# Book Notes: Flexible Imputation of Missing Data -- Ch3 Univariate Missing Data

### Notations

• In this chapter, we assume that there is only one variable having missing values. We call this variable $$y$$ the target variable.

• $$y_\text{obs}$$: the $$n_1$$ observed data in $$y$$
• $$y_\text{mis}$$: the $$n_0$$ missing data in $$y$$
• $$\dot{y}$$: imputed values in $$y$$
• Suppose $$X$$ are the variables (covariates) in the imputation model.

• $$X_\text{obs}$$: the subset of $$n_1$$ rows of $$X$$ which $$y$$ is observed
• $$X_\text{mis}$$: the subset of $$n_0$$ rows of $$X$$ which $$y$$ is missing

# Imputation under the Normal Linear Model

### Four methods to impute under the normal linear model

1. Regression imputation: Predict (bad!). Fit a linear model on the observed data and get the OLS estimates $$\hat{\beta}_0, \hat{\beta}_1$$. Impute with the predicted values $\dot{y} = \hat{\beta}_0 + X_\text{mis} \hat{\beta}_1$
• In mice package, this method is norm.predict
2. Stochastic regression imputation: Predict + noise (better, but still bad). Also add a random drawn noise from the estimated residual normal distribution $\dot{y} = \hat{\beta}_0 + X_\text{mis} \hat{\beta}_1 + \dot{\epsilon}, \quad \dot{\epsilon} \sim \text{N}(0, \hat{\sigma}^2)$
• In mice package, this method is norm.nob

### Method 3: Bayesian multiple imputation

• Predict + noise + parameter uncertainty $\dot{y} = \dot{\beta}_0 + X_\text{mis} \dot{\beta}_1 + \dot{\epsilon}, \quad \dot{\epsilon} \sim \text{N}(0, \dot{\sigma}^2)$

• Under the priors (where the hyper-parameter $$\kappa$$ is fixed at a small value, e.g., $$\kappa = 0.0001$$) $\beta \sim \text{N}(0, \mathbf{I}_p/\kappa), \quad p(\sigma^2) \propto 1/\sigma^2$ We draw $$\dot{\beta}$$ (including both $$\dot{\beta}_0$$ and $$\dot{\beta}_1$$), $$\dot{\sigma^2}$$ from the posterior distribution

• In mice package, this method is norm

### Method 4: Bootstrap multiple imputation

• Predict + noise + parameter uncertainty $\dot{y} = \dot{\beta}_0 + X_\text{mis} \dot{\beta}_1 + \dot{\epsilon}, \quad \dot{\epsilon} \sim \text{N}(0, \dot{\sigma}^2)$ where $$\dot{\beta}_0$$, $$\dot{\beta}_1$$, and $$\dot{\sigma^2}$$ are OLS estimates calculated form a bootstrap sample taken from the observed data

• In mice package, this method is norm.boot

### A simulation study, to impute MCAR missing in $$y$$

• Missing rate $$50\%$$ in $$y$$, and number of imputations $$m = 5$$.
• From coverage, norm, norm.boot, and listwise deletion are good
• From CI width, listwise deletion is better than multiple imputation here, but it’s not always this case, especially when the number of covariates is large.
• RMSE is not imformative at all!

### A simulation study, to impute MCAR missing in $$x$$

• Missing rate $$50\%$$ in $$x$$, and number of imputations $$m = 5$$.
• norm.predict is severely biased; norm is slightly biased
• From coverage, norm, norm.boot, and listwise deletion are good
• Again, RMSE is not imformative at all!

### Impute from a (continuous) non-normal distributions

• Optional 1: mean predictive matching

• Optional 2: model the non-normal data directly
• E.g., impute from a t-distribution
• The GAMLSS package: extends GLM and GAM

# Predictive Mean Matching

### Predictive mean matching (PMM), general principle

• For each missing entry, the method forms a small set of candidate donors (3, 5, or 10) from completed cases whose predicted values closest to the predicted value for the missing entry

• One donor is randomly drawn from the candidates, and the observed value of the donor is taken to replace the missing value

### Advantages of predictive mean matching (PMM)

• PMM is fairly robust to transformations of the target variable

• PMM can also be used for discrete target variables

• PMM is fairly robust to model misspecification
• In the following example, the relationship between age and BMI is not linear, but PMM seems to preserve this relationship better than linear normal model

### How to select the donors

• Once the metric has been defined, there are four ways to select the donors.
• Let $$\hat{y}_i$$ denote the predicted values of rows with observed $$y_i$$
• Let $$\hat{y}_j$$ denote the predicted values of rows with missing $$y_j$$
1. Pre-specify a threshold $$\eta$$, take all $$i$$ such that $$\left| \hat{y}_i - \hat{y}_j\right| < \eta$$ as donors, and randomly sample one donor to impute
2. Choose the closest candidate as the donor (only 1 donor), also called (nearest neighbor hot deck)
3. Pre-specify a number $$d$$, take the $$d$$ closest candidate as donors, and randomly sample one donor to impute. Usually, $$d = 3, 5, 10$$
4. Sample one donor with a probability that depends on the distance $$\left| \hat{y}_i - \hat{y}_j\right|$$
• Implemented by the midastouch method in mice, and also the midastouch package

### Types of matching

• Type 0: $$\hat{y} = X_\text{obs} \hat{\beta}$$ is matched to $$\hat{y}_j = X_\text{mis} \hat{\beta}$$
• Bad: it ignores the sampling variability in $$\hat{\beta}$$
• Type 1: $$\hat{y} = X_\text{obs} \hat{\beta}$$ is matched to $$\dot{y}_j = X_\text{mis} \dot{\beta}$$
• Here, $$\dot{\beta}$$ is a random draw from the posterior distribution
• Good. The default in mice
• Type 2: $$\dot{y} = X_\text{obs} \dot{\beta}$$ is matched to $$\dot{y}_j = X_\text{mis} \dot{\beta}$$
• Not very ideal, when model is small, the same donors get selected too often
• Type 3: $$\dot{y} = X_\text{obs} \dot{\beta}$$ is matched to $$\ddot{y}_j = X_\text{mis} \ddot{\beta}$$
• Here, $$\dot{\beta}$$ and $$\ddot{\beta}$$ are two different random draws from the posterior distribution
• Good

### Number of donors $$d$$

• $$d=1$$ is too low (bad!). It may select the same donor over and over again

• The default in mice is $$d=5$$. Also, $$d = 3, 10$$ are also feasible

### Pitfalls of PMM

• If the data is small, or if there is a region where the missing rate is high, then the same donors may be used for too many times.

• Mis-specification of the impute model

• PMM cannot be used to extrapolate beyond the range of the data, or to interpolate within the region where data is sparse

• PMM may not perform well with small datasets

# Imputation under CART

### Multiple imputation under a tree model

• missForest: single imputation with CART is bad

• Multiple imputation under a tree model using the bootstrap:
1. Draw a bootstrap sample among the observed data, and fit a CART model $$f(X)$$
2. For each missing value $$y_j$$, find it’s terminal node $$g_j$$. All the $$d_j$$ cases in this node are the donors
3. Randomly select one donor to impute

• When fitting the tree, it may be useful to pre-set the size of nodes to be 5 or 10
• We can also use random forest instead of CART

# Imputing Categorical and Other Types of Data

### Imputation under Bayesian GLMs

• Binary data: logistic regression (logreg method in mice)
• In case of data separation, use a more informative Bayesian prior
• Categorical variable with $$K$$ unordered categories: multinomial logit model (polyreg method in mice package) $P(y_i = k\mid X_i, \beta) = \frac{\exp(X_i \beta_k)}{\sum_{j=1}^K \exp(X_i \beta_j)}$

• Categorical variable with $$K$$ ordered categories: ordered logit model (polr method in mice package) $P(y_i \leq k\mid X_i, \beta, \tau_k) = \frac{\exp(\tau_k - X_i \beta)}{1 + \exp(\tau_k - X_i \beta)}$
• For identifiability, set $$\tau_1 = 0$$
• When impute from these GLM models, make sure to not use the MLE of parameters, but either a draw from posterior, or a bootstraped estimate.

### Categorical variables are harder to impute than continuous ones

• Empirically, the GLM imputations do not perform well
• If missing rate exceeds 0.4
• If the data is imbalanced
• If there are many categories
• GLM imputation is found inferior than CART or latent class models

### Imputation of count data

• Option 1: predictive mean matching
• Option 2: ordered categorical imputation
• Option 3: (zero-inflated) Poisson regression
• Option 4: (zero-inflated) negative binomial regression

### Imputation of semi-continuous data

• Semi-continuous data: has a high mass at one point (often zero) and a continuous distribution over the remaining values

• Option 1: model the data in two parts: logistic regression + regression
• Option 2: predictive mean matching

### References

• Van Buuren, S. (2018). Flexible Imputation of Missing Data, 2nd Edition. CRC press.