Indirect Effects and the Mediation Model

🧠Refresher🧠

In regression, we often ask: does \(X\) predict \(Y\)? But sometimes the more interesting question is how or why \(X\) affects \(Y\). That is where mediation comes in.

A mediator is a variable \(M\) that transmits the effect of \(X\) to \(Y\). Instead of \(X\) affecting \(Y\) directly, \(X\) first affects \(M\), and then \(M\) affects \(Y\).

A classic example:

  • \(X\) = exercise
  • \(M\) = stress reduction
  • \(Y\) = depression

Exercise may reduce depression partly because it reduces stress first. Stress reduction is the mediator.

The Paths in a Mediation Model

A mediation model involves three regression equations and three key paths:

Path Description
\(a\) Effect of \(X\) on \(M\)
\(b\) Effect of \(M\) on \(Y\), adjusting for \(X\)
\(c\) Total effect of \(X\) on \(Y\) (without examining a mediator)
\(c'\) Direct effect of \(X\) on \(Y\), adjusting for \(M\)

The three equations:

Equation 1 (predicting \(M\)):

\[ \hat{M} = int_1 + aX \]

Equation 2 (predicting \(Y\) with both \(X\) and \(M\)):

\[ \hat{Y} = int_2 + c'X + bM \]

Equation 3 (predicting \(Y\) with only \(X\)):

\[ \hat{Y} = int_3 + cX \]

Calculating the Indirect Effect

The indirect effect is the portion of the total effect that “travels through” the mediator.

It is computed as:

\[ \text{Indirect Effect} = a \times b \]

You multiply the \(a\) path (X → M) by the \(b\) path (M → Y). This is sometimes called the product of coefficients method.

Example:

  • \(a = 0.5\) (exercise reduces stress)
  • \(b = -0.4\) (stress increases depression)

\[ \text{Indirect Effect} = 0.5 \times (-0.4) = -0.20 \]

This means that exercise reduces depression by 0.20 units through the stress-reduction pathway.

Direct, Indirect, and Total Effects

🧠Refresher🧠

Once you have the indirect effect, you can decompose the total effect into its parts.

The Fundamental Decomposition

\[ c = c' + (a \times b) \]

In words:

\[ \text{Total Effect} = \text{Direct Effect} + \text{Indirect Effect} \]

  • Total effect \(c\): how much \(Y\) changes for a 1-unit increase in \(X\), ignoring \(M\)
  • Direct effect \(c'\): how much \(Y\) changes for a 1-unit increase in \(X\), while adjusting for \(M\)
  • Indirect effect \(a \times b\): how much of that total effect “travels through” \(M\)

Example:

  • Total effect \(c = -0.35\)
  • Indirect effect \(a \times b = -0.20\)
  • Direct effect \(c' = c - (a \times b) = -0.35 - (-0.20) = -0.15\)

You can verify: \(-0.15 + (-0.20) = -0.35\) ✓

Finding the Missing Piece

If you know any two of the three values, you can always find the third:

\[ c' = c - (a \times b) \]

\[ a \times b = c - c' \]

\[ c = c' + (a \times b) \]

📝Homework Problems📝

M.1.1

A researcher models whether social support (\(X\)) reduces anxiety (\(Y\)) through self-efficacy (\(M\)).

From the regression output:

  • \(a = 0.6\)
  • \(b = -0.5\)

What is the indirect effect of social support on anxiety through self-efficacy?

M.1.2

Using the same study:

  • Total effect \(c = -0.45\)
  • Indirect effect \(a \times b = -0.30\)

What is the direct effect \(c'\)?

M.1.3

A study finds:

  • Direct effect \(c' = 0.10\)
  • Indirect effect \(a \times b = 0.25\)

What is the total effect \(c\)?

M.1.4

In a mediation model, the total effect \(c\) equals:

A. \(a \times b\)
B. \(c' - (a \times b)\)
C. \(c' + (a \times b)\)
D. \(a + b + c'\)

M.1.5

A researcher reports:

  • \(a = 0.4\)
  • \(b = 0.3\)
  • \(c = 0.28\)

What is the direct effect \(c'\)?

Percentage Breakdown of Effects

🧠Refresher🧠

The raw values of the indirect and direct effects are useful, but researchers often want to know what proportion of the total effect each pathway accounts for. This gives a cleaner picture of how much of the X → Y relationship is “explained” by the mediator.

Proportion Mediated

The proportion mediated (also called the percentage mediated) is:

\[ \text{Proportion Mediated} = \frac{a \times b}{c} \]

Multiply by 100 to express as a percentage:

\[ \% \text{ Mediated} = \frac{a \times b}{c} \times 100 \]

Example:

  • Indirect effect \(a \times b = -0.20\)
  • Total effect \(c = -0.35\)

\[ \% \text{ Mediated} = \frac{-0.20}{-0.35} \times 100 = 57.1\% \]

About 57% of the total effect of \(X\) on \(Y\) is explained by the mediator.

Proportion Direct

The proportion direct is simply the remainder:

\[ \% \text{ Direct} = \frac{c'}{c} \times 100 \]

Or equivalently:

\[ \% \text{ Direct} = 100\% - \% \text{ Mediated} \]

These two percentages must always add up to 100%.

A Note on Interpretation

  • If the proportion mediated is close to 100%, we say there is complete (or full) mediation: \(X\) affects \(Y\) almost entirely through \(M\), and the direct path becomes negligible.
  • If the proportion mediated is small, we say there is partial mediation: the mediator explains some, but not most, of the total effect.
  • If the direct and indirect effects have opposite signs, the proportion mediated can exceed 100% or be negative. This is called inconsistent mediation and requires careful interpretation.

📝Homework Problems📝

M.2.1

A study finds:

  • Indirect effect \(a \times b = 0.18\)
  • Total effect \(c = 0.30\)

What percentage of the total effect is mediated?

M.2.2

Using the same study above, what percentage of the total effect is direct (i.e., not mediated)?

M.2.3

A mediation analysis reports:

  • \(a = 0.5\)
  • \(b = 0.4\)
  • \(c = 0.40\)

What is the proportion mediated? Round to one decimal place.

M.2.4

A researcher finds a total effect of \(c = 0.50\), and reports that 60% of the effect is mediated.

What is the indirect effect \(a \times b\)?

M.2.5

Which of the following is always true about the proportion mediated and proportion direct?

A. They are always both positive
B. They always sum to 100%
C. The proportion mediated is always larger
D. They are equal when the total effect is zero

M.2.6

In the model below, what does the path labeled \(c'\) represent?

\[ X \xrightarrow{a} M \xrightarrow{b} Y, \quad X \xrightarrow{c'} Y \]

A. The total effect of \(X\) on \(Y\)
B. The effect of \(X\) on \(M\)
C. The direct effect of \(X\) on \(Y\), adjusting for \(M\)
D. The indirect effect of \(X\) on \(Y\) through \(M\)