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.