The Gaussian Mixture Copula for Inference with Missing Data

Joe Feldman 7/10/2022

Installation

# library(devtools)
# remotes::install_github('jfeldman396/GMCImpute')

library(GMCImpute)

Background: Missing Data

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] = NA

We 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 Gaussian Mixture Copula with Margin Adjustment

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.

Fitting the model

To use the function, the user can specify a number of properties of the model:

• nImp: The number of imputations to create
• H: The upper bound for the number of clusters in the truncated DP mixture
• k.star: The dimension of the latent factors, defult is ceiling(0.7*p)
• nsamp: number of interations in the MCMC
• burn: burn-in before posterior samples are saved
• hyperparams:
  • 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 10
  • k.star: change to increase or decrease dimension of latent factors
  • a_alpha
  • b_alpha
  • nu_mix
  • kappa_0
  • nu
  • a1
  • a2
  • a.sigma
  • b.sigma
  • D_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.

Plotting Results

Visualizing Imputations:

## Warning: Removed 1 rows containing missing values (geom_point).

Plotting posterior samples from the margin adjustment

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)

Simulating Predictive Data Sets

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)