Statistical matching using fractional imputation 5. Measurement error modelsStatistical matching using fractional imputation 5. Measurement error models

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We now consider an application of statistical matching to the problem of measurement error models. Suppose that we are interested in the parameter $\theta$ in the conditional distribution $f\left(y_2|y_1\mathrm{<mo>;</mo>}\theta \right).$ In the original sample, instead of observing $\left(y_{1i}\mathrm{,}{y}_{2i}\right),$ we observe $\left(x_i\mathrm{,}{y}_{2i}\right),$ where $x_i$ is a contaminated version of $y_{1i}.$ Because inference for $\theta$ based on $\left(x_i\mathrm{,}{y}_{2i}\right)$ may be biased, additional information is needed. One common way to obtain additional information is to collect $\left(x_i\mathrm{,}{y}_{1i}\right)$ in an external calibration study. In this case, we observe $\left(x_i\mathrm{,}{y}_{1i}\right)$ in Sample A and $\left(x_i\mathrm{,}{y}_{2i}\right)$ in Sample B, where Sample A is the calibration sample, and Sample B is the main sample. Guo and Little (2011) discuss an application of external calibration.

The external calibration framework can be expressed as a statistical matching problem. Table 5.1 makes the connection between statistical matching and external calibration explicit. An instrumental variable assumption permits inference for $\theta$ based on data with the structure of Table 1.1. In the notation of the measurement error model, the instrumental variable assumption is

$$f\left(y_{2i}|y_{1i}\mathrm{,}{x}_i\right)\mathrm{<mo>=</mo>}f\left(y_{2i}|y_{1i}\right)\text{and}f\left(y_{1i}|x_i\mathrm{<mo>=</mo>}a\right)\ne f\left(y_{1i}|x_i\mathrm{<mo>=</mo>}b\right)\mathrm{,}(5.1)$$

for some $a\ne b.$ The instrumental variable assumption may be judged reasonable in applications related to error in covariates because the subject-matter model of interest is $f\left(y_{2i}|y_{1i}\right)\mathrm{,}$ and $x_i$ is a contaminated version of $y_{1i}$ that contains no additional information about $y_{2i}$ given $y_{1i}.$

For fully parametric $f\left(y_{2i}|y_{1i}\right)$ and $f\left(y_{1i}|x_i\right),$ one can use parametric fractional imputation to execute the EM algorithm. This method requires evaluating the conditional expectation of the complete-data score function given the observed values. To evaluate the conditional expectation using fractional imputation, we first express the conditional distribution of $y_1$ given $\left(x\mathrm{,}{y}_2\right)$ as,

$$f\left(y_1|x\mathrm{,}{y}_2\right)\propto f\left(y_1|x\right)f\left(y_2|y_1\right)\mathrm{.}(5.2)$$

We let an estimator ${\widehat{f}}_a\left(y_{1i}|x_i\right)$ of $f\left(y_{1i}|x_i\right)$ be available from the calibration sample (Sample A). Implementation of the EM algorithm via fractional imputation involves the following steps:

1. For each $i\in B,$ generate $y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}$ from ${\widehat{f}}_a\left(y_1|x_i\right),$ for $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m.$
2. Compute the fractional weights
3. $$w_{ij\left(t\right)}^{*}\propto f\left(y_{2i}|y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{<mo>;</mo>}{\widehat{\theta }}_t\right)$$
4. with ${\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij\left(t\right)}^{*}\mathrm{<mo>=</mo>\; 1.}$

3. Update $\theta$ by solving
4. $${\displaystyle \sum _{i\in B}}{w}_{ib}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij\left(t\right)}^{*}S\left(\theta \mathrm{<mo>;</mo>}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right)\mathrm{<mo>=</mo>\; 0,}$$
5. where $S\left(\theta \mathrm{<mo>;</mo>}{y}_1\mathrm{,}{y}_2\right)\mathrm{<mo>=</mo>}\partial \log f\left(y_2|y_1\mathrm{<mo>;</mo>}\theta \right)/\partial \theta .$

4. Go to Step 2 until convergence.

The method above requires generating data from $f\left(y_1|x\right).$ For some nonlinear models or models with non-constant variances, simulating from the conditional distribution of $y_1$ given $x$ may require Monte Carlo methods such as accept-reject or Metropolis Hastings. The simulation of Section 6.2 exemplifies a simulation in which the conditional distribution of $y_1|x$ has no closed form expression. In this case, we may consider an alternative approach, which may be computationally simpler. To describe this approach, let $h\left(y_1|x\right)$ be the “working” conditional distribution, such as the normal distribution, from which samples are easily generated. We assume that estimates ${\widehat{f}}_a\left(y_1|x\right)$ and ${\widehat{h}}_a\left(y_1|x\right)$ of $f\left(y_1|x\right)$ and $h\left(y_1|x\right),$ respectively, are available from Sample A. Implementation of the EM algorithm via fractional imputation then involves the following steps:

1. For each $i\in B,$ generate $x_i^{\mathrm{<mo>*</mo>}\left(j\right)}$ from ${\widehat{h}}_a\left(y_1|x_i\right),$ for $j\mathrm{<mo>=</mo>\; 1,}\dots \mathrm{,}m.$
2. Compute the fractional weights
3. $$w_{ij\left(t\right)}^{*}\propto f\left(y_{2i}|y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{<mo>;</mo>}{\widehat{\theta }}_t\right){\widehat{f}}_a\left(y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}|x_i\right)/ {\widehat{h}}_a\left(y_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}|x_i\right)(5.3)$$
4. with ${\displaystyle {\sum }_{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij\left(t\right)}^{*}\mathrm{<mo>=</mo>\; 1.}$

3. Update $\theta$ by solving
4. $${\displaystyle \sum _{i\in B}}{w}_{ib}{\displaystyle \sum _{j\mathrm{<mo>=</mo>\; 1}}^m}{w}_{ij\left(t\right)}^{*}S\left(\theta \mathrm{<mo>;</mo>}{y}_{1i}^{\mathrm{<mo>*</mo>}\left(j\right)}\mathrm{,}{y}_{2i}\right)\mathrm{<mo>=</mo>\; 0.}$$

4. Go to Step 2 until convergence.

Variance estimation is a straightforward application of the linearization method in Section 3. The hot-deck fractional imputation method described in Section 3 with fractional weights defined in (3.3) also extends readily to the measurement error setting.

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