0.07 minus 0.22 and divide it by 0.22 which is akin,

which would be that negative 0.15 over 0.22

that is equal to negative 0.68 that is 68 percent reduction.

Another way to think about it is just if we had a ratio that becomes

0.32 over one that is our actual relative risks.

So if we look into the percentage decrease that is relative to the denominator

one is 0.032 minus one over one is negative 0.68,

so 68 percent reduction in relative terms.

So what gives here we've got two different numbers and they're both

based on the same two p hat just different computations with them.

So how do we interpret these or what's the difference in interpretation?

Because both measures used

exactly the same information but it give seemingly different results.

With the risk difference we say there's a 15 percent reduction in HIV transmission,

and with the relative risk we say there's a 68 percent reduction.

It sounds a lot more when we talk about relative terms.

Notice that both agree in terms of the direction of the association,

both show that the risk in the first group,

the children whose mothers were given AZT,

is smaller than the rest in the second group,

the group of children whose mothers were given the PLACEBO.

So, what can we do here to rectify this differs?

Well, let's talk about ways we can interpret these and

think about what they mean similarly and differently.

So, one way to think about using this number

substantively is it can be interpreted as

the impact (assuming causation) on a fixed number of persons.

In other words, if we had a certain number of HIV positive pregnant women,

how many fewer transmissions

could we expect if we were to treat these women with AZT versus not treat them?

So, if we were working in a city where every year

we had about a 1,000 HIV pregnant women,

we'd expect to see

15 percent fewer mother/child transmission if the 1,000 women were given

AZT during pregnancy and that would result in

a 150 fewer transmissions out of the 1,000 women,

so that will be a substantial decrease numerically.

We worked in a large city or at the country or county level,

we had a group of 50,000 HIV positive pregnant woman.

We'd expect to see 7,500 fewer mother/child transmissions

if the 50,000 women were given AZT during pregnancy compared to if they were not treated.

Whereas the relative risk can be interpreted

as a way of communicating impact at the "individual level".

Both of these can be used at the individual level but talks about the reduction in

relative risk for an individual compared to not be treated versus being treated.

So, for example, the risk that an HIV positive mother who takes AZT during pregnancy

transmit HIV to her child is 0.32 times the risk if she did not take AZT.

So far we're casting a woman about the potential benefits of taking AZT during pregnancy.

Another way to put this is that you could reduce

your personal risk of transmitting HIV to your child by 60 percent.

In general, the risk that an HIV positive mother transmits HIV to her child is

68 percent lower if she takes AZT during

pregnancy as compared to if she did not take AZT.

Something to note about these two measures is they

will always agree in terms of the direction of association.

If in the comparison we're making, if p_1,

the first group has lower risk than the second group,

then the difference in the two will be less than

zero and the relative risk will be less than one,

still positive but less than one.

If the first group in the comparison has a greater risk than the second group,

then the risk difference will be greater than

zero and the relative risk will be greater than one.

If the two risks are equal in the two samples being compared,

then the risk difference will be equal

to zero and the relative risk will be exactly equal to one.

So they will always agree in their general conclusions.

However, the two quantities can appear different in terms of their magnitudes.

Something to think about from a scientific perspective is it's possible to

see a "large" effect with one measure and a "small" effect with the other.

So, for example, if the underlying risk in

the two risks being prepared are generally small,

let's say it's 0.001 in the first group and

0.004 in the second group or 0.1 and 0.4 percent respectively,

then p_1 minus p_2 is equal to negative 0.003 or negative 0.3 percent,

an absolute decrease of 0.3 percent.

Which doesn't look that large although if this were something

we can enact at a population level and give to a large number of people,

there still could be a sizeable reduction in the number of people who have the outcome.

However, if we were to compare this in

the relative scale and take the point 0.001 divided by the point 0.004,

that ratio is equal to 0.25.

That's a relative decrease of 75 percent.

So certainly on the relative scale this is quite an impact even if it doesn't seem to be

on the absolute risk difference scale.

So, you can see in some of the situations we've looked at the increase or

decrease as measured by the relative risk

looks more dramatic than that measured by the risk difference.

So you can imagine which one of these tends to

appear more often in news articles for example.

So you always want to think when you're reading the results of a study

whether it be in a journal article or a news publication,

you want to think about what are they presenting

here in terms of the measure of risk comparison.

So something to think about having only one of these numbers or the other,

will not give the full story.

If we have the relative comparison we don't have a sense of the magnitude of

the individual proportions nor do we have that if we have the absolute difference.

So we need both together to get a sense of that or at least the two proportions to start.

Marilyn Vos Savant, who was a newspaper columnist,

who takes questions about all kinds of things and she answers them.

Sometimes they're riddles submitted by readers,

sometimes they're questions about the meaning of life,

sometimes they're real scientific questions and I actually really unimpressed with how

she handled the response to this and we'll talk about what she's getting at.

So this letter says,

"I'm a middle-aged woman on hormone replacement therapy."

So back in the early 2000s,

there was a large trial on

hormone replacement therapy done on post-menopausal women in the US.

The trials suspended or cancelled early because of what they saw,

an increased risk of heart disease in those who were given the replacement.

So what this woman is asking is about this.

So she 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 meditation.

Yet I also read that the increase was slight. Which is it?

What do the numbers really mean?"

Well, here are the results of the study from the actual publication that came out.

This looks at the incidence of

heart disease in the women who were given

hormone replacement therapy versus the women who were given a placebo,

and all of the women were free of coronary heart disease at the beginning of the trial.

So out of the 8,508 women who were

randomized to have received hormone replacement therapy,

163 developed coronary heart disease for a percentage of 0.019 or 1.9 percent.

Out of the 8,102 women who were given placebo,

122 developed coronary heart disease for a percentage of 0.015 or 1.5 percent.

So, here are the risk difference and the relative risk.

A risk difference is can we compare the proportion who

developed coronary heart disease in

the hormone replacement therapy group compared to the placebo?

It's 0.019 minus 0.015 or 0.04,

a difference of 0.4 percent.

A 0.4 percent absolute greater risk of

coronary heart disease among the women who got hormone replacement therapy.

However, the relative risk looks very different if we take

this ratio of 0.019 to 0.014 is 1.27.

On the relative scale,

this is a 27 percent increase in the risk and so this is what the woman who's letter was

getting read that she saw both these numbers

reported and was confused and Marylin does a very good job of breaking this down.

What I want to note is that actually this is

kind of not usual because a lot of times the number

that the media is conceived upon would be the one that looks sexy or more dramatic.

So it's important to know that

a 27 percent increase on the relative scales is substantial but you still want to

have some sense of what the absolute study values are to figure out what that means.

How do we compare more than two groups with these risk differences and relative risk?

Well, here is an example of a study.

A randomized intervention through a health insurance plan,

the hope was to increase colon cancer

screening on part of the participants and the designs

of four-group parallel design

where patients in the health were randomized to one of four groups.

They were either given the usual care or standard care that was used up to that point.

They were given electronic health record linked-messages,

automatic emails for example.

Automated messages plus telephone assistance or automated messages and

assisted telephone assistance plus nurse navigation to testing completion or refusal.

So they have four different groups where

the persons were randomized and what they did was they looked at

the proportion in each of those groups who got

screening within two years of the start of the study.

What they found were the proportions in the respective groups were varied

a lot and the group that got the usual care only a little

more than a quarter past 26.3 percent.

The group that take automated messages on top of that,

the percentage almost doubled to 50.8 percent.

Those who got the additional staff of assisted care,

telephone assistance increased to 57.5 percent.

Those who got everything posted as navigation,

the completion rate or string rate was 64.37 percent.

So, it's pretty impressive the gains that were made

especially this jump from usual care to automated.

So, for both the risk difference and relative risk comparisons,

if you wanted to give them the numerical comparison,

we can designate one of the four groups as a reference group and then

compute the comparisons for each of the other three groups compared to this reference.

So here are the risk differences and

the relative risks for each of

the other three groups each compared to the same reference group.

You can see this is a situation where the gains on both scales are very unable.

So in summary, the risk difference and

the relative risk are two different estimates of the magnitude and

the direction of association when comparing binary outcomes between groups.

These two estimates are based on the exact same inputs and will always agree in

terms of the direction of association but not necessarily the magnitude.

The risk difference helps to quantify

the potential impact of a treatment or exposure for a group of

individuals and the relative risk helps quantify

the potential impact or treatment for an individual as well.

Neither estimate alone is sufficient to tell the entire "story".

So it's important to have,

even if you have only one of these,

to have at least one of the proportions in the two groups being

compared otherwise you're only getting a piece of the story.