we'll compare and contrast these with the estimates we would get from the Kaplan-Meier approach.
