So these types of questions are not new and scientists have

addressed these for many years using tools from econometrics,

that's simultaneous equations models and psychology,

the more general structural equation models.

Unlike the example above,

these models were typically used in conjunction with

observational studies and many more variables than the three above were considered.

However, it suffices to consider a simple example with

one mediator to illustrate the issues involved in using these types of models.

So in the following treatment, I'll follow Holland,

who first carefully studied the use of these models for

causal inference using the potential outcomes framework,

and to keep matters at their simplest,

we shall as in Holland assume treatment as randomized,

and we'll ignore covariates x.

In a linear structural equation model,

one would model the three variables; treatment assignment Z,

study-time m; m for mediator,

and test score Y;

Y for outcome, using a two equation model.

So the first equation is the equation for the mediator.

The second equation is the equation for the outcome Y.

You'll notice that it involves both z the treatment and m, the mediator.

We'll make the usual assumptions to identify the model about the errors.

So now, one of the things we can do is substitute the equation for m. You see

into the equation for Y and that yields what economists call the reduced form equation,

which I've written out below.

It's just simple algebra, arithmetic.

So from the reduced form equation,

and recalling that we're assuming treatment assignment is randomized,

we can see that the difference in the expectations

in the treatment and control group and it's randomized,

so we can see that those parameters,

that some parameters is the average treatment effect.

So, sometimes we call it the total effect in this context.

So the total effect consists of a direct effect,

which is gamma y of z on y plus the so-called indirect effect Beta y Beta m of z on y

through m. You will notice that I put direct effect in parentheses and then the other,

the indirect effects so-called,

we'll look at other ways to operationalize that nomenclature later.

So similarly, due to randomization.

The Beta coefficient, Beta m is equal

to the average effect of the treatment on the mediator.

So, the indirect effect may now be written as the product of the direct effect of

M on y with the average effect of z on m. Now,

it is useful to look more carefully at these effects.

The direct effect which is gamma y,

so just a couple of lines of algebra, no big deal,

so the first line is just what it is.

The next line we try to put in some potential outcomes.

Then in the third line,

we note that, well,

because after all, treatment is randomized,

we can take the z out.

We see a problem pretty quickly.

In general, the direct effect gamma y is not a causal parameter because it

compares y zero under the condition that when treatment is not assigned,

the outcome M, M zeros m,

as compared with Y one,

I had this when treatment is assigned under the condition that m

one also equals m but these are really two different conditions.

Because if you think about the expectation,

you're comparing two different subpopulations,

a subpopulation when M one is m with the sub-population when m zeros M,

and those are two distinct sub-populations in general.

They're not distinct, you can see below if encouragement doesn't affect study time.

But in general, if study time is affected,

then those are two different conditions, two different sub-populations.

So we need to compare under the same condition or same sub-populations.

So basically, that happens when either treatment doesn't affect the outcome.

That is the intermediate outcome here,

or you see also from the previous equation that if

the mediators under the mediator Mi zero is independent of y zero,

the mediator Mi one is independent Y one.

Then you see, you can take that out.

You just look back at the equation and we all have

the expected value of Y one given M one is the expected value of Y one.

Similarly with y zero.

In a similar way, you can do this for yourself,

the direct effect of M on y,

which is Beta y that's not a causal parameter either for the same reasons.

Okay, so what have we learned?

In short, in a randomized study or in an observational study,

where unconfounded and this holds conditional and covariates.

If you go ahead and you condition on an intermediate outcome,

in general, this yields estimands that no longer worn at causal interpretation.

Rosenbaum pointing this out in article in 1984.

It doesn't mean that such estimands are uninterpretable or not worthy of interest,

but in order for them to warrant the causal interpretation usually given,

additional conditions must hold.

Now, in the next lesson,

we're going to reformalize this and develop potential outcomes notation for the setup

above and give identification conditions under which

parameters such as those above actually do worn a causal interpretation.

Well, having given those conditions and investigator that may then decide

whether or not these conditions seem reasonable for his or her application,

and we'll also have some discussion of that ourselves.