Regression analysis of interval-censored failure time data under semiparametric transformation models with missing covariates

Appendix A: Imputation algorithm for situations of q > 1

In this Appendix, we extend the algorithm given in Section 4 for the situation of q = 1 to the case of q > 1. For simplicity, we focus on the case of q = 2 and the case with larger q can be generalized similarly. First, let X = ( X 1 , X 2 ) T denote the covariates with missing, and the missing indicator denoted by χ = ( χ 1 , χ 2 ) T . Based on the MAR assumptions in Section 2, we impute X ₁ and X ₂ in turn, then perform the analysis using the imputed X ₁ and X ₂.

Similar to the definition in Section 4, with W ̃ , we fit two linear/generalized linear regression models (Model H _1,(1)(⋅) and H _1,(2)(⋅)) using W ̃ as the covariate for the complete cases to obtain the predictive scores for the values of X ₁ and X ₂, denoted as ζ ̃ ( 1 ) and ζ ̃ ( 2 ) , respectively. Simultaneously, we fit logistic regression models (Model H _2,(1)(⋅) and H _2,(2)(⋅)) using W ̃ as the covariate to predict the missing indicators χ ₁ and χ ₂, denoted as ζ ̄ ( 1 ) and ζ ̄ ( 2 ) , respectively.

The detailed multiple imputation procedures are outlined as follows.

[Step 1] Bootstrap Sampling.

• Obtain a bootstrap sample B ( s ) = B 1 ( s ) , … , B n ( s ) from the original dataset O. The complete cases within this bootstrap sample are denoted as B obs ( s ) = B [ 1 ] ( s ) , … , B [ n ( s ) ] ( s ) .

[Step 2] Self-Consistency Algorithm.

1. Apply Turnbull’s self-consistency algorithm to B ^(s) to obtain W ̃ ( s ) .

2. Perform the same procedure on O to obtain W ̃ .

[Step 3] Model Fitting.

1. Fit models H _1,(1)(⋅) and H _1,(2)(⋅), as well as H _2,(1)(⋅) and H _2,(2)(⋅) using B obs ( s ) and B ^(s), respectively. This will yield the estimates H ̂ 1 , ( 1 ) ( s ) ( ⋅ ) , H ̂ 1 , ( 2 ) ( s ) ( ⋅ ) , H ̂ 2 , ( 1 ) ( s ) ( ⋅ ) and H ̂ 2 , ( 2 ) ( s ) ( ⋅ ) .

2. Calculate the initial predictive scores ζ ̃ ( 1 ) 0 ( s ) and ζ ̃ ( 2 ) 0 ( s ) for each individual in B obs ( s ) and ζ ̄ ( 1 ) 0 ( s ) and ζ ̄ ( 2 ) 0 ( s ) for each individual in B ^(s).

3. Standardize the scores by subtracting the sample mean and dividing by the standard deviation, resulting in standardized scores ( ζ ̃ ( 1 ) ( s ) , ζ ̃ ( 2 ) ( s ) , ζ ̄ ( 1 ) ( s ) , ζ ̄ ( 2 ) ( s ) ) for the individual in B obs ( s ) .

[Step 4] Impute Missing Values for X ₁.

1. For each missing value X _1,k associated with individual O _k in O, apply the estimated working models H ̂ 1 , ( 1 ) ( s ) ( ⋅ ) and H ̂ 2 , ( 1 ) ( s ) ( ⋅ ) to compute its initial predictive score.

2. Standardize these scores by subtracting the sample mean of the bootstrap predictive scores and dividing by the standard deviation of the bootstrap predictive scores, resulting in standardized scores ( ζ ̃ k ( 1 ) , ζ ̄ k ( 1 ) ) .

3. Use the pre-specified ω ₁ to calculate the distance D _k,(1) = {D(k, [1]), …, D(k, [n ^(s)])} between O _k and the individuals in B obs ( s ) as

   D ( k , [ j ] ) = ω 1 ζ ̃ k ( 1 ) − ζ ̃ [ j ] ( 1 ) ( s ) 2 + ( 1 − ω 1 ) ζ ̄ k ( 1 ) − ζ ̄ [ j ] ( 1 ) ( s ) 2 1 / 2 , j = 1 , … , n ( s ) .

4. Define a neighborhood set consisting of nearest neighbor (NN) subjects from B ^(s) by sorting distances in ascending order. Then, randomly draw a pseudo observation value for X _1,k from the observed values of X 1 ( s ) within this neighborhood set.

[Step 5] Impute Missing Values for X ₂.

1. For each missing value X _2,c associated with individual O _c in O, apply the estimated working models H ̂ 1 , ( 2 ) ( s ) ( ⋅ ) and H ̂ 2 , ( 2 ) ( s ) ( ⋅ ) to compute its initial predictive score.

2. Standardize these scores by subtracting the sample mean of the bootstrap predictive scores and dividing by the standard deviation of the bootstrap predictive scores, resulting in standardized scores ( ζ ̃ c ( 2 ) , ζ ̄ c ( 2 ) ) .

3. Use the pre-specified ω ₁ to calculate the distance D _c,(2) between O _c and the individuals in B obs ( s ) similarly.

4. Define the neighborhood set from B ^(s), and randomly draw a pseudo observation value for X _2,c from the observed values of X 2 ( s ) within this neighborhood set.

[Step 6] Final Estimation.

After completing the imputation for all missing covariates, proceed with the sieve maximum likelihood procedure using the log-likelihood function to obtain the final estimator θ ̂ ( s ) = ( α ̂ ( s ) , β ̂ ( s ) , Λ ̂ 0 ( s ) ) .

The entire imputation procedure will be repeated K times, generating K multiple imputed data sets. The overall estimates and their variances will be obtained by applying Rubin’s combination rule, similar to the method described in Section 4.

Appendix B: Additional details about numerical studies

We also consider a scenario with q > 2. In this case, we maintain Z as previously defined, while letting X = ( X 1 , X 2 ) T . The variable X ₁ was generated from the normal distribution with the mean Z ₁ and variance of 1, and X ₂ is generated from a Bernoulli distribution with success probability given by 1/{1 + exp(−0.3 + 0.9Z ₁) + exp(0.3 − 0.6Z ₂)}. The missing mechanisms for X ₁ and X ₂ are defined as P(χ ₁ = 1∣O) = 1/{1 + exp(−2.5 − 0.5Z ₁ + 0.5L + 0.5R)} and P(χ ₂ = 1∣O) = 1/{1 + exp(−2.0 − 0.5Z ₁ + 0.5L + 0.5R)}, respectively, leading to missing rates of approximately 25 % for X ₁ and 35 % for X ₂, resulting in a total missing rate of around 50 %. The other settings remain the same as those in Section 5, and the results for n = 200 under the PO model are presented in Table 7. These results confirm the same conclusions as before.