So in this section, we'll extend the same ideas we looked at

in lecture seven A regarding the

central limit theorem and creating confidence intervals.

And here we'll do the same thing, not for means, but

for summaries of binary data, proportions, and timed event data incidence rates.

So what we're going to learn in this section is how

to estimate a 95% confidence interval for a population proportion

based on the results of a single sample from the population.

And, we'll do the same thing, estimate a 95% confidence interval for a population

incidence rate, based on the results of

a single sample from the population of interest.

So, the drill is going to be very similar to what we did with means here

because we laid out the logic in lecture six and the beginning of lecture seven A.

So let's

just jump in head first.

So recall the Kaggle data were we looked at the response to therapy

in a random sample of 1000 HIV

positive patients from a citywide clinical population.

And you may recall, out of the thousand subjects in this sample, 206 responded to

give us a sample-based estimate of 20.6% or 0.206 responding

in this sample as our best estimate for the population.

So how are we going to actually create an interval statement about the unknown true

proportion of people in this population who

would respond in this citywide clinical population?

Well, we've got an estimate here, it's, it's our best

estimate based on the information we have, this p hat.

And what we're going to do is exactly the

same drill, to create a 95% confidence interval,

we're going to add and subtract two standard errors.

And again, we're going to have to estimate the standard error of

the sample proportion from the single sample of data we have.

So you may recall that the standard error of a sample proportion, the formula, is a

function of our estimated proportion itself, times 1 minus the proportion.

So it would be the proportion who respond

in this sample, times the proportion that did not, divided by the size

of the sample. So with these data, the estimated standard

error looks like this. It's square root of

0.206 times 1 minus 0.206, so 79.4%

or 0.794 of the sample did not respond divided by

the sample size of a thousand. If you do

the math, this is approximately equal to 1.3%

or 0.13. So to do our computations

to get the confidence interval, we take this sample proportion

estimate, the p hat of 0.206, and add and subtract two standard

errors that we just computed of 0.013 or 1.3%.

So if you do the math on this, you get a confidence interval of 0.18

to, and I'll just be, we could round this, but this is, if you do the

math directly, you get 0.232. So I would probably

present this as 0.18 to 0.23 or

18% to 23.2%. So now, we've quantified

the rate of response in this population, both by our

best estimate of 20.6% or .206, and now we've

given a range of possibilities for the true response

rate 18% to 23.2%. This plus or minus two standard error

piece for a proportion is frequently called its margin of error.

And many of you have probably heard this phrase in

the news when the results from a poll are being reported.

Like this poll was conducted with a margin of error of plus or minus 3%.

And what they're telling you is the piece that you would add and subtract

to their estimate, to get a confidence

interval for the true proportion that's being estimated.

Let's look at our, another example that we've been working with,

this seminal study on maternal and infant HIV transmission, where they ultimately

had data, followup data on 663 infants born to 363 women who were HIV positive.

And what they found within 18 months of the birth, 53 of these 363 infants

contracted HIV, or estimated, if we round, it's roughly 15% overall.

So again, this is just an estimate of the overall rate

of contraction in this study population based on these 363 observations.

So, we wanted

to quantify the uncertainty in this. We

could estimate the standard error for this

overall proportion by taking the square

root of sample proportion, 0.15

times 1 minus that sample proportion of

0.85, divided by sample size. And this is, with

rounding, approximately equal to 0.019 or 1.9%.

So we could actually estimate the confidence interval for this population of

women who were HIV infected and pregnant. And it's a mixed population,

some were treated with AZT and some weren't.

And so, the resulting confidence interval estimate would be the overall estimated

proportion of 15% plus or minus 2 times the 0.019.

If you do this out, you get a confidence interval of, of 11% or 0.11 to

0.19 or 19%.

So this quantifies the burden of transmission in this mixed

population of some women who were treated and some weren't.

Here's our example of colorectal screening,

and remember, from the results section,

we've sort of mentioned this before, and now we'll bring it in.

This is the study where they

actually compared automated information, intervention with

stepped increases in support to increase the uptake of colorectal cancer screening.

And they found that with increased intensity with stepped increases in

support, we saw increased response or uptake of colorectal screening.

And they reported in each group,

the proportion who actually got screened within

two years after the study started, and

a 95% confidence interval for that proportion.

So they did it for the usual care group,

the one that was ostensibly given standard of care.

The automated care group, so the estimated proportion in

this sample who got the automated care was about half, 50.8% were screened for

colorectal cancer, but a 95% confidence interval, and that was 47.3% to 54.4%.

And you can go through this and look and see the estimates that

we reported in the section on binary

outcomes, now coupled with their confidence intervals.

And here are the proportions in each of the four groups.

And now

let's just focus on the usual care group.

So this would sort of describe if people went with

the status quo, if there were no changes in how

we treated colorectal cancer screening marketed to people, this is

what we'd expect to see in terms of people getting screened.

It would be about a quarter, slightly over

a quarter, of the population would get screened.

And our estimate of that is 26.3%,

based on the sample we have at hand.

So this describes what we'd expect if no changes were

made, if everyone was given the usual standard of care.

But of course, this is an imperfect estimate, because

it's only based on a subsample of about 1,166 persons.

So, let's put confidence limits on this to get a sense

of how much response we can expect at the population level.

So,

I'll leave this for the review exercises to actually formally do.

But if you actually do the routine p hat plus or minus 2

estimated standard errors, p hat, the confidence interval

for the proportion of who could get the screen, is between

0.237 and 0.289. So roughly 0.24

to 0.29, 24% to 29%.

So, this tells us that on the whole, we would expect somewhere around a quarter

of the population to get screened for colorectal cancer if

we continued to give the usual standard of care.

In the next section we'll show how to compare these proportions across the

usual care groups versus the intervention groups, to get a sense of how

much better this outcome would be, how much higher the proportion would be,

if the intervention were enacted, if some level of the intervention were enacted.

What about for incidence rates?

Well,

the drill is the same in terms of taking

our estimate and adding and subtracting two standard errors.

So here's the example from the Mayo clinic

and the primary biliary cirrhosis randomized clinical trial.

And if we wanted to just get a sense of the burden of

death in the entire population from which the sample were

taken, mixed between those who got treated with DPCA and those who got a placebo.

We can think of this a mixed population where some

people got treatment and some didn't, and of course more

interesting will be the comparison between those who got treated

and those who didn't, but this will just get started.

The overall incidence rate of death in these data was

125 deaths per 1715 years of follow up, and we

showed how to compute that in lecture five, or an

incidence rate of 0.073 deaths per person year of follow-up time.

So how do we get the standard error for this?

Well, you may remember from the previous section, the standard error of

an incidence rate can be computed by taking the square root of the number of

events we saw in the sample, divided by the total person follow-up time.

So for these PBC data, there were 125 deaths over a total 1,715

person years giving us a standard error estimate of the square root of

125, divided by the 1,715

person years, is a standard error of 0.0065 deaths per person year.

So, in order to quantify the uncertainty, put, put

an interval statement on the true incidence rate in this

population, this mixed population of some who got treated

and some who didn't, we take our estimated incidence rate.

Add and subtract two estimated standard errors.

By that same logic, we worked out in lecture seven A and in lecture

six, regarding creating an interval of possible

values for an unknown truth using the

logic from the central limit theorem. So to do this, we take our

estimated incidence rate in these data 0.73 and add and

subtract two standard errors to get, ultimately

get a confidence interval that goes from 0.06 deaths per year to

0.086 deaths per year. So of course, just like

the estimated rates can be re-scaled to different reference time periods, so can

the confidence interval. So if we scaled this up to per 1000

years of follow up time, the sample estimate

expressed in this would be 73 deaths per a 1000

person-years. And the confidence interval

would be.

So look at one more example for incidence rates, the maternal vitamin

supplementation and infant mortality data, this

is amongst the sample of Nepali children.

So amongst the entire sample of 10,295 Nepali children, what we saw,

there were, there were 644 deaths for 1,627,725 days of follow-up

time, for an estimated incidence rate, we presented as in lecture five, of

644 deaths per 1,627,725 days or 0.0004 deaths per day.

So if we were to actually estimate the standard error of this incidence rate,

we take the square root of the number of deaths, the square root of 644,

and then divide by that total follow up time of over 1.6 million days.

And we get a standard error of 0.000016

deaths per day. So in order to create a confidence

interval for this mortality rate, the incidence rate of mortality in

the six months following birth in this population of Nepali children,

we'd take our estimate from the sample, 0.0004 deaths per day, and

subtract two standard errors, 2 times 0.000016 deaths per day.

And we get a confidence

interval of 0.000037 deaths per day, up to 0.000043 deaths per day.

And of course, we could present these on

a different scale, per year, per 100 years, etc.,

depending on the venue.

So, in summary, what we've done in this section is more of the same.

We've done, to get a confidence interval for either

a proportion or an incidence rate, we've taken our

best estimate from a sample, and added and subtracted two

estimated standard errors to get a 95% confidence interval.

So this is 95% CI.

For proportions, the estimated standard error

is a function of the proportion itself and the

sample size. For an incidence rate,

the standard error is estimated by

taking the square root of the number of

events total in the sample, divided

by the total follow-up time. And if

we wanted to create other levels of interval, whether it be,

say, a 99% confidence interval or a 90%, we could alter

this formula slightly. For the 99% confidence

interval, we'd add and subtract 2.58 standard errors.

Estimated standard errors.

And for the 90%, we could add and subtract 1.65 standard errors.

For this course and in most of your

research life, you will exclusively use 95% confidence

intervals, but I just wanted to remind you

that in theory, that level is arbitrary and we

could present any level of confidence based on what we know about

the relationship between area under the

normal curve and number of standard errors.

Okay, so we've done in these past two sections, we've shown the mechanics

of creating confidence intervals, and we've talked a bit about what they mean.

In the next section, we're going to hone in on thinking

about what this really means and what the ramifications are for understanding

what's going in, on in the larger population from which we've sampled.

Lecture 7B: Confidence Intervals for Sample Proportions and Rates

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

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