Marginal effect interpretation

[Kethan Chowdary]

Hello Everyone

My dependent variable Y (shareholder return) is a continuous variable. My independent variable X (CEO duality) is a dummy variable (1 if the CEO was also board chair, 0 otherwise). My moderator is z (board independence) is a continuous variable. My model also includes 4 controls. My hypotheses are: (1) X is negatively associated with Y, and (2) Z negatively moderates the relationship between X and Y (i.e., Z weakens the negative relationship between X and Y)

I used GEE estimations.
xtgee y L.x L.z L.xz L.c1 L.c2 L.c3 L.c4,corr(exchangeable) link(iden)family(gauss)

Please see my results below. It is clear that X is negatively associated with Y in the absence of Z (coefficient = -.152). I relied on this post to interpret the sign of the interaction coefficient: https://www.statalist.org/forums/for...action-results

My first question is: The interaction coefficient is positive (.520), thus Z makes the impact of X larger, less negative (-2 is less negative than -1 for illustration). Is this correct? Does this result mean Z weakens the negative impact of X on Y?

To illustrate the interaction term, I plotted the marginal effect of CEO duality (see figure below), following Brambor etal (2005) "Understanding interaction models: Improving empirical analyses." Political analysis 14: 63-82. According to them, the solid sloping line in this figure indicates how the marginal effect of X changes with different values of Z.

My second question is: what exactly this figure tells me? Does the interaction term is positive or negative? Does Z increase or decrease the “negative impact of X”?

Similar code to plot marginal effect can be found here: https://www.statalist.org/forums/for...ual-inspection

Thank you very much for your help.


------------------------------------------------------------------------------
y | Coef. Std. Err. z P>|z| [95% Conf. Interval]
-------------+----------------------------------------------------------------
x |
L1. | -.1527303 .0616545 -2.48 0.013 -.273571 -.0318896
|
z |
L1. | -.2229808 .2097017 -1.06 0.288 -.6339885 .1880269
|
xz |
L1. | .5201228 .2969235 1.75 0.080 -.0618366 1.102082

[Clyde Schechter]

My first question is: The interaction coefficient is positive (.520), thus Z makes the impact of X larger, less negative (-2 is less negative than -1 for illustration). Is this correct? Does this result mean Z weakens the negative impact of X on Y?

-2 is more negative than -1. But that's not your situation here anyway. The XZ coefficient here is 0.52, so when Z = 1, for example, the marginal effect of X is -.1527303 + .5201228 which is a positive number. So it is not only less negative than when Z = 0, it is not negative at all: it is positive, and actually a larger positive than the negativity of the effect of X when Z = 0. If Z is positive, but smaller than 1, then the marginal effect of X could be negative, but less negative than the marginal effect of X when Z = 0. If Z < 0 is a possibility, then in those circumstances the marginal effect of X would be more negative than when Z = 0.

I think you do not fully understand your model. Even the phrasing of your research question,

(1) X is negatively associated with Y

is inconsistent with your model. In an interaction model, you cannot make a braod statement about how X is negatively associated with Y, because the association between X and Y is different for different values of Z. There is no one effect of X on Z. And in any interaction model, the effect of X on Y will be sometimes negative and sometimes positive, depending on the value of Z we're talking about. (Although as a practical matter, in some situations the actually observed values of Z all lie in a range where the effect of X on Y is only positive, or only negative. But in principle, in this kind of model, the effect of X on Y can be anything, and there is always a value of Z that will get you there.)

Now, you have correctly interpreted the coefficient of X in your model to be the effect of X when Z = 0 (which I'm assuming is what you mean by "the absence of Z."

The graph you show has Z on the horizontal axis, and the vertical axis shows the marginal effect of X on Y at each value of Z between Z = -.4 and Z = +.6. This type of graph is often called an interaction plot. The main graph is a straight line sloping up to the left gives the relationship between the marginal effect of X on Y and the value of Z. The curve above and below the line show the upper and lower confidence bounds, or perhaps +/- 1 standard error, depending on the code you used.

What this graph shows is that the effect of X on Y is negative whenever Z < about 0.3, and positive when Z >= that value. Moreover, the larger Z is, the greater (in the sense of either less negative or more positive depending on where we are) the effect of X on Y.

Note that if you are using a recent version of Stata (version 12 or later for sure, perhaps even 11, I don't recall) you could have done this more easily:

xtgee y L.(i.x##c.z c1 c2 c3 c4), corr(exchangeable) link(iden)family(gauss)
margins, dydx(L.x) at(L.z = (-0.4(0.1)0.6))
marginsplot

(In fact, simpler still you could have omitted the -link()- and -family()- options, as the ones you specified are the defaults for -xtgee-.)

Recommended reading:
1. -help fvvarlist- and the linked manual section.
2. the excellent Richard Williams' https://www3.nd.edu/~rwilliam/stats/Margins01.pdf for a very clear introduction to the -margins- command.

[Kethan Chowdary]

Thank you so much Clyde. Your code made it very very clear to me.

Your code shows that the marginal impact of X on Y is significant and negative only if Z<=.1. Also, the marginal impact becomes larger (more positive or less negative) when Z increases (Z is a percentage, 0 is a possibility and its range in my sample is -.41, +.62).

My supervisor needs me to only consider the significant results. Statistically speaking, in my situations, is there any difference between these two hypotheses? Or both are correct and can be tested in my situations? :

H1: The negative impact of X becomes weaker when Z increases.

H2: The negative impact of X becomes stronger when Z is low.

Thank you very much for your help.

[Clyde Schechter]

I would say that hypotheses H1 and H2 are equivalent, just different ways of saying the same thing.

When I write up my results, I avoid words like "weaker" and "stronger" because they can be confusing. For example, if the impact of X (i.e. marginal effect) goes from -.5 to + .7, it has changed sign, but also increased its magnitude. Is it now "stronger" because of the increased magnitude? Or is it "weaker" because it is no longer negative at all? So I tend to write these using strictly mathematical terminology. "When Z increases, the marginal effect of X on Y increases, changing from -whatever to + whatever_else as Z goes from first_Z_value to second_z_value." Yes, it's a bit long-winded, but it is completely unambiguous. If the listener/reader wishes to interpret this terminology in terms of his or her own concept of what "stronger" or "weaker" means, that is up to him or her. Verbosity in the defense of clarity is no vice!

My supervisor needs me to only consider the significant results.

This is most unfortunate, because it completely distorts the results of the modeling. In an interaction model with a continuous effect modifier, there is always some value of Z that corresponds to a marginal effect of X on Y equal to zero. And in some interval around that value of Z, the marginal effect of X on Y will be close enough to zero that it is not statistically significant. But, as you saw from the output of -marginsplot-, the relationship betweeen Z and the marginal effect of X is a continuous one. It is simply nonsense to focus on the range of Z values for which the marginal effect of X is large enough to be "statistically significant" and act as if X "has an effect" in those two regions (one positive, the other negative) but "has no effect" in between. [The exception to this argument occurs if the value of Z which corresponds to a zero marginal effect of X on Y lies sufficiently far outside the range of actually observed values of Z so that within the range of values of Z that actually occur, the marginal effect of X on Y is always positive and statistically significant, or always negative and statistically significant. I won't even say that this is a rare occurrence; it happens fairly often in real life. But it appears not to be your situation.]

[Kethan Chowdary]

Thank you very much for your clarification.
Very much appreciated Clyde