Joe Feldman 7/10/2022
# library(devtools)
# remotes::install_github('jfeldman396/GMCImpute')
library(GMCImpute)Missing data is commonplace in survey and fused data sets, necessitating sophisticated methods for dealing with missingness when deriving inference. First, these data sets may be comprised of mixed data types, such as continuous, count, and nominal variables. Second, the missing data may bias certain properties of the data set. As a result, a complete case analysis, where the analyst drops any observations with missing values, would lead to misleading insights on the data.
Consider the following simulated example of a continuous variable X1, a count variable X2, and a binary variable X3, where missing values in X2 and X3 are correlated to large values of X1 in absolute value.
set.seed(47)
num= 500
X1<-rnorm(num)
X2<- rpois(num,5*abs(X1))
X3<- as.factor(rbernoulli(num,pnorm(-.5+scale(X2))))
X<- data.frame(X1,X2,X3)
R = t(sapply(1:num, function(x)rbernoulli(2, p = pnorm(-.5 + .5*abs(X1[x]))))) # missingness mechanism
X_noMis = X
X[which(R[,1] == T),2] = NA
X[which(R[,2] == T),3] = NAWe can visualize bias that the mechanism creates in X2, X3 with the following plots.
## Warning: Removed 211 rows containing missing values (geom_point).
As you can see, both margins are affected. We show a comparison of the empirical cdfs of X2 before and after inputting missing values, while the incidence of positive indicators is greatly reduced for X3
The function GMC_Impute allows users to fit a Gaussian mixture copula
to data comprised of unordered categorical, binary, count and continuous
data types with missing values. This is done through utilization of the
extended rank-probit likelihood, which enables copula estimation on the
aforementioned data types. The function then produces a user specified
number of multiple imputations.
Key to the Gaussian mixture copula are the marginal distributions of each variable in the data, as latent variables are linked to the observed scale using the inverse marginal distribution function. Previous work estimates these margins empirically, which is problematic given that the missing data clearly biases these estimates. The margin adjustment corrects these biases, yielding proper inference with missing data.
To use the function, the user can specify a number of properties of the model:
nImp: The number of imputations to createH: The upper bound for the number of clusters in the truncated DP mixturek.star: The dimension of the latent factors, defult isceiling(0.7*p)nsamp: number of interations in the MCMCburn: burn-in before posterior samples are savedhyperparams:delta: to the precision of the prior covariance. This parameter has been the most influential in the discovery of new clusters. Lower to find more clusters. Default value is 10k.star: change to increase or decrease dimension of latent factorsa_alphab_alphanu_mixkappa_0nua1a2a.sigmab.sigmaD_0: k.star dimensional identity
Default values are included in the function documentation, but we recommend altering δ to improve model fit. The function is called below:
hyperparams = list(delta = 10,
k.star = 2,
plugin.threshold = 100,
a_alpha = 1,
b_alpha = 3,
nu_mix = 4,
kappa_0 = .001,
nu = 3,
a1 = 2,
a2 = 3,
a.sigma = 1,
b.sigma = .3,
D_0 = diag(1,2))
mcmc<-GMC.mcmc(Data = X, nImp = 5,hyperparams = hyperparams, burn = 1500,nsamp = 2000, seed = 47)GMC_Impute returns nImp imputations, as well as posterior samples of
Copula parameters which may be used for simulation of posterior
predictive data sets or posterior inference. See documentation for
format.
## Warning: Removed 1 rows containing missing values (geom_point).
We can plot posterior samples of the marginal distribution of X2, as well as point-wise posterior means:
par(mar = c(5,6,4,2))
range = range(mcmc$Support[[3]][1,(2:dim(mcmc$Support[[3]])[2])]) # get support
quantiles = mcmc$Quantiles[[3]] #get quantiles
plot(range[1]:range[2],quantiles[1,2:(range[2]+2)],
col = "gray",
type = 'b',
xlab = expression(X[2]),
ylab = expression(P(X[2] <= x)),
main = "Posterior Samples of F_X2",
cex.lab = 1.5,
cex.main = 2)
sapply(2:500,function(x)points(range[1]:range[2],quantiles[x,2:(range[2]+2)], col ='gray', type = 'b'))
lines(ecdf(X_noMis$X2), cex = 2)
points(range[1]:range[2],colMeans(quantiles[2:500,2:(range[2]+2)]),pch =2, cex = 2)
lines(ecdf(X$X2), col = 2, cex = 2)
legend("bottomright",c("ECDF w/o Mis","ECDF w/ Mis","Posterior Mean"), pch = c(16,16,2),col = c(1,2,1),bty = 'n', cex = 1.3, text.font = 2)
Finally, we can use the posterior samples to generative a fixed quantity of posterior predictive data sets for checks and inference. This is done by using the returned samples from GMC.mcmc. Here we create two predictive data sets.
#get predictive data sets
pred<- get_predictive_Y(mcmc, # GMC mcmc object
nobs = dim(X)[1], # number of observations in predictive data set; we use n = dim(X)[1]
nsets = 2, # number of predictive data sets to create
seed = 10)
#plot results
grid.arrange(ggplot(pred$Y_pred[[1]], aes(x = X1, y = X2, color = X3))+
geom_point() +
ggtitle("Posterior Predictive Data Set #1")+
xlab(expression(X[1]))+
ylab(expression(X[2]))+
scale_color_discrete(name = expression(X[3]),
labels = c(0,1))+
theme(axis.text = element_text(size = 12, face = "bold"),
axis.title = element_text(size = 14, face = "bold"),
legend.title= element_text(size = 14, face = "bold"),
plot.title = element_text(size = 16, face = "bold")),
ggplot(pred$Y_pred[[2]], aes(x = X1, y = X2, color = X3))+
geom_point() +
ggtitle("Posterior Predictive Data Set #2")+
xlab(expression(X[1]))+
ylab(expression(X[2]))+
scale_color_discrete(name = expression(X[3]),
labels = c(0,1))+
theme(axis.text = element_text(size = 12, face = "bold"),
axis.title = element_text(size = 14, face = "bold"),
legend.title= element_text(size = 14, face = "bold"),
plot.title = element_text(size = 16, face = "bold")), ncol = 1)