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)

!!!!INSERT TABLE

# 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 these traumatic brain injury (TBI) data: http://wiki.socr.umich.edu/index.php/SMHS_MissingData#Raw_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")]

Motivation

In complete-case analysis using multiple regression modeling, response results may be missing may involve automatically excluding the cases with missing response value. This leads to restricting the amount of information available in the analysis, especially if the model has many parameters that need to be estimated and many responses are potentially missing. Alternatively, missing outcomes in a regression can be handled by modeling the outcome variable to impute missing values at each iteration.

A more challenging situation in regression analysis involves missing values in predictor variables. The options here are to remove the missing values, impute them, or analytically model them by supplying distributions for the input variables.

Theory

Types of data missingness

Knowing why and how is data missing is impoetant for determining hte appropriate protocol for handling the data. There are several categories of data-missingness.

• Missingness completely at random (MCAR). A variable is missing completely at random if the probability of missingness is the same for all units. For instance, if each respondent decides whether to answer an income question by rolling a (fair) die and refusing to answer if a a number > 3 turn up. Inference based on imputing data missing completely at random, by throwing out cases with missing data, is unbiased.

• Missingness at random (MAR). Most missingness is not completely random. For example, different non-response rates for whites and minorities on income question may be due to socioeconomic factors. Missing at random is a more general assumption where the probability a missing variable depends only on available information. Suppose demographic variables (e.g., age, gender, race, education) are recorded for all the people in the study. Then income will be missing at random if the probability of non-response to this question depends only on these other, fully recorded variables. This process many be modeled by logistic regression with the outcome variable (Y) representing indicator of missingness ($Y=1$ for observed cases and $Y=0$ for missing cases). When an outcome variable is missing at random, a regression model can exclude the missing cases (treat them as NA’s), if it controls for all the variables that affect the probability of missingness for the outcome. In our case, regression models of income would have to include predictors for ethnicity to avoid possible non-response bias.

• Non-random missingness: When the data missingness depends on unobserved predictors, this indicates non-random gaps in the observed data, dues to information that may not be available, which may also be predictive of the missing values. For instance, highly-educated (high-income?) people may be less likely to respond to income questions. So, college degree may have predictive value for income. Another example is a new clinical treatment that causes side-effects which may cause attrition of participants (patients drop out of study) dependent on their level of ability to deal with the side effects. Non-random missingness has to be explicitly modeled, otherwise, bias would creep into the scientific inference and impact the results of the study.

• Missingness that depends on the missing value itself. If the probability of missingness depends on the (potentially missing) variable itself the situation is a bit more interesting. For example, higher earners may be less likely to reveal their income. In these situations, missing-data have to be modeled or accounted for by including more predictors in the missing-data model to bringing it closer to missing at random situation. For example, whites and highly-educated participants may have higher-than-average incomes and we can control for such predictors to correct for higher rates of non-response (missingness) among higher-income people. Sometime, the missing data situation may require predictive models extrapolating beyond the range of the observed data.

Example

Suppose we are interested in identifying patterns, relations and associations between demographic, clinical and cognitive variables in a cohort of traumatic brain injury (TBI) patients. The table below shows the raw data. Notice the missing values in this table. Imputing the missing data would allow us to use all cases in our analysis of the multivariate relations using the completed dataset.

Raw TBI data

The variables in the table include: id=participant ID; age=age; sex=gender; mechanism=type of TBI injury; field.gcs=field Glasgow Coma Scale; er.gcs=emergency room Glasgow Coma Scale; icu.gcs=intensive care unit Glasgow Coma Scale; worst.gcs=lowest Glasgow Coma Scale; 6m.gose=Extended Glasgow Outcome Scale score at 6-month follow up; 2013.gose=Extended Glasgow Outcome Scale score in 2013; skull.fx=skull fracture criterion for impact-induced head injury; temp.injury=injury characteristic; surgery=index of surgery; spikes.hr=EEG spikes per hour; min.hr=min EEG per hour; max.hr=max EEG spikes per hour; acute.sz=indicator of seizure at acute state; late.sz=indicator of seizure at chronic state; ever.sz=indicator of seizures at any time. Period, ".", indicates missing data.

id	age	sex	mechanism	field.gcs	er.gcs	icu.gcs	worst.gcs	6m.gose	2013.gose	skull.fx	temp.injury	surgery	spikes.hr	min.hr	max.hr	acute.sz	late.sz	ever.sz
1	19	Male	Fall	10	10	10	10	5	5	0	1	1	.	.	.	1	1	1
2	55	Male	Blunt	.	3	3	3	5	7	1	1	1	168.74	14	757	0	1	1
3	24	Male	Fall	12	12	8	8	7	7	1	0	0	37.37	0	351	0	0	0
4	57	Female	Fall	4	4	6	4	3	3	1	1	1	4.35	0	59	0	0	0
5	54	Female	Peds_vs_Auto	14	11	8	8	5	7	0	1	1	54.59	0	284	0	0	0
6	16	Female	MVA	13	7	7	7	7	8	1	1	1	75.92	7	180	0	1	1
7	21	Male	Fall	3	3	6	3	3	3	1	0	1	.	.	.	0	0	0
8	25	Male	Fall	3	4	3	3	3	3	0	1	0	5.26	0	88	0	1	1
9	30	Male	GSW	3	9	3	3	3	5	1	1	1	43.88	0	367	0	1	1
10	38	Male	Fall	3	6	6	3	3	3	1	1	1	45.6	4	107	0	1	1
11	43	Male	Blunt	8	7	7	7	6	7	1	0	1	7.76	0	72	0	0	0
12	40	Male	Fall	12	14	14	12	7	8	0	1	1	26.64	0	125	0	0	0
13	21	Male	MVA	12	13	9	9	7	7	1	0	1	.	.	.	0	1	1
14	35	Female	MVA	6	5	6	5	5	7	1	1	0	65.14	0	655	1	1	1
15	59	Male	Peds_vs_Auto	14	14	0	0	8	8	1	1	0	.	.	.	0	0	0
16	32	Male	MCA	5	6	3	3	4	5	1	0	0	.	.	.	0	0	0
17	31	Male	MVA	7	3	9	3	5	7	1	0	0	3.82	0	28	0	0	0
18	57	Male	MVA	4	3	7	3	3	3	0	1	1	.	.	.	0	1	1
19	18	Male	Blunt	4	3	6	3	5	3	1	1	1	.	.	.	0	1	1
20	48	Male	Bike_vs_Auto	3	8	7	3	5	7	0	0	0	.	.	.	0	0	0
21	19	Male	GSW	15	15	3	3	.	6	1	0	1	.	.	.	1	1	1
22	22	Male	Fall	3	3	3	3	2	2	1	1	1	9.7	0	80	0	1	1
23	20	Male	Peds_vs_Auto	15	14	13	13	5	8	1	1	1	.	.	.	0	1	1
24	41	Male	MVA	3	3	6	3	3	7	1	0	0	.	.	.	0	0	0
25	27	Male	MCA	15	13	6	6	6	7	1	0	1	.	.	.	0	0	0
26	23	Male	MVA	14	14	7	7	4	7	1	1	1	.	.	.	0	0	0
27	36	Male	MCA	3	3	3	3	5	6	0	0	0	.	.	.	0	1	1
28	83	Female	Fall	14	14	9	9	.	5	0	1	1	208.92	42	641	1	1	1
29	26	Male	MCA	5	7	5	5	6	7	0	1	0	.	.	.	0	0	0
30	21	Male	Fall	14	14	14	14	5	7	0	1	1	294	30	1199	1	1	1
31	23	Male	MCA	12	13	13	12	.	7	1	0	1	.	.	.	0	0	0
32	45	Male	MCA	6	6	6	6	3	6	0	0	1	.	.	.	0	0	0
33	18	Male	Bike_vs_Auto	8	8	7	7	7	7	0	0	0	7.14	0	20	0	1	1
34	34	Male	MVA	7	7	3	3	4	6	0	1	1	47.73	0	226	0	1	1
35	19	Male	MVA	3	7	7	3	7	8	0	0	0	97.43	0	300	0	0	0
36	77	Female	Peds_vs_Auto	3	6	3	3	3	3	1	1	0	7.09	0	31	0	1	1
37	75	Male	Bike_vs_Auto	.	.	.	.	.	8	1	0	0	5.9	0	42	0	1	1
38	25	Male	Fall	14	.	6	6	8	8	0	0	1	29.61	0	175	1	0	1
39	62	Female	Fall	12	8	8	8	3	3	0	1	1	6.16	0	33	0	1	1
40	41	Male	MCA	7	3	7	3	5	5	1	1	1	1.66	0	23	0	1	1
41	60	Male	Bike_vs_Auto	3	12	7	3	3	5	1	1	0	3.8	0	12	0	1	1
42	29	Female	Peds_vs_Auto	9	14	3	3	8	7	1	0	1	.	.	.	0	1	1
43	48	Male	Blunt	12	12	11	11	6	7	0	0	1	5.39	0	43	0	0	0
44	41	Male	Peds_vs_Auto	3	3	3	3	2	2	1	1	0	1.28	0	15	1	1	1
45	34	Male	Fall	6	8	3	3	3	3	1	1	1	213.84	3	824	1	1	1
46	25	Female	MVA	6	8	3	3	.	7	0	1	0	1.7	0	36	0	0	0

R imputation script

The R code below illustrates the imputation of a raw data.

###########################################
# example of multiple imputation (R MI package)
# See Docs: http://www.stat.ucla.edu/~yajima/Publication/mipaper.rev04.pdf
# 
# If we don't have real or observed or derived data, we can simulate fake data
# See this SOCR Data/Activity:  Predictive Big Data Analytics, Modeling and Visualization of Clinical, Genetic and Imaging Data for Parkinson’s Disease
#
# Alternatively use the example below.
# set.seed(123)
# 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),100)[sample(1:n, n)]); x5 <- rep(letters[1:10],10)[sample(1:n, n)]; x6 <- trunc(runif(n, 1, 10)); x7 <- rnorm(n); x8 <- factor(rep(seq(1,10),10)[sample(1:n, n)]); x9 <- runif(n, 0.1, .99); x10 <- rpois(n, 4); y <- x1 + x2 + x7 + x9 + rnorm(n)
# fakedata <- cbind.data.frame(y, x1, x2, x3, x4, x5, x6, x7, x8, x9, x10)
# randomly create missing values
# dat <- mi:::.create.missing(fakedata, pct.mis=30)
##########################################################
#
library("mi")
# copy-paste the (raw) data from the table into a plain text file "EpiBioSData.csv"
EpiBiosData <- read.csv("~/EpiBioSData.csv", na.strings=c("",".","NA"))
# get information matrix of the data
# (1) Convert to a missing_data.frame
mdf <- missing_data.frame(EpiBiosData) # warnings about missingness patterns
show(mdf)
# (2) change things
mdf <- change(mdf, y = "spikes.hr", what = "transformation", to = "identity")
# (3) look deeper
summary(mdf)
hist(mdf)
image(mdf)
#
imputations <- mi(mdf)
hist(imputations)
data.frames <- complete(imputations, 5)
lapply(data.frames, summary)

# To cast the imputed numbers as integers (not necessary, but may be useful)
indx <- sapply(data.frames\([[5]]\), is.numeric)  # get the indices of numeric columns
data.frames\([[5]]\)[indx] <- lapply(data.frames\([[5]]\)[indx], function(x) as.numeric(as.integer(x))) # cast each value as integer

#
# save results out
# write.mi(data.frames\([[5]]\), format = "csv")
write.csv(data.frames\([[5]]\), "C:\\Users\\Dinov\\Desktop\\EpiBioS_MIData.csv")
# 
# Complete Data analytics: ression functions:
# lm.mi(); glm.mi(); polr.mi(); bayesglm.mi(); bayespolr.mi(); lmer.mi(); glmer.mi()
#
fit <- lm.mi(ever.sz ~ surgery + worst.gcs + sex + age, IMP)
# fit1 <- lm.mi(ever.sz.1 ~ surgery.1 + worst.gcs.1 + sex.1 + age.1, IMP)
# fit2 <- lm.mi(ever.sz.2 ~ surgery.2 + worst.gcs.2 + sex.2 + age.2, IMP)
display(fit)

Imputed Complete Data

The table below shows 3 alternative imputation results for the same TBI data (generated using the R script above). Variable indices (ID, ..., ever.sz; ID.1, ..., ever.sz.1; ID.2, ..., ever.sz.2) indicate the results of the 3 complementary multiple imputations.

ID	age	sex	mechanism	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	ID.1	age.1	sex.1	mechanism.1	field.gcs.1	er.gcs.1	icu.gcs.1	worst.gcs.1	X6m.gose.1	X2013.gose.1	skull.fx.1	temp.injury.1	surgery.1	spikes.hr.1	min.hr.1	max.hr.1	acute.sz.1	late.sz.1	ever.sz.1	ID.2	age.2	sex.2	mechanism.2	field.gcs.2	er.gcs.2	icu.gcs.2	worst.gcs.2	X6m.gose.2	X2013.gose.2	skull.fx.2	temp.injury.2	surgery.2	spikes.hr.2	min.hr.2	max.hr.2	acute.sz.2	late.sz.2	ever.sz.2
1	19	Male	Fall	10	10	10	10	5	5	0	1	1	31.33408337	17.64389893	329.4906597	1	1	1	1	19	Male	Fall	10	10	10	10	5	5	0	1	1	96.74023746	18.35575322	474.5394709	1	1	1	1	19	Male	Fall	10	10	10	10	5	5	0	1	1	134.3949936	13.93977114	357.0245178	1	1	1
2	55	Male	Blunt	8.949465016	3	3	3	5	7	1	1	1	168.74	14	757	0	1	1	2	55	Male	Blunt	4.583430169	3	3	3	5	7	1	1	1	168.74	14	757	0	1	1	2	55	Male	Blunt	6.759506638	3	3	3	5	7	1	1	1	168.74	14	757	0	1	1
3	24	Male	Fall	12	12	8	8	7	7	1	0	0	37.37	0	351	0	0	0	3	24	Male	Fall	12	12	8	8	7	7	1	0	0	37.37	0	351	0	0	0	3	24	Male	Fall	12	12	8	8	7	7	1	0	0	37.37	0	351	0	0	0
4	57	Female	Fall	4	4	6	4	3	3	1	1	1	4.35	0	59	0	0	0	4	57	Female	Fall	4	4	6	4	3	3	1	1	1	4.35	0	59	0	0	0	4	57	Female	Fall	4	4	6	4	3	3	1	1	1	4.35	0	59	0	0	0
5	54	Female	Peds_vs_Auto	14	11	8	8	5	7	0	1	1	54.59	0	284	0	0	0	5	54	Female	Peds_vs_Auto	14	11	8	8	5	7	0	1	1	54.59	0	284	0	0	0	5	54	Female	Peds_vs_Auto	14	11	8	8	5	7	0	1	1	54.59	0	284	0	0	0
6	16	Female	MVA	13	7	7	7	7	8	1	1	1	75.92	7	180	0	1	1	6	16	Female	MVA	13	7	7	7	7	8	1	1	1	75.92	7	180	0	1	1	6	16	Female	MVA	13	7	7	7	7	8	1	1	1	75.92	7	180	0	1	1
7	21	Male	Fall	3	3	6	3	3	3	1	0	1	97.81993169	-0.213863843	38.62625185	0	0	0	7	21	Male	Fall	3	3	6	3	3	3	1	0	1	57.41204562	22.47601237	-152.4300823	0	0	0	7	21	Male	Fall	3	3	6	3	3	3	1	0	1	-88.08486298	-17.52260473	-126.0982147	0	0	0
8	25	Male	Fall	3	4	3	3	3	3	0	1	0	5.26	0	88	0	1	1	8	25	Male	Fall	3	4	3	3	3	3	0	1	0	5.26	0	88	0	1	1	8	25	Male	Fall	3	4	3	3	3	3	0	1	0	5.26	0	88	0	1	1
9	30	Male	GSW	3	9	3	3	3	5	1	1	1	43.88	0	367	0	1	1	9	30	Male	GSW	3	9	3	3	3	5	1	1	1	43.88	0	367	0	1	1	9	30	Male	GSW	3	9	3	3	3	5	1	1	1	43.88	0	367	0	1	1
10	38	Male	Fall	3	6	6	3	3	3	1	1	1	45.6	4	107	0	1	1	10	38	Male	Fall	3	6	6	3	3	3	1	1	1	45.6	4	107	0	1	1	10	38	Male	Fall	3	6	6	3	3	3	1	1	1	45.6	4	107	0	1	1
11	43	Male	Blunt	8	7	7	7	6	7	1	0	1	7.76	0	72	0	0	0	11	43	Male	Blunt	8	7	7	7	6	7	1	0	1	7.76	0	72	0	0	0	11	43	Male	Blunt	8	7	7	7	6	7	1	0	1	7.76	0	72	0	0	0
12	40	Male	Fall	12	14	14	12	7	8	0	1	1	26.64	0	125	0	0	0	12	40	Male	Fall	12	14	14	12	7	8	0	1	1	26.64	0	125	0	0	0	12	40	Male	Fall	12	14	14	12	7	8	0	1	1	26.64	0	125	0	0	0
13	21	Male	MVA	12	13	9	9	7	7	1	0	1	-139.5137892	-33.84480983	19.23090912	0	1	1	13	21	Male	MVA	12	13	9	9	7	7	1	0	1	125.7529181	19.14860131	291.9026199	0	1	1	13	21	Male	MVA	12	13	9	9	7	7	1	0	1	-12.93052956	-11.43761507	39.10220788	0	1	1
14	35	Female	MVA	6	5	6	5	5	7	1	1	0	65.14	0	655	1	1	1	14	35	Female	MVA	6	5	6	5	5	7	1	1	0	65.14	0	655	1	1	1	14	35	Female	MVA	6	5	6	5	5	7	1	1	0	65.14	0	655	1	1	1
15	59	Male	Peds_vs_Auto	14	14	0	0	8	8	1	1	0	104.0330205	28.99974405	718.6130268	0	0	0	15	59	Male	Peds_vs_Auto	14	14	0	0	8	8	1	1	0	158.5435588	11.79083406	788.0040332	0	0	0	15	59	Male	Peds_vs_Auto	14	14	0	0	8	8	1	1	0	70.32014857	-4.21119291	-20.90694906	0	0	0
16	32	Male	MCA	5	6	3	3	4	5	1	0	0	99.20158013	-17.25973621	351.0875441	0	0	0	16	32	Male	MCA	5	6	3	3	4	5	1	0	0	-217.7065141	3.020094958	-464.0898326	0	0	0	16	32	Male	MCA	5	6	3	3	4	5	1	0	0	-33.098168	-0.137931315	487.9378096	0	0	0
17	31	Male	MVA	7	3	9	3	5	7	1	0	0	3.82	0	28	0	0	0	17	31	Male	MVA	7	3	9	3	5	7	1	0	0	3.82	0	28	0	0	0	17	31	Male	MVA	7	3	9	3	5	7	1	0	0	3.82	0	28	0	0	0
18	57	Male	MVA	4	3	7	3	3	3	0	1	1	-149.9422408	-30.42385829	-302.9962493	0	1	1	18	57	Male	MVA	4	3	7	3	3	3	0	1	1	56.13694368	18.8199818	-62.49856966	0	1	1	18	57	Male	MVA	4	3	7	3	3	3	0	1	1	45.55341087	4.468479615	-213.454323	0	1	1
19	18	Male	Blunt	4	3	6	3	5	3	1	1	1	40.39410742	-36.87895336	97.37341326	0	1	1	19	18	Male	Blunt	4	3	6	3	5	3	1	1	1	-106.5749596	5.559437069	-1149.613796	0	1	1	19	18	Male	Blunt	4	3	6	3	5	3	1	1	1	51.18634141	-1.937828826	-342.6008879	0	1	1
20	48	Male	Bike_vs_Auto	3	8	7	3	5	7	0	0	0	118.8000347	32.36213678	-73.13271061	0	0	0	20	48	Male	Bike_vs_Auto	3	8	7	3	5	7	0	0	0	-87.77082486	3.717360906	-4.392630662	0	0	0	20	48	Male	Bike_vs_Auto	3	8	7	3	5	7	0	0	0	108.2372319	0.187220054	519.1251498	0	0	0
21	19	Male	GSW	15	15	3	3	9.25042468	6	1	0	1	40.13687855	-14.22895294	736.3143588	1	1	1	21	19	Male	GSW	15	15	3	3	7.223305755	6	1	0	1	18.46733942	25.54649386	-315.4151426	1	1	1	21	19	Male	GSW	15	15	3	3	3.828147759	6	1	0	1	111.5912998	17.80387347	200.2606779	1	1	1
22	22	Male	Fall	3	3	3	3	2	2	1	1	1	9.7	0	80	0	1	1	22	22	Male	Fall	3	3	3	3	2	2	1	1	1	9.7	0	80	0	1	1	22	22	Male	Fall	3	3	3	3	2	2	1	1	1	9.7	0	80	0	1	1
23	20	Male	Peds_vs_Auto	15	14	13	13	5	8	1	1	1	143.8993295	9.086586358	410.1307881	0	1	1	23	20	Male	Peds_vs_Auto	15	14	13	13	5	8	1	1	1	68.95037612	-3.634136214	745.7692261	0	1	1	23	20	Male	Peds_vs_Auto	15	14	13	13	5	8	1	1	1	17.52584031	4.326014737	307.7486916	0	1	1
24	41	Male	MVA	3	3	6	3	3	7	1	0	0	36.38713558	-7.70043125	397.1825116	0	0	0	24	41	Male	MVA	3	3	6	3	3	7	1	0	0	16.44376591	-3.227208192	318.7734905	0	0	0	24	41	Male	MVA	3	3	6	3	3	7	1	0	0	39.56564601	2.069381684	695.0459207	0	0	0
25	27	Male	MCA	15	13	6	6	6	7	1	0	1	89.67182293	-5.545616243	290.1814949	0	0	0	25	27	Male	MCA	15	13	6	6	6	7	1	0	1	17.66215864	4.730421601	191.3041522	0	0	0	25	27	Male	MCA	15	13	6	6	6	7	1	0	1	67.26462511	8.869208934	468.9527863	0	0	0
26	23	Male	MVA	14	14	7	7	4	7	1	1	1	46.02145191	-13.18053664	91.69057998	0	0	0	26	23	Male	MVA	14	14	7	7	4	7	1	1	1	39.36793917	12.35289402	405.9565656	0	0	0	26	23	Male	MVA	14	14	7	7	4	7	1	1	1	154.8598602	30.03873428	384.849982	0	0	0
27	36	Male	MCA	3	3	3	3	5	6	0	0	0	57.6777688	2.679615385	664.0843475	0	1	1	27	36	Male	MCA	3	3	3	3	5	6	0	0	0	-90.21498545	-17.3682012	-26.4769941	0	1	1	27	36	Male	MCA	3	3	3	3	5	6	0	0	0	-50.90586656	-6.319652137	352.0473406	0	1	1
28	83	Female	Fall	14	14	9	9	4.636755103	5	0	1	1	208.92	42	641	1	1	1	28	83	Female	Fall	14	14	9	9	6.242101503	5	0	1	1	208.92	42	641	1	1	1	28	83	Female	Fall	14	14	9	9	2.05329258	5	0	1	1	208.92	42	641	1	1	1
29	26	Male	MCA	5	7	5	5	6	7	0	1	0	-13.37057197	-14.89381792	152.0008772	0	0	0	29	26	Male	MCA	5	7	5	5	6	7	0	1	0	91.7509579	-0.870090941	529.0874936	0	0	0	29	26	Male	MCA	5	7	5	5	6	7	0	1	0	-0.534176227	-9.783851967	490.89377	0	0	0
30	21	Male	Fall	14	14	14	14	5	7	0	1	1	294	30	1199	1	1	1	30	21	Male	Fall	14	14	14	14	5	7	0	1	1	294	30	1199	1	1	1	30	21	Male	Fall	14	14	14	14	5	7	0	1	1	294	30	1199	1	1	1
31	23	Male	MCA	12	13	13	12	2.860415458	7	1	0	1	74.98397791	13.22365165	155.338988	0	0	0	31	23	Male	MCA	12	13	13	12	3.086836617	7	1	0	1	-11.96623577	-7.703251349	530.2622312	0	0	0	31	23	Male	MCA	12	13	13	12	5.904815799	7	1	0	1	16.65737111	2.374428875	436.7728378	0	0	0
32	45	Male	MCA	6	6	6	6	3	6	0	0	1	104.8995511	-16.52390582	483.006168	0	0	0	32	45	Male	MCA	6	6	6	6	3	6	0	0	1	-80.67017517	-15.81483739	116.5352423	0	0	0	32	45	Male	MCA	6	6	6	6	3	6	0	0	1	71.76729005	-1.066523637	893.1056629	0	0	0
33	18	Male	Bike_vs_Auto	8	8	7	7	7	7	0	0	0	7.14	0	20	0	1	1	33	18	Male	Bike_vs_Auto	8	8	7	7	7	7	0	0	0	7.14	0	20	0	1	1	33	18	Male	Bike_vs_Auto	8	8	7	7	7	7	0	0	0	7.14	0	20	0	1	1
34	34	Male	MVA	7	7	3	3	4	6	0	1	1	47.73	0	226	0	1	1	34	34	Male	MVA	7	7	3	3	4	6	0	1	1	47.73	0	226	0	1	1	34	34	Male	MVA	7	7	3	3	4	6	0	1	1	47.73	0	226	0	1	1
35	19	Male	MVA	3	7	7	3	7	8	0	0	0	97.43	0	300	0	0	0	35	19	Male	MVA	3	7	7	3	7	8	0	0	0	97.43	0	300	0	0	0	35	19	Male	MVA	3	7	7	3	7	8	0	0	0	97.43	0	300	0	0	0
36	77	Female	Peds_vs_Auto	3	6	3	3	3	3	1	1	0	7.09	0	31	0	1	1	36	77	Female	Peds_vs_Auto	3	6	3	3	3	3	1	1	0	7.09	0	31	0	1	1	36	77	Female	Peds_vs_Auto	3	6	3	3	3	3	1	1	0	7.09	0	31	0	1	1
37	75	Male	Bike_vs_Auto	13.73432996	14.02844584	12.08707554	6.608860636	5.076496724	8	1	0	0	5.9	0	42	0	1	1	37	75	Male	Bike_vs_Auto	1.439149068	9.833777842	1.828027302	-0.395951549	8.130388643	8	1	0	0	5.9	0	42	0	1	1	37	75	Male	Bike_vs_Auto	4.627381629	11.70604238	4.818391501	2.101228568	7.755182097	8	1	0	0	5.9	0	42	0	1	1
38	25	Male	Fall	14	18.45286032	6	6	8	8	0	0	1	29.61	0	175	1	0	1	38	25	Male	Fall	14	7.292553675	6	6	8	8	0	0	1	29.61	0	175	1	0	1	38	25	Male	Fall	14	9.021238862	6	6	8	8	0	0	1	29.61	0	175	1	0	1
39	62	Female	Fall	12	8	8	8	3	3	0	1	1	6.16	0	33	0	1	1	39	62	Female	Fall	12	8	8	8	3	3	0	1	1	6.16	0	33	0	1	1	39	62	Female	Fall	12	8	8	8	3	3	0	1	1	6.16	0	33	0	1	1
40	41	Male	MCA	7	3	7	3	5	5	1	1	1	1.66	0	23	0	1	1	40	41	Male	MCA	7	3	7	3	5	5	1	1	1	1.66	0	23	0	1	1	40	41	Male	MCA	7	3	7	3	5	5	1	1	1	1.66	0	23	0	1	1
41	60	Male	Bike_vs_Auto	3	12	7	3	3	5	1	1	0	3.8	0	12	0	1	1	41	60	Male	Bike_vs_Auto	3	12	7	3	3	5	1	1	0	3.8	0	12	0	1	1	41	60	Male	Bike_vs_Auto	3	12	7	3	3	5	1	1	0	3.8	0	12	0	1	1
42	29	Female	Peds_vs_Auto	9	14	3	3	8	7	1	0	1	43.10920277	-12.88149901	504.4628083	0	1	1	42	29	Female	Peds_vs_Auto	9	14	3	3	8	7	1	0	1	206.9761351	13.04583926	-237.7973505	0	1	1	42	29	Female	Peds_vs_Auto	9	14	3	3	8	7	1	0	1	117.2635425	-5.753348151	233.8244434	0	1	1
43	48	Male	Blunt	12	12	11	11	6	7	0	0	1	5.39	0	43	0	0	0	43	48	Male	Blunt	12	12	11	11	6	7	0	0	1	5.39	0	43	0	0	0	43	48	Male	Blunt	12	12	11	11	6	7	0	0	1	5.39	0	43	0	0	0
44	41	Male	Peds_vs_Auto	3	3	3	3	2	2	1	1	0	1.28	0	15	1	1	1	44	41	Male	Peds_vs_Auto	3	3	3	3	2	2	1	1	0	1.28	0	15	1	1	1	44	41	Male	Peds_vs_Auto	3	3	3	3	2	2	1	1	0	1.28	0	15	1	1	1
45	34	Male	Fall	6	8	3	3	3	3	1	1	1	213.84	3	824	1	1	1	45	34	Male	Fall	6	8	3	3	3	3	1	1	1	213.84	3	824	1	1	1	45	34	Male	Fall	6	8	3	3	3	3	1	1	1	213.84	3	824	1	1	1
46	25	Female	MVA	6	8	3	3	8.135915557	7	0	1	0	1.7	0	36	0	0	0	46	25	Female	MVA	6	8	3	3	7.477618429	7	0	1	0	1.7	0	36	0	0	0	46	25	Female	MVA	6	8	3	3	6.093298745	7	0	1	0	1.7	0	36	0	0	0

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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