Missing Data and Imputation

Introduction

• Missing Data

  • Common occurrence among datasets
    • Not widely discussed how to resolve
  • Alone, not a problem
    • Becomes problem when analysis shows bias or lacks power

Issues with Missing Data

• Within experiments and analyses, missing data can lead to

  • Inaccurately distributed data and calculations
  • Skewed visuals
  • Inaccurate conclusions

Types of Missing Data

• Missing data categorized into one of three types:
  • Missing Completely at Random (MCAR)
  • Missing at Random (MAR)
  • Missing Not at Random (MNAR)

Missing Completely at Random (MCAR)

• Probability that missing data not related to either
  • Specific value supposed to be obtained
  • Set of obtained values
• MCAR data is unbiased

Missing Completely at Random (MCAR), continued

• Examples:
  • Equipment failure
  • Samples lost in transit
  • Unsatisfactory samples

Missing at Random (MAR)

• Probability that missing responses are
  • Dependent upon set of observed responses
  • Not related to specific expected values
• Most realistic option for missing data
• Missing data is knowable, missingness is predictable
  • Missingness not at random; is random condition on observed values from entire dataset
  • Estimates determined
  • Bias recovered

Missing at Random (MAR), continued

• Example: Men less likely to fill out depression survey
  • Reason: because of society
    • Not because of lack of depression symptoms

Missing Not at Random (MNAR)

• Missing data not classified as either MCAR or MAR
  • No bias is observed
  • Power is affected
    • Larger Standard Error (SE) due to reduced sample size
  • Least desirable missing data scenario
• When MNAR occurs, research subjects affect variables
• Example: Subjects don’t disclose accurate information for fear or shame; forego providing data altogether

Types of Missing Data Solutions

• Missing Imputation (MI)
• Multivariate Imputation by Chained Equation (MICE)
• Single Center Imputation from Multiple Chained Equation (SICE)

Multiple Imputation (MI)

• Most common method
• Results similar to using complete datasets
  • Resolves issue of too small or too large standard errors
  • Large standard error (SE) - results acquired lack precision
  • Small standard error (SE) - results acquired with overestimation of precision

Multivariate Imputation by Chained Equation (MICE)

• Assumes data is Missing at Random (MAR)

• Purpose: Consider uncertainty of missing data

• Uses multiple imputation methods to provide better results

Single Center Imputation from Multiple Chained Equation (SICE)

• Alternative to MICE
• Improves upon MICE by creating hybrid of single and multiple imputation techniques
• Uses the respective SICE variant (categorical or numeric)
  • Missing data values corrected using thorough approach to find more accurate value to use
  • Uses predicted values imputed from MI approach

Single Center Imputation from Multiple Chained Equation(SICE), continued

• Computes mean or mode for imputed values (depending on data type)
• Replaces original imputed value with respective central measure
• Has lowest computation time all all missing data solutions
  • Purpose: Replace predicted imputed values computed using MI with central measure computed using SICE (Khan and Hoque 2020)

Methods

• Dataset: 21 specific chronic illnesses of Medicare beneficiaries, from U.S. Department of Health & Human Services (Medicare & Medicaid Services 2018).

• Goal: Fill in missing data (2244 missing entries, about 5%)

• Approach: Imputation using 3 approaches: Predictive Means Matching, Classification and Regression Trees, Lasso Regression, and Random Forest

  • These are all forms of Multiple Imputation: perform single imputation several times to create multiple data sets, analyze and compute the error of each set, then combine the data into a single, final data set (Little et al. 2014).

Figure 1: Multiple Imputation Process using 5 sets

Figure 1: Multiple Imputation Process using 5 sets

Classification and Regression Trees(CART)

• Machine Learning

• Robust

• Flexible

• Straightforward(in R)

  • Automates variable selection, missing values, outliers, variable interaction, and nonlinear relationships

• In practice works similarly to PMM with tree instead of regression

Random Forest(Miss Forest)

• Subset of CART

• Uses CART and Rubin’s rules automatically to yield a single dataset

• Does not account for uncertainty and increases P values

• Usually similar to average of all predictions from the CART models

• Better predictions and accuracy than a single CART model

Analysis and Results

• Prior to implementing MICE method

  • Utilized a Missing Map to create a visualization

  • Verified three variables missing entries: Provisional Income(408), Total Medicare Standardized Payment(918), and Total Medicare Payment(908)

  • Total of 2244 missing data points

Code

# loading packages

library(ggplot2)

library(dplyr)

library(mice)

library(missForest)

library(VIM)

library(ggmice)

library(xlsx)

library(readxl)

library(knitr)

Figure 2: Missing Map

## reading data file from github


Chronic_Conditions <- read_excel("Chronic_Conditions.xlsx")

#Calculate pattern of missing data

Chronic_Conditions <- Chronic_Conditions %>%

select( PrvInc , Stdzd_Pymt_PC, Pymt_PC)

## Display table of missing entries per variable(complete columns were omitted)

plot_pattern(Chronic_Conditions)

Figure 3: Histogram of Missing Data

#Histogram of that same missingness data

aggr_plot <- aggr(Chronic_Conditions, col=c('navyblue','red'),

numbers=TRUE, sortVars=FALSE, labels=names(data), cex.axis=.7, gap=3,

ylab=c("Histogram of missing data","Pattern"))

MICE Package

• Following the missing map, we ran the MICE package in R.

# Imputes missing data using the three selected methods 
mice_imputed <- data.frame(
  original = Chronic_Conditions$PrvInc,
 imputed_pmm = complete(mice(Chronic_Conditions, method = "pmm", printFlag = FALSE))$PrvInc,
  imputed_cart = complete(mice(Chronic_Conditions, method = "cart", printFlag = FALSE))$PrvInc,
  imputed_lasso = complete(mice(Chronic_Conditions, method = "lasso.norm", printFlag = FALSE))$PrvInc)



#Imputation using miss Forest 
Chronic_Conditions.mis <- prodNA(Chronic_Conditions, noNA = 0.1)

Figure 4: Missing data percentages per variable

# Plots missing data percentages per variable

aggr(Chronic_Conditions.mis, col=c('navyblue', 'yellow' ),

numbers=TRUE, sortVars=FALSE,

labels=names(Chronic_Conditions.mis), cex.axis=.7,

gap=3, ylab=c("Missing Data", "Pattern"))

     PrvInc Stdzd_Pymt_PC Pymt_PC     
2707      1             1       1    0
320       1             1       0    1
296       1             0       1    1
762       1             0       0    2
513       0             1       1    1
64        0             1       0    2
52        0             0       1    2
200       0             0       0    3
        829          1310    1346 3485

Miss Forest

#Missingness data for miss Forest

plot_pattern(Chronic_Conditions.mis)

Figure 5: Table of Missing Values - Miss Forest

#Summarizes the POOLED data after miss Forest imputation

summary(Chronic_Conditions.mis)

     PrvInc       Stdzd_Pymt_PC      Pymt_PC      
 Min.   :0.0000   Min.   :    0   Min.   :     0  
 1st Qu.:0.0394   1st Qu.:18725   1st Qu.: 18500  
 Median :0.1114   Median :23292   Median : 23141  
 Mean   :0.1733   Mean   :24099   Mean   : 24142  
 3rd Qu.:0.2767   3rd Qu.:28251   3rd Qu.: 28356  
 Max.   :0.7588   Max.   :89235   Max.   :100028  
 NA's   :829      NA's   :1310    NA's   :1346    

Figure 6: Density Plot

# Density plot to compare to original data

densityplot(imputed_Data)

Conclusion

• Missing data is common

• Multiple Imputation(MI) is the act of completing data sets where there are missing data points

• Less biased and more accurate outcome

• Multivariate Imputation by Chained Equations(MICE) is an effective way of imputing data

Conclusion, continued

• Benefits

  • Reduced uncertanty

  • Reduced Bias

  • Multiple methods of imputation in MICE package (PMM, lasso, and CART)