graph_a

Diagram (a): - Is there confounding? Yes, W is a confounder because it affects both X and Y. - Bias Direction: Upward bias. Since X decreases W, and W decreases Y, the bias is positive (negative × negative = positive).

graph_b

Diagram (b): - Is there confounding? Yes, Z is a confounder because it affects X and W, and W influences Y. - Bias Direction: Downward bias. Z increases X, Z increases W, but W decreases Y, so the bias is negative.

graph_c

Diagram (c): - Is there confounding? No confounding. The paths Z → W → Y and X → Y are independent. - Bias Direction: No bias since the relationship between X and Y is unaffected.

graph_d

Diagram (d): - Is there confounding? No confounding. The path X → W → Y is a single mediating pathway without confounders. - Bias Direction: No bias because the relationship between X and Y is direct and unconfounded. ”