1:13

You may get two identical studies for

those drug trials submitted to regulatory agency for approval.

For example, for a drug to be licensed in the United States,

the drug company have to do two identical trials and submit the results to the FDA.

In that case, you may get two identical studies.

But for most part, what you're going to have, for

your systematic review and meta-analysis,

are different studies done by different investigators in different places.

We may or may not know for

sure whether these characteristics are actually related to the effect size.

For example, you may argue that vitamin D studies done in the United States

are going to be different from those studies done in Asia.

Because some countries may be more closer to equator,

such that they will have more sun exposure.

So their Vitamin D level, the baseline level in the participants, are different.

3:02

So this study look at the association between the duration of breastfeeding and

the risk of childhood obesity.

So the hypothesis is that the duration is associated with the childhood overweight.

And here is what the review group found.

Our 21 studies were all cohort studies, of which eight were in the US,

nine in Europe and four in Asia, Australia or the Middle East.

The studies analyzed breastfeeding duration ranges from as little as

0-16 weeks to as much as greater than 12 month.

And the sample sizes range from a little over 300 to over 117,000.

And the study dropout rates prior to follow up ranges is from 5% to 52%.

Again, since all of these studies are brought together in a systematic

review of meta-analysis, we are answering a clinical question.

Which is, is there any association between the duration of breastfeeding and

the risk of childhood obesity?

But each study is different from another study.

And, we are concerned about whether, for example, the duration for

breastfeeding is associated with the association you are going to see, okay?

8:58

Here is another way to look at it.

Here, the Y3 can be written as the mu plus zeta 3 plus epsilon 3.

Now, the distance between the overall mean, and

the observed effect, in any given study, consists of two distinct parts.

The true variation in effect sizes, which is the zeta i's.

And, the sampling arrow epsilon i's within the study.

More generally, the observed effect Yi for

any given study can be written as a grand mean plus the deviation

of the study's true effect from the grand mean and the sampling error in that study.

So now, because we have assumed all the circles, again,

going back to the circles which are the true effect in each individual studies,

there's a distribution, okay?

So that there are two sources of variance.

The first source, if we just focus one study, there's within study variance.

So the distance from the theta i, the circle,

to the Yi, the square, depends on the within study variance.

So, the variance is the random errors within that study.

That's the within study variance and we have that.

We have exactly the same within study variance from our fixed effect model.

10:14

On top of that,

there's another level of variance which is the between-study variance, okay?

So the distance form the mu, the triangle, to each theta i, the circles, depends

on the variance of the distribution of the true effects across studies.

And we call that variance tau square.

Because all of the circles, again, the circles on this plot

are the true effects in each study because they don't line up together,

they don't coincide with each other, there's a distribution.

And that distribution is the between-study variance.

So under a random effects model, we have to capture both sources of variance,.

The within study variance, as well as the between-study variance.

So here's a nice contrast, of the fixed effect model and a random effects model.

The different assumptions you are making.

Again if there's only one thing you are going to take away from this section of

the lecture, is this slide.

The different assumptions underneath, two different models for meta-analysis.

On the left-hand side, all three figures.

You have seen all of them and they are the assumptions for the fixed effects model.

There's only one assumption and they're showing you three different ways.

Under the fixed effect model,

we assume all studies share a common true effect size.

That's why all the three circles, lying on top of each other, so

there's only one identical common effect size.

However, under the random effects model,

if you look at the figure on the upper right-hand corner, here's a distribution.

The three circles no longer align together.

There's a distribution of effect size, okay?

And because of that distribution, we added one more level of variability,

which is the between-study variance.

And that's why you have to capture them

in your analysis using the two slightly different equations.

Under the fixed effect model, Yi equals theta plus your error term within study.

However, under the random-effects model,

the Yi equals the mu of the grand mean plus the zi.

That captures the distance of each circle from that triangle,

plus the epsilon i, the error term.

Again, the difference is, under the fixed effect model,

there's one source of variance.

If you look at the last set of figures, right, there's only

one source of variance, which is captured by that normal curve for each study.

However, under the random effects model, we actually have one more layer,

which is the variance between studies.

That's why you have four normal curves instead of three,

which still have that within-study variance.

But however, underneath the last plot,

that little curve shows you the variability.

The distribution of the true effect size.

That's your between-study variance.

Now let's take a pause for a moment.

Well, what are we trying to do?

What are you going to observe from the study?

You have three studies, right?

And the data you observe are actually, what?

Always the same.

You will get an risk ratio estimate, odds ratio, plus some variance from that study.

So, you observe that the amount of information data you have in your

hand will stay the same, regardless of which model you are trying to use.

And the purpose of doing a meta-analysis.

You're trying to use your data you have in your hands and trying to guess where that

center of the distribution or where that common effect size is.

That's what you are trying to do in meta-analysis.

And, we are saying,

there are two different ways to get that number, get your meta-analytical results.

Either assume the studies are identical,

they're the same then, we are going to use a fixed-effect model.

If we cannot make that assumption, then we're going to use a random effects model

by assuming, well, the studies are slightly different from one another.

We're going to assume the true effect sizes are not the same, but

there's a distribution.

That's what you're doing.

You're taking the data you have, you collected from each individual study,

and trying to make a best guess where the common effect is.

And that guess depends on how different or how similar the studies are.

If they are identical, then go ahead, use the fixed effects model.

If you can not make that assumption,

then you're better off with the random effects model.

That leads us to the second session of the random effects model,

which we are going to show you how to do it.