Represent bias parameters — bias

bias_params is one of two different options to represent bias assumptions for bias adjustment. The multibias_adjust() function will apply the assumptions from these models and use them to adjust for biases in the observed data. It takes one input, a list, where each item in the list corresponds to the necessary models for bias adjustment. See below for bias models.

For each of the following bias models, the variables are defined:

X = True exposure
X* = Misclassified exposure
Y = True outcome
Y* = Misclassified outcome
C = Known confounder
j = Number of known confounders
U = Uncontrolled confounder
S = Selection indicator

Uncontrolled confounding: logit(P(U=1)) = α₀ + α₁X + α₂Y + α_2+jC_j
Exposure misclassification: logit(P(X=1)) = δ₀ + δ₁X* + δ₂Y + δ_2+jC_j
Outcome misclassification: logit(P(Y=1)) = δ₀ + δ₁X + δ₂Y* + δ_2+jC_j
Selection bias: logit(P(S=1)) = β₀ + β₁X + β₂Y
Uncontrolled Confounding & Exposure Misclassification (Option 1): logit(P(U=1)) = α₀ + α₁X + α₂Y
logit(P(X=1)) = δ₀ + δ₁X* + δ₂Y + δ_2+jC_j
Uncontrolled Confounding & Exposure Misclassification (Option 2): log(P(X=1,U=0)/P(X=0,U=0)) = γ_1,0 + γ_1,1X* + γ_1,2Y + γ_1,2+jC_j
log(P(X=0,U=1)/P(X=0,U=0)) = γ_2,0 + γ_2,1X* + γ_2,2Y + γ_2,2+jC_j
log(P(X=1,U=1)/P(X=0,U=0)) = γ_3,0 + γ_3,1X* + γ_3,2Y + γ_3,2+jC_j
Uncontrolled Confounding & Outcome Misclassification (Option 1): logit(P(U=1)) = α₀ + α₁X + α₂Y
logit(P(Y=1)) = δ₀ + δ₁X + δ₂Y* + δ_2+jC_j
Uncontrolled Confounding & Outcome Misclassification (Option 2): log(P(U=1,Y=0)/P(U=0,Y=0)) = γ_1,0 + γ_1,1X + γ_1,2Y* + γ_1,2+jC_j
log(P(U=0,Y=1)/P(U=0,Y=0)) = γ_2,0 + γ_2,1X + γ_2,2Y* + γ_2,2+jC_j
log(P(U=1,Y=1)/P(U=0,Y=0)) = γ_3,0 + γ_3,1X + γ_3,2Y* + γ_3,2+jC_j
Uncontrolled Confounding & Selection Bias: logit(P(U=1)) = α₀ + α₁X + α₂Y + α_2+jC_j
logit(P(S=1)) = β₀ + β₁X + β₂Y
Exposure Misclassification & Outcome Misclassification (Option 1): logit(P(X=1)) = δ₀ + δ₁X* + δ₂Y* + δ_2+jC_j
logit(P(Y=1)) = β₀ + β₁X + β₂Y* + β_2+jC_j
Exposure Misclassification & Outcome Misclassification (Option 2): log(P(X=1,Y=0) / P(X=0,Y=0)) = γ_1,0 + γ_1,1X* + γ_1,2Y* + γ_1,2+jC_j
log(P(X=0,Y=1) / P(X=0,Y=0)) = γ_2,0 + γ_2,1X* + γ_2,2Y* + γ_2,2+jC_j
log(P(X=1,Y=1) / P(X=0,Y=0)) = γ_3,0 + γ_3,1X* + γ_3,2Y* + γ_3,2+jC_j
Exposure Misclassification & Selection Bias: logit(P(X=1)) = δ₀ + δ₁X* + δ₂Y + δ_2+jC_j
logit(P(S=1)) = β₀ + β₁X* + β₂Y + β_2+jC_j
Outcome Misclassification & Selection Bias: logit(P(Y=1)) = δ₀ + δ₁X + δ₂Y* + δ_2+jC_j
logit(P(S=1)) = β₀ + β₁X + β₂Y* + β_2+jC_j
Uncontrolled Confounding, Exposure Misclassification, and Selection Bias (Option 1): logit(P(U=1)) = α₀ + α₁X + α₂Y
logit(P(X=1)) = δ₀ + δ₁X* + δ₂Y + δ_2+jC_j
logit(P(S=1)) = β₀ + β₁X* + β₂Y + β_2+jC_j
Uncontrolled Confounding, Exposure Misclassification, and Selection Bias (Option 2): log(P(X=1,U=0)/P(X=0,U=0)) = γ_1,0 + γ_1,1X* + γ_1,2Y + γ_1,2+jC_j
log(P(X=0,U=1)/P(X=0,U=0)) = γ_2,0 + γ_2,1X* + γ_2,2Y + γ_2,2+jC_j
log(P(X=1,U=1)/P(X=0,U=0)) = γ_3,0 + γ_3,1X* + γ_3,2Y + γ_3,2+jC_j
logit(P(S=1)) = β₀ + β₁X* + β₂Y + β_2+jC_j
Uncontrolled Confounding, Outcome Misclassification, and Selection Bias (Option 1): logit(P(U=1)) = α₀ + α₁X + α₂Y
logit(P(Y=1)) = δ₀ + δ₁X + δ₂Y* + δ_2+jC_j
logit(P(S=1)) = β₀ + β₁X + β₂Y* + β_2+jC_j
Uncontrolled Confounding, Outcome Misclassification, and Selection Bias (Option 2): log(P(U=1,Y=0)/P(U=0,Y=0)) = γ_1,0 + γ_1,1X + γ_1,2Y* + γ_1,2+jC_j
log(P(U=0,Y=1)/P(U=0,Y=0)) = γ_2,0 + γ_2,1X + γ_2,2Y* + γ_2,2+jC_j
log(P(U=1,Y=1)/P(U=0,Y=0)) = γ_3,0 + γ_3,1X + γ_3,2Y* + γ_3,2+jC_j
logit(P(S=1)) = β₀ + β₁X + β₂Y* + β_2+jC_j

Usage

bias_params(coef_list)

Arguments

coef_list: List of coefficient values from the above options of models. Each item of the list is an equation. The left side of the equation identifies the model (i.e., "u" for the model predicting the uncontrolled confounder). For the multinomial models, specify the value here based on the numerator (i.e., "x1u0", "x0u1", "x1u1" for the three multinomial models in Uncontrolled Confounding & Exposure Misclassification, Option 2) The right side of the equation is the vector of values corresponding to the model coefficients (from left to right).

Examples

list_for_uc <- list(
  u = c(-0.19, 0.61, 0.70, -0.09, 0.10, -0.15)
)

bp_uc <- bias_params(coef_list = list_for_uc)

list_for_em_om <- list(
  x1y0 = c(-2.18, 1.63, 0.23, 0.36),
  x0y1 = c(-3.17, 0.22, 1.60, 0.40),
  x1y1 = c(-4.76, 1.82, 1.83, 0.72)
)

bp_em_om <- bias_params(coef_list = list_for_em_om)