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

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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.

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Module 3B: Sampling Variability and Confidence Intervals

The concepts from the previous module (3A) will be extended create 95% CIs for group comparison measures (mean differences, risk differences, etc..) based on the results from a single study.

- John McGready, PhD, MSAssociate Scientist, Biostatistics

Bloomberg School of Public Health

So in this section we're going to show how to estimate confidence

intervals for incidence ratios based on

samples from two populations that we're comparing.

So upon completion of this lecture section you

will by able to estimate and interpret a 95%

or other level confidence interval for an incidence rate

ratio comparing time to event outcomes between two populations.

So this process is analogous to creating confidence intervals

for relative risks and odds ratios for strictly binary outcomes.

All computations are done on the natural log scale.

And then the results are antilogged, or exponentiated to get

the interval, the confidence interval on the incidence rate ratio scale.

So let's go back to one of the studies we first looked at when we were

coming, describing incidence rate ratios, the famous primary

biliary cirrhosis trial conducted at the Mayo Clinic.

Where a sample of over 300 patients with primary biliary

cirrhosis were randomized to receive the drug DPCA, or a placebo.

And the primary research question of this study was how does mortality,

and hence survival, for primary biliary cirrhosis patients randomized to receive

DPCA compared to survival for such

patients randomized to receive the placebo.

So amongst the 158 persons randomized to receive DPCA

there were 65 deaths and 872.5 years of followup for an incidence rate of 0.075

deaths per year. In the 154 persons randomized to the

placebo group, the incidence rate of mortality was 0.071 deaths per year.

So when we actually compared the incidence rates in ratio

format between the DPCA group and the placebo is 1.06.

So we actually saw an elevated risk, at least in the samples,

of death in the DPCA group, about 1.06 times

the risk of the mortality in the placebo group.

In other words, subjects in the DPCA group had 6% higher estimated risk of

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

But of course that's just a sample estimate.

So we need to put uncertainty bounds on it before making a formal conclusion about

the nature of the association between mortality in DPCA in

the population of all such patients with primary biliary cirrhosis.

So I don't think this will come as a

surprise but we're going to do our computations on the log

scale because we have a ratio and then exponentiate the

results to get back to the incidence rate ratio scale.

So our estimated incident rate ratio is 1.06.

The log

of this is 0.06.

So the way we're going to do the confidence interval in

the log scale is take the log over estimated units in

its rate ratio, the 0.06, and add and subtract two estimated

standard errors of the log of the estimated incidence rate ratio.

So now we just have to consider how we're going to estimate the standard error.

And it turns out the standard error of the log of

the estimated incidence rate ratio is easier to compute perhaps than

the standard errors for relative risks and odds ratios from binary data.

For incidence rate ratios, the standard error of

the log for estimating incidence rate ratio is a

function of the number of outcomes in the

two groups that are being compared by the ratio.

So the formula is one over the number of events in the first

group plus one over the number of events in the second group.

So for these data, we have the two groups,

the DPCA groups, in which we saw 65 outcomes

or deaths, because that was our event of interest

and in the placebo group, we saw 60 deaths.

So the estimated standard error, the log over estimated incidence rate ratio

is the square root of one over 65 plus one over 60 and when

the dust settles this turns out to be 0.18.

So going back to our log scale.

The log of our estimated incidence rate ratio is 0.06.

To get a confidence limits on the log of the true incidence rate

ratio we'll take that estimated 0.06 plus or minus 2 estimated standard errors.

So plus or minus 2 times 0.18 and we get

confidence interval for the log of the true

incidence rate ratio between negative 0.3 and positive 0.42.

So this confidence interval for the log

of the true incidence rate ratio includes zero.

And so we know when we exponentiate, and get the confidence

level for the true incidence rate ratio, our results will include one.

So what we get is the confidence interval for the true

[INAUDIBLE]

ratio rate of mortality for the population of PVC patients where they're given the

drug DCPA compared to the same population given a placebo of

0.74 to 1.52. So let's talk about how to interpret this.

How we might write this up.

So we could say something like, in this

study, the 158 subjects with primary biliary cirrhosis

randomized to receive the drug DPCA had a slightly elevated risk of

death when compared to the 154 such subjects randomized to the placebo group.

And then we'd report our estimated incidence rate ratio, 1.06.

After accounting for sampling variability, however, there is no evidence of an

association between DPCA and death in the population of patients with PBC.

And that's

because, I say that because the 95% confidence interval includes the null

value of one. And literally,

this confidence interval,

0.74 to 0.152 means

the DPCA could be associated

with anywhere from a 26%

reduction in risk and mortality

to a 52% increase in

the risk of mortality when

compared to patients who

didn't receive the treatment. So there's no strong

conclusion here about the nature of the association between DPCA and mortality.

Could either increase or decrease and hence there's no specific

population level association found. Look at the study with the

1,700 plus sero-discordant, HIV sero-discordant couples.

Who were randomized for the HIV positive partner

to receive either early anti retroviral therapy or standard.

And want to see what the association was that between these

different approaches to treatment, and the partner to partner transmission of HIV.

And what they found in these couples,

over 1,700 couples, is there were 28 overall linked transmissions.

Partner to partner transmissions.

And only one occurred in the early therapy group.

So they report something called a hazard ratio

which is nearly synonymous with an incidence rate ratio.

We'll get into the distinction between the two in the second

term of this course, but functionally they have the same interpretation.

Their estimated incidence

rate ratio was 0.04, with a 96% confidence interval for the ratio 0.01

to 0.27, so we can see that does not include the null value 1.

In fact, all possibilities are far from 1.

So let's just look at how these results panned out.

So what they've shown is their estimated association

compared the incidence rate of partner to partner transmission

in the early anti-retro therapy group to the

late, or standard, therapy group and it was 0.04.

So again, a way to interpret this would be to say

HIV discordant at baseline couples in which the HIV positive partner

was given early anti retroviral therapy, had

0.04 times the risk to within couple transmission

when compared to couples in which the

HIV positive partner was given the standard therapy.

In other words, HIV discordant at baseline

couples had 96% lower risk of within couple

transmission when the partner was given the

early ART therapy, compared to couples in which

the HIV partner was given standard. So, quite a reduction.

Quite an observed reduction in risk in this study.

But of course, we understand that this was done

on a, a sample of couples from the larger population.

So we want to account for that in detail in the final conclusion.

So, our observed incidence rate ratio was 0.04, we take

the log of that, the natural log is negative 3.22.

We compute the standard error of the log

of the observed incidence rate ratio, it's theoretical

variation from study to study of the same

size with random samples from the same population.

Standard error is computed by taking the square root of 1 over the 1

event in the early group, plus 1 over the 27 events in the standard group.

Standard errors 1.02. So if we get our confidence interval

first for the log of the incidence rate ratio it's negative 3.22

plus or minus 2 times 1.02. If you do the math on

this you get a 95% confidence interval for the log of the true incidence rate ratio.

I get a 5.26 and I get 1.18. So of course that's the

confidence interval for the log of the true incidence rate ratio, though we so it

does not include zero. So when we exponentiate that to get

confidence interval for the incidence rate ratio exponentiate

the end points of e negative 5.26, e to the negative

1.18, the confidence interval goes from, from around

here, from 0.01 to 0.31.

So, notice as you would expect the interval does

not include the value of 1, a null value.

So, let's just parse these results in a substantive context.

I'm going to use the results I computed.

They differed slightly from what was shown by the

articles author, because a slight difference in the computations.

But the effective message is exactly the same.

So in a study of 1,763 HIV sero-discordant

couples the risk of partner-to-partner transmission among the 866

randomized to receive early anti-retro therapy was 96% lower than among

the 877 randomized to receive standard antiretroviral therapy.

After accounting for sampling variability, the early antiretroviral therapy

could reduce the risk of partner transmission, partner to partner

transmission, from anywhere from 69% to 99% at the population level.

So even in the quote un-quote worst case scenario

the potential degree of association is reduction of nearly 70%.

So these findings are very strong.

Obviously the 96% reduction in the sample is very strong

but this holds up after accounting for the sampling variability.

And this was quite a result even though there was

a lot of uncertainty in that estimate, even in the worst

case scenario we're dealing with a reduction on the order

of nearly 70% in the risk of partner to partner transmission.

Finally we'll look again at our

maternal vitamin supplementation and infant mortality study.

And this was where a large number of pregnant women in Nepal were randomized

to receive pre-natal vitamin supplementation of either Vitamin

A, Beta-carotene or a placebo to see if

there was any association between vitamin supplementation and

infant mortality in the six months following birth.

So we'll just cut to the chase here, but what we

did in the previous lecture, lecture five we'll just build on here.

We, we designated the placebo group to be the group

we'll compare the vitamin groups to, in terms of mortality.

And so we estimated the incidence rates ratio mortality for the Vitamin A group

compared to the placebo group to be 1.05. So in this study, the infant mortality

rate among infants born to mothers who got vitamin A in the

prenatal period was slightly higher than among those who got the placebo.

When we compare the infants rate ratio for the beta carotene

group to the placebo group, the estimates were exactly the same.

for the incidence rate in this ratio was one.

Effectively no difference in the estimated mortality between the two groups.

This was a large study so we do want to, and we do want to put sampling bounds on

our estimates to reflect the uncertainty and I'll

just cut to the chase and do it here.

But, if we did that 95% confidence interval

for the incidence rate ratio, for the ratio comparing

vitamin A to placebo, it goes from 0.87 to

0.28, and certainly includes the null value of one.

If we did the same thing for beta-carotene, compared to the placebo,

our estimate was one.

So of course, it's going to be in the interval.

And the interval goes from 0.84 to 1.25.

So the ultimate conclusion of the study with

regards to vitamin supplementation in the pre-natal period was

that it had no impact on infant mortality, because

these results showed no associations at the population level.

So in

summary, the 95% confidence interval for an incidence rate ratio can be computed by

creating a 95% confidence interval for the log of the incidence rate ratio, and

exponentiating, ie anti-logging the results.

This 95% confidence interval for the incidence rate ratio gives a range of

possible values for the true incidence

rate ratio for the populations being compared

by the two samples.

As with the relatives risks and odds ratios, and you can think of the

incidence rate ratios being a relative risk

that includes information about time at risk.

Incorporates that into its computation. As with the relative risk

and odds ratios, the null value for the incidence rates ratio is also one.

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