# ILSE: simulated examples

#### 2022-01-31

The package can be loaded with the command:

library("ILSE")

# ILSE can handle (non)missing data with continuous variables

First, we generate a small simulated data.

  set.seed(1)
n <- 100
p <- 6
X <- MASS::mvrnorm(n, rep(0, p), cor.mat(p, rho=0.5))
beta0 <- rep(c(1,-1), times=3)
Y <- -2+ X %*% beta0 + rnorm(n, sd=1)

## A special case: without missing values

Then, we fit the linear regression model without missing values based on ILSE.

ilse1 <- ilse(Y~X)
print(ilse1)

We can also create a (data.frame) object as input for ILSE.

dat <- data.frame(Y=Y, X=X)
ilse1 <- ilse(Y~., data=dat)
print(ilse1)
Coef(ilse1) # access the coefficients
Fitted.values(ilse1)[1:5]
Residuals(ilse1)[1:5]

Check the significant variables by bootstratp.

s1 <- summary(ilse1)
s1

## Handle data with missing values

First, we randomly remove some entries in X.

mis_rate <- 0.3
set.seed(1)
na_id <- sample(1:(n*p), n*p*mis_rate)
Xmis <- X
Xmis[na_id] <- NA
ncomp <- sum(complete.cases(Xmis))
message("Number of complete cases is ", ncomp, '\n')

Second, we use lm to fit linear regression model based on complete cases, i.e., CC analysis. We can not detect any siginificant covariates.

lm1 <- lm(Y~Xmis)
s_cc <- summary.lm(lm1)
s_cc

Third, we use ILSE to fit the linear regression model based on all data. We can fit a linear regression model without intercept by setting formula:

ilse2 <- ilse(Y~Xmis+0, data=NULL, verbose=T)
print(ilse2)

Then, we fit a linear regression model with intercept by following command

ilse2 <- ilse(Y~Xmis, data=NULL, verbose=T)
print(ilse2)

Fourth, Bootstrap is applied to evaluate the standard error and p-values of each coefficients estimated by ILSE. We observe four significant coefficients.

s2 <- summary(ilse2, Nbt=20)
s2

In ILSE package, we also provide Full Information Maximum Likelihood for Linear Regression fimlreg. We show how to use it to handle the above missing data.

fimllm <- fimlreg(Y~Xmis)
print(fimllm)

We also use bootstrap to evaluate the standard error and p-values of each coefficients estimated by ILSE. We observe only one significant coefficients.

s_fiml <- summary(fimllm, Nbt=20)
s_fiml

## Visualization

We visualize the p-vaules of each methods , where red line denotes 0.05 in y-axis and blue line 0.1 in y-axis.

pMat <- cbind(CC=s_cc$coefficients[,4], ILSE=s2[,4], FIML=s_fiml[,4]) library(ggplot2) df1 <- data.frame(Pval= as.vector(pMat[-1,]), Method =factor(rep(c('CC', "ILSE", "FIML"),each=p)), covariate= factor(rep(paste0("X", 1:p), times=3))) ggplot(data=df1, aes(x=covariate, y=Pval, fill=Method)) + geom_bar(position = "dodge", stat="identity",width = 0.5) + geom_hline(yintercept = 0.05, color='red') + geom_hline(yintercept = 0.1, color='blue') # ILSE can handle missing data with continuos and categorical variables Base on the above data, we add a new column, a categorical variable (Sex), into the data.frame. This variable is not associated with the outcome variable. dat <- data.frame(Y=Y, X=Xmis) dat$Sex <- factor(rep(c('male', 'female'), times=n/2))
dat$Sex[sample(1:n, n*mis_rate)] <- NA ilse1 <- ilse(Y~., data=dat, verbose = T) We can change the bootstrap times in calculate the standard errors, Z value and p-values of coefficients. s3 <- summary(ilse1, Nbt=40) s3 # ILSE can correctly identify the important variables ## generate data First, we generate data from a linear regression model with three inportant variables(1,3,5) and three unimportant variables(2,4,6). set.seed(10) n <- 100 p <- 6 X <- MASS::mvrnorm(n, rep(0, p), cor.mat(p, rho=0.5)) beta0 <- rep(c(1,0), times=3) Y <- -2+ X %*% beta0 + rnorm(n, sd=1) message("The true regression coefficients are: ", paste0(beta0, ' ')) We randomly assign missing values in the design matrix. mis_rate <- 0.3 set.seed(1) na_id <- sample(1:(n*p), n*p*mis_rate) Xmis <- X Xmis[na_id] <- NA Next, we use ILSE to fit model. dat <- data.frame(Y=Y, X=Xmis) ilse1 <- ilse(Y~., data=dat, verbose = T) s3 <- summary(ilse1) s3 Fit model by using lm and FIML, finally compare ILSE with these two methods. lm1 <- lm(Y~Xmis) s_cc <- summary.lm(lm1) fimllm <- fimlreg(Y~Xmis) s_fiml <- summary(fimllm) ## Visualization We visualize the p-vaules of each methods , where red line denotes 0.05 in y-axis. Under significance level 0.05, we found both ILSE and FIML can identify all important variables (X1, X3 and X5), while CC method only identified X1 and X5. library(ggthemes) pMat <- cbind(CC=s_cc$coefficients[,4], ILSE=s3[,4], FIML=s_fiml[,4])
df1 <- data.frame(Pval= as.vector(pMat[-1,]),
Method =factor(rep(c('CC', "ILSE", "FIML"),each=p)),
covariate= factor(rep(paste0("X", 1:p), times=3)))
ggplot(data=df1, aes(x=covariate, y=Pval, fill=Method)) + geom_bar(position = "dodge", stat="identity",width = 0.5) + geom_hline(yintercept = 0.05, color='red') + scale_fill_economist()

# ILSE can handle data with high missing rate

Here, we generate a data with 80% missing values, then use ILSE to fit model.


# generate data from linear model
set.seed(10)
n <- 100
p <- 6
X <- MASS::mvrnorm(n, rep(0, p), cor.mat(p, rho=0.5))
beta0 <- rep(c(1,-1), times=3)
Y <- -2+ X %*% beta0 + rnorm(n, sd=1)

# generate missing values
mis_rate <- 0.8
set.seed(1)
na_id <- sample(1:(n*p), n*p*mis_rate)
Xmis <- X
Xmis[na_id] <- NA
# retain 4 complete cases.
Xmis[1:4,] <- X[1:4, ]
sum(complete.cases(Xmis))

CC method will failed.

lm1 <- lm(Y~Xmis)
summary.lm(lm1)

However, ILSE can still work.

ilse2 <- ilse(Y~Xmis, verbose = T)
s2 <- summary(ilse2)
s2

# ILSE can handle large-scale data

We generate a large-scale data with n=1000 and p = 50

n <- 1000
p <- 50
X <- MASS::mvrnorm(n, rep(0, p), cor.mat(p, rho=0.5))
beta0 <- rep(c(1,-1), length=p)
Y <- -2+ X %*% beta0 + rnorm(n, sd=1)

mis_rate <- 0.3
set.seed(1)
na_id <- sample(1:(n*p), n*p*mis_rate)
Xmis <- X
Xmis[na_id] <- NA

Xmis[1:10,] <- X[1:10,]
lm1 <- lm(Y~Xmis)
lm1
system.time(ilse2 <- ilse(Y~Xmis, data=NULL, verbose=T))

# Session information

sessionInfo()
#> R version 4.0.3 (2020-10-10)
#> Platform: x86_64-w64-mingw32/x64 (64-bit)
#> Running under: Windows 10 x64 (build 22000)
#>
#> Matrix products: default
#>
#> locale:
#>  LC_COLLATE=C
#>  LC_CTYPE=Chinese (Simplified)_China.936
#>  LC_MONETARY=Chinese (Simplified)_China.936
#>  LC_NUMERIC=C
#>  LC_TIME=Chinese (Simplified)_China.936
#>
#> attached base packages:
#>  stats     graphics  grDevices utils     datasets  methods   base
#>
#> other attached packages:
#>  ILSE_1.1.7
#>
#> loaded via a namespace (and not attached):
#>   Rcpp_1.0.7      digest_0.6.28   R6_2.5.1        jsonlite_1.7.2
#>   magrittr_2.0.1  evaluate_0.14   pbapply_1.5-0   rlang_0.4.11
#>   stringi_1.7.5   jquerylib_0.1.4 bslib_0.3.1     rmarkdown_2.11
#>  tools_4.0.3     stringr_1.4.0   parallel_4.0.3  xfun_0.29
#>  yaml_2.2.2      fastmap_1.1.0   compiler_4.0.3  htmltools_0.5.2
#>  knitr_1.37      sass_0.4.0