SMHS MissingData

Scientific Methods for Health Sciences - Missing Data

Questions

Why is data usually incomplete?

What are the best strategies for dealing with missing data? Ignore cases, replace them by some population derived values, or impute them?

What is the impact of data manipulations on the core scientific inference?

Overview

Many research studies encounter incomplete (missing) data that require special handling (e.g., processing, statistical analysis, visualization). There are a variety of methods (e.g., multiple imputation) to deal with missing data, detect missingness, impute the data, analyze the completed data-set and compare the characteristics of the raw and imputed data.

Multiple imputation involves 3 steps.

1) Impute: Create sets of plausible values for the missing observations that reflect uncertainty about the non-response model. Each of these sets of plausible values can be used to “fill-in” or complete the data-set.

2) Analyze: process each of these imputed data-sets using complete-data methods.

3) Combine: synthesize the results accounting for the uncertainty within each imputation round.

In a general regression setting, let’s denote the scalar, or vector valued, outcomes by Y, and the corresponding vector of predictors by X. For a given case (e.g., subject, unit), these quantities are either observed (obs) or missing (mis). Thus, Y_obs and X_obs represent the observed component of the outcome and the predictors; and Y_mis and X^mis denote the unobserved components of the outcome and predictors, respectively. Imputation involves the estimation of the regression parameters β governing the conditional distribution of Y given X: f(Y|X,β). The efficiency, bias and precision of the estimates are important in this process.

The type of missingness in the data is an important factor in the imputation process. The basic patterns of missingness include:

• Missing completely at random (MCAR) assumes that the missing data is not related to any factor, known or unknown, in the study.

• Missingness at random (MAR) assumes that the missingness depends only on observed quantities, which may include outcomes and/or predictors.

• Non-ignorable missingness occurs when the missing data depends on unobserved quantities.

Multiple Imputation Protocol

(1) Imputation: Generate a set of m > 1 plausible values for Z_mis= Y_(mis,X_mis).

The imputation step relies upon assumptions regarding the cause of missingness in the dataset. The goal of the imputation is to account for the relationships between the unobserved and observed variables, while taking into account the uncertainty of the imputation. The commonly made MAR assumption for missing data is untestable without additional information. With MAR assumption we can generate imputations (Z_{1},Z_{2},...Z_{m}) from the distribution f(Z_mis |Z_obs, since after conditioning on Z_obs, the missingness is assumed to be random.

The missingness is monotone when the data matrix can be rearranged so that there is a hierarchy of missingness where observing a particular variable Z_b for a subject implies that Z_a is observed, for a < b. In monotone condition settings many imputation methods may be employed including (for continuous variables) propensity methods, predictive mean matching, and (for discrete variables) discriminant analysis or logistic regression. For non-monotonic missingness, Markov Chain Monte Carlo (MCMC) approaches may be used.

Each method has its own assumptions. For instance, predictive mean matching and MCMC approaches require multivariate normality. Predictive mean matching approach employs linear regression for the distribution of a partially observed variable, conditional on other factors. To impute the data using the predictive mean matching approach for a variable Z_i with missing values, we fit a model using complete observations for Z₁, . . . , Z_(i-1) (not monotonicity assumption!):

E[Z_i |φ] = φ₀ + φ₁ Z₁ + φ₂ Z₂ + · · · + φ_(i-1) Z_(i-1)

Next, new parameters φ^* are randomly drawn from the distribution of the parameters (since these values are estimated instead of exactly known). For the 1≤l≤m imputation, the missing values are replaced by:

Z_i^l = φ₀^* + φ₁^* Z₁ + φ₂^* Z₂ + · · · + φ_(i-1)^* Z_(i-1)+σ^{* ϵ},

where σ^* is the estimate of variance from the model and ϵ is a Gaussian, N(0,1). This is the simple regression method. We can impute the observed value of Z_i that is closest to predicted Z _i=Z_i^l in the dataset – this is the predictive mean matching method.

The propensity score method uses an alternative model for imputation where the values are imputed from observations that are equally likely to be missing, by fitting a logistic regression model for the missingness indicators. In some situations, propensity score imputation may lead to serious bias for missing covariates. For discrete incomplete variables, discriminant analysis or dichotomous logistic regression may be employed to impute values based on the estimated probability that a missing observation takes on a certain value.

(2) Analysis: Analyze the m datasets using complete-case methods. This second step in the protocol involves carrying out the analysis of interest for each of the m imputed complete-observation datasets and storing the parameter vectors (e.g., effect-size coefficients, β’s) and their standard error estimates

(3) Combination: Combine the results from the m analyses and calculate the estimates of the within imputation and between imputation variability. If the imputation model is correct (assumptions are valid), these statistics account for the variability of the imputations and provide consistent estimates of the parameters and their standard errors.

Examples

Simulated example

set.seed(123)
# create MCAR missing-data generator
create.missing <- function (data, pct.mis = 10)
{
n <- nrow(data)
J <- ncol(data)
if (length(pct.mis) == 1) {
n.mis <- rep((n * (pct.mis/100)), J)

}
else {
if (length(pct.mis) < J) 
stop("The length of missing does not equal to the column of the data")
n.mis <- n * (pct.mis/100)

}
   for (i in 1:ncol(data)) {
       if (n.mis[i] == 0) { # if column has nothing missing, do nothing.
           data[, i] <- data[, i]
       }
       else {
           data[sample(1:n, n.mis[i], replace = FALSE), i] <- NA
# For each given column (i), sample the row indices (1:n), 
# a number of indices to replace as “missing”, n.mis[i], “NA”,
 	  # without replacement
       }
   }
   return(as.data.frame(data))
}

# Simulate some real multivariate data sim_data={y,x1,x2,x3,x4,x5,x6,x7,x8,x9,x10}
# 
n <- 1000; u1 <- rbinom(n, 1, .5); v1 <- log(rnorm(n, 5, 1)); x1 <- u1*exp(v1)
u2 <- rbinom(n, 1, .5); v2 <- log(rnorm(n, 5, 1)); x2 <- u2*exp(v2)
x3 <- rbinom(n, 1, prob=0.45); x4 <- ordered(rep(seq(1, 5),n)[sample(1:n, n)]); x5 <- rep(letters[1:10],n)[sample(1:n, n)]; x6 <- trunc(runif(n, 1, 10)); x7 <- rnorm(n); x8 <- factor(rep(seq(1,10),n)[sample(1:n, n)]); x9 <-   runif(n, 0.1, .99); x10 <- rpois(n, 4); y <- x1 + x2 + x7 + x9 + rnorm(n)

# package the simulated data as a data frame object
sim_data <- cbind.data.frame(y, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)

# randomly create missing values
sim_data_30pct_missing <- create.missing(sim_data, pct.mis=30); 
head(sim_data_30pct_missing); summary(sim_data_30pct_missing)

# install.packages("mi")
# install.packages("betareg")
library("betareg"); library("mi")

# get show the missing information matrix			
mdf <- missing_data.frame(sim_data_30pct_missing) 
show(mdf); mdf@patterns; image(mdf)   # remember the visual pattern of this MCAR

# Next try to impute the missing values …
imputations <- mi(sim_data_30pct_missing, n.iter=10, n.chains=3, verbose=TRUE)
hist(imputations)

# Extracts several multiply imputed data.frames from “imputations” object
data.frames <- complete(imputations, 3)

# Compare the summary stats for the original data (prior to introducing missing
# values) with missing data and the re-completed data following imputation
summary(sim_data); summary(sim_data_30pct_missing); summary(data.frames1);
lapply(data.frames, summary)

Original
	y		x1	x2	x3	x4	x5	x6	x7		x8	x9	x10
Min	:-3.046	Min	:0.000	Min.	:0.000	Min	:0.000	0.180555556	a	:100	Min	:1.000	Min.	:-2.880570	1	:100	Min.	:0.1024	Min.	:0.000
1st Qu.:2.788	1st Qu.:0.000	1st Qu.:0.000	1st Qu.:0.000	0.222222222	b	:100	1st Qu.:3.000	1st Qu.:-0.687578	2	:100	1st Qu.:0.3356	1st Qu.:2.000
Median :5.613	Median :3.186	Median :2.024	Median :0.000	0.263888889	c	:100	Median :5.000	Median :0.000836	3	:100	Median :0.5546	Median :4.000
Mean	:5.658	Mean	:2.626	Mean	:2.480	Mean	:0.459	0.305555556	d	:100	Mean	:4.998	Mean	:-0.020541	4	:100	Mean	:0.5542	Mean	:3.898
3rd Qu.:8.525	3rd Qu.:5.101	3rd Qu.:4.962	3rd Qu.:1.000	0.347222222	e	:100	3rd Qu.:7.000	3rd Qu.:0.635832	5	:100	3rd Qu.:0.7829	3rd Qu.:5.000
Max.	:17.263	Max.	:8.774	Max.	:8.085	Max	:1.000	f	:100	Max.	:9.000	Max.	:3.281171	6	:100	Max.	:0.9887	Max.	:12.000
Missing Data
	y		x1	x2	x3	x4	x5	x6	x7		x8	x9	x10	6	:100
Min.	:-3.046	Min	:0.000	Min.	:0.000	Min.	:0.0000	1	:144	h	:76	Min.	:1.000	Min.	:-2.8806	8	:77	Min.	:0.1024	Min.	:0.000
1st Qu.:2.768	1st Qu.:0.000	1st Qu.:0.000	1st Qu.:0.0000	2	:128	a	:72	1st Qu.:3.000	1st Qu.:-0.6974	4	:75	1st Qu.:0.3504	1st Qu.:2.000
Median :5.650	Median :3.127	Median :0.000	Median :0.0000	3	:143	d	:72	Median :5.000	Median :-0.0490	9	:72	Median :0.5590	Median :4.000

# Check imputation convergence (details provided below)
round(mipply(imputations, mean, to.matrix = TRUE), 3)
Rhats(imputations, statistic = "moments") # assess the convergence of MI algorithm

plot(imputations); hist(imputations); image(imputations); summary(imputations)

# Finally, pool over the m = 5 imputed datasets when we fit the “model”
# Pool from across the 4 chains – in order to estimate a linear regression model
model_results <- pool(y ~ x1+x2+x3+x4+x5+x6+x7+x8+x9+x10, data=imputations,  m=5 )
display (model_results); summary (model_results)  
# Report the summaries of the imputations
data.frames <- complete(imputations, 3)  	# extract the first 3 chains
lapply(data.frames, summary)

TBI Data Example

See [ttp://wiki.socr.umich.edu/index.php/SMHS_MissingData#Raw_TBI_data these traumatic brain injury (TBI) data]:

Ind	ID	Age	Sex	TBI	field.gcs	er.gcs	icu.gcs	worst.gcs	X6m.gose	X2013.gose	skull.fx	temp.injury	surgery	spikes.hr	min.hr	max.hr	acute.sz	late.sz	ever.sz
1	1	19	M	Fall	10	10	10	10	5	5	0	1	1	NA	NA	NA	1	1	1
2	2	55	M	Blunt	NA	3	3	3	5	7	1	1	1	168.74	14	757	0	1	1
3	3	24	M	Fall	12	12	8	8	7	7	1	0	0	37.37	0	351	0	0	0
4	4	57	F	Fall	4	4	6	4	3	3	1	1	1	4.35	0	59	0	0	0
5	5	54	F	Peds_vs_Auto	14	11	8	8	5	7	0	1	1	54.59	0	284	0	0	0
6	6	16	F	MVA	13	7	7	7	7	8	1	1	1	75.92	7	180	0	1	1

# Load the (raw) data from the table into a plain text file "08_EpiBioSData_Incomplete.csv"
TBI_Data <- read.csv("https://umich.instructure.com/files/720782/download?download_frd=1", na.strings=c("",".","NA"))
summary(TBI_Data)

# Get information matrix of the data
# (1) Convert to a missing_data.frame
# library("betareg"); library("mi")			
mdf <- missing_data.frame(TBI_Data) # warnings about missingness patterns
show(mdf); mdf@patterns

# (2) change things: mi::change() method changes the family, imputation method,
# size, type, and so forth of a missing variable. It’s called 
# before calling mi to affect how the conditional expectation of each 
# missing variable is modeled.
mdf <- change(mdf, y = "spikes.hr", what = "transformation", to = "identity")

Arg	Description
y	A character vector naming one or more missing variables within the missing_data.frame specified by the data argument, or a vector of integers or a logical vector indicating which missing_variables to change.
what	A character string naming what is to be changed, such as "family", "imputation_method", "size","transformation", "type", "link", or "model."
to	A character string naming what y should be changed to, such as one of the admissible families, imputation methods, transformations, or types. If missing, then possible choices for the "to" argument will be helpfully printed on the screen.

# (3) examine missingness patterns
summary(mdf); hist(mdf); 
image(mdf)

# (4) Perform initial imputation
imputations <- mi(mdf, n.iter=30, n.chains=5, verbose=TRUE)
hist(imputations)

# (5) Extracts several multiply imputed data.frames from “imputations” object
data.frames <- complete(imputations, 5)

# (6) Report a list of “summaries” for each element (imputation instance)
lapply(data.frames, summary)

# (6.a) To cast the imputed numbers as integers (not necessary, but may be useful)
indx <- sapply(data.frames5, is.numeric)  # get the indices of numeric columns
data.frames5[indx] <- lapply(data.frames5[indx], function(x) as.numeric(as.integer(x))) 		    # cast each value as integer

# (7) Save results out
write.csv(data.frames5, "C:\\Users\\Dinov\\Desktop\\TBI_MIData.csv")

# (8) Complete Data analytics functions:
# library("mi")
#lm.mi(); glm.mi(); polr.mi(); bayesglm.mi(); bayespolr.mi(); lmer.mi(); glmer.mi()

# (8.1) Define Linear Regression for multiply imputed dataset – Also see Step (10)
##linear regression for each imputed data set - 5 regression models are fit
fit_lm1 <- glm(ever.sz ~ surgery + worst.gcs + factor(sex) + age, data.frames$`chain:1`, family = "binomial"); summary(fit_lm1); display(fit_lm1)

# Fit the appropriate model and pool the results (estimates over MI chains)
model_results <- pool(ever.sz ~ surgery + worst.gcs + factor(sex) + age, family = "binomial", data=imputations,  m=5)
display (model_results); summary (model_results)  

# Report the summaries of the imputations
data.frames <- complete(imputations, 3)  	# extract the first 3 chains
lapply(data.frames, summary)

# (9) Validation: we now verify whether enough iterations were conducted. 
# Validation criteria demands that the mean of each completed variable should
# be similar for each of the k chains (in this case k=5).
# mipply is wrapper for sapply invoked on mi-class objects to compute the col means
round(mipply(imputations, mean, to.matrix = TRUE), 3)

# Rhat convergence statistics compares the variance between chains to the variance
# across chains. 
# Rhat Values ~ 1.0 indicate likely convergence, 
# Rhat Values > 1.1 indicate that the chains should be run longer 
# (use large number of iterations)
Rhats(imputations, statistic = "moments") # assess the convergence of MI algorithm

# When convergence is unstable, we can continue the iterations for all chains, e.g.
imputations <- mi(imputations, n.iter=20) # add additional 20 iterations

# To plot the produced mi results, for all missing_variables we can generate
# a histogram of the observed, imputed, and completed data.
# We can compare of the completed data to the fitted values implied by the model
# for the completed data, by plotting binned residuals. 
# hist function works similarly as plot. 
# image function gives a sense of the missingness patterns in the data
plot(imputations); hist(imputations); image(imputations); summary(imputations)

# (10) Finally, pool over the m = 5 imputed datasets when we fit the “model”
# Pool from across the 4 chains – in order to estimate a linear regression model
# and impact ov various predictors

model_results <- pool(ever.sz ~ surgery + worst.gcs + factor(sex) + age, data =  imputations,  m =  5 ); display (model_results); summary (model_results)  

# Report the summaries of the imputations
data.frames <- complete(imputations, 3)  	# extract the first 3 chains
lapply(data.frames, summary)			# report summaries

Parkinson's Disease Case Study

Background: See: http://wiki.socr.umich.edu/index.php/SOCR_Data_PD_BiomedBigMetadata

#imputation for logistic regression
# load the data: 08_PPMI_GSA_clinical.csv
ppmi_new<-read.csv("https://umich.instructure.com/files/330401/download?download_frd=1",header=TRUE)

# install.packages("psych")
library(psych)
# report descriptive statistics

ppmi_new$\$$ResearchGroup <- ifelse(ppmi_new$\$$ResearchGroup == "Control", "Control", "Patient")
ppmi_new$\$$ResearchGroup <- as.factor(ppmi_new$\$$ResearchGroup): table(ppmi_new$\$$ResearchGroup)

# Reduce the Dataset: extract only the columns of interest (to ensure real time calculations)
ppmi_new_1 <- as.data.frame(ppmi_new)[,   c("ResearchGroup","R_fusiform_gyrus_Curvedness","Sex","Age","chr12_rs34637584_GT","UPDRS_Part_I_Summary_Score_Baseline","UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Baseline","UPDRS_Part_III_Summary_Score_Baseline",  "UPDRS_Part_IV_Summary_Score_Baseline")]

descript <- describe(ppmi_new_1); descript	# reports means, SE’s, etc.

dim(ppmi_new); dim(ppmi_new_1)	# check dimensions before 

# imputation 1
# install.packages("mi"); install.packages("betareg")
library(mi); library(betareg)

mdf<-missing_data.frame(ppmi_new_1[-1,])
apply(ppmi_new_1[-1,],1,mean)
imputations<-mi(mdf, seed=900);   summary(imputations)
ppmi_complete<-complete(imputations)

# Save the imputation
# again you may want to cast imputed numerical values as integers, perhaps
# (indx <- sapply(data.frames5, is.numeric)  # get the indices of numeric columns
# data.frames5[indx] <- lapply(data.frames5[indx], function(x) as.numeric(as.integer(x)))   # cast each value
# write.csv(ppmi_complete,"C:\\Users\\Dinov\\Desktop\\ppmi_new_1_complete.csv")
write.csv(ppmi_complete,"ppmi_new_1_complete.csv")

# Get the imputations of 4 iterations
ppmi_complete_1<- ppmi_complete1; ppmi_complete_2<- ppmi_complete2; 
ppmi_complete_3<- ppmi_complete3; ppmi_complete_4<- ppmi_complete4

# average the imputations 
ppmi_updrs <- (ppmi_complete_1+ppmi_complete_2+ppmi_complete_4+ppmi_complete_3)/4
# ppmi_top<-cbind(ppmi_new_1[-1,1:335], ppmi_updrs) #delete the first observation
# colnames(ppmi_top)<-colnames(ppmi_new[,1:393])

# Save results – complete dataset
#  write.csv(ppmi_top,"C:\\Users\\Dinov\\Desktop\\ppmi_top_complete_1.csv")
# write.csv(ppmi_top,"ppmi_top_complete_1.csv")

# Now, run some Data analytics on complete data:
# library("mi")
#lm.mi(); glm.mi(); polr.mi(); bayesglm.mi(); bayespolr.mi(); lmer.mi(); glmer.mi()
# Define Linear Regression for multiply imputed dataset
##linear regression for each imputed data set - 5 regression models are fit

model_results <- pool(ResearchGroup ~  R_fusiform_gyrus_Curvedness + factor(Sex) + Age + 
factor(chr12_rs34637584_GT) + UPDRS_Part_I_Summary_Score_Baseline + UPDRS_Part_II_Patient_Questionnaire_Summary_Score_Baseline + UPDRS_Part_III_Summary_Score_Baseline, family = "binomial", data=imputations,  m=3)
display (model_results); summary (model_results)

See also

• The SOCR Simulated HELP data activity provides additional missing data management examples.
• Multiple Imputation FAQs.

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• SOCR Home page: http://www.socr.umich.edu

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