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 2A: Summarization and Measurement

Module 2A consists of two lecture sets that cover measurement and summarization of continuous data outcomes for both single samples, and the comparison of two or more samples. Please see the posted learning objectives for these two lecture sets for more detail.

- John McGready, PhD, MSAssociate Scientist, Biostatistics

Bloomberg School of Public Health

So measuring associations is paramount in

quantifying the relationship between exposures and outcomes.

In this section, we'll show how to quantify the association between

a continuous outcome across two groups and also display that visually.

In this section, we'll talk about some

ways to compare distributions of continuous data.

And we'll first focus on some visual tools that we developed in the last sections,

and we'll show that those are great for exploratory purposes and display

purposes, and then we'll sort of segue into looking at a numerical

comparison, something that allows us to encapsulate a lot of the information

about distributional differences into a single

quantity for any two groups we're comparing.

So upon completion of this lecture section you should be able to suggest

some graphical approaches to comparing distributions of

continuous data between two or more samples.

And explain why a difference in sample means can be used to

quantify In a single number summary,

differences in distributions of continuous data.

Frequently, in public health and medicine,

science, etc, researchers or practitioners are interested

in comparing two or more populations via

data collected on samples from these populations.

Such comparisons can be used to investigate questions such as.

How does the weight change differs between those who are on

a low fat diet compared to those on a low carbohydrate diet?

How do salaries differ between males and females?

How do cholesterol levels differ across weight groups?

Et cetera, et cetera.

While these comparisons can be done visually,

it's also useful to have a numerical summary.

Theoretically, this numerical summary could be many things.

It could be a difference in medians between

the two samples from the two populations we're comparing.

It could be a ratio of means, it could be a difference in the 95th

percentiles of the samples, the ratio of the

standard deviations, I could go on and on.

However,

what is commonly used for reasons that we'll

try and elaborate on in this lecture section.

And certainly get to very shortly in the course.

What is commonly used is the difference in sample means.

And we'll show here, when comparing sample

distributions, this can be a reasonable measure.

Of the overall differences in these distributions as an estimate

of the underlying mean differences in the populations we're comparing.

Okay, and now you'll see by shortly, I meant really

shortly because I couldn't wait to tell you about this.

The reason we use mean differences as opposed to other comparison metrics

so frequently in statistics is because there's a lot of statistical theory.

That will develop throughout the rest of the course to actually help us understand

the sampling behavior of means across multiple

random samples that we didn't actually observe.

And this helps us get a grip on the uncertainty in our

estimates as they relate to the populations that we're sampling from.

We don't have the same predictable type of metrics

for quantifying uncertainty for other quantities like medians and percentiles.

Ergo the most frequently-used measure of association is this mean difference.

Here I have some data that we'll be looking

at in more detail as we go along this course.

What I have is actually the weight, on a random

sample, of 239 12 month old children, from a group of Nepalese children.

And I'm interested in looking at their weight distribution in kilograms.

And, actually I'm interested in comparing it between boys and girls.

So one way to start by doing this from a visual perspective might

be to actually show boxplots of the weight distributions between the boys and girls.

And

I think side by side box plots are

a really wonderful way to actually compare visually.

The distribution of the continuous outcome between multiple groups.

So what do you see in these pictures here.

Well, we all may see different things but I think it's pretty clear

that the median value of weight for the boys is higher than girls.

And the whole distribution is shifted up.

Relative to those in the girls, so we

might say visually speaking, boys tend to weigh more than girls at 12 months old.

That doesn't mean, obviously, you can see there's a lot of cross over.

Not every boy weighs more than every girl, but on the whole, boys tend to weigh more.

We could also look at stacked histograms to do

this, just another way in presenting the distributions, but I

think the side by side box plots are more

sustained, personally, but you don't have to agree with me.

And what I've shown here is the mean, the

sample mean of the boy's weight and the girl's weight.

So for the males, the average weight.

Was 7.4 kilograms as compared to the girl's, which was 6.7.

So you can see numerically now

a number that shows boys tend to weigh more on average than girls.

And let me plot those means on these histograms so you can

see if you look at this histogram's stacked on top of each other.

You can sort of see that the distribution.

the weights in each of the two groups is relatively similar, roughly symmetric.

Not completely filled out, because these are samples.

But you can see that the distribution for boys, like we saw with the

box plot, sort of shifts over or shifts up a little bit, and so

that mean for the boys shifts over relative to the females.

Yet a common numerical comparison is the difference

in means, and we could express this weight

difference by taking the mean difference in weights

for the males and females in the sample.

And if we compare the males to the

females, the weight difference is 0.7 kilograms, on average.

So we could say on average male children, weigh more by 0.7

kilograms than female children, or male children have larger

weights of 0.7 kilograms on average, compared to female children.

There's a couple different ways of verbally expressing it, but this

compares the average between the two groups in a single number summary.

Direction is arbitrary.

We could've compared females to males, but it's important to know.

So for example we could've compared the females to males in

our mean difference instead that we'd get 6.7 minus 7.4 kilograms.

The absolute value would be exactly the same, but the

sign would be different, so it's negative 0.7 kilograms here.

Meaning that the females are smaller on average in

terms of weight, so on average females weigh less

than males by about .7 kilograms.

Or, or on average, this is akin to saying what we said on the previous slide, on

average males weigh more 5.7 kilograms, the females

weigh less by 0.7, males weigh more by that.

So as long as know the direction of the comparison, we can express it in any way.

Now, why does this single number summary, why

does this help us characterize the difference in distribution?

Well, I'll go back to the stacked histograms again, and sort of

emphasize what I said before, is if the distributions of the measures

we're looking at are similar in the groups we're comparing, but shifted

up or down, then this difference in means really captures that shift.

And it's a nice way, a single number summary to do that.

Let's look at the length of stay data by age of first claim.

For the data that we previously looked at in some of

the prior lectures, and we actually have the length of stay here.

What I've done is broken it out by two groups to start,

those whose first claim in the health system, this is the Heritage

Health data, was greater than 40 years, whereas those whose first claim

was when they were less than or equal to 40 years old.

And I've got these stacked histograms here.

And remember, these length of stay data will vary right.

So it, it may not be that apparent

visually what's going on by comparing these stacked histograms.

But we can summarize these values by saying the average.

And I'll give you the average length of stay for the older

group, those greater than 40 years old when they had their first claim.

Was 4.9 days.

As compared to the group who was younger when they had their first claim.

Whose average

length of stay was 2.7 days.

So the mean difference if we compare those who were older to the first claim to those

that were younger, is that 4.9 days minus the

2.7 day, so the average diference of 2.2 days.

And what this does is we've got these two

heavily right tailed, right skewed distributions length of stay data.

Similar shapes, but this really shows us numerically that the group that was older.

When they had their first claim, their average length

of stay has shifted over and is larger than that

of the group that was smaller, and that's a

one number way of summarizing that shift in the distributions.

So just to reiterate, when comparing distributions

of similar shapes, this mean difference really does

encapsulate the shift in the entire distribution and

its set of values, either up or down.

And even when we're actually end up comparing distributions

that aren't so similar in shape, at least the mean

difference gives some quantification of where the measure, or

one measure of central tendency differs between the two samples.

Let's look at another example from the literature here.

This is an interesting study.

That came out in the American Journal of

Public Health on menu labeling and calorie intake.

And what the author sought to do, and I'll just read you from

the abstract here, is they wanted to

assess the impact of restaurant menu calorie

labels on food choices and intake, and this has led to this kind of

work has been done, and there's also been public policy in certain places made,

to display these things.

So they wanted to assess what was going on.

And what they did is they, they had a study dinner, where they invited

a bunch of people to come to one sitting, and they randomly assigned people

upon their presentation of the meal, either

a menu without calorie labels, a menu

with calorie labels, or a menu with

calorie labels and a label stating recommended

daily caloric intake for an average adult.

They call that the calorie labels plus information group.

And what they were actually looking at, so they

randomly assigned these, roughly 100 people per group, and they

wanted to see if there was any difference in

the average consumption or the consumption distributions between these groups.

So we have three groups we're trying to compare now.

So I'll just give you a little piece of the result section here.

What they say when calories consumed

during and after the study dinner were combined,

participants in the calorie labels plus information group.

Consumed an average of 250 fewer calories than those in the other group.

So here's an example of a mean

difference in action, representing the results in that.

Here's actually a graphic from the article itself, just comparing the

average calories ordered Then the average calories consumed and the average

calories consumed during and after the meal

between the three groups and so this, the

darkest bar here represents those who we're,

got, did not receive any information about calories.

the second darkest, the gray, represents those who got the labels

but not the extra information, and then the unshaded bars represent.

Those who got the calorie labels plus the information.

So if we were to actually

quantify this, in terms of mean differences, let's focus here on

this last section, the total calories consumed during and after the meal.

the way we may present this in terms of the mean

differences, iswe might designate one of the three groups to be

our basis for comparison, and then compare the other groups to

that same group, so that those differences themselves were comparable as well.

So for example, we might

actually designate the no power labels group, the group

that received no, if you will, intervention at all.

As our reference, and then actually compute mean differences in

consumption between the other two groups in that same reference.

So for example, we might our first mean difference might be between group,

I'll call it group the calorie labels minus, the no calorie labels group.

We might take that mean of 1,625 and subtract the

mean calorie consumption of 1,630, and we get a mean difference of of negative five,

which indicates that when all the dust settled here, the group that

got the calorie labels with nothing else, consumed on average five calories less.

During and after the meal, so we haven't

even begun to talk about the uncertainty in that

estimate the, qualitatively speaking, the difference of five

calories doesn't mean anything in terms of overall consumption.

If we did the same comparison but for the

group that received the calorie labels plus information and compared

them to the same reference group of those that recieved

nothing, no calorie labels, no information, the difference would be.

So this is my representation here,

x bar of the calorie label's information, the CLI, minus x bar of NCL.

I aplogize for the handwriting, but it does give it a human twist, right?

So we will look at this for that it was 1,380

calories consumed on average for those that got the full information ensemble.

Minus 1,630 on average for those who got nothing.

And that difference was negative

250 calories on average.

So we could say that those who got the full

on intervention with the information about the average calorie consumption.

for persons and calorie labeling, consumed on average, 250 calories less during the

whole meal and after the meal than those who got none of the information.

Here's another study that came out.

In 2012 comparing, and it was very focused on

comparing gender, or sex based differences in academic physician's salaries.

And so, they go in the abstract to say, the mean

salary of the academic physicians, this is US physicians in our cohord.

Was $167,669 on average, that's US dollars, in 2012.

For women, and $200,443 on average for men.

So we could actually look at this mean difference right here, right now.

If we actually compared,

what we call the unadjusted mean

difference between the male and female salaries.

It would be that 200,433 for men

minus the 167,669 for women.

What they really focused on in this discrepancy

is something we'll get to later in the course

is, is actually making the comparison more comparable

in terms other characteristics than the men and women.

One of the potential criticisms of reporting this

number would be that well.

It's not a pure comparison of sex based differences cause

the men and women could have different types of positions.

They could have been working for different years etc.

So they go on and actually and we'll get to adjustment in the

second part of the course but just to give you this mean difference take.

They say that male gender was associated with higher

salary, plus $13,399, that's the mean difference between males and

females even after adjustment.

And they go on to say they adjusted for a

bunch of potential differences that could have fueled that original difference.

So the mean difference decreased relative to what it was before.

They considered those other factors, but it was still, still substantially larger.

Almost 14,000 US dollars per year higher for men than compared.

They also go, went on in this article to look at other associations

with salary, that were kind of interesting to look at.

they looked at physicians across the United States.

And the, they looked at things like

the institution's record of funding from the National

Institutes of Health the current institution region in

the U-S, and a bunch of other things.

Pair of salaries.

I am just going to blow up a part of this Table

2 here, and sometime we are interested because of differences in cost

of living, et cetera, on regional differences.

And so, what we're going to look at here,

are, they actually reported the average salary by region.

Breaking the United States into four different regional groups.

No other differences were considered in this

analysis, so the average salary of all

physicians pulled together who worked in institutions

in the western United States was $194,474.

So we could actually report in something that would be done

to actually do regional comparisons of salaries might be something like this.

We would report, we would choose one of the regional groups to be our reference.

The group that we compared all other regions to.

And then compute the differences between those

other regions, and that same reference point.

So I'm going to just arbitrarily.

Choose the western region as our reference group.

And again, the average salary in the western region given by that table that we

just looked at. It was $194,474.

So if we wanted to actually compute mean differences, if we did

Midwest, and I'll leave it to you, to check the math on this.

If we did Midwest minus West, we took the average

salary for the Midwestern group and subtracted that for the

West, it would turn out to be

$4,416, so those physicians, academic physicians in the

Midwest, made an average of over $4,000 more per year than those in the west.

We compared the south to the west, using that as the same reference.

We take that difference,

it's actually, a

measly if you will, $35, $35 less on

average for Southern academic physicians compared to Western.

And then if we actually did Northeast to West, we'd

get, and again I'll let you check the math, a difference, an average difference

of negative $2,322.

So you can see

that the Midwest, not considering, you

know, the other characteristics for academic

physicians in the Midwest do the best, in terms of average salaries.

And because all of these mean differences are filled through to

the same reference comparison, they're comparable to one another, as well.

We know since the Southern states make less on average than the western states.

But by a smaller magnitude than Midwestern states compared to the Western.

We know that salaries are higher in

Southern states compared with the Midwest, for example.

because they're all filtered through the same comparison region.

So, let me just summarize what we've done.

While the distributions of continuous data can

be compared between samples in many ways.

Some key approaches include visual comparisons for descriptive

purposes, such as side by side box plots.

That's a great way to kind of give the whole picture of what's going on.

But then when we want to get to numerical

comparisons, which will be critical for a couple reasons.

A, because it gives us something to sink our teeth into numerically.

Secondly, because we'll ultimately, in many situations,

want to recompute these adjusted for other

differences and the groups we're comparing, and

finally because we're going to want to bring in

the role of uncertainty to the estimated

differences and put limits on it, describing

what could actually be going on in

the population from which the samples were taken.

The mean difference between any two groups of samples really helps

us characterize how the distributions shift up or down

relative to one another in a single numerical statement.

So in the next section, you'll get an opportunity to sort of

review and practice the ideas we've developed in lecture sections A through D.

On continuous data measures and summarization.

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