A note on multiply robust predictive mean matching imputation with complex survey data
Section 4. Simulation study

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To assess the performance of the proposed method in terms of bias and efficiency, we conducted a limited simulation study. We generated $B\mathrm{<mo>=</mo>}$ 2,000 finite populations, each of size $N\mathrm{<mo>=</mo>}$ 20,000. First, the explanatory variables $x_1$ - $x_4$ were generated from a multivariate standard normal distribution. Then, given $x_1$ - $x_4,$ we generated the survey variable $y$ according to the following outcome regression models:

(M1).      $y\mathrm{<mo>=</mo>}1+x_1+x_2+x_3+x_4+\epsilon ,$ where $\epsilon ~N\left(0,1\right).$

(M2).      $y\mathrm{<mo>=</mo>}1+x_1^2+x_2^2+x_3+x_4+x_3{x}_4+\epsilon ,$ where $\epsilon ~N\left(0,1\right).$

Note that both (M1) and (M2) are linear models based on the explanatory variables $x_1$ - $x_4,$ except that (M2) includes quadratic terms and an interaction term.

From each finite population, a probability sample $S$ was selected according to probability proportional-to-size (PPS) systematic sampling based on the size variable $z_i\mathrm{<mo>=</mo>}\log \left(0.1\left|y_i+v_i\right|+4\right),$ where $v_i~N\left(0,1\right).$ The first-order inclusion probabilities are given by ${\pi }_i\mathrm{<mo>=</mo>}n{z}_i/{\displaystyle {\sum }_{i\mathrm{<mo>=</mo>\; 1}}^N{z}_i}$ with $n\mathrm{<mo>=</mo>}$ 200, 500 and 1,000.

In each sample, the response indicators $r_i$ were generated from a Bernoulli distribution with probability $p_i,$ where

$$p_i\mathrm{<mo>=</mo>}\text{0}\text{.1}+\text{0}\text{.9}\times \frac{\exp \left({\alpha }_0+{\alpha }_1{x}_{1i}+{\alpha }_2{x}_{2i}+{\alpha }_3{x}_{3i}+{\alpha }_4{x}_{4i}\right)}{1+\exp \left({\alpha }_0+{\alpha }_1{x}_{1i}+{\alpha }_2{x}_{2i}+{\alpha }_3{x}_{3i}+{\alpha }_4{x}_{4i}\right)}\mathrm{.}(4.1)$$

We used two sets of values for $\left({\alpha }_0\mathrm{,}{\alpha }_1\mathrm{,}{\alpha }_2\mathrm{,}{\alpha }_3\mathrm{,}{\alpha }_4\right):$ $\left(\mathrm{0,}\mathrm{1,}\mathrm{1,}\mathrm{1,}1\right)$ and $\left(\text{1}\text{.38}\mathrm{,}\mathrm{1,}\mathrm{1,}\mathrm{1,}1\right).$ These led to response rates approximately equal to 70%, and 50%, respectively.

We computed the following estimators of $\theta$

We computed the Monte Carlo relative bias (MCRB), the Monte Carlo relative standard error (MCRSE) and the Monte Carlo relative root mean squared error (MCRMSE), defined respectively as

$$\text{MCRB}\mathrm{<mo>=</mo>}\frac{{\text{2,000}}^{-1}{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^{\text{2,000}}\left({\widehat{\theta }}_b-{\theta }_b\right)}}{{\theta }_{\text{MC}}}\mathrm{,}$$

$$\text{MCRSE}\mathrm{<mo>=</mo>}\frac{\sqrt{{\left(B-1\right)}^{-1}{{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B\left({\widehat{\theta }}_b-{\widehat{\theta }}_{\text{MC}}\right)}}^2}}{{\theta }_{\text{MC}}}$$

and

$$\text{MCRMSE}\mathrm{<mo>=</mo>}\frac{\sqrt{{\left(B-1\right)}^{-1}{{\displaystyle {\sum }_{b\mathrm{<mo>=</mo>\; 1}}^B\left({\widehat{\theta }}_b-{\theta }_{\text{MC}}\right)}}^2}}{{\theta }_{\text{MC}}}\mathrm{,}$$

where ${\theta }_b$ denotes the population mean in the $b^{\text{th}}$ population, ${\widehat{\theta }}_b$ denotes the estimator $\widehat{\theta }$ in the $b^{\text{th}}$ sample, $b\mathrm{<mo>=</mo>}\mathrm{1,}\dots \mathrm{,}$ 2,000, and

$${\theta }_{\text{MC}}\mathrm{<mo>=</mo>}\frac{1}{\text{2,000}}{\displaystyle \sum _{b\mathrm{<mo>=</mo>\; 1}}^{\text{2,000}}{\theta }_b}\mathrm{,}{\widehat{\theta }}_{\text{MC}}\mathrm{<mo>=</mo>}\frac{1}{\text{2,000}}{\displaystyle \sum _{b\mathrm{<mo>=</mo>\; 1}}^{\text{2,000}}{\widehat{\theta }}_b}\mathrm{.}$$

The results are presented in Tables 4.1 and 4.2. The naive estimator exhibited a significant bias in all the scenarios, as expected. When the true model was given by (M1), we note from Table 4.1 that linear regression imputation performed very well in terms of bias, as expected. Both PMM and the proposed method showed negligible bias for $n\mathrm{<mo>=</mo>}$ 1,000 and a slight bias for $n\mathrm{<mo>=</mo>}$ 500 and $n\mathrm{<mo>=</mo>}$ 200. For instance, for $n\mathrm{<mo>=</mo>}$ 200 and a response rate of 70%, the value of RB was equal to 2.4% for PMM, New1 and New2. In terms of efficiency, linear regression imputation slightly outperformed both PMM and the proposed methods, as expected. For instance for $n\mathrm{<mo>=</mo>}$ 1,000 and a response rate of 70%, the value of RMSE was equal to 7.5% for linear regression imputation and equal to 8.0% for both PMM, New1 and New2. It is worth pointing out that both PMM and the proposed methods exhibited almost identical performances in all the scenarios presented in Table 4.1. Therefore, incorporating two additional models did not seem to affect the efficiency of the resulting estimator (New2).

When the true model was given by (M2), we note from Table 4.2 that both linear regression imputation and PMM led to significant biases in all the scenarios, as expected. Being a parametric imputation procedure, linear regression imputation is vulnerable to model misspecification. On the other hand, PMM showed smaller biases than linear regression imputation, suggesting some robustness against model misspecification. For instance, for $n\mathrm{<mo>=</mo>}$ 1,000 and a response rate of 70%, the value of RB was equal to -9.2% for linear regression imputation and -3.7% for PMM. The proposed methods outperformed both linear regression imputation and PMM in terms of bias, standard error and mean square error in all the scenarios. Finally, both New1 and New2 exhibited almost identical performances.

Acknowledgements

S. Chen was supported by the National Institute on Minority Health and Health Disparities (NIMHD) at National Institutes of Health (NIH) (1R21MD014658-01A1) and the Oklahoma Shared Clinical and Translational Resources (U54GM104938) with an Institutional Development Award (IDeA) from National Institute of General Medical Sciences. The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. The work of D. Haziza was supported by grants from the Natural Sciences and Engineering Research Council of Canada.

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