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 2B: Summarization and Measurement

Module 2B includes a single lecture set on summarizing binary outcomes. While at first, summarization of binary outcome may seem simpler than that of continuous outcomes, things get more complicated with group comparisons. Included in the module are examples of and comparisons between risk differences, relative risk and odds ratios. Please see the posted learning objectives for these this module for more details.

- John McGready, PhD, MSAssociate Scientist, Biostatistics

Bloomberg School of Public Health

So let's explore this dichotomy further in

terms of the difference in interpretation and why they can be different numerically.

Let's focus on another example.

Here's the results from a randomized trial involves giving Aspirin,

low dose Aspirin or not,

or placebo to women of a certain age,

45 years or older.

And this study, they randomized almost 40000 initially healthy women,

people who did not have any evidence of cardiovascular disease.

Upon enrollment, they either received 100 milligrams of

Aspirin on alternate days or then placebo.

And these women were followed for up to 10 years after enrolling in the study.

And what they wanted to look at was who succumbed to a first major cardiovascular event.

OK? And during this 10 year follow-up,

there were 477 major cardiovascular events in

the Aspirin group as compared with 522 in the placebo group.

So in order to make sense with those raw counts,

we're going to have to standardize them by the number of women in each group.

In other words, calculate our sample proportions.

So here's a two-by-two table laying out the results here.

So roughly half, 19,934 women are randomized to receive Aspirin,

and within the 10-year follow-up 477 of them had

a major cardiovascular event or considered to have cardiovascular disease.

477 out of that denominator of 19,457 women is 0.024, 2.4 percent.

If we looked at the same summary measure for the placebo group,

the 522 out of the total 19,942 women,

the proportion is slightly larger,

0.026 or 2.6 percent.

So if we were to actually quantify this as a risk difference,

we would take, say for example,

Aspirin compared to placebo,

we would take that 0.024,

that 2.4 percent of

cardiovascular outcomes in the Aspirin group and

subtract that 2.6 percent in the placebo group.

And that would give us a difference of -0.002

which translates to a risk difference of -0.2 percent.

So we could say there was a 0.2 percent absolute reduction in the 10-year risk of

CVD - cardiovascular disease - for woman on

low dose Aspirin therapy compared to women who were on the placebo.

The way we could think about this is,

what would be the impact on a group of women?

That suggests, assuming causality in a population of 100,000 women 45 years and older,

we would expect to see 0.02 times 100,000,

or 200 fewer cases of cardiovascular disease within 10 years if

the woman were given low dose Aspirin therapy as opposed to not treated.

How about the relative risk?

Well, the relative risk here is risk ratio would be obtained by taking that 2.4 percent

in the Aspirin group and dividing by that

2.6 percent in the placebo group, a relative risk of 0.92.

So the 10-year risk we could say of cardiovascular disease development for women on

the low-dose Aspirin regimen is 0.92 times the risk for women given placebo.

In other words, a woman can reduce her personal risk of cardiovascular disease by

eight percent if she takes a low dose of Aspirin every other day intermittently,

daily as compared to not doing so.

Where did I get that from?

Well, if you take that 0.024,

that difference we had before that 0.026,

divide it by that 0.026,

2.6 percent of the placebo group at -0.002 divided

by that- It's in the placebo group that's a reduction,

a relative reduction of 0.08 or eight percent.

So let's recap about similarities and differences in these two quantities.

So we've talked about the differences, substitute interpretations.

We've seen that they can differ in values,

but the general themes will be the same.

If the risk difference is positive,

the first group in the comparison has

higher proportion with the outcome than the second group.

Then the relative risk in the same direction will be greater than one.

So P1 hat over P2 hat will be greater than one.

If the relative risk is less than zero,

mean that the first group has lower risk than the second group,

then the relative ris P1 hat over P2 hat, will be less than one.

And if the risks are equivalent the estimated risks are equivalent,

and the risk difference is zero.

Then the relative risk,

the two proportions will be the same,

the relative risk will be exactly equal to one.

So they will always agree with each other in terms of

the direction or the lack of difference if you will.

So on the whole they'll come to the same conclusions.

However, as we've seen,

and let's investigate this a little more,

the two quantities compared can appear different in terms of magnitude.

So it is possible to see a large effect with

one measure and a small effect with the other.

So for example, suppose

our underlying proportions of outcomes in two groups we're comparing are small.

Each of them is small.

So in the first group maybe the proportion who have

the event is one in a thousand or 0.1 percent.

And the proportion of have in the second group is 3 in a 1000 or 0.3 percent.

So if we look at the absolute reduction is small,

0.001 minus 0.003 is -0.002, -0.2 percent.

So it's an absolute decrease of 0.2 percent.

That doesn't look very dramatic but think about it.

The the lowest we could go.

Suppose the first group had no outcomes.

You effectively cured whatever this outcome was if it were a bad outcome and so,

the sample portion was zero.

Even in that case the risk difference wouldn't look that

impressive because it would be a reduction from 0.3 percent.

The most our reduction could be here is 0.3 percent

because that's the burden of the outcome in the group that has more.

So we're constrained with this measure by how much of a reduction we can see.

And we could talk about similar examples and with an increase.

But if we instead reported the relative comparison

0.001 divided by 0.003 is 0.33.

We could phrase this in,

and I'll let you do the math,

but as a relative decrease of 67 percent,

that sounds a lot more dramatic than a decrease of 0.2 percent.

Doesn't it? But these are based on the same numbers.

So let me give you some context for this in the media.

And Marilyn Vos Savant,

the world smartest person who has a weekly column in Parade magazine and other outlets,

takes a lot of questions from readers.

Sometimes they're important, sometimes they're trivial.

But I do credit her with doing a really nice job of

handling difficult statistical questions. And something that came up.

There was a lot of confusion over the results from

the hormone replacement therapy trial and it centered around

this difference in values being reported for relative versus absolute decrease.

So this letter to C says "I'm a middle aged woman on hormone replacement therapy,

HRT, and the news about HRT is very confusing.

For example, I read that heart disease increased

by almost a third as a result of the medication.

And I also read the increase was slight.

Which is it?

Well, let's parse the results as Marilyn does this nicely in her response.

Here are the results taken at face value from

the trial so are about 17000 less than 70000 women who

participate in this trial and who are

randomized to either receive hormone replacement therapy or placebo.

So post-menopausal women who were randomized to one of these two groups.

And the incidents over the years average of 5.2 years of follow up,

the proportion of women who developed coronary heart disease in the group that got

hormone replacement therapy was 163 out of 8508 women or 0.0019, 1.9 percent.

The proportion, the placebo group was 122 out of 8102 or point 0.15, 1.5 percent.

So you can see that the observed proportion in

the hormone replacement therapy group was higher.

So if we were to actually compute this on both of our summary measures,

it can look very different.

The risk difference is 0.019 minus 0.015 or 0.004.

So we could say that hormone replacement therapy was

associated with 0.4 percent increase in

the absolute risk of heart disease over that five year follow up period.

But we do the relative risk,

0.019 divided by 0.015 is 1.27.

And that could be reported as a 27 percent increase risk of coronary heart disease.

For those women who were given hormone replacement therapy.

So which value do you think made more headlines?

No, I'm not suggesting that this is a trivial result per se,

but it's important to know that that 27 percent increase was

an increase on something that was

relatively low risk to begin with over the five year period.

And so, for women weighing the costs and benefits of

taking are enrolling in hormone replacement therapy,

it would be vital not only to know

that 27 percent increase but also to know that the baseline for

that comparison was 1.5 percent-

give some estimate of the burden of the outcome over that five year period.

Here's one more example that will actually recap within our review questions.

Just looking at how we can compare multiple samples instead of two, only two at a time.

We would actually compare multiple samples.

Here is a study from Health Affairs looking at

the out-of-pocket spending medical adherence

among dialysis patients in 12 countries so there's 12 samples of data.

Okay. And we'll get to this in

the review exercise and actually have you quantify some of these things.

But with more than two categories a common practice for making comparisons like

the risk difference and relative risk is to designate one of

the categories as the reference group and present categories,

all other categories compared to this reference.

And the choice of the reference group is arbitrary,

but in many cases has chosen to highlight the substance of emphasis.

So for example, for this article written in the US Published Journal,

the primary question of interest might be,

how the other 11 countries that were not United States compare to

the United States with secondary interest in how these countries compared to one another.

So we'll set this up in the review exercises and give you

some practice of computing comparisons across more than two groups.

So just to summarize. We've talked about a lot here but this is

a big deal and I'm going to emphasize this type

of thinking throughout the rest of the course when we're dealing

with binary and other types of outcomes,

where we can have different types of comparisons using the same numbers.

But risk difference, the difference in

the proportions of having an outcome between any two groups we're comparing,

P1 hat minus P2 hat is a generic representation.

And the relative risk, which is instead of taking the differences,

is the ratio of P1 divided by P2 hat are two different estimates of

the magnitude and direction of association for binary outcomes between two groups.

And these two estimates are based on the exact same inputs.

And we'll always agree in terms of

the direction the association but not necessarily the magnitudes as we've seen.

The risk difference helps to quantify the potential impact of a treatment or

exposure when applied to a group of individuals or removed.

The relative risk helps to quantify the potential impact

of a treatment or exposure for any one person,

sort of the clinical measure.

[inaudible] estimate alone is sufficient to tell the whole story`.

So just keep that in mind when you're hearing

the reports in the media where they talk about

relative increases or decreases but don't necessarily ground that in the baseline risk.

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