Vignette_1_simulated

This tutorial describes how to use the mvnimpute package. We will go through an example generated by the data.generation function in the package.

Data generation

#### data generation
n <- 500; p <- 6
# set.seed(133)
m <- c(1, 1, 2, 4, 3, 5)
v <- clusterGeneration::genPositiveDefMat(p, "eigen")$Sigma

miss.var <- c(1, 2)
censor.var <- c(3, 5)

example.data <- data.generation(num_ind = n,
                                mean_vec = m,
                                cov_mat = v,
                                miss_var = miss.var,
                                miss_mech = "MAR",
                                miss_prob = NULL,
                                censor_var = censor.var,
                                censor_type = "interval",
                                censor_param = 0.5)

names(example.data)
#> [1] "full.data"    "observe.data" "bounds"       "indicator"

We generate a dataset with sample size 500, with specified mean vector consists of six specified values, and a random positive definite covariance matrix. The output includes four objects which include, full.data, the data matrix that would have been observed without missing and censoring information; observe.data, the truly observed data matrix with missing and censoring information; bounds, the bounds information of the variables that have missing or censored values. Specifically, we take missing as a special type of interval censoring with no censoring limits (censoring limits are set as \(\pm \infty\)); indicator, the matrix that indicating the data type: 0 as the missing data, 1 as the observed data, and 2 as the censored data. Now, let us take a look at the dataset

# data
full <- example.data$full.data
observed <- example.data$observe.data
censoring.bounds <- example.data$bounds
indicator <- example.data$indicator
# summary
tail(observed) # observed data
#>               y1         y2          y3       y4       y5       y6
#> [495,] -1.547567         NA         NaN 2.315895      NaN 3.907107
#> [496,]        NA -0.2994182         NaN 4.492385 6.786569 2.777854
#> [497,]  1.952071  0.9228392  3.27131928 4.488331      NaN 6.952747
#> [498,] -2.256520 -1.4893098 -0.06679569 4.408467      NaN 2.853682
#> [499,]  2.890608 -2.3451872 -5.92138055 6.106857 4.939494 4.159683
#> [500,]  2.510733  2.0193359         NaN 4.387197 3.525996 6.315236
tail(censoring.bounds[[1]]);tail(censoring.bounds[[2]]) # censoring bounds information
#>               y1         y2          y3       y4        y5       y6
#> [495,] -1.547567         NA  0.15034900 2.315895 1.4321201 3.907107
#> [496,]        NA -0.2994182  0.85767392 4.492385 6.7865694 2.777854
#> [497,]  1.952071  0.9228392  3.27131928 4.488331 3.1594592 6.952747
#> [498,] -2.256520 -1.4893098 -0.06679569 4.408467 0.0873068 2.853682
#> [499,]  2.890608 -2.3451872 -5.92138055 6.106857 4.9394944 4.159683
#> [500,]  2.510733  2.0193359  0.26097282 4.387197 3.5259965 6.315236
#>               y1         y2          y3       y4       y5       y6
#> [495,] -1.547567         NA  0.69562111 2.315895 6.794582 3.907107
#> [496,]        NA -0.2994182  3.20116375 4.492385 6.786569 2.777854
#> [497,]  1.952071  0.9228392  3.27131928 4.488331 5.118294 6.952747
#> [498,] -2.256520 -1.4893098 -0.06679569 4.408467 4.010630 2.853682
#> [499,]  2.890608 -2.3451872 -5.92138055 6.106857 4.939494 4.159683
#> [500,]  2.510733  2.0193359  5.08191846 4.387197 3.525996 6.315236
tail(indicator) # indicator matrix
#>        y1 y2 y3 y4 y5 y6
#> [495,]  1  0  4  1  4  1
#> [496,]  0  1  4  1  1  1
#> [497,]  1  1  1  1  4  1
#> [498,]  1  1  1  1  4  1
#> [499,]  1  1  1  1  1  1
#> [500,]  1  1  4  1  1  1

From the observed data matrix, the missing values are denoted as NA, the censored values are denoted as NaN. The missing data has censoring limits set as \(\pm 10,000\) (we set \(\pm 10,000\) as proxies for \(\pm \infty\)). The indicator matrix labels the missing value as 0 and the interval censored value as 4. Then the plot that visually shows the percentages of observed, missing and censored values can be generated by the visual.plot function.

visual.plot(indicator, title = NULL)

Multiple Imputation

For doing multiple imputation, we take a Normal-Inverse-Wishart distribution for joint prior distribution of the mean vector and covariance matrix. We run MCMC simulation for 1,000 iterations.

### prior specifications
prior.spec <- list(
  mu.0 = rep(0, p),
  Lambda.0 = diag(10, p),
  kappa.0 = 100,
  nu.0 = p * (p + 1) / 2
)

start.vals <- list(
  mu = rep(1, p),
  sigma = diag(p)
)
### MCMC simulation
iter <- 1000
sim.res <- multiple.imputation(
  censoring.bounds,
  prior.spec,
  start.vals,
  iter,
  TRUE
)

Convergence plots

conv.plot(sim.res$simulated.mu, 0, iter, title = "convergence plot of the mean values")

conv.plot(sim.res$simulated.sig, 0, iter, title = "convergence plot of the variance values")

Autocorrelation plots

title = paste("acf: variable", 1:p)
acf.calc(sim.res$simulated.mu, title = title)

acf.calc(sim.res$simulated.sig, title = title)

From the convergence plots and the autocorrelation plots, we see that the posterior distributions of both the mean and variance seem to converge to equilibrium in first few iterations.

Marginal plot

We draw and compare the marginal density plots using the full data, the CC data and the imputed data from the last iteration of MCMC using the mvnimpute package.

for (i in c(1, 2, 3, 5)) {
  plot(density(full[, i]), main = colnames(full)[i])
  lines(density(observed[!is.na(observed[, i]), i]), col = 6, lty = 2)
  lines(density(sim.res$imputed.data[[iter]][, i]), col = 4, lty = 3)
}

The black line is the density plot of the full data, the blue dotted line is the imputed data using the mvnimpute pacakge, the pink dashed line is the CC data. We see that the imputed data density align with the full data density quite well.

Linear Regression