Moderation Effect

Large N & Leeuwenhoek (70700173)

Yue Hu

Overview

Moderation in Theory

Specialty
Wrong ways
Right way

Moderation in Practice

Theory
Data
Interpretation

1 Moderation in Theory

1.1 Moderation

Conditional effect: the contribution of X on the variance of Y varies when Z has different values

\[Y = \beta_0 + \beta_1X + \beta_2Z + \beta_3XZ + \epsilon.\]

nonlinear effect

One unit change of X does not lead to β unit change in Y.

H₀: If effects of X depends on Z (β₃ = 0).

H₀: If X has an effect when Z varies (β₁ + β₃Z = 0).

1.2 Two Versions

Dichotomous Z

Let (X|Z=0) = X⁰; (X|Z=1) = X¹, then

\[\begin{align} \hat Y =& \hat\beta_0 + \hat\beta_1X + \hat\beta_2Z + \hat\beta_3X\times Z;\\ \Leftrightarrow \hat{\tilde{Y}} =& \hat{\tilde\beta_0} + \hat{\tilde\beta_1}X^0 + \hat{\tilde\beta_2}Z + \hat{\tilde\beta_3}X^1. \end{align}\]

When Z = 0, \(H_0: \tilde\beta_1 = 0;\)
When Z = 1, \(H_0: \tilde\beta_3 = 0.\)

Continuous/Ordinal Z

Effect of X:

\[\frac{\partial Y}{\partial X} = \beta_1 + \beta_3Z.\]

Statistics:

\[\frac{\hat\beta_1 + \hat\beta_3Z}{SE(\hat\beta_1 + \hat\beta_3Z)}\sim t_{n - 4}\]

2 How Things Get Wrong?

2.1 Not Including the Interaction

☠️ “I don’t care about the interaction, and so just control the moderator.”

\[\begin{align} Y =& \beta_0 + \color{red}{\beta_1}X + \beta_2Z + \beta_3XZ + \epsilon,\\ =& \beta_0 + (\beta_1 + \beta_3Z) X + \beta_2Z + \epsilon.\\ \text{Z increases c, } Y =& \beta_0 + [\beta_1 + \beta_3(Z + c)] X + \beta_2(Z + c) + \epsilon,\\ =& (\beta_0 + \beta_2c) + \color{red}{(\beta_1 + \beta_3c + \beta_3Z)}X + \beta_2Z + \epsilon. \end{align}\]

Consequence

The coefficient of X changes by changing Z.

2.2 No β₁

☠️☠️ “I just care about the interaction.”

\[Y = \beta_0 + \beta_2Z + \beta_3X\times Z + \epsilon',\]

which means \(\epsilon' = \beta_1X + \epsilon\). Then

\[\begin{align} E(u'|X) \neq& 0,\\ E[u'(X,Z)] \neq& 0. \end{align}\]

…unless β₁X or XZ is zero → β₃ is biased and meaningless.

(What is this problem called?)

2.3 Only β₃

☠️☠️☠️ “The interaction is significant”

When testing the X’s effect, \(\frac{\partial Y}{\partial X} = \beta_1 + \beta_3Z.\)

The standard error: \(SE_{\frac{\partial Y}{\partial X}} = \sqrt{var(\hat{\beta_1}) + Z^2var(\hat{\beta_3}) + 2Zcov(\hat{\beta_1}, \hat{\beta_3})} > SE_{\beta_3}\)

Implication

It’s possible for the contribution of X on Y to be statistically significant for certain values of Z, even if all of the model parameters are insignificant.
- One cannot infer whether X has a meaningful conditional effect on Y simply from the magnitude and significance of either β₁ or β₃.
Point estimation may not be reliable, if link function (e.g., logit and probit) is involved
Substantive significance of the conditional effect highly relates to the distribution of the conditioning variable (viz., Z in the above example).

2.4 How to Interpret Right

“Sins”:

Not including XZ ☠️
Not including X or Z ☠️☠️
Interpreting based on the signifance of the interaction ☠️☠️☠️

Solution:
- Using the full model
- Plotting the marginal effects and their CIs.¹
- Presenting the frequency distribution of Z, esp., when the effect trend goes across the zero point.

2.5 Beyond Moderation Effect of Interest

As a “control”:

\[\begin{align} Pr(\text{Civil War}) \sim& \beta_0 + \beta_1Inequality \\ &+ \beta_2Regime + \beta_3 Inequality \times Regime \\ &+ \beta_4GDP + \beta_5Inequality \times GDP + \epsilon. \end{align}\](Beiser-McGrath & Beiser-McGrath 2020).

What happened if Inequality × GDP is not in the model?

3 Moderation in Practice

3.1 Context: Meritocracy and Inequality

Low-income → Meritocracy
↑
Income inequality

Self-reproducing

H₀: Poor people living with high inequality believe in meritocracy less.

Self-negating

H₁: Poor people living with high inequality believe in meritocracy more.

3.2 Examination

\[\begin{align} Reject\thinspace Meritocracy_{ij} =\boldsymbol{X\gamma} &+ \gamma_{10}Income_{ij} + \gamma_{01}Inequality_{j}\\ &+ \gamma_{11}Inequality_{j} \times Income_{ij} + \epsilon_{ij}. \end{align}\]

Findings

Newman, Johnston, & Lown(2015, AJPS)