Front-door Criterion: Explaining Causal Paths Through Mediators • CausalInvestData

🚪 Front-door Criterion in Causal Inference

The front-door criterion allows causal identification even when there is an unmeasured confounder between the treatment and outcome — provided we can observe a mediator that fully transmits the causal path.

1. 🎯 Simulate Front-door Data

set.seed(42)
n <- 1000

U <- rnorm(n)                # Unobserved confounder
X <- 0.5 * U + rnorm(n)      # Treatment influenced by U
M <- 1.5 * X + rnorm(n)      # Mediator influenced by X
Y <- 2 * M + 0.5 * U + rnorm(n)  # Outcome influenced by M and U

df <- data.frame(X, M, Y)
head(df)

##            X            M          Y
## 1  3.0105377  4.766384643  9.5325868
## 2  0.2417731  0.084735594 -0.9055924
## 3  1.1522976  0.003710699 -0.2180185
## 4  0.6934047 -0.966597894 -2.7654351
## 5 -0.7937992 -2.482507183 -3.6471197
## 6 -0.6505452 -0.609979528 -2.1524781

2. ⚠️ Naive Regression is Biased

naive_model <- lm(Y ~ X, data = df)
summary(naive_model)

## 
## Call:
## lm(formula = Y ~ X, data = df)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -7.8508 -1.5715  0.0027  1.4748  7.9338 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -0.03721    0.07284  -0.511     0.61    
## X            3.21389    0.06561  48.987   <2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 2.303 on 998 degrees of freedom
## Multiple R-squared:  0.7063, Adjusted R-squared:  0.706 
## F-statistic:  2400 on 1 and 998 DF,  p-value: < 2.2e-16

3. 🧠 Estimate Causal Effect Using Front-door Formula

model_M_X <- lm(M ~ X, data = df)
model_Y_M <- lm(Y ~ M, data = df)

beta_M_X <- coef(model_M_X)["X"]
beta_Y_M <- coef(model_Y_M)["M"]

frontdoor_effect <- beta_M_X * beta_Y_M
cat("Estimated Front-door Effect:", round(frontdoor_effect, 2))

## Estimated Front-door Effect: 3.13

4. 📈 Visualize Causal Chain

ggplot(df, aes(x = X, y = M))

  geom_point(alpha = 0.4)

## geom_point: na.rm = FALSE
## stat_identity: na.rm = FALSE
## position_identity

  geom_smooth(method = "lm", se = FALSE)

## geom_smooth: na.rm = FALSE, orientation = NA, se = FALSE
## stat_smooth: na.rm = FALSE, orientation = NA, se = FALSE, method = lm
## position_identity

  labs(title = "Path A: Treatment to Mediator", x = "Treatment (X)", y = "Mediator (M)")

## $x
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## 
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## [1] "Mediator (M)"
## 
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## [1] "Path A: Treatment to Mediator"
## 
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ggplot(df, aes(x = M, y = Y))

  geom_point(alpha = 0.4)

## geom_point: na.rm = FALSE
## stat_identity: na.rm = FALSE
## position_identity

  geom_smooth(method = "lm", se = FALSE)

## geom_smooth: na.rm = FALSE, orientation = NA, se = FALSE
## stat_smooth: na.rm = FALSE, orientation = NA, se = FALSE, method = lm
## position_identity

  labs(title = "Path B: Mediator to Outcome", x = "Mediator (M)", y = "Outcome (Y)")

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##  $ axis.ticks.y.left               : NULL
##  $ axis.ticks.y.right              : NULL
##  $ axis.ticks.theta                : NULL
##  $ axis.ticks.r                    : NULL
##  $ axis.minor.ticks.x.top          : NULL
##  $ axis.minor.ticks.x.bottom       : NULL
##  $ axis.minor.ticks.y.left         : NULL
##  $ axis.minor.ticks.y.right        : NULL
##  $ axis.minor.ticks.theta          : NULL
##  $ axis.minor.ticks.r              : NULL
##  $ axis.ticks.length               : 'simpleUnit' num 2.75points
##   ..- attr(*, "unit")= int 8
##  $ axis.ticks.length.x             : NULL
##  $ axis.ticks.length.x.top         : NULL
##  $ axis.ticks.length.x.bottom      : NULL
##  $ axis.ticks.length.y             : NULL
##  $ axis.ticks.length.y.left        : NULL
##  $ axis.ticks.length.y.right       : NULL
##  $ axis.ticks.length.theta         : NULL
##  $ axis.ticks.length.r             : NULL
##  $ axis.minor.ticks.length         : 'rel' num 0.75
##  $ axis.minor.ticks.length.x       : NULL
##  $ axis.minor.ticks.length.x.top   : NULL
##  $ axis.minor.ticks.length.x.bottom: NULL
##  $ axis.minor.ticks.length.y       : NULL
##  $ axis.minor.ticks.length.y.left  : NULL
##  $ axis.minor.ticks.length.y.right : NULL
##  $ axis.minor.ticks.length.theta   : NULL
##  $ axis.minor.ticks.length.r       : NULL
##  $ axis.line                       : list()
##   ..- attr(*, "class")= chr [1:2] "element_blank" "element"
##  $ axis.line.x                     : NULL
##  $ axis.line.x.top                 : NULL
##  $ axis.line.x.bottom              : NULL
##  $ axis.line.y                     : NULL
##  $ axis.line.y.left                : NULL
##  $ axis.line.y.right               : NULL
##  $ axis.line.theta                 : NULL
##  $ axis.line.r                     : NULL
##  $ legend.background               : list()
##   ..- attr(*, "class")= chr [1:2] "element_blank" "element"
##  $ legend.margin                   : 'margin' num [1:4] 5.5points 5.5points 5.5points 5.5points
##   ..- attr(*, "unit")= int 8
##  $ legend.spacing                  : 'simpleUnit' num 11points
##   ..- attr(*, "unit")= int 8
##  $ legend.spacing.x                : NULL
##  $ legend.spacing.y                : NULL
##  $ legend.key                      : list()
##   ..- attr(*, "class")= chr [1:2] "element_blank" "element"
##  $ legend.key.size                 : 'simpleUnit' num 1.2lines
##   ..- attr(*, "unit")= int 3
##  $ legend.key.height               : NULL
##  $ legend.key.width                : NULL
##  $ legend.key.spacing              : 'simpleUnit' num 5.5points
##   ..- attr(*, "unit")= int 8
##  $ legend.key.spacing.x            : NULL
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##  $ legend.ticks.length             : 'rel' num 0.2
##  $ legend.axis.line                : NULL
##  $ legend.text                     :List of 11
##   ..$ family       : NULL
##   ..$ face         : NULL
##   ..$ colour       : NULL
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##   ..$ hjust        : NULL
##   ..$ vjust        : NULL
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##   ..$ inherit.blank: logi TRUE
##   ..- attr(*, "class")= chr [1:2] "element_text" "element"
##  $ legend.text.position            : NULL
##  $ legend.title                    :List of 11
##   ..$ family       : NULL
##   ..$ face         : NULL
##   ..$ colour       : NULL
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##   ..$ vjust        : NULL
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##   ..$ inherit.blank: logi TRUE
##   ..- attr(*, "class")= chr [1:2] "element_text" "element"
##  $ legend.title.position           : NULL
##  $ legend.position                 : chr "right"
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##  $ legend.justification            : chr "center"
##  $ legend.justification.top        : NULL
##  $ legend.justification.bottom     : NULL
##  $ legend.justification.left       : NULL
##  $ legend.justification.right      : NULL
##  $ legend.justification.inside     : NULL
##  $ legend.location                 : NULL
##  $ legend.box                      : NULL
##  $ legend.box.just                 : NULL
##  $ legend.box.margin               : 'margin' num [1:4] 0cm 0cm 0cm 0cm
##   ..- attr(*, "unit")= int 1
##  $ legend.box.background           : list()
##   ..- attr(*, "class")= chr [1:2] "element_blank" "element"
##  $ legend.box.spacing              : 'simpleUnit' num 11points
##   ..- attr(*, "unit")= int 8
##   [list output truncated]
##  - attr(*, "class")= chr [1:2] "theme" "gg"
##  - attr(*, "complete")= logi TRUE
##  - attr(*, "validate")= logi TRUE

✅ Assumptions for Front-door Identification

All directed paths from X to Y go through M
No unmeasured confounders of X → M and M → Y
X must have a causal effect on M

📖 References

Pearl, J. (2009). Causality: Models, Reasoning, and Inference
Textor et al. (2016). dagitty: Graphical Analysis of Structural Causal Models