Missing data lecture 2

Notation Meaning Domain
\(Y\) \(n\times K\) matrix, collection of measurements of interest \(\R^{n \times K}\)
\(\tilde{y}\) Realization of \(Y\) \(\R^{n \times K}\)
\(Y_{ij}\) Element of \(Y\), random variables \(\R\)
\(\tilde{y}_{ij}\) Particular realization of \(Y_{ij}\) \(\R\)
\(Y_{i}\) \(i^\mathrm{th}\) row of \(Y\), random variables \(\R^K\)
\(\tilde{y}_{i}\) Particular realization of \(Y_{i}\) \(\R^K\)
\(y_{i}\) Arbitrary element of \(\R^K\), for use in density functions related to \(Y_i\) \(\R^K\)
\(M\) \(n\times K\) binary matrix of missingness indicators \(\{0,1\}^{n \times K}\)
\(\tilde{m}\) Realization of \(M\) \(\{0,1\}^{n \times K}\)
\(M_{ij}\) Element of \(M\), random variable \(\{0,1\}\)
\(M_{i}\) \(i^\mathrm{th}\) row of \(M\), random variables \(\{0,1\}^{K}\)
\(\tilde{m}_{ij}\) Realization of \(M_{ij}\) \(\{0,1\}\)
\(\tilde{m}_{i}\) Realization of \(M_{i}\) \(\{0,1\}^K\)
\(m_{i}\) Arbitrary element of \(\{0,1\}^K\) for use in PMFs for \(M_i\) \(\{0,1\}^K\)
\(R_i\) Number of observed values for row \(i\) in \(Y\), \(\sum_{j=1}^K (1 - M_{ij})\) \(\mathbb{N}\)
\(Y_{(0)i}\) Observed elements of \(Y_i\) \(\R^{R_i}\)
\(\tilde{y}_{(0)i}\) Realization of \(Y_{(0)i}\) \(\R^{R_i}\)
\(y_{(0)i}\) Arbitrary element of \(\R^{R_i}\) \(\R^{R_i}\)
\(Y_{(1)i}\) Unobserved or missing elements of \(Y_i\) \(\R^{K - R_i}\)
\(\tilde{y}_{(1)i}\) Realization of \(Y_{(1)i}\) \(\R^{K - R_i}\)
\(y_{(1)i}, y^\star_{(1)i}\) Arbitrary elements of \(\R^{K - R_i}\) \(\R^{K - R_i}\)

Recap

Missingness mechanisms

Mechanisms relate to the distribution: \[ f_{M_i \mid Y_i}(M_i = \tilde{m}_i \mid Y_i = y_i, \phi), \] where \(\phi\) are the parameters that govern the missingness mechanism.

Missing completely at random (MCAR)

Data are said to be MCAR if the following holds for all \(i\), \(y_i\), \(y^\star_i\), and \(\phi\): \[ f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_i = y_i, \phi) = f_{M\mid Y}(M_i = \tilde{m}_i \mid Y_i = y^\star_i, \phi). \]

Missing at random (MAR)

Missing at random data are characterized by the following equality for all \(i\), \(y_{(1)i}\), \(y^\star_{(1)i}\), and \(\phi\): \[ f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_{(0)i} = \tilde{y}_{(0)i}, Y_{(1)i} = y_{(1)i}, \phi) = f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_{(0)i} = \tilde{y}_{(0)i}, Y_{(1)i} = y^\star_{(1)i}, \phi) \]

Univariate MAR data

Univariate MAR data is useful to consider because it elucidates a key assumption that can be used in MAR analyses.

Suppose we have a single measurement on \(n\) individuals, \(Y_i\) and \(r\) of the individuals in our sample have observations \(\tilde{y}_i\) while \(n-r\) individuals have missing values for \(Y_i\) (i.e. \(\tilde{y}_{(1)i} = \tilde{y}_{i}\), \(\tilde{y}_{(0)i} = \emptyset\)). Suppose we also have iid measurements and an iid missingness process.

Writing out the missingness mechanism for this scenario gives us: \[ \begin{aligned} f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_{(0)i} = \tilde{y}_{(0)i}, Y_{(1)i} = y_{(1)i}, \phi) & = f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_{(0)i} = \tilde{y}_{(0)i}, Y_{(1)i} = y^\star_{(1)i}, \phi) \\ & = f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid \phi) \end{aligned} \] This implies something about the relationship between the units the have observed data and those that do not: \[ \begin{aligned} f_{Y_i \mid M_i}(Y_{i} = y_{i} \mid M_i = \tilde{m}_i, \theta) & = \frac{f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_{i} = y_i, \phi) f_{Y_i}(Y_i = y_i \mid \theta)}{f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid \phi)} \\ & = f_{Y_i}(Y_i = y_i \mid \theta) \end{aligned} \] This implies that \[ \begin{aligned} f_{Y_i \mid M_i}(Y_{i} = y_{i} \mid M_i = 1, \theta) & = f_{Y_i \mid M_i}(Y_{i} = y_{i} \mid M_i = 0, \theta) \end{aligned} \] Thus, we can use the distribution we learn from the complete cases as the distribution for the cases that are missing. This idea holds for more general missingness patterns with more measurements. We’ll see this later in the course.

Continuing with our example from last lecture

The following example is adapted from Mealli and Rubin (2015): Suppose we’re analyzing data from that Crohn’s disease trial and \(Y_i\) has two components: \(Y_{i1}\) is CDAI at visit 1 and \(Y_{i2}\) is CDAI at visit 2. For patient \(i\) suppose that \(m_i = (0, 1)\), or that \(Y_{i2}\) is missing but \(Y_{i1}\) is observed.

Bringing this example into line with our previous notation, \(Y_{(0)i} \equiv Y_{i1}\) and \(Y_{(1)i} \equiv Y_{i2}\). Then \(\tilde{y}_{(0)i} \equiv \tilde{y}_{i1}\).

Consider two scenarios:

  1. \(Y_{i2}\) is missing because \(Y_{i1} > \phi\). Given that \(0^0 = 1\), in this scenario the missingness mechanism can be translated as: \[ f_{M_{i2}\mid Y_i}(M_{i2} = m_{i2} \mid Y_{(0)i} = \tilde{y}_{(0)i}, Y_{(1)i} = y_{(1)i}, \phi) = \ind{\tilde{y}_{(0)i} > \phi}^{m_{i2}}(1 - \ind{\tilde{y}_{(0)i} > \phi})^{1 - m_{i2}} \]

  2. \(Y_{i2}\) is missing because \(Y_{i2} > \phi\). This mechanism is translated as \[ f_{M_{i2}\mid Y_i}(M_{i2} = m_{i2} \mid Y_{(0)i} = \tilde{y}_{(0)i}, Y_{(1)i} = y_{(1)i}, \phi) = \ind{y_{(1)i} > \phi}^{m_{i2}}(1 - \ind{y_{(1)i} > \phi})^{1 - m_{i2}} \] In scenario 1 the data are MAR because the mass function is a function of \(\tilde{y}_{(0)i}\) only, (i.e it depends only on \(Y_{i1}\)), while in scenario 2 the data do not satisfy the definition of MAR.

Generalizing the example

Let’s make this example more general. The following is from Little and Rubin (2019, 23). Again consider the bivariate case with \(y_{i1}, y_{i2}\). There are 4 possible missing data patterns: \[ (m_{i1}, m_{i2}) \in \{(0,0), (0,1), (1,0), (1,1)\} \] We’ll need to define \(f_{M \mid Y}(m_{i1} = r, m_{i2} = s \mid y_{i1}, y_{i2}, \phi)\). To simplify the notation, let \[ g_{rs}(y_{i1}, y_{i2}, \phi) = f_{M \mid Y}(m_{i1} = r, m_{i2} = s \mid y_{i1}, y_{i2}, \phi) \] The MAR assumption implies the following: \[ \begin{aligned} g_{11}(y_{i1}, y_{i2}, \phi) & = g_{11}(\phi) \\ g_{01}(y_{i1}, y_{i2}, \phi) & = g_{01}(y_{i1}, \phi) \\ g_{10}(y_{i1}, y_{i2}, \phi) & = g_{10}(y_{i2}, \phi) \\ g_{00}(y_{i1}, y_{i2}, \phi) & = 1 - g_{10}(y_{i2}, \phi) - g_{01}(y_{i1}, \phi) - g_{11}(\phi) \end{aligned} \] Thus the probability that \(y_{ij}\) is missing can depend only on \(y_{i(-j)}\), which is a bit odd.

Little and Rubin (2019) proposes the following modification:

\[ \begin{aligned} g_{11}(y_{i1}, y_{i2}, \phi) & = g_{1+}(y_{i1}, \phi)g_{+1}(y_{i2}, \phi) \\ g_{01}(y_{i1}, y_{i2}, \phi) & = (1 - g_{1+}(y_{i1}, \phi))g_{+1}(y_{i2}, \phi) \\ g_{10}(y_{i1}, y_{i2}, \phi) & = g_{1+}(y_{i1}, \phi)(1 - g_{+1}(y_{i2}, \phi)) \\ g_{00}(y_{i1}, y_{i2}, \phi) & = (1 - g_{1+}(y_{i1}, \phi))(1 - g_{+1}(y_{i2}, \phi)) \end{aligned} \] While this is maybe more realistic, though it does make an assumption that \(m_{i1}\) and \(m_{i2}\) are conditionally independent given \(y_{i1}, y_{i2}\), it is also hard to estimate, because we won’t observe missing values of \(y_{i1}\) and \(y_{i2}\).

This is a scenario called missing-not-at-random, or MNAR. This is defined in the next subsection.

Missing-not-at-random (MNAR)

MNAR data is characterized by the following relationship:

\[ f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_{(0)i} = \tilde{y}_{(0)i},Y_{(1)i} = y_{(1)i}, \phi) \neq f_{M_i\mid Y_i}(M_i = \tilde{m}_i \mid Y_{(0)i} = \tilde{y}_{(0)i},Y_{(1)i} = y^\star_{(1)i}, \phi) \] for some \(\phi\) and \(y_{(1)i} \neq y^\star_{(1)i}\) (Mealli and Rubin 2015).

The version using conditional dependence is called missing not always at random, or MNAAR, which is

\[ f_{M_i\mid Y_i}(M_i = m_i \mid Y_{(0)i} = y_{(0)i},Y_{(1)i} = y_{(1)i}, \phi) \neq f_{M_i\mid Y_i}(M_i = m_i \mid Y_{(0)i} = y_{(0)i},Y_{(1)i} = y^\star_{(1)i}, \phi) \]

for some \(\phi, m_i, y_{(0)i}\) and \(y_{(1)i} \neq y^\star_{(1)i}\)(Mealli and Rubin 2015).

Our textbook mentions that sometimes MAR can yield better results than MNAR, citing Rubin, Stern, and Vehovar (1995). This is something we’ll explore later on in the course, namely how we would determine whether it was worth it to fit an MNAR model vs. a MAR model. Not surpisingly, like most things in statistics, the answer is “It depends.”

Complete case analysis

The most common approach to missing data analysis is complete case analysis, or CCA. Given the matrix \(M\) and its realization \(\tilde{m}\), the complete cases are those rows \(i\) such that \(\sum_{j=1}^K \tilde{m}_{ij} = 0\). Let these indices be denoted \(\mathcal{R} = \{i \mid \sum_{j=1}^K \tilde{m}_{ij} = 0, i = 1, \dots, n\}\). Let the size of this set be \(r = \abs{\mathcal{R}}\).

Complete case analysis is, in general, only valid when the missingness mechanism is MCAR, though depending on what your estimand of interest is, there can be exceptions for MAR and even MNAR missingness1.

The first issue with complete case analysis is that, even if your estimator is unbiased, your estimator will not be efficient. A simple example demostrates the problem.

Suppose we have two measurements of interest, \(Y_{i1}\) and \(Y_{i2}\) for \(n\) individuals and that they are jointly multivariate normal: \[ (Y_{i1}, Y_{i2}) \sim \text{MultiNormal}\lp(\mu_1, \mu_2), \begin{bmatrix} \sigma^2_1 & \sigma_{12} \\ \sigma_{12} & \sigma_2^2 \end{bmatrix} \rp \] We’ll consider the scenario in which \(\mu_1, \mu_2\) are the estimands of interest, and Further, suppose that \(r\) of those individuals have complete observations, (i.e. \(\tilde{y}_{(0)i} = (\tilde{y}_{i1}, \tilde{y}_{i2})\)) and \(n-r\) of those individuals have missing values for \(Y_{i2}\) (i.e. \(\tilde{y}_{(0)i} = (\tilde{y}_{i1})\)).

The parameters of interest are all of the unknown parameters of the multivariate normal distribution. Using only the \(r\) complete cases will allow us to estimate the mean of \(\mu_1\). The sample mean, which we’ll denote \(\bar{y}^{\mathrm{CC}}_{1}\), has variance \(\sigma_1^2/r\). We could have used all \(n\) observations to estimate the mean, which we’ll call the efficient estimator, or \(\bar{y}^\mathrm{eff}_1\) which would instead have variance \(\sigma_1^2 / n\). We can represent the variance of the complete case estimator as a multiple of the variance of the efficient estimator: \[ \text{Var}\lp \bar{y}^{\mathrm{CC}}_{1}\rp = \text{Var}\lp \bar{y}^{\mathrm{eff}}_{1}\rp(1 + \Delta^\mathrm{CC}) \] In this case, \(\Delta^{\mathrm{CC}} = \frac{n - r}{r}\).

For the mean of \(Y_{i2}\), \(\mu_2\), the efficiency loss is: \[ \Delta^{\mathrm{CC}} \approx \frac{(n-r)\rho^2}{n(1 - \rho^2) + r \rho^2} \] We’ll prove this later in the class when we work through likelihood theory for incomplete data.

Bias of CCA

We showed earlier that the bias of the complete case analysis was dependent on the covariance between the missingness indicator and the response; another way to decompose the bias is via the following representation of the sample mean. Recall that \(\mathcal{R} = \{i \mid \sum_{j=1}^K \tilde{m}_{ij} = 0, i = 1, \dots, n\}\); we can define the complement of this set \(\mathcal{R}^\comp\). Then we can represent the sample mean as: \[ \bar{y} = \frac{r}{n}\bar{y}^\mathrm{CC} + (1 - \frac{r}{n})\bar{y}^{\mathcal{R}^\comp} \] Note that the sample mean may be unobservable because some units may be missing \(y_i\). Then, conditional on the missingness pattern that was observed, the bias of the

\[ \Exp{\bar{y}^\mathrm{CC} - \bar{y} \mid M = \tilde{m}} = \lp 1 - \frac{r}{n}\rp(\Exp{\bar{y}^\mathrm{CC} \mid M = \tilde{m}} - \Exp{\bar{y}^{\mathcal{R}^\comp} \mid M = \tilde{m}}) \]

Weighting estimators

An alternative to straight CCA is the class of weighting estimators. These estimators are asymptotically unbiased under the assumption of MAR.

The idea behind weighting estimators is similar to weighted estimators used in survey analysis. A weighting estimator you may have encountered before is called the Horvitz-Thompson estimator. Suppose we have a populaton of size \(N\) and we wish to estimate the mean of some fixed, but unknown, scalar quantity, \(Y_i\). If create a sample of size \(n\) from the population with known weights \(\pi_i\), we can compute an unbiased estimator of the mean with \(\bar{y}^\mathrm{HT}\): \[ \bar{y}^\mathrm{HT} = \frac{1}{N} \sum_{j=1}^n \frac{y_j}{\pi_j} \] This estimator can be written as a sum over the population of size \(N\) with the introduction of an inclusion indicator, \(I_i\) which equals \(1\) if the \(i\) is sampled and \(0\) otherwise: \[ \bar{y}^\mathrm{HT} = \frac{1}{N} \sum_{i=1}^N \frac{y_i I_i}{\pi_i} \] Treating \(y_i\) as fixed unknown quantities and \(I_i\) as a random quantity gives the following unbiased estimator for the population average: \[ \begin{aligned} \ExpA{\bar{y}^\mathrm{HT}}{I_i} & = \frac{1}{N} \sum_{i=1}^N \ExpA{\frac{y_i I_i}{\pi_i}}{I_i} \\ & = \frac{1}{N} \sum_{i=1}^N \frac{y_i \pi_i}{\pi_i}\\ & = \frac{1}{N} \sum_{i=1}^N y_i \end{aligned} \]

Inverse probability estimators

Simple estimator

This same idea can be used for missing data and is called the inverse probability weighted complete case estimator, or IPWCC for short.

Suppose that we have a sample of pairs \((y_i, m_i), i = 1, \dots, n\).

For the next section, we’ll make the more stringent assumption that there are completely observed covariates, \(X_i\) such that the following holds \(M_i \indy Y_i \mid X_i\) and that we have specified the correct model for \(M_i\): \[ P(M_i = 0 \mid X_i = x, \beta) = \pi(x,\beta) \]

\[ \bar{y}^\mathrm{IPWCC} = \frac{1}{n} \sum_{i=1}^n \frac{y_i (1 - M_i)}{\pi(x_i, \beta)} \overset{p}{\to} \ExpA{\frac{Y_i (1 - M_i)}{\pi(X_i, \beta)}}{(Y_i, M_i, X_i)} \] where the convergence in probability follows from the weak law of large numbers.

\[ \begin{aligned} \ExpA{\frac{Y_i (1 - M_i)}{\pi(X_i, \beta)}}{(Y_i, M_i, X_i)} & = \ExpA{\ExpA{\frac{Y_i (1 - M_i)}{\pi(X_i, \beta)}}{M_i \mid Y_i, X_i}}{(Y_i, X_i)} \\ & = \ExpA{\frac{Y_i}{\pi(X_i, \beta)}\ExpA{ (1 - M_i)}{M_i \mid Y_i, X_i}}{(Y_i, X_i)} \\ & = \ExpA{\frac{Y_i}{\pi(X_i, \beta)}\pi(X_i, \beta)}{(Y_i, X_i)} \\ & = \ExpA{Y_i}{(Y_i, X_i)} \end{aligned} \]

There are two drawbacks with the IPWCC estimator:

  • It requires that \(\pi(X_i, \beta) > 0\) for all \(X_i\) in the population; this may not be a good assumption.

  • Even if \(\pi(X_i, \beta) > 0\) is strictly true, small values of \(\pi(X_i, \beta)\) may occur (the interpretation is that the \(i^\mathrm{th}\) observation will represent an inordinate number of missing cases because its probability of being missing is so high). If this happens, these estimators can have very large variance

  • In practice, we don’t know \(\pi(X_i, \beta)\) and typically have to estimate \(\beta\) with \(\hat{\beta}\). If the sample is small, even if the form \(\pi(X_i, \beta)\) is right, there may be high variance. If the model for \(M_i \mid X_i\) is wrong, then we have no guarantees that our IPWCC estimator will be unbiased.

IPW-Z-estimators

Most estimators of interest can be formulated as solutions in \(\theta\) to the following system of equations: \[ \frac{1}{n} \sum_{i=1}^n \psi(\theta, y_i, x_i) = 0 \] The key property of \(\psi(\theta, Y_i, X_i)\) is that the following identity holds: \[ \ExpA{\psi(\theta^\dagger, Y_i, X_i)}{(Y_i, X_i \mid \theta^\dagger)} = 0 \] We can use the same idea as above: fit the complete cases with weights: \[ \frac{1}{n} \sum_{i=1}^n \frac{(1 - m_i)}{\pi(x_i, \beta)} \psi(\theta, y_i, x_i) = 0 \] Given the inverse weighting, we’ll end up with an unbiased estimator of \[ \ExpA{\psi(\theta, Y_i, X_i)}{(Y_i, X_i \mid \theta^\dagger)} \] which we can then use to find \(\hat{\theta}\).

Augmented inverse probability weighted estimators

In order to help with the third bullet above, enter augmented inverse probability weighted complete case estimators or AIPWCCs. These models require building a model for \(\ExpA{Y_i \mid X_i}{Y_i \mid X_i}\) and \(\ExpA{M_i \mid X_i}{M_i \mid X_i}\). If either of these models is correct, the estimator will correctly estimate \(\Exp{Y_i}\). Let’s write the model for \(\Exp{Y_i \mid X_i}\) as \(\mu(X_i, \gamma)\) and the model for \(P(M_i = 0 \mid X_i)\) as above, or \(\pi(X_i, \beta)\).

The AIPWCC estimator is as follows:

\[ \bar{y}^\mathrm{AIPWCC} = \frac{1}{n} \sum_{i=1}^n \frac{y_i (1 - m_i)}{\pi(x_i, \hat{\beta})} - \lp\frac{(1 - m_i) - \pi(x_i, \hat{\beta})}{\pi(x_i, \hat{\beta})}\rp\mu(x_i, \hat{\gamma}) \]

We can add and subtract the term \(\frac{(1 - m_i) - \pi(x_i, \hat{\beta})}{\pi(x_i, \hat{\beta})} y_i\) to give us:

\[ \begin{aligned} \frac{1}{n} & \sum_{i=1}^n \frac{y_i (1 - m_i)}{\pi(x_i, \hat{\beta})} - \lp\frac{(1 - m_i) - \pi(x_i, \hat{\beta})}{\pi(x_i, \hat{\beta})}\rp\mu(x_i, \hat{\gamma}) \\ & = \frac{1}{n} \sum_{i=1}^n \frac{y_i (1 - m_i)}{\pi(x_i, \hat{\beta})} - \lp\frac{(1 - m_i) - \pi(x_i, \hat{\beta})}{\pi(x_i, \hat{\beta})}\rp(y_i - \mu(x_i, \hat{\gamma})) - \lp\frac{(1 - m_i) - \pi(x_i, \hat{\beta})}{\pi(x_i, \hat{\beta})}\rp y_i \\ & = \frac{1}{n} \sum_{i=1}^n y_i - \lp\frac{(1 - m_i) - \pi(x_i, \hat{\beta})}{\pi(x_i, \hat{\beta})}\rp(y_i - \mu(x_i, \hat{\gamma})) \end{aligned} \] Of course, the first line is computable, while the last line isn’t because we don’t observe \(y_i\) when \(m_i = 1\). However, the version of the estimator below can be used to show that the estimator will be asymptotically unbiased when either \(\mu(X_i, \gamma)\) or \(\pi(X_i, \beta)\) is correctly specified.

We’ll go through both cases. For both cases, suppose that \(\ExpA{Y_i \mid X_i}{Y_i \mid X_i} = \mu(X_i, \gamma^\dagger)\) and \(P(M_i = 0) = \pi(X_i, \beta^\dagger)\).

Correctly specified \(\pi(x_i,\beta)\)

Suppose that \(\pi(X_i, \beta)\) is correctly specified, but \(\mu(x_i, \gamma)\) is misspecified. Then \(\hat{\beta} \overset{p}{\to} \beta^\dagger\) but \(\hat{\gamma} \overset{p}{\to} \gamma^\prime\) such that \(\gamma^\prime \neq \gamma^\dagger\). By the WLLN, \(\bar{y}^\mathrm{AIPWCC} \overset{p}{\to}\) the expression: \[ \ExpA{Y_i}{(Y_i, M_i, X_i)} - \ExpA{\lp\frac{(1 - M_i) + \pi(X_i, \beta^\dagger)}{\pi(X_i, \beta^\dagger)}\rp(Y_i - \mu(x_i, \gamma^\prime))}{(Y_i, M_i, X_i)} \] The second term simplifies: \[ \begin{aligned} & \ExpA{\lp\frac{(1 - M_i) - \pi(X_i, \beta^\dagger)}{\pi(X_i, \beta^\dagger)}\rp(Y_i - \mu(x_i, \gamma^\prime))}{(Y_i, M_i, X_i)} \\ & = \ExpA{\ExpA{\lp\frac{(1 - M_i) - \pi(X_i, \beta^\dagger)}{\pi(X_i, \beta^\dagger)}\rp(Y_i - \mu(x_i, \gamma^\prime))}{M_i \mid Y_i, X_i}}{(Y_i, X_i)} \\ & = \ExpA{(Y_i - \mu(x_i, \gamma^\prime))\frac{1}{\pi(X_i, \beta^\dagger)}\ExpA{(1 - M_i) - \pi(X_i, \beta^\dagger)}{M_i \mid Y_i, X_i}}{(Y_i, X_i)} \\ & = 0 \end{aligned} \]

Correctly specified \(\mu(x_i,\gamma)\)

Suppose that \(\mu(X_i, \gamma)\) is correctly specified, but \(\pi(x_i, \beta)\) is misspecified. Then \(\hat{\gamma} \overset{p}{\to} \gamma^\dagger\) but \(\hat{\beta} \overset{p}{\to} \beta^\prime\) such that \(\beta^\prime \neq \beta^\dagger\). By the WLLN, \(\bar{y}^\mathrm{AIPWCC} \overset{p}{\to}\) the expression: \[ \ExpA{Y_i}{(Y_i, M_i, X_i)} - \ExpA{\lp\frac{(1 - M_i) + \pi(X_i, \beta^\prime)}{\pi(X_i, \beta^\prime)}\rp(Y_i - \mu(x_i, \gamma^\dagger))}{(Y_i, M_i, X_i)} \] The second term simplifies: \[ \begin{aligned} & \ExpA{\lp\frac{(1 - M_i) - \pi(X_i, \beta^\prime)}{\pi(X_i, \beta^\prime)}\rp(Y_i - \mu(x_i, \gamma^\dagger))}{(Y_i, M_i, X_i)} \\ & = \ExpA{\ExpA{\lp\frac{(1 - M_i) - \pi(X_i, \beta^\prime)}{\pi(X_i, \beta^\prime)}\rp(Y_i - \mu(x_i, \gamma^\dagger))}{Y_i \mid M_i, X_i}}{(M_i, X_i)} \\ & = \ExpA{\lp\frac{(1 - M_i) - \pi(X_i, \beta^\prime)}{\pi(X_i, \beta^\prime)}\rp\ExpA{(Y_i - \mu(x_i, \gamma^\dagger))}{Y_i \mid M_i, X_i}}{(M_i, X_i)} \\ & = 0 \end{aligned} \]

This is great if you get one of the forms right. However, if both are wrong, there are no guarantees about the bias. Kang and Schafer (2007) investigates, via simulation study, how poorly AIPWCC estimators perform when both models are wrong. They find that regression imputation estimators, which are what we’ll study in the next section, perform better, both in terms of coverage and MSE, even when the regression equation is misspecified.

The key is that the AIPWCC estimators have high variance because of the denominator \(\pi(X_i, \beta)\).

AIPWCC-Z-estimators

The same idea can be used for Z-estimators; \(\hat{\theta}^\mathrm{AIPWCC}\) solves the following systems of equations: \[ \frac{1}{n} \sum_{i=1}^n \frac{1 - m_i}{\pi(x_i, \hat{\beta})}\psi(\theta, y_i, x_i) - \lp\frac{(1 - m_i) - \pi(x_i, \hat{\beta})}{\pi(x_i, \hat{\beta})}\rp\ExpA{\psi(y_i,x_i, \theta) \mid X_i = x_i}{Y_i \mid X_i} = 0 \]

Poststratified estimators

These estimators directly exploit the law of total expectation. Suppose we have \((y_i, x_i, m_i), i = 1, \dots, n\) where \(m_i = 1\) indicates missing \(y_i\). The target of inference is \(\ExpA{Y_i}{(Y_i, X_i, M_i)}\). Assuming univariate MAR conditional on covariates yields the following: \[ \ExpA{M_i \mid X_i = x, Y_i = y}{M_i \mid X_i, Y_i} = \ExpA{M_i \mid X_i = x}{M_i \mid X_i} \] Which equates to: \[ \ExpA{Y_i \mid X_i = x, M_i = 1}{Y_i \mid X_i, M_i} = \ExpA{Y_i \mid X_i = x, M_i = 0}{Y_i \mid X_i, M_i} \] This motivates the following estimator: \[ \bar{y}^\mathrm{ps} = \int_{\mathcal{X}} \mu(x, \hat{\gamma}) d\hat{P}(x) \] where \[ \mu(x, \gamma^\dagger) = \ExpA{Y_i \mid X_i = x, M_i = 0}{Y_i \mid X_i, M_i} \]

This technique is also subject to model misspecification issues; if \(\mu(x,\gamma) \neq \ExpA{Y_i \mid X_i = x, M_i = 0}{Y_i \mid X_i, M_i}\), then our estimator will not be consistent.

References

Kang, Joseph D. Y., and Joseph L. Schafer. 2007. “Demystifying Double Robustness: A Comparison of Alternative Strategies for Estimating a Population Mean from Incomplete Data.” Statistical Science 22 (4). https://doi.org/10.1214/07-STS227.
Little, Roderick JA, and Donald B Rubin. 2019. Statistical Analysis with Missing Data. John Wiley & Sons.
Mealli, Fabrizia, and Donald B. Rubin. 2015. “Clarifying Missing at Random and Related Definitions, and Implications When Coupled with Exchangeability: Table 1.” Biometrika 102 (4): 995–1000. https://doi.org/10.1093/biomet/asv035.
Rubin, Donald B, Hal S Stern, and Vasja Vehovar. 1995. “Handling ‘Don’t Know’ Survey Responses: The Case of the Slovenian Plebiscite.” Journal of the American Statistical Association 90 (431): 822–28.

Footnotes

  1. One exception is regression with missing predictors; scenarios in which the predictors have MNAR missingness that doesn’t depend on the outcome can be analyzed with CCA.↩︎