# Book Notes: Statistical Analysis with Missing Data -- Ch3 Complete Case Analysis and Weighting Methods

### Complete-case (CC) analysis

• Complete-case (CC) analysis: use only data points (units) where all variables are observed

• Loss of information in CC analysis:

• Loss of precision (larger variance)
• Bias, when the missingness mechanism is not MCAR. In this case, the complete units are not a random sample of the population
• In this notes, I will focus on the bias issue

• Adjusting for the CC analysis bias using weights
• This idea is closed related to weighting in randomization inference for finite population surveys

# Weighted Complete-Case Analysis

### Notations

• Population size $$N$$, sample size $$n$$
• Number of variables (items): $$K$$
• Data: $$Y=(y_{ij})$$, where $$i = 1, \ldots, N$$ and $$j = 1, \ldots, K$$
• Design information (about sampling or missingness): $$Z$$
• Sample indicator: $$I = (I_1, \ldots, I_N)'$$; for unit $$i$$, $I_i = \mathbf{1}_{\{\text{unit } i \text{ included in the sample}\}}$

• Sample selection processes can be characterized by a distribution for $$I$$ given $$Y$$ and $$Z$$.

### Probability sampling

• Properties of probability sampling

1. Unconfounded: selection doesn’t depend on $$Y$$, i.e., $f(I \mid Y, Z) = f(I \mid Z)$

2. Every unit has a positive (known) probability of selection $\pi_i = P(I_i = 1 \mid Z) > 0, \quad \text{for all } i$

• In equal probability sample design, $$\pi_i$$ is the same for all $$i$$

### Stratified random sampling

• $$Z$$ is a variable defining strata. Suppose Stratum $$Z=j$$ has $$N_j$$ units in total, for $$j= 1, \ldots, J$$

• In Stratum $$j$$, stratified random sampling takes a simple random sample of $$n_j$$ units

• The distribution of $$I$$ under stratified random sampling is $f(I \mid Z) = \prod_{j=1}^J {N_j \choose n_j}^{-1}$

### Example: estimating population mean$$\bar{Y}$$

• An unbiased estimate is the stratified sample mean $\bar{y}_{\text{st}} = \frac{\sum_{j=1}^J N_j \bar{y}_j}{N}$ where $$\bar{y}_j$$ is the sample mean in stratum $$j$$

• Sampling variance approximation $v(\bar{y}_{st}) \approx \frac{1}{N^2} \sum_{j=1}^J N_j^2 \left(\frac{1}{n_j} - \frac{1}{N_j} \right)s_j^2$ where $$s_j$$ is the sample variance of $$Y$$ in stratum $$j$$

• A large sample 95% confidence interval for $$\bar{Y}$$ is $\bar{y}_{\text{st}} \pm 1.96 \sqrt{v(\bar{y}_{st})}$

### Weighting methods

• Main idea: A unit selected with probability $$\pi_i$$ is “representing” $$\pi_i^{-1}$$ units in the population, hence should be given weights $$\pi_i^{-1}$$.

• For example, in stratified random sample

• A selected unit $$i$$ in stratum $$j$$ represents $$N_j/n_j$$ population units
• Thus by Horvitz-Thompson estimate, the population mean can be estimated by the weighted sum $\bar{y}_w = \frac{1}{n}\sum_{i=1}^n w_i y_i, \quad \pi_i = \frac{n_j}{N_j}, \quad w_i = n \cdot \frac{\pi_i^{-1}}{\sum_k \pi_k^{-1}}$
• It is not hard to show that $\bar{y}_w = \bar{y}_{\text{st}}$

### Weighting with nonresponses

• If the probability of selecting unit $$i$$ is $$\pi_i$$, and the probability of response for unit $$i$$ is $$\phi_i$$, then $P(\text{unit } i \text{ is observed}) = \pi_i \phi_i$

• Suppose there are $$r$$ units observed (respondents). Then the weighted estimate for $$\bar{Y}$$ is $\bar{y}_w = \frac{1}{r} \sum_{i=1}^r w_i y_i, \quad w_i = r \cdot \frac{(\pi_i \phi_i)^{-1}}{\sum_k (\pi_k \phi_k)^{-1}}$

• Usually $$\phi_i$$ is unknown and thus needs to be estimated

### Weighting class estimator

• Weighting class adjustments are used primarily to handle unit nonresponse

• Suppose we partition the sample into $$J$$ “weighting classes”. In the weighting class $$C = j$$:

• $$n_j$$: the sample size
• $$r_j$$: number of observed samples
• A simple estimator for $$\phi_j$$ is $$\hat{\phi}_j = \frac{r_j}{n_j}$$
• For equal probability designs, where $$\pi_i$$ is constant, the weighting class estimator is $\bar{y}_{\text{wc}} = \frac{1}{n}\sum_{j=1}^J n_j \bar{y}_{j\text{R}}$ where $$\bar{y}_{j\text{R}}$$ is the respondent mean in class $$j$$

• The estimate is unbiased under the following form of MAR assumption (Quasirandomization): data are MCAR within weighting class $$j$$

• Pros: handle bias with one set of weights for multivariate $$Y$$

• Cons: weighting is inefficient and can increase in sampling variance, if $$Y$$ is weakly related to the weighting class variable $$C$$

• How to choose weighting class adjustments: weighting is only effective for outcomes ($$Y$$) that are associated with the adjustment cell variable ($$C$$). See the right column in the table below.

### Propensity weighting

• The theory of propensity scores provides a prescription for choosing the coarsest reduction of $$X$$ to a weighting class variable $$C$$ so that quasirandomization is roughly satisfied

• Let $$X$$ denote the variables observed for both respondents and nonrespondents

• Suppose data are MAR, with $$\phi$$ being unknown parameters about missing mechanism $P(M \mid X, Y, \phi) = P(M \mid X, \phi)$ Then quasirandomization is satisfied when $$C$$ is chosen to be $$X$$

### Response propensity stratification

• Define response propensity for unit $$i$$ as $\rho(x_i, \phi) = P\left(m_i = 0 \mid \rho(x_i, \phi), \phi\right)$ i.e., respondents are a random subsample within strata defined by the propensity score $$\rho(X, \phi)$$

• Usually $$\phi$$ is unknown. So a practical procedure is

1. Estimate $$\hat{\phi}$$ from a binary regression of $$M$$ on $$X$$, based on respondent and nonrespondent data
2. Let $$C$$ be a grouped variable by coarsening $$\rho\left(X, \hat{\phi}\right)$$ into 5 or 10 values
• Thus, within the same adjustment class, all respondents and nonrespondents have the same value of the grouped propensity score

### An alternative procedure: propensity weighting

• An alternative procedure is to weight respondents $$i$$ directly by the inverse propensity score $$\rho\left(X, \hat{\phi}\right)^{-1}$$

• This method removes nonresponse bias

• But it may yield estimates with extremely high sampling variance because respondents with very low estimated response propensities receive large nonresponse weights

• Also, weighting directly by inverse propensities place may reliance on correct model specification of the regression of $$M$$ on $$X$$

### Example: inverse probability weighted generalized estimating equations (GEE)

• Let $$x_i$$ be covariates of GEE, and $$z_i$$ be a fully observed vector that can predict missing mechanism

• If $$P(m_i = 1 \mid x_i, y_i, z_i, \phi) = P(m_i = 1 \mid x_i, \phi)$$, then the unweighted completed case GEE is unbiased $\sum_{i=1}^r D_i(x_i, \beta)\left[y_i - g(x_i, \beta)\right] = 0$

• If $$P(m_i = 1 \mid x_i, y_i, z_i, \phi) = P(m_i = 1 \mid x_i, z_i, \phi)$$, then the inverse probability weighted GEE is unbiased $\sum_{i=1}^r w_i(\hat{\alpha}) D_i(x_i, \beta)\left[y_i - g(x_i, \beta)\right] = 0, \quad w_i(\hat{\alpha}) = \frac{1}{p(x_i, z_i \mid \hat{\alpha})}$ where $$p(x_i, z_i \mid \hat{\alpha})$$ is the probability of being a complete unit, based on logistic regression of $$m_i$$ on $$x_i, z_i$$

### Poststratification

• The weighting class estimator $\bar{y}_{\text{wc}} = \frac{1}{n}\sum_{j=1}^J n_j \bar{y}_{j\text{R}}$ uses the sample proportion $$n_j/n$$ to estimate the population proportion $$N_j/N$$.

• If from an external resource (e.g., census or a large survey), we know the population proportion of weighting classes, then we can use the post stratified mean to estimate $$\bar{Y}$$: $\bar{y}_{\text{ps}} = \frac{1}{N}\sum_{j=1}^J N_j \bar{y}_{j\text{R}}$

### Summary of weighting methods

• Weighted CC estimates are often simple to compute, but the appropriate standard errors can be hard to compute (even asymptotically)

• Weighting methods treat weights as fixed and known, but these nonresponse weights are computed from observed data and hence are subject to sampling uncertainty

• Because weighted CC methods discard incomplete units and do not provide an automatic control of sampling variance, they are most useful when

• Number of covariates is small, and
• Sample size is large

# Available-Case Analysis

### Available-case (AC) analysis

• Available-case analysis: for univariate analysis, include all unites where that variable is present

• Sample changes from variable to variable according to the pattern of missing data
• This is problematic if not MCAR
• Under MCAR, AC can be used to estimate mean and variance for a single variable
• Pairwise AC: estimates covariance of $$Y_j$$ and $$Y_k$$ based on units $$i$$ where both $$y_{ij}$$ and $$y_{ik}$$ are observed

• Pairwise covariance estimator: $s_{jk}^{(jk)} = \sum_{i \in I_{jk}} \left( y_{ij} - \bar{y}_j^{(jk)} \right) \left( y_{ik} - \bar{y}_k^{(jk)} \right)/ \left( n^{(jk)} - 1 \right)$ where $$I_{jk}$$ is the set of $$n^{(jk)}$$ units with both $$Y_j$$ and $$Y_k$$ observed

### Problems with pairwise AC estimators on correlation

• Correlation estimator 1: $r_{jk}^* = \frac{s_{jk}^{(jk)}}{\sqrt{s_{jj}^{(j)} s_{kk}^{(k)}}}$

• Problem: it can lie outside of $$(-1, 1)$$
• Correlation estimator 2 corrects the previous problem: $r_{jk}^{(jk)} = \frac{s_{jk}^{(jk)}}{\sqrt{s_{jj}^{(jk)} s_{kk}^{(jk)}}}$

• Under MCAR, all these estimators on covariance and correlation are consistent

• However, when $$K > 3$$, both correlation estimators can yield correlation matrices that are not positive definite!

• An extreme example: $$r_{12} = 1, r_{13} = 1, r_{23} = -1$$

### Compare CC and AC methods

• When data is MCAR and correlations are mild, AC methods are more efficient than CC

• When correlations are large, CC methods are usually better

### References

• Little, R. J., & Rubin, D. B. (2019). Statistical Analysis with Missing Data, 3rd Edition. John Wiley & Sons.