Mediation Basics

ENT5587B - Research Design & Theory Testing II

Brian S. Anderson, Ph.D.

Assistant Professor

Department of Global Entrepreneurship & Innovation

andersonbri@umkc.edu

Â© 2017 Brian S. Anderson

- How goes it?
- IRB update
- Paper meetings
- The curse of Baron & Kenny
- Mediation with observed variables
- Bootstrap analysis
- Lab 9 March â€“ Mediation Assessment

Why Baron and Kenny (1986) did so much to help clarify mediation analysis, and how misuse of their method resulted in, well, some of the worst *science* in our fieldâ€¦

This is the classical graphical depiction for mediation, drawn from Baron and Kenny (1986). Lets walk through the logic.

There is ample discussion of just what a mediator is, but there are generally two schools of thought. Both of them are really talking about the same thing, but theory construction is subtly different.

The first perspective is that a mediator is an intervening mechanism connecting X to Y. By mechanism, I mean the theoretical device that explains *why* Y changes as a function of a change in X.

For example, a firmâ€™s entrepreneurial behaviors (Y) increases as a function of prior firm growth (X) because the firmâ€™s knowledge base (M) expands as a result of positive growth. A larger knowledge base is the impetus for future opportunity exploitation through the firmâ€™s entrepreneurial behaviors.

By the way, I donâ€™t care for this perspective, even though Iâ€™ve done it. A lot.

Two challenges I have with the mechanism perspectiveâ€¦

- The problem of the infinitesimal
- Theoretical specificity (or lack thereof)

The second perspective is the causal chain approach. X causes a change in M, which in turn causes a change in Y.

We can use our earlier example and talk about the growth story in the same way.

Firm growth (X) causes the firmâ€™s knowledge base (M) to expand. This expansion illuminates new opportunities for value creation using the firmâ€™s existing and newly acquired resources. The presence of new value creating opportunities is a key antecedent to enacting entrepreneurial behaviors (Y).

Ok, while I like this perspective better, we still have challenges. Lets go back to my logic and lay out just all of the steps necessary to specify this causal chainâ€¦

Oh, Iâ€™ve missed something *really* important in my logic. What is it?

Go ahead, model that well.

We also have a theoretical specificity issue with the causal chain perspectiveâ€”what would it be?

Yep, that pesky eliminating alternate causes problem.

As we will explore, modeling mediation is frighteningly simple. Modeling mediation well is, well, pretty darn hard.

I think causal mediation models offer tremendous opportunity for knowledge creation in our field. I also think that while difficult to model well, the process for causal mediation modeling is generally easier than that for moderation, because endogeneity is easier to address in a mediation model.

Yes, Iâ€™m serious.

Lets go back and revisit Baron and Kennyâ€¦

You can break down this picture into a series of equationsâ€¦

\(m = \alpha + {\beta}x + \epsilon\)

\(y = \alpha + {\beta}x + {\beta}m + \epsilon\)There is technically another equation in the B&K method, but weâ€™re going to ignore that because, well, they were wrong.

Why are they wrong you say? Take a look at this graphic againâ€¦

If the path (called the \(c'\) path) from X to Y remains statistically significant in the presence of M, why would this model **always** be misspecified?

Infinitesimals. Annoying little things.

So depending on who you ask, the *correct* way to graphically depict a mediation model is like thisâ€¦

Note though that we still use the same equations to specify our model, why? Hintâ€¦what would happen if we dropped \(x\) from the second equation?

\(m = \alpha + {\beta}x + \epsilon\)

\(y = \alpha + {\beta}x + {\beta}m + \epsilon\)So hereâ€™s the kickerâ€¦

Our theoryâ€”whether mechanism or causal chainâ€”specifies these equations as a *system*. The two equations do not exist independently of each other, because the value of \(m\) is a consequence in the first model, and **the same** observation is an antecedent to the next.

This means that to evaluate our model, we must use an estimator that allows for *simultaneous equations*.

Failing to take into consideration the joint dependence of the error term (because the two equations share some of the same terms) will yield biased coefficient estimates.

All the time.

This means that you can safely ignore mediation studies on observational dataâ€”and even some experimental designsâ€”that test for mediation using separately specified equations. Why?

Now, layer in the concern with alternate causes, omitted variables, measurement error, selection effects, simultaneity, and all of the other threats to causal inference, and itâ€™s easy to see that most mediation studies using observational data are, well, not so good.

So Iâ€™m not going to bother teaching you the B&K method, and Iâ€™m only going to briefly touch on the bootstrapping method that you read about today. While I like bootstrapping, I donâ€™t think it buys you much over a 2SLS estimator with robust standard errorsâ€¦

The reason being is that I agree with Antonakis et al (2010) that there is never a reason to specify a mediation model, particularly with observational data, that does not account for endogeneity.

Before we get to instruments though, lets start from the top.

Take another look at our modelâ€¦

The path between X â€“> M is usually termed \(a\). The M â€“ > Y path is termed \(b\).

In a mediation model, we are interested in the *indirect effect* of X on Y that passes through M. Whether M is the mechanism or the causal chain, the â€˜effectâ€™ of X on Y must transfer through M.

You canâ€™t get a change in Y as a function of a change in X without also having an intermediary change in M. Make sense?

We calculate the indirect effect by multiplying \(a\) x \(b\).

Whatâ€™s so important about multiplying these values together?

Remember, a mediation model suggests that M is *the* critical piece connecting X and Y together.

So an easy way to think about the importance of the \(ab\) path is to consider effect sizes. Take a small effect of \(a\), say \(\beta\) = .2. Now imagine another small effect of \(b\), again \(\beta\) = .2. Both of these paths are statistically significant.

But when you look at the indirect effect of X on Y through M, youâ€™re only talking about a total effect of .04 (.2 * .2). Thatâ€™s not very big, and may not be statistically significant.

How would we interpret this kind of result?

One way to think about it is that X â€“> M is important, and M â€“> Y is important, but X â€“> M â€“> Y isnâ€™t all that important.

M isnâ€™t the critical piece we thought it would be to connect X and Y together. M is still in the nomological network for X and Y, but M does not meaningfully connect those two (X and Y) nomological networks together.

A (brief) note about the Sobel test and the proportion of effect mediatedâ€¦

Back in the day, you would run two (or three in the way back in the day) different regressions, grab the coefficient estimates for \(a\) and \(b\), multiply them together, and construct whatâ€™s called the Sobel test to evaluate the strength and statistical significance of the indirect effect.

The problem is that it depends on \(ab\) being normally distributed, which it *almost never is* (remember our normality discussion from moderation?). So Sobel, while still found in our literatureâ€”unfortunatelyâ€”isnâ€™t a valid approach.

Also back in the day folks would calculate the ratio of the *ab* path to the â€˜totalâ€™ effect. The total effect equals to \(c' + ab\), with \(c'\) being the effect of X on Y controlling for the presence of M. You will sometimes see the total effect written as just \(c\).

The logic being that for a mediation effect to be meaningful, the indirect effect should account for proportionally more variance in Y than the effect of X on Y absent M.

But we donâ€™t think mediation in those terms anymore, because of the fallacy of partial mediation.

Quick quizâ€¦why do we not care **AT ALL** about \(c\) or \(c'\)?

Ok, thatâ€™s not entirely true. Observing a statistically significant \(c'\) path if we included it does tell us that we have an omitted variable problem, why?

So we canâ€™t use independent regression equations, and we canâ€™t use the Sobel method, whatâ€™s a researcher to do?

Well, the choice of modeling approach *may* vary depending on whether you have an observational design or an experimental design.

Today weâ€™re going to tackle the observational design, and next week the experimental design.

Training time out.

What are the three necessary and sufficient conditions to establish causality?

What is the condition that we **NEVER** have in an observational design?

So what must we do then when modeling observational data?

So we need an estimator that can estimate simultaneous equations, AND incorporate instrumental variables into the model.

There are a couple of options, including Three Stage Least Squares, which you will sometimes see in our literature.

My preferred option though is SEM. Yes, structural equation modeling.

There are a lot of SEM tools in R. My go-to is the `lavaan`

package, and itâ€™s the one that I base my SEM course on (just FYI).

Weâ€™re just going to be working with observed variables today, but the logic extends to latent variables as well.

Lets start by getting some dataâ€¦

```
library(tidyverse)
my.ds <- read_csv("http://a.web.umkc.edu/andersonbri/ENT5587.csv")
my.df <- as.data.frame(my.ds) %>%
filter(SGR > -50) %>%
na.omit()
```

The model that weâ€™re going to work with today posits that innovativeness (M) is a causal mechanism connecting the firmâ€™s long-term strategic orientation (X) to its strategic risk taking (Y).

In equation formâ€¦

\(Innovativeness = \alpha + {\beta}LongRangePlanning + \epsilon\)

\(RiskTaking = \alpha + {\beta}LongRangePlanning + {\beta}Innovativeness + \epsilon\)Lets start with a simple mediation model using `lavaan`

.

```
library(lavaan)
mediation.model <- 'Innovativeness ~ a * LRP # a path
RiskTaking ~ b * Innovativeness # b path
ab := a*b # Indirect effect'
mediation.fit <- sem(mediation.model, data = my.df)
summary(mediation.fit)
```

```
## lavaan (0.5-23.1097) converged normally after 13 iterations
##
## Number of observations 109
##
## Estimator ML
## Minimum Function Test Statistic 1.247
## Degrees of freedom 1
## P-value (Chi-square) 0.264
##
## Parameter Estimates:
##
## Information Expected
## Standard Errors Standard
##
## Regressions:
## Estimate Std.Err z-value P(>|z|)
## Innovativeness ~
## LRP (a) 0.316 0.073 4.318 0.000
## RiskTaking ~
## Innovtvnss (b) 0.460 0.065 7.063 0.000
##
## Variances:
## Estimate Std.Err z-value P(>|z|)
## .Innovativeness 1.805 0.245 7.382 0.000
## .RiskTaking 0.978 0.132 7.382 0.000
##
## Defined Parameters:
## Estimate Std.Err z-value P(>|z|)
## ab 0.145 0.039 3.684 0.000
```

The estimate of the standard error for the indirect effect (\(ab\)) comes from the Delta method.

But, weâ€™re not done.

Lets talk about bootstrapping.

This approach has become all the rage, and there is nothing wrong with it. In fact, itâ€™s a good thing, and I recommend integrating it with your mediation analysis.

Butâ€¦the non-parametric approach that you see often has largely been replaced by a Monte Carlo approach.

Yes, this is from the same Kris Preacher from the Preacher & Hayes (2004) paper that you read about today. While non-parametric methods are also quite good, the Monte Carlo method is generally more computationally efficient, and is easier to use.

Weâ€™re going to make use of the `semTools`

package for this, which is an add-on to `lavaan`

.

```
library(semTools)
indirect.effect <- 'a*b'
monteCarloMed(indirect.effect, object = mediation.fit, rep = 10000, CI = 95)
```

```
## $`Point Estimate`
## [1] 0.1454109
##
## $`95% Confidence Interval`
##
## LL 0.0744
## UL 0.2280
```

We interpret this result the same way as any other confidence interval, with the critical concern being whether the interval contains zero. If it did, how would we interpret this finding?

Hereâ€™s the thing though, particularly with observational designs. Because you canâ€™t rule out omitted variable bias from either the \(a\) path or the \(b\) path, whatâ€™s the point of bootstrapping the standard errors for a biased indirect effect?

No matter how many iterations of the bootstrap you run, any inference drawn on the indirect effect will be incorrect. Still, assuming that you *have* an endogeneity correction, bootstrapping is a good idea, although itâ€™s up in the air whether bootstrapping is superior to just robust standard errors.

So, we need to talk about endogeneity, but thatâ€™s for next week!

Wrap-up.

Lab 9 March â€“ Mediation assessment

Seminar 13 March â€“ Isolating causal estimates in mediation