# 3  Missing Data

## 3.1 Types of Missing Data

• Missing completely at random (MCAR) A
• Missing at random (MAR)1
• Informative missing (non-ignorable non-response; missing not at random; MNAR)

1 “Although missing at random (MAR) is a non-testable assumption, it has been pointed out in the literature that we can get very close to MAR if we include enough variables in the imputation models” Harel & Zhou (2007).

See Carpenter & Smuk (2021), Schafer & Graham (2002), Donders et al. (2006), Harel & Zhou (2007), Allison (2001), White et al. (2011), Stef Buuren (2012) for an introduction to missing data and imputation concepts. See Hazewinkel et al. (2023) for a way to indirectly detect MNAR by comparing variances of response variables between treatments.

## 3.2 Prelude to Modeling

• Quantify extent of missing data B
• Characterize types of subjects with missing data
• Find sets of variables that are missing on same subjects

## 3.3 Missing Values for Different Types of Response Variables

• Serial data with subjects dropping out (not covered in this C course2
• $$Y$$=time to event, follow-up curtailed: covered under survival analysis3
• Missing time-dependent covariates in longitudinal data: Mamouris et al. (2023)
• Often discard observations with completely missing $$Y$$ but sometimes wasteful4
• Characterize missings in $$Y$$ before dropping obs.

2 Twisk et al. (2013) found instability in using multiple imputation of longitudinal data, and advantages of using instead full likelihood models.

3 White & Royston (2009) provide a method for multiply imputing missing covariate values using censored survival time data.

4 $$Y$$ is so valuable that if one is only missing a $$Y$$ value, imputation is not worthwhile, and imputation of $$Y$$ is not advised if MCAR or MAR.

## 3.4 Problems With Simple Alternatives to Imputation

Deletion of records—

• Badly biases parameter estimates when the probability of a D case being incomplete is related to $$Y$$ and not just $$X$$ .
• Deletion because of a subset of $$X$$ being missing always results in inefficient estimates
• Deletion of records with missing $$Y$$ can result in biases but is the preferred approach under MCAR5
• von Hippel (2007) found advantages to a “use all variables to impute all variables then drop observations with missing $$Y$$” approach (but see Sullivan et al. (2015))
• Lee & Carlin (2012) suggest that observations missing on both $$Y$$ and on a predictor of major interest are not helpful
• MCAR can be justified
• Rarely missing predictor of overriding importance that can’t be imputed from other data
• Fraction of obs. with missings small and $$n$$ is large
• No advantage of deletion except savings of analyst time
• Making up missing data better than throwing away real data
• See Knol et al. (2010)

5 Multiple imputation of $$Y$$ in that case does not improve the analysis and assumes the imputation model is correct.

Adding extra categories of categorical predictors—

• Including missing data but adding a category missing’ causes E serious biases
• Problem acute when values missing because subject too sick
• Difficult to interpret
• Fails even under MCAR
• May be OK if values are “missing” because of “not applicable”6
• May be OK for pure prediction where interpretation of coefficients is of no interest and patterns of missingness will be the same in future data as in the training data

6 E.g. you have a measure of marital happiness, dichotomized as high or low, but your sample contains some unmarried people. OK to have a 3-category variable with values high, low, and unmarried—Paul Allison, IMPUTE list, 4Jul09.

Likewise, serious problems are caused by setting missing continuous predictors to a constant (e.g., zero) and adding an indicator variable to try to estimate the effect of missing values.

Two examples from Donders et al. (2006) using binary logistic regression, $$N=500$$.

Results of 1000 Simulations With $$\beta_{1}=1.0$$ with MAR and Two Types of Imputation F

Imputation $$\hat{\beta}_{1}$$ S.E. Coverage of
Method 0.90 C.I.
Single 0.989 0.09 0.64
Multiple 0.989 0.14 0.90

Now consider a simulation with $$\beta_{1}=1, \beta_{2}=0$$, $$X_{2}$$ correlated with $$X_{1} (r=0.75)$$ but redundant in predicting $$Y$$, use missingness indicator when $$X_{1}$$ is MCAR in 0.4 of 500 subjects. This is also compared with grand mean fill-in imputation.

Results of 1000 Simulations Adding a Third Predictor Indicating Missing for $$X_{1}$$} G

Imputation $$\hat{\beta}_{1}$$ $$\hat{\beta}_{2}$$
Method
Indicator 0.55 0.51
Overall mean 0.55

In the incomplete observations the constant $$X_{1}$$ is uncorrelated with $$X_{2}$$.

## 3.5 Strategies for Developing an Imputation Model

The goal of imputation is to preserve the information and meaning of the non-missing data.

There is a full Bayesian modeling alternative to all the methods presented below. The Bayesian approach requires more effort but has several advantages .

Exactly how are missing values estimated?

• Predictive model for each target uses any outcomes, all predictors in the final model other than the target, plus auxiliary variables not in the outcome model
• Could ignore all other information — random or grand mean H fill-in
• Can use external info not used in response model (e.g., zip code for income)
• Need to utilize reason for non-response if possible
• Use statistical model with sometimes-missing $$X$$ as response variable
• Model to estimate the missing values should include all variables that are either I
1. related to the missing data mechanism;
2. have distributions that differ between subjects that have the target variable missing and those that have it measured;
3. associated with the sometimes-missing variable when it is not missing; or
4. included in the final response model
• Ignoring imputation results in biased $$\hat{V}(\hat{\beta})$$
• transcan function in Hmisc library: “optimal” transformations of all variables to make residuals more stable and to allow non-monotonic transformations
• aregImpute function in Hmisc: good approximation to full Bayesian multiple imputation procedure using the bootstrap
• transcan and aregImpute use the following for fitting imputation models: J
1. initialize NAs to median (mode for categoricals)
2. expand all categorical predictors using dummy variables
3. expand all continuous predictors using restricted cubic splines
4. optionally optimally transform the variable being predicted by expanding it with restricted cubic splines and using the first canonical variate (multivariate regression) as the optimum transformation (maximizing $$R^2$$)
5. one-dimensional scoring of categorical variables being predicted using canonical variates on dummy variables representing the categories (Fisher’s optimum scoring algorithm); when imputing categories, solve for which category yields a score that is closest to the predicted score
• aregImpute and transcan work with K fit.mult.impute to make final analysis of response variable relatively easy
• Predictive mean matching
• Recursive partitioning with surrogate splits — handles case where a predictor of a variable needing imputation is missing itself. But there are problems even with completely random missingness.
• White et al. (2011) discusses an alternative method based on choosing a donor observation at random from the $$q$$ closest matches ($$q=3$$, for example)

### 3.5.1 Interactions

• When interactions are in the outcome model, oddly enough it may L be better to treat interaction terms as “just another variable” and do unconstrained imputation of them

## 3.6 Single Conditional Mean Imputation

• Can fill-in using unconditional mean or median if number of M missings low and $$X$$ is unrelated to other $$X$$s
• Otherwise, first approximation to good imputation uses other $$X$$s to predict a missing $$X$$
• This is a single “best guess” conditional mean
• $$\hat{X}_{j} = Z \hat{\theta}, Z = X_{\bar j}$$ plus possibly auxiliary variables that precede $$X_{j}$$ in the causal chain that are not intended to be in the outcome model.
Cannot include $$Y$$ in $$Z$$ without adding random errors to imputed values as done with multiple imputation (would steal info from $$Y$$)
• Recursive partitioning can sometimes be helpful for nonparametrically estimating conditional means

## 3.7 Predictive Mean Matching

• Developed by Little & Rubin (2002)
• Replace missing value with observed value of subject having closest predicted value to the predicted value of the subject with the NA. Key considerations are how to
1. model the target when it is not NA
2. match donors on predicted values
3. avoid overuse of “good” donors to disallow excessive ties in imputed data
4. account for all uncertainties
• No distributional assumptions; nicely handles target variables with strange distributions
• Predicted values need only be monotonically related to real predictive values
• PMM can result in some donor observations being used repeatedly N
• Causes lumpy distribution of imputed values
• Address by sampling from multinomial distribution, probabilities = scaled distance of all predicted values to predicted value ($$y^{*}$$) of observation needing imputing
• Tukey’s tricube function is a good weighting function (used in loess): $$w_{i} = (1 - \min(d_{i}/s, 1)^{3})^{3}$$,
$$d_{i} = |\hat{y_{i}} - y^{*}|$$
$$s = 0.2\times\text{mean} |\hat{y_{i}} - y^{*}|$$ is a good default scale factor
scale so that $$\sum w_{i} = 1$$

### 3.7.1 Predictive Mean Matching With Constraints

• PMM is empirical so it is relatively easy to add constraints
• Need sample size to be large enough so that there are several donor observations meeting the constraints
• Implemented in Hmisc 5.1-1
• For each variable to be imputed the user can specify an R expression that is a logical TRUE/FALSE condition specifying which observations qualify
• Constraint can involve relationships between potential donor observations (variables prefixed by d$) and the recipient (target) observation (variables prefixed by r$) having the missing value
• PMM on the qualifying donor observations
• Example
• time 0 corresponds to the day a patient is admitted to the hospital
• day of discharge is missing for some patients
• every patient had at least one follow-up visit
• constrain the day of discharge to be before the day of the first follow-up visit
Code
a <- aregImpute(~ age + day_dis + day_follow_up, data=d,
constraint=list(day_dis=expression(d$day_dis < r$day_follow_up)))

## 3.8 Multiple Imputation

• Single imputation could use a random draw from the conditional O distribution for an individual
$$\hat{X}_{j} = Z \hat{\theta} + \hat{\epsilon}, Z = [X\bar{j}, Y]$$ plus auxiliary variables
$$\hat{\epsilon} = n(0, \hat{\sigma})$$ or a random draw from the calculated residuals
• bootstrap
• approximate Bayesian bootstrap : sample with replacement from sample with replacement of residuals
• Multiple imputations ($$M$$) with random draws
• Draw sample of $$M$$ residuals for each missing value to be imputed
• Average $$M$$ $$\hat{\beta}$$
• In general can provide least biased estimates of $$\beta$$
• Simple formula for imputation-corrected var($$\hat{\beta}$$)
Function of average “apparent” variances and between-imputation variances of $$\hat{\beta}$$
• Even when the $$\chi^2$$ distribution is a good approximation when data have no missing values, the $$t$$ or $$F$$ distributions are needed to have accurate $$P$$-values and confidence limits when there are missings
• BUT full multiple imputation needs to account for uncertainty in the imputation models by refitting these models for each of the $$M$$ draws
• transcan does not do that; aregImpute does
• Note that multiple imputation can and should use the response variable for imputing predictors
• aregImpute algorithm P
• Takes all aspects of uncertainty into account using the bootstrap
• Different bootstrap resamples used for each imputation by fitting a flexible additive model on a sample with replacement from the original data
• This model is used to predict all of the original missing and non-missing values for the target variable for the current imputation
• Uses flexible parametric additive regression models to impute
• There is an option to allow target variables to be optimally transformed, even non-monotonically (but this can overfit)
• By default uses predictive mean matching for imputation; no residuals required (can also do more parametric regression imputation)
• By default uses weighted PMM; many other matching options
• Uses by default van~Buuren’s “Type 1” matching to capture the right amount of uncertainty by computing predicted values for missing values using a regression fit on the bootstrap sample, and finding donor observations by matching those predictions to predictions from potential donors using the regression fit from the original sample of complete observations
• Uses Rubin’s rule to estimate variances of $$\hat{\beta}$$, involving between- and within-imputation variance combination
• When a predictor of the target variable is missing, it is first imputed from its last imputation when it was a target variable
• First 3 iterations of process are ignored (“burn-in”)
• Compares favorably to R MICE approach
• Example:
Code
a <- aregImpute(~ age + sex + bp + death + heart.attack.before.death,
data=mydata, n.impute=5)
f <- fit.mult.impute(death ~ rcs(age,3) + sex +
rcs(bp,5), lrm, a, data=mydata)

See Barzi & Woodward (2004) for a nice review of multiple imputation with detailed comparison of results (point estimates and confidence limits for the effect of the sometimes-missing predictor) for various imputation methods. Barnes et al. (2006) have a good overview of imputation methods and a comparison of bias and confidence interval coverage for the methods when applied to longitudinal data with a small number of subjects. Horton & Kleinman (2007) have a good review of several software packages for dealing with missing data, and a comparison of them with aregImpute. Harel & Zhou (2007) provide a nice overview of multiple imputation and discuss some of the available software. White & Carlin (2010) studied bias of multiple imputation vs. complete-case analysis. White et al. (2011) provide much practical guidance.

Caution: Methods can generate imputations having very reasonable distributions but still not having the property that final response model regression coefficients have nominal confidence interval coverage. It is worth checking that imputations generate the correct collinearities among covariates.

• With MICE and aregImpute we are using the chained Q equation approach R
• Chained equations handles a wide variety of target variables to be imputed and allows for multiple variables to be missing on the same subject
• Iterative process cycles through all target variables to impute all missing values
• Does not attempt to use the full Bayesian multivariate model for all target variables, making it more flexible and easy to use
• Possible to create improper imputations, e.g., imputing conflicting values for different target variables
• However, simulation studies demonstrate very good performance of imputation based on chained equations

## 3.9 Likelihood Ratio Tests and Multiple Imputation

• For frequentist-based analysis Wald tests are ultimately unsatisfying for reasons detailed in Section 9.4
• Also it was never clear that simple averaging of $$\hat{\beta}$$ over completed datasets is the best approach
• Chan & Meng (2022) have developed a promising approach to obtaining approximate LRT (likelihood ratio tests) in the presence of missing data
• Instead of averaging $$M$$ $$\hat{\beta}$$ from multiple separate analyses of completed datasets, stacks all completed datasets into one large dataset and computes maximum likelihood estimates on the one large stacked dataset
• Divide LR $$\chi^2$$ by $$M$$ to get a rough estimate of the more correct LR described below; need to multiply by a discounting factor to take imputation into account
• Individual fit LRs are used to derive imputation correction factors and the fraction of missing information
• Chan and Meng have a formula that provides more accuate $$p$$-values by solving for denominator degrees of freedom for an $$F$$ distribution
• The degrees of freedom must be computed separately for each hypothesis test
• Accounts for heavier tail problem mentioned earlier
• $$F$$ test denominator d.f. is $$\frac{k (M - 1)}{\hat{f}^{2}}$$ where $$k$$ is the numerator d.f. (number of parameters being tested simultaneously), the fraction of missing information $$\hat{f} = \frac{\hat{r}}{1+\hat{r}}$$, $$\hat{r} = \max(0, \frac{M+1}{k(M-1)} (\bar{d} - \hat{d}))$$, where $$\bar{d}$$ is the mean over imputed datasets of the individual LR statistics and $$\hat{d}$$ is LR on the stacked dataset, divided by $$M$$
• Use large $$M$$ to make the $$\chi^2$$ approximation better by making the denominator d.f. larger. E.g., if $$k=1$$ and $$\hat{f}=0.5$$, $$M=26$$ already yields more than 100 denominator d.f. for $$F$$, making it hard to distinguish from a $$\chi_{1}^{2}$$ distribution. If the test statistic is 4.0, the $$p$$-value $$\chi_{1}^{2}$$ is 0.0455 and is 0.048 from $$F_{1, 100}$$.
• Hmisc fit.mult.impute function run with lrt=TRUE runs and saves all the needed LR tests to allow rms processMI to compute everything else
• processMI computes $$\hat{f}$$, the $$F$$ test denominator d.f., and the $$\chi^2$$ discount factor and uses these to get the final imputation-adjusted LR tests
• fit.mult.impute computes approximate Wald statistics by multiplying the variance-covariance matrix computed on the stacked data by $$M$$ if individual completed dataset analyses are not done (not recommended)
• Chan and Meng approach uses $$\hat{\beta}$$ computed once on stacked data as the MLE in the multiple imputation context; fit.mult.impute computes the final $$\hat{\beta}$$ this way if stacking is done; however the variance-covariance matrix is estimated from Rubin’s rule as in the Wald approach first used in this chapter
• Imputation-adjusted LRTs can be a large computational burden for large datasets—for $$M$$ imputations, $$j$$ effects being tested requires $$(M+1) j$$ model fits. For each imputation, the rms anova function has very litle overhead though, due to it only computing the design matrix once, and calling low-level fitting functions for all LRTs once the overall model is fitted with a high-level function.

Example

Code
require(rms)
set.seed(1)
n <- 500
x1 <- runif(n)
x2 <- runif(n)
L  <- x1 + 2*x2
y  <- rbinom(n, 1, plogis(L))
x2[1:300] <- NA   # make 300 observations have missing x2
d <- data.frame(x1, x2, y)
a <- aregImpute(~ y + x1 + x2, n.impute=10, data=d, pr=FALSE)
f <- fit.mult.impute(y ~ x1 + x2, lrm, a, data=d, pr=FALSE)
f

Logistic Regression Model

fit.mult.impute(formula = y ~ x1 + x2, fitter = lrm, xtrans = a,
data = d, pr = FALSE)

Model Likelihood
Ratio Test
Discrimination
Indexes
Rank Discrim.
Indexes
Obs 500 LR χ2 48.78 R2 0.150 C 0.725
0 93 d.f. 2 R22,500 0.089 Dxy 0.451
1 407 Pr(>χ2) <0.0001 R22,227.1 0.183 γ 0.451
max |∂log L/∂β| 3×10-7 Brier 0.138 τa 0.137
β S.E. Wald Z Pr(>|Z|)
Intercept  -0.2395  0.4788 -0.50 0.6169
x1   1.6851  0.4804 3.51 0.0005
x2   2.2036  0.8701 2.53 0.0113

In the above results, the LR $$\chi^2$$ did not account for imputation and is too high. So are the $$R^2$$ measures derived from it.

Compute Wald $$\chi^2$$ test statistics accounting for imputation

Code
anova(f)
Wald Statistics for y
χ2 d.f. P
x1 12.30 1 0.0005
x2 6.41 1 0.0113
TOTAL 15.96 2 0.0003

Now compute LR test statistics accounting for imputation

Code
# lrt=TRUE makes fit.mult.impute set up all the arguments needed for LR tests
h <- fit.mult.impute(y ~ x1 + x2, lrm, a, data=d, lrt=TRUE, pr=FALSE)
h

Logistic Regression Model

fit.mult.impute(formula = y ~ x1 + x2, fitter = lrm, xtrans = a,
data = d, lrt = TRUE, pr = FALSE)

Model Likelihood
Ratio Test
Discrimination
Indexes
Rank Discrim.
Indexes
Obs 500 LR χ2 45.74 R2 0.142 C 0.724
0 93 d.f. 2 R22,5000 0.087 Dxy 0.449
1 407 Pr(>χ2) <0.0001 R22,2271.1 0.182 γ 0.449
max |∂log L/∂β| 7×10-14 Brier 0.139 τa 0.136
β S.E. Wald Z Pr(>|Z|)
Intercept  -0.2047  0.4788 -0.43 0.6690
x1   1.6607  0.4804 3.46 0.0005
x2   2.1519  0.8701 2.47 0.0134
Code
as <- processMI(h, which='anova')  # LR tests accounting for imputation
as
Likelihood Ratio Statistics for y
χ2 d.f. P
x1 10.23 1 0.0014
x2 6.30 1 0.0121
TOTAL 16.01 2 0.0003
Code
prmiInfo(as)
Imputation penalties
Test Missing
Information
Fraction
Denominator
d.f.
χ2 Discount
x1 0.336 79.5 0.664
x2 0.788 14.5 0.212
TOTAL 0.650 42.6 0.350

The Chan & Meng $$F$$ approximation (processMI uses the $$\chi^2$$ version of it) is needed for x2 as the denominator d.f. of 14.5 is too low for $$p$$-values from the $$\chi^2$$ distribution to well approximate those from the $$F$$ distribution. Increase the number of imputations to increase the denominator d.f.

Code
a <- aregImpute(~ y + x1 + x2, n.impute=50, data=d, pr=FALSE)
h <- fit.mult.impute(y ~ x1 + x2, lrm, a, data=d, lrt=TRUE, pr=FALSE)
h

Logistic Regression Model

fit.mult.impute(formula = y ~ x1 + x2, fitter = lrm, xtrans = a,
data = d, lrt = TRUE, pr = FALSE)

Model Likelihood
Ratio Test
Discrimination
Indexes
Rank Discrim.
Indexes
Obs 500 LR χ2 40.49 R2 0.126 C 0.712
0 93 d.f. 2 R22,25000 0.078 Dxy 0.425
1 407 Pr(>χ2) <0.0001 R22,11355.3 0.163 γ 0.425
max |∂log L/∂β| 2×10-13 Brier 0.141 τa 0.129
β S.E. Wald Z Pr(>|Z|)
Intercept  -0.0783  0.3986 -0.20 0.8443
x1   1.5982  0.4640 3.44 0.0006
x2   1.9308  0.7341 2.63 0.0085
Code
as <- processMI(h, which='anova')
as
Likelihood Ratio Statistics for y
χ2 d.f. P
x1 12.10 1 0.0005
x2 6.74 1 0.0094
TOTAL 17.51 2 0.0002
Code
prmiInfo(as)
Imputation penalties
Test Missing
Information
Fraction
Denominator
d.f.
χ2 Discount
x1 0.153 2097.1 0.847
x2 0.724 93.4 0.276
TOTAL 0.568 304.2 0.432

What happens if we use an extremely high number of imputations?

Code
a <- aregImpute(~ y + x1 + x2, n.impute=500, data=d, pr=FALSE)
h <- fit.mult.impute(y ~ x1 + x2, lrm, a, data=d, lrt=TRUE, pr=FALSE)
h

Logistic Regression Model

fit.mult.impute(formula = y ~ x1 + x2, fitter = lrm, xtrans = a,
data = d, lrt = TRUE, pr = FALSE)

Model Likelihood
Ratio Test
Discrimination
Indexes
Rank Discrim.
Indexes
Obs 500 LR χ2 42.41 R2 0.132 C 0.717
0 93 d.f. 2 R22,250000 0.081 Dxy 0.433
1 407 Pr(>χ2) <0.0001 R22,113553 0.170 γ 0.433
max |∂log L/∂β| 5×10-12 Brier 0.140 τa 0.131
β S.E. Wald Z Pr(>|Z|)
Intercept  -0.1123  0.3900 -0.29 0.7734
x1   1.5950  0.4688 3.40 0.0007
x2   2.0122  0.7207 2.79 0.0052
Code
as <- processMI(h, which='anova')
as
Likelihood Ratio Statistics for y
χ2 d.f. P
x1 11.12 1 0.0009
x2 7.85 1 0.0051
TOTAL 19.45 2 <0.0001
Code
prmiInfo(as)
Imputation penalties
Test Missing
Information
Fraction
Denominator
d.f.
χ2 Discount
x1 0.218 10542.2 0.782
x2 0.702 1011.3 0.298
TOTAL 0.541 3405.6 0.459

## 3.10 Diagnostics

• MCAR can be partially assessed by comparing distribution of S non-missing $$Y$$ for those subjects with complete $$X$$ vs. those subjects having incomplete $$X$$
• Yucel and Zaslavsky (Yucel & Zaslavsky, 2008; see also He & Zaslavsky, 2012)
• Interested in reasonableness of imputed values for a sometimes-missing predictor $$X_{j}$$
• Duplicate entire dataset
• In the duplicated observations set all non-missing values of $$X_{j}$$ to missing; let $$w$$ denote this set of observations set to missing
• Develop imputed values for the missing values of $$X_{j}$$
• In the observations in $$w$$ compare the distribution of imputed $$X_{j}$$ to the original values of $$X_{j}$$
• Bondarenko & Raghunathan (2016) present a variety of useful diagnostics on the reasonableness of imputed values.

## 3.11 Summary and Rough Guidelines

T

The following contains crude guidelines. Simulation studies are needed to refine the recommendations. Here $$f$$ refers to the proportion of observations having any variables missing.

• $$f < 0.03$$: U It doesn’t matter very much how you impute missings or whether you adjust variance of regression coefficient estimates for having imputed data in this case. For continuous variables imputing missings with the median non-missing value is adequate; for categorical predictors the most frequent category can be used. Complete case analysis is also an option here. Multiple imputation may be needed to check that the simple approach “worked.”
• $$f \geq 0.03$$: Use multiple imputation with number of imputations7 equal to $$\max(5, 100f)$$. Fewer imputations may be possible with very large sample sizes. See statisticalhorizons.com/how-many-imputations. Type 1 predictive mean matching is usually preferred, with weighted selection of donors. Account for imputation in estimating the covariance matrix for final parameter estimates. Use the $$t$$ distribution instead of the Gaussian distribution for tests and confidence intervals, if possible, using the estimated d.f. for the parameter estimates.
• Multiple predictors frequently missing: V More imputations may be required. Perform a “sensitivity to order” analysis by creating multiple imputations using different orderings of sometimes missing variables. It may be beneficial to initially sort variables so that the one with the most NAs will be imputed first. aregImpute cycles in the order the analyst specifies variables in the formula.

7 White et al. (2011) recommend choosing $$M$$ so that the key inferential statistics are very reproducible should the imputation analysis be repeated. They suggest the use of $$100f$$ imputations. See also . von Hippel (2016) finds that the number of imputations should be quadratically increasing with the fraction of missing information.

Reason for missings more important than number of missing values.

Extreme amount of missing data does not prevent one from using multiple imputation, because alternatives are worse .

### 3.11.1 Effective Sample Size

It is useful to look at examples of effective sample sizes in the presence of missing data. If a sample of 1000 subjects contains various amounts and patterns of missings what size $$n_c$$ of a complete sample would have equivalent information for the intended purpose of the analysis?

1. A new marker was collected on a random sample of 200 of the subjects and one wants to estimate the added predictive value due to the marker: $$n_{c}=200$$ W
2. Height is missing on 100 subjects but we want to study association between BMI and outcome. Weight, sex, and waist circumference are available on all subjects: $$n_{c}=980$$
3. Each of 10 predictors is randomly missing on $$\frac{1}{10}$$ of subjects, and the predictors are uncorrelated with each other and are each weakly related to the outcome: $$n_{c}=500$$
4. Same as previous but the predictors can somewhat be predicted from non-missing predictors: $$n_{c}=750$$
5. The outcome variable was not assessed on a random $$\frac{1}{5}$$ of subjects: $$n_{c}=800$$
6. The outcome represents sensitive information, is missing on $$\frac{1}{2}$$ of subjects, and we don’t know what made subjects respond to the question: $$n_{c}=0$$ (serious selection bias)
7. One of the baseline variables was collected prospectively $$\frac{1}{2}$$ of the time and for the other subjects it was retrospectively estimated only for subjects ultimately suffering a stroke and we don’t know which subjects had a stroke: $$n_{c}=0$$ (study not worth doing)
8. The outcome variable was assessed by emailing the 1000 subjects, for which 800 responded, and we don’t know what made subjects respond: $$n_{c}=0$$ (model will possibly be very biased—at least the intercept)

## 3.12 Bayesian Methods for Missing Data

• Multiple imputation developed as an approximation to a full X Bayesian model
• Full Bayesian model treats missings as unknown parameters and provides exact inference and correct measures of uncertainty
• See this case study for an example
• The case study also shows how to do “posterior stacking” if you want to avoid having to specify a full model for missings, and instead use usual multiple imputations as described in this chapter. See Zhou & Reiter (2012).
• Run a multiple imputation algorithm
• For each completed dataset run the Bayesian analysis and draw thousands of samples from the posterior distribution of the parameters
• Pool all these posterior draws over all the multiple imputations and do posterior inference as usual with no special correction required
• Made easy by the Hmisc package aregImpute function and the rms stackMI` function as demonstrated in the Titanic case study later in the notes.

## 3.13 Study Questions

Section 3.1

1. What is the problem with doing ordinary analysis of data from survey responders?

Section 3.4

1. What problem is always present when doing complete-case analysis when missing values exist in the data?
2. What problem is often present?
3. Why does imputation not help very much when a variable being imputed is a main variable of interest in the analysis?
4. What is a major reason that adding a new category for a predictor for missings doesn’t work?
5. Why does inserting a constant for missing values of a continuous predictor primarily fail?

Section 3.5

1. What is a more accurate statement than “imputation boosts the sample size”?
2. What does single-value fill-in of missings almost always damage?
3. What is a general way to describe why predictive mean matching (PMM) works?
4. What is a general advantage of PMM?

Section 3.6

1. Why does single conditional mean imputation result in biased regression coefficients?

Section 3.8

1. Why can multiple imputation use Y in predicting X?
2. What are the sources of uncertainty that a multiple imputation algorithm must take into account for final standard errors to not be underestimated?

Section 3.9

1. Explain the Yucel-Zaslavsky diagnosic and what it is checking for.

Section 3.10

1. What is the only reason not to always do 100 or more imputations?

Section 3.11

1. If using multiple imputation but within a Bayesian framework, what is a major advantage of posterior stacking over what we have been doing in the frequentist domain?