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 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 ratios can be a little bit tricky to interpret because of a numerical

quirk about them.

As we noted in these previous lectures, when comparing two groups,

either by an odds ratio or relative risk, if the first group has lower

risk than the second group, then the ratios will between zero and one.

If the opposite is true, if the first group is higher risk

than the second group, then these ratios will be between one and positive infinity.

Notice that the range of potential values for

associations in which the first group has lower risks

versus associations in which the first group has higher risk, are not equal.

And this could lead to some misunderstanding when in

terms of comparing effect sizes in such.

So we're going to discuss that and a remedy for that in this lecture section.

Welcome back!

In this section I just want to talk a little bit about some properties of

ratios and we're just going to introduce it here.

And it's a theme that will come up in several other junctures in the course,

so I'll reinforce it at various other points in the course.

Upon completion of this lecture section you will be able to,

understand that the scaling of ratios is not symmetric around the value of one.

Which would indicate equal values in the numerator and

denominator, and we'll go into detail on that.

You'll be able to consider the implications of this point about scaling

when interpreting the size of associations,

comparing them across different factors.

And then you'll be able to understand that on a log scale and

we're going to use the natural log, sometimes written as ln,

the values of log ratios are symmetric about the value 0.

That doesn't only apply to natural logs.

That applies to logs of any base but we will be using the natural log.

So let's go back to the maternal/infant HIV transmission example that we've used

several times just to illustrate this point.

And here are the results.

And, again, in this two by two table format you'll recall that we had 363 HIV

pregnant women randomized to receive AZT or placebo during their pregnancy.

And here's at 18 months after birth,

here's the breakdown of the numbers of children who're HIV

positive versus not born until the mothers in each of the groups.

And you'd recall, we've looked at measures of association before such as the risk

difference and when we've compared it AZT to placebo with the 7% proportion

who contracted HIV within 18 months in the AZT group minus the 22% in the placebo.

The risk difference is negative 0.15 or negative 15%.

The relative risk of developing HIV for those in the AZT group or

to mothers in the AZT group compared to the placebo group was 0.32.

And the odds ratio was 0.27, and we went through these in detail previously.

And here are the interpretations we discussed, we have the risk difference

can be interpreted as 15% absolute decrease in HIV transmission risk for

mothers to infants when the mothers take AZT during pregnancy.

That relative risk of 0.32 could be characterized as any individual woman,

reduces her risk of transmitting to her child by about 68%.

And the odds ratio of 0.27 showed a 73% relative reduction in HIV transmission

odds, and again, this is for AZT group, compared to those in the placebo group.

However, [INAUDIBLE] directions [INAUDIBLE] comparison is arbitrary,

admittedly because we're interested in the efficacy of AZT,

on the outcome it makes a lot of sense to present the results as the AZT or

treated compared to the untreated.

But, there's no reason we couldn't set it up as to where we compare

the placebo to AZT.

So, lets look at what happens when we do this.

So, all I'm doing here is reversing the direction of comparison.

So, for example, with the risk difference now,

I'm taking the proportion of children who contracted HIV.

Born to mothers who received the placebo minus the proportion who contracted HIV

born to mothers who got AZT.

And you see it's just the opposite of what we had before.

Instead of taking the direction we had before, we're reversing it,

we're taking the 22% minus the 7%.

And so our risk difference is the same absolute value of 0.15 or

15% but now it's positive.

Because the direction of what we're comparing is to the higher

probability group to the lower.

That's probably pretty intuitive but lets look at what happens when we do ratios.

We do the relative risk in this direction.

The relative risk of contracting

HIV for infants born to mothers with placebo compared to AZT.

We do the numbers here.

The relative risk turns out to be 3.1.

And if we do the odds ratio, and then actually is

the reciprocal of the previous odds ratio, it's 3.7.

You can verify the missing step there on your own.

So how would we, if we were to actually reveal these results in terms of

the no treatment or placebo group compared to the AZT group,

how would this come out linguistically or semantically?

Well, a risk difference now of positive 0.15 or

15% suggest there's a 15% absolute increase or

greater risk of HIV transmission amongst those in the placebo as compared to AZT.

This relative risk of 3.1 indicates that there's a 210% increase.

In the HIV transmission risk for

mothers who are untreated compared to mothers who are treated with AZT.

And the odds ratio of 3.7 says that there's

a 270% remember if we took 3.7 to 1.

And we wanted the percent increase we take 3.7.

Minus the comparison of 1.

Divide by 1 is 2.7 or

a 270% increase in the HIV transmission odds for mothers

who were untreated compared to children born to mothers who were given AZT.

So just to recap what we've seen here, we've used the same two numbers

to create a whole world of other numbers that's the magic of binary outcome.

So, when we did things in the AZT to placebo direction.

Let's just compare that with what happens when we reversed the direction,

placebo to AZT.

So in terms of our risk difference, difference in proportions,

when we compare AZT to placebo, that was that -0.15 or -15%.

And when we did in the opposite direction it was positive 15%.

That makes some sense.

So if one group, the AZT group has, the difference is such that they have 15%

lower, that implies that the other group conversely has 15% higher than the AZT.

So these two are the same absolute value, just different directions.

The ratios look a little different because they're reciprocals, not differences.

Ratios themselves are quotients or divisions.

And to get from one to the other direction, take the reciprocal.

So when we did AZT to placebo, the relative risk was 0.32.

What we did it in the opposite direction, 1 over 0.32 is 3.1.

And the same thing could be said for

the odds ratio when we did it in the original direction, it was 0.27,

in the opposite direction it was 3.7.

So why, if you go back and look at the verbal interpretations of these things,

you'll see that the percent decrease when we do the relative risk or

odds ratios when we're comparing the AZT to placebo,

are not the same as the percent increase as when we go in the other direction.

And this is a property of ratios that's a little bit tricky.

Let's just work this through, so why do the ratio based associations seem

different in magnitude if the direction of comparison was reversed?

Well, this comes down to an interesting fact.

If you think about, in terms of the ratios we're making here, when

the association between the group we're comparing to the other group is negative.

In other words with the AZT, there was a reduced risk of transmission

in the numerator group, the AZT compared to placebo.

The range of possible values for negative associations lies between 0 and

1 On the ratio scale.

So if the first group has lower risk than the second group, the ratio

at its smallest could be 0 if there were no risk in that numerator group.

And at most could be slightly less than 1 because as soon as those two

proportions are equal, the ratio is going to be 1.

So the range of values on the ratio scale for

negative associations is between 0 and 1.

For positive associations if the numerator is greater than the denominator

in terms of the risk the range of possible values theoretically is from

just above 1 all the way up to positive infinity.

There's an inequality here, in terms of the range of possible values for

associations, depending on the direction.

So when we switch directions,

we've changed the potential range for our ratio values.

It turns out, if we take this and rescale it, and take this to the log scale,

if we take the log of values that are between 0 and 1,

that maps them to the values, the log of 1 in any base is 0.

And the log of zero in any base is essentially negative infinity.

So we stretch this range out from 1 to 0 on the original scale,

down to 0, down to the left half of the entire number line.

And when we do the same thing for the positive associations.

We mapped that from 1 to positive infinity, the log of 1 again is 0.

So we just set this over here.

And the range goes from 0 to positive infinity.

So by taking logs, we've actually equalized the range of possibilities for

a positive and negative association.

So let's look at how this shakes down in the example we just did with the AZT and

HIV transmission.

So, if we look at the relative risk we've talked about multiple times now, we know

that it's formed by taking the risk of proportion of infants who contracted HIV

born to mothers who were given AZT during pregnancy, divided by the proportion of

infants who contracted HIV among mothers who weren't treated.

That 0.07 over 0.22.

Now, lets just take the log of the ratio, the log of this relative risk.

Is equal to the log of this fraction we talked about before.

But here is a neat, little property of logarithms, mathematically speaking,

the log of a ratio can be equally expressed as the log of

the numerator minus the log of the denominator.

So, it'd be the log of

0.07 minus the log of 0.2.

So now let's go over to the other group.

Or other comparison when we compare the placebo to AZT

we just switched the numerator and denominator.

It turns out when we do the log of this ratio.

The log of the risk in placebo group, children born to mothers in placebo group

compared to children born in mothers in the AZT group which is just the log of

that numerator minus the log of that denominator.

All we've done here on the log scale is reverse the direction over difference.

So this will be the log of 0.22 minus the log of 0.07.

So notice that these differences on the log scale,

the log of these ratios are the same in magnitude.

But just in the opposite direction.

So I just wanted to plant a seed in your head on something that we're going to look

at multiple times that will get you to think, what are the implications of this?

This is the first of many times we will discuss this issue over terms one and

two of Statistical Reasoning.

But think about this, there's the potential implications, things that don't

look equal in terms of the effect size on the original scale, the ratio scale.

What happens when we took the log of those ratios?

Well we got the log of the ratios were equal in absolute value.

But just differed in sign when we reversed the direction of comparison.

So this actually has potential implications for

displaying ratio based associations in a manner that scales estimates.

So the magnitudes of associations can be appropriately compared.

And we'll see some examples through this down the line.

But for example if you actually look at the ratios head on and

we're comparing associations say with HIV outcomes to things like treatment and

then sex of the child.

Maybe we want to compare to see if there's any risk difference between sex

of the child, age groups of the mother etc,

well some of these may be positively associated.

Some like AZT may be negatively and if we look at them on the original ratio scales,

it may be hard to determine which of those associations is strongest, numerically.

Because of this imbalance of range of possible values for

positive associates versus negative.

So, when we put things on the long scale everything becomes equalized and

is comparable.

Moreover, this will have implications for how statistical imprints, which we'll get

to shortly, creating what are called confidence intervals and finding p values.

This will implications for how this is done with these ratios.

We're also going to see in the next unit that we'll be dealing with

ratios once again.

Of event rates and the same principles will hold.

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