A conceptual and interpretive public health approach to some of the most commonly used methods from basic statistics.

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From the course by Johns Hopkins University

Statistical Reasoning for Public Health 1: Estimation, Inference, & Interpretation

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Johns Hopkins University

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A conceptual and interpretive public health approach to some of the most commonly used methods from basic statistics.

From the lesson

Module 2C: Summarization and Measurement

This module consists of a single lecture set on time-to-event outcomes. Time-to-event data comes primarily from prospective cohort studies with subjects who haven to had the outcome of interest at their time of enrollment. These subjects are followed for a pre-established period of time until they either have there outcome, dropout during the active study period, or make it to the end of the study without having the outcome. The challenge with these data is that the time to the outcome is fully observed on some subjects, but not on those who do not have the outcome during their tenure in the study. Please see the posted learning objectives for each lecture set in this module for more details.

- John McGready, PhD, MSAssociate Scientist, Biostatistics

Bloomberg School of Public Health

So as with relative risks, incidence ratios do

not fully encapsulate the information in our data.

And if we only look at the incidence rate

ratios we don't have an understanding of the sort

of order of magnitude of the risks or how

it unfolds over time in the two groups we're comparing.

So, as we did with binary data where we present a relative risk

and a risk difference are analogue to the risk difference for tying to event

data will actually we tracing the survival over experience over

time for the two groups we're comparing separately via Kaplan-Meier curves.

Okay now let's talk about comparing time to

event data between two or more samples visually.

So, upon completion of this lecture section hopefully you will be able

to visually compare time to event data across two or more samples,

explain how survival proportions across time can remain relatively

high, and alternatively, the cumulative probability of having the event.

Relatively low, even if only a small proportion of the original study

is around or at risk at the end of the study period.

So let's go back to our example

with the primary biliary cirrhosis randomized clinical

trial, comparing death among patients with primary biliary cirrhosis.

Who take the drug DPCA versus those who were randomized to placebo group.

So this is this famous study conducted at the Mayo

clinic, and we saw a numerical summary of the association between

the treatment and death was such that the relative risk or

incidence-rate ratio was 1.06, indicating, at least in this study sample,

those in the drug group, the DPCA group, had 6% higher risk of

death in the followup period when compared to those in the placebo group.

This number is helpful, but it would be

nice to also get a sense of what proportion

of persons are still alive, as we progressed across

the followup period in each of the two groups.

So a nice way to visually assess and give some background on that incidence rate

ratios is to actually look at the Kaplan-Meier curves for

the placebo and drug groups plotted on the same graph.

You can see on this graph here that's what we've done.

This blue line, tracks the proportion of patients who are

still alive in the placebo group over the follow up period.

And, this tracks the proportion of patients

who are still alive, given time in

the drug group.

You can see visually these curves are sort of neck and neck, there's a little bit

of separation here, but on the whole, they're

neck and neck across the entire follow-up period.

So, that's why we don't get a large difference between the survival rates.

When we compare it with that ratio it's 1.06.

And the reason the drug proof has slightly higher risk than

the placebo group over the entire follow up here you can see

is mostly because of this period here, where the proportion is still alive.

It's lesser than the drug group compared to the placebo.

So this helps us establish some things.

So for example, this gives a face to the percentage still surviving.

So if we look at the followup period of about five years

we can see that almost 75% in both groups are still around.

That's information that doesn't get

supplied with an incidence rate ratio of 1.06.

So this visual presentation adds some flavor and information to the story.

Remember, we could also plot this as the, the complementary plot.

Instead of the percentage surviving, we could look at the

percentage who have had the event by a certain time.

It's just 1 minus the values in the previous curve.

And you see this is basically the same story, but just

presented in the opposite direction. So the five year measurement is 25%.

Which corresponds to 25% of the persons having died by five years.

Which also corresponds to the 75% we saw who hadn't

had the event by five years in the previous presentation.

How about the maternal vitamin supplementation

and infant mortality examples we gave?

Here we had

three groups.

Placebo, Vitamin A, and beta carotene, and we computer the associations between

the two different vitamin groups relative

to the placebo through incidence rate ratios.

And the relative incidence rate ratio of vitamin A

mortality, to placebo mortality was 1.05 in the study sample.

So those children, whose mothers got vitamin A during

the study, were at slightly higher risk of dying

in the following period compared to those on placebo.

And the risks were essentially equal in the beta carotene, and placebo groups.

But again, this just tells us about the relative comparison.

Doesn't give us sort of the absolute percentage of children.

Who died in the followup period in any of these groups.

Nor does it show it how it unfolds

over time.

So let's look at the Kaplan-Meier curves for

each of these groups, plotted on the same graphic.

And you can see that these three curves, for

the three groups, are very similar, and that's because

we saw already from the incidence rate ratios that

relatively speaking, the risks were very similar as well.

So this brownish group is the placebo group.

The blue group hidden in here is the beta-carotene.

And the green curve is for the vitamin A.

So you can see that the beta-carotene placebo groups are

pretty much neck-and-neck after a certain point in the follow-up period.

Whereas the green group falls a bit lower.

And, that's why we get the slightly elevated

risk ratio of 1.06 for the vitamin A group,

which is the green group compared to the, the

placebo group.

But, by looking at this graph we can learn about the burden of mortality in

this sample, which we don't get, we

only get the relative comparison of those numbers.

So we can see that 50 days of followup roughly, 95% of the samples

are still alive, children are still alive in each of the samples we looked at.

The three different groups,

which means that roughly 5% died within the

first two months after birth, which is substantial.

And now, this gives us a sense of the burden of

infant mortality in this population, as estimated through the sample statistics.

Alternatively, we can present it in this other method.

Where we take the complement of what we just saw, instead of

tracking the proportion who have, still haven't died, by a certain time.

Still haven't had the event we actually track the proportion

that has had the event, or died by a given time.

And you can see it's pretty much the same picture, just in reverse.

Let's look at our anti-retroviral therapy in partner to partner HIV transition

study that we looked at before, and let's look at the outcomes.

Remember these were discordant couples where one was

HIV positive and the other was HIV negative,

were randomize to receive either early anti-retroviral therapy

or the traditional, time to give retroantiviral therapy later.

In the progression of HIV, and we want to see, the researchers

wanted to see what differences, if any, there were, in terms of partner infection.

So, we already assessed this with the incidence rate ratios and

we solved that the incidence rate ratio strongly favor the early.

And retroviral therapy group that, the instance rate, ratio

of transmission for those couples where the HIV positive

partner was given early anti-retrovirals relative to those, the

group where it was, the standard therapy was 0.04.

Indicating that transmission

was 96% less. In the group.

They got the early anti-retrovirals.

So, a strong association between the

early anti-retroviral therapy, and reduced transmission.

We actually wanted to assess this visually to get a sense of how much transmission

there was in each of the groups and how it unfolded over the follow-up period.

It would be helpful to look at

some graphics, and thankfully the authors included some.

In their article to nicely

illustrate this.

So thankfully they do provide a nice visual.

They give the cumulative probability version of

the Kaplan-Meier curve tracking the proportion of couples

who have experienced transmission over the followup period.

And they actually do this two versions they in,

they show the overall graph and then this inset here.

Is actually the same graph, but with a truncated vertical

axis to blow up the picture of what's going on.

But the reason they do that, and you can see even with the

blown up version, you can't see the curve for the early group because

there are so few transmissions.

There's only one transmission in the entire group over the follow up period.

So this curve basically stays at, or

very close to zero, and isn't visually detectable.

For the delayed group however, the standard therapy you can

see that this human interproportion sort of increases over time.

An we hit a point here where it stops and stays flat which,

indicates after that point, all observations where

censored toward the body end of the study.

In other words, the proportion didn't change after

three and a half years of follow up.

And this puts a face on this we'll show us that really, that incidence rate we

show, 0.04 means that the proportion of actually

experience transmission in the delay group, never exceeded 10%.

So we can argue about whether that's high or not, but that incidence rate ratio

was so strong, and so small, because there were hardly

any transmissions in the early group. So the relative

ratio of nearly 0% over time to something that accumulates to

10% was still a very small ratio in terms of

the relative rates of transmission.

They want you to notice here and just think about in terms of

this study is this study statrted at time zero, they randomized these couples.

There were 893 couples.

Randomize the early transmission group in 882 to the

standard of the delay transmission. By year one, there were only 658 people

at risk in the early transmission group.

We started with the 893, there were only 60-, 658 at

risk, and we know there was only one actual realization of transmission.

One event in the entire early group.

So the majority, maybe all, because not sure when that event occurred.

Of these, the difference between these two numbers is because of censoring.

And similarly, between year one and year

two, lost a bunch more people to censoring.

Bunch more people to censoring, bunch more, and

then it sort of stayed at the end of

the study, there were still 24 couples around

who had not yet had partner to partner transmission.

So you can see even though we started with a lot of people, and by

the end of the study there were very few at risk, the majority of the

loss of persons was due to censoring, not the event, and as such the overall

proportion experience in the event in the early

transmission group was very, very close to zero.

And you can see sort of the same thing for

the delayed group.

We ended up with 22 people at risk at the end of

the study, versus the 882 we started with, but because the cumulative proportion.

A person, or couple's experience in transmission was on the order of 10%.

The majority of this discrepancy here, and there's some

details in the interim numbers, is because of sensory.

So in summary, plotting Kaplan-Meier survival curve estimates, or

the cumulative event probability version of those for multiple samples

on the same graphic gives a nice overall visual

comparison of the time to event experience across different groups.

Kaplan-Meier curves nicely comps, compliment the

incidents ratio, rate ratio estimates, and

to provide a little more detail about how the events unfold over time.

Like the incidence rate ratios however the Kaplan-Meier

curve estimates are sample statistics, and hence they

estimate the underlying unknown true survival curves in

the populations from which the samples are taken.

So just like all summary statistics we discussed before.

The Kaplan-Meier curves of estimates of some underlying

truth that we can't directly observe.

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