This is likelihood approach is done after the slopes are estimated,

after the baseline hazard function,

the general shape of risk over time for any of the groups that we're looking at based on

their x values is estimated via another process which

also generates estimates as uncertainty for this estimated function.

So, in general, always this mode of

estimation has to be done with the computer, it's complicated.

So, we get standard error estimates for our slopes,

but we also get which will be used when

we put confidence limits on predicted survival curves,

some estimates of the uncertainty of that intercept function over time.

So, the values chosen for

the slope estimates are just estimates based on a single sample,

and as with everything we've done,

had we taken different samples from the same population estimated the same model,

the estimates of our slopes that

best fit the data in any given sample would vary across the sample,

just by chance variation.

So, all regression slopes have an associated standard error that can be used

to make statements about the true relationship between the time to event outcome,

and a given particular x based on a single sample.

So, this method of partial likelihood as I said before,

yields standard error estimates for the slope estimates,

and we can use the estimated slope and its standard error to create confidence intervals.

So, the standard errors allow for the computation of

95 percent confidence intervals in p-values for the slopes,

and as with all other regressions,

we looked at the random sampling of regression slopes is approximately normal.

So, it's "business as usual" for getting

95 percent confidence intervals and do hypothesis tests for the slopes.

But again, the slope is on the log-hazard ratio scale,

so ultimately the confidence intervals that are

created on the slope scale are the results will be

exponentiated to get the confidence intervals for the adjusted hazard ratio estimates.

So, just as we've done each time,

it's always good to remind you that for any given slope estimate,

if we were to run a study over and over again,

and estimate the same Cox regression model with the same predictors on,

multiple random samples of the same size for any one of our slope,

so you have multiple slopes,

the variation in the slope estimates would be,

if we did a histogram for any given slope,

the estimates across multiple models for multiple samples in a histogram,

the distribution of our estimates would be approximately

normal and on average equal the truth,

the unobservable true value of the association, the adjusted association.

So again, we'll get one estimate under

this curve that describes the distribution of all estimates,

it could be way out here,

it could be here, it may even be right on top of the truth,

but we'll never know for sure.

But because the distribution is normal,

we know that most of the estimates we get would fall within

two standard errors of the unknown truth.

So, if we start with most of the estimates we get,

it will fall in two standard errors.

So for most of the estimates we get,

if we start by adding and subtracting two standard errors,

the interval will include the unknown truth.

Same philosophy, same result has been working

every time we've talked about sampling variability confidence intervals.

In terms of hypothesis testing,

the null value on the slope scale of no association between a given predictor x sub i,

and the time to event outcome after accounting for the other predictors,

is that the slope for that predictor is equal to

zero which corresponds to a hazard ratio of one.

So, what we do is we pretend that that's the truth,

we start by assuming that's the truth and then we measure

how far our particular slope estimate is from zero,

and if it's far enough such that less than five percent of results

that could have occurred by chance from the null are as likely or more extreme than ours,

then we would reject the null,

otherwise we would not.

So, it's the same story as we've been working with in terms

of hypothesis testing since we started it in the first part of the course.

So, the intercept is a function of time:

uncertainty estimates for the value of this function over time

are estimated and are used for putting confidence limits

on predicting survival curves based on the Cox results,

and in a subsequent section,

we'll show the idea of looking at predicted

survival curves starting from the Cox regression,

and we'll talk about these confidence intervals.

But for any single slope,

the processes is as we've seen before,

where we get the 95 percent confidence interval for the slope,

we just take the estimate plus or minus two estimated standard errors,

and then exponentiate the end points to get

95 percent confidence interval for the corresponding adjusted hazard ratio.

In order to get a p-value for testing the null of

no association between the time to event outcome in that particular predictor x,

when predictor can be described by a single x,

or modeled by a signal x,

what we would do is the nulls that the slope

for that particular predictor zero at the population level,

corresponding to the adjusted hazard ratio of one,

and the way we do this for each of our slopes we take the estimated slope itself,

and divide by its standard errors which would measure how far our estimate

from zero is what we'd expect it to be under the null in terms of standard errors,

and this distance would then be translated into a p-value.

So, let's look at the results we looked at for

Cox regression results for predictors of mortality in

the PBC or primary biliary cirrhosis trial of the drug D-Penicillamine or DPCA.

Let's look at the results for model two which includes as potential predictors the drug,

age of the patient at the time of enrollment in the study,

their bilirubin levels in milligrams per deciliter at the time of enrollment,

and their sex, their biological sex.

Here's the resulting underlying Cox regression model for these data.

These slopes, you could get them from

the previous table simply by taking the natural logarithm of

each of those adjusted hazard ratio estimates presented for model two.

So, the slope for treatment is the log of

the adjusted hazard ratio for treatment of long of 1.10 which is just 0.1, etc.

So, we have the age quartiles here,

the slope for bilirubin,

and the slope for sex,

where sex is a 1 for females.

So, I'll just give you the standard error for two of these slopes.

The standard error for the slope of treatment which was the estimate was 0.10, was 0.19.

The standard error for the slope of bilirubin which was Beta five equal to 0.15 is 0.013.

So, if we wanted to get

a 95 percent confidence interval for the slope of treatment, in other words,

the adjusted hazard ratio mortality for

those in the DPCA group compared to those in the placebo group.

We take the estimated slope or log of the adjusted hazard ratio and subtract

two standard errors and we get a confidence interval for

the log of the hazard ratio that goes from negative 0.28-0.48.

So, notice on the slope scale and includes the null value of

zero and we know the result is not statistically significant.

But to present it on the hazard ratio scale,

we first take our estimate of the log hazard ratio 0.1 and

exponentiate it to get an adjusted hazard ratio estimate of 1.10.

Then, they get the confidence interval for this on the hazard ratio scale.

We'd exponentiate the end points of

the confidence interval for the slope negative 0.28 and 0.48.

This will give us a confidence interval for the hazard ratio,

the adjusted hazard ratio of treatment of 0.77-1.6.

Similarly, we could do the same thing for the bilirubin association

and all the others in the model but can only fit two two the slide here.

So, we take the slope for bilirubin,

take 0.15 and slope for bilirubin and subtract two as to standard errors,

get a confidence interval for the true slope or the true log

adjusted hazard ratio with bilirubin of 0.124-0.176.

Notice this confidence interval for the slope does not

include the null value on the slope scale of zero.

Now, to get this on the hazard ratio scale,

the adjusted hazard ratio scale,

we would first to the estimate e_0.15 this one 1.16,

that's the adjusted hazard ratio,

and then we'd exponentiate

the end points to

the confidence interval for the slope to get the endpoints on the hazard ratio scale,

the confidence interval 1.3-1.19.