Table 2. Data generating mechanism (DGM) and missingness mechanism used in the simulation study in scenarios 1—4.

Seen.	DGM for Y	DGM for X	DGM for C	Partially observed variablesa	Missingness meehanismb
1	Y is continuous and depends on X, X2 and C: Y = Y = -0.4 + 0.4 X + 0.8 C + ϕ X2 + εY where ϕ = 0.1, 0.6 or 1.0	X is continuous and depends on C: X = C + εX	C is binary, with probability 0.5 of a value of 0 or 1	C or Y	Missingness depends on X: logit{P(RΔ = 1)} = τ (α + X)
2	Y is continuous and depends on X, C and C2: Y = -0.4 + 0.4 X + 0.8 C + 0.6 C2 + εY	X is continuous and depends on C: X = C + εX	C is normally distributed, with mean 0.5, variance 1	X or Y	Missingness depends on C: logit{P(RΔ = 1)} = τ (α + C)
3	Y is continuous and depends on X and C: Y = -0.4 + 0.4 X + 0.8 C + εY	X is continuous and depends on C2: X = C2 + εX	C is normally distributed, with mean 0.5, variance 1	X or Y	Missingness depends on C: logit{P(RΔ = 1)} = τ (α + C)
4	Y is binary and depends on X, X2 and C: logit{P(Y = 1)} = -0.4 + 0.4 X + 0.8 C + 0.5 X2	X is continuous and depends on C: X = C + εX	C is binary, with probability 0.5 of a value of 0 or 1	C or Y	Missingness depends on X: logit{P(RΔ = 1)} = τ (α + X)

Abbreviation: logit, logistic function.

In each scenario, we assumed the error terms ε_Y and ε_X were uncorrelated, with standard normal distributions (mean 0, variance 1).

For each missingness mechanism, α was chosen empirically to give approximately 70% observed values (and additionally 50% and 90% observed values in scenario 1 when ϕ = 1.0), for each strength of missingness association (τ), τ = 0.1, 1, 3 or 5.

^aTwo separate situations were considered in each scenario: (i) the partially observed variable was directly involved in the mis-specified relationship and (ii) the partially observed variable was not directly involved. Values were set to missing for one variable only in each situation.

^bP(R_Δ = 1) denotes the probability that a value (of the partially observed variable) is observed, with Δ = X, C or Y.