## Example 2: Continuous Y, additional simulation 1 in the Supporting Material

### Install/load `SynDI`

``````#if(!("SynDI" %in% rownames(installed.packages()))) install.packages("SynDI")
library(SynDI)
``````

Install and load these other packages to complete the tutorial:

``````library(mvtnorm)
library(mice)
library(arm)
library(dplyr)
library(StackImpute)
library(randomForest)
library(boot)
library(broom)
``````

### Settings

``````k = 2  #total number of external models
n = 100  #internal study size
nrep = 3 #when generating the synthetic data, replicate the observed X
#nrep times
m1 = 10000  #size of external study 1
m2 = 10000  #size of external study 2
p1 = 1 #length of X for external calculator 1, excluding the intercept
p2 = 2 #length of X for external calculator 2, excluding the intercept
q = 5 #length of (X,B) in the full model, including the intercept
M = 100 #number of multiple imputation, denoted as M in the manuscript

gamma.S0.true = c(-1, rep(-1,4))   #true target model parameter for the internal
# population
gamma.S1.true = c(1, rep(-1,4))    #true target model parameter for external
# population 1
gamma.S2.true = c(3, rep(-1,4))    #true target model parameter for external
# population 2
``````

### Create external models

``````###obtain beta estimates from external model 1: Y~X1
###obtain beta estimates from external model 2: Y~X1+X2

set.seed(2333)

#Population 1 has (Y,X1)
data.m1 = data.frame(matrix(ncol = 5, nrow = m1))
names(data.m1) = c('Y', 'X1', 'X2','B1','B2')
data.m1[,c('X1', 'X2','B1')] = MASS::mvrnorm(m1, rep(0,3), diag(0.7,3)+0.3)
data.m1[,c('X1', 'X2', 'B1')] = apply(data.m1[,c('X1', 'X2', 'B1')], 2,
function(x) x-mean(x))
data.m1\$B2 = rbinom(m1, 1, arm::invlogit(0.1*data.m1\$X1 + 0.2*data.m1\$X2 +
0.3*data.m1\$B1))
data.m1\$Y = rnorm(m1, mean = data.matrix(cbind(1,
data.m1[,c('X1','X2','B1','B2')])) %*% matrix(gamma.S1.true, q, 1), sd = 1)

#Population 2 has (Y,X1,X2)
data.m2 = data.frame(matrix(ncol = 5, nrow = m2))
names(data.m2) = c('Y', 'X1', 'X2','B1','B2')
data.m2[,c('X1', 'X2','B1')] = MASS::mvrnorm(m2, rep(0,3), diag(0.7,3)+0.3)
data.m2[,c('X1', 'X2', 'B1')] = apply(data.m2[,c('X1', 'X2', 'B1')], 2,
function(x) x-mean(x))
data.m2\$B2 = rbinom(m2, 1, arm::invlogit(0.1*data.m2\$X1 + 0.2*data.m2\$X2 +
0.3*data.m2\$B1))
data.m2\$Y = rnorm(m2, mean = data.matrix(cbind(1,
data.m2[,c('X1','X2','B1','B2')])) %*% matrix(gamma.S2.true, q, 1), sd = 1)

fit.E1 = glm(Y ~ X1, data = data.m1, family=gaussian())
fit.E2 = glm(Y ~ X1 + X2, data = data.m2, family=gaussian())

#Calculator 1
beta.E1 = coef(fit.E1)
names(beta.E1) = c('int', 'X1')
sigma.E1 = sigma(fit.E1)

#Calculator 2
beta.E2 = coef(fit.E2)
names(beta.E2) = c('int', 'X1', 'X2')
sigma.E2 = sigma(fit.E2)

betaHatExt_list = list(Ext1 = beta.E1, Ext2 = beta.E2)
sigmaHatExt_list = list(Ext1 = sigma.E1, Ext2 = sigma.E2)
``````

### Create internal data set

``````datan = data.frame(matrix(ncol = 6, nrow = n))
names(datan) = c('Y', 'X1', 'X2', 'B1', 'B2','S')
datan[,c('X1', 'X2', 'B1')] = MASS::mvrnorm(n, rep(0,3), diag(0.7,3)+0.3)
datan[,c('X1', 'X2', 'B1')] = apply(datan[,c('X1', 'X2', 'B1')], 2,
function(x) x-mean(x))
datan\$B2 = rbinom(n, 1, prob = expit(0.1*datan\$X1 + 0.2*datan\$X2 +
0.3*datan\$B1))
datan\$Y = rnorm(n, mean = data.matrix(cbind(1,
datan[,c('X1', 'X2', 'B1','B2')])) %*% matrix(gamma.S0.true, q, 1), sd = 1)
``````

### Step 1: convert the external model information into the synthetic data

``````#### Function Create.Synthetic() can create synthetic data for both
# parametric model 1 and model 2
data.combined = Create.Synthetic(datan=datan,
nrep=nrep,
Y='Y',
XB = c('X1','X2','B1','B2'),
Ytype='continuous',
parametric = c('Yes','Yes'),
betaHatExt_list=betaHatExt_list,
sigmaHatExt_list=sigmaHatExt_list)
``````

### Step 2: Multiple imputation

``````### Impute missingness ignoring the outcome Y
pred_matrix = mice::make.predictorMatrix(data.combined)
pred_matrix[c('Int',"Y",'X1','S'),] = 0
pred_matrix[,c('Int','Y','S')] = 0
imp_method = mice::make.method(data.combined)
#choose your desired imputation method for each missingness
imp_method[c('X2','B1','B2')] = c('norm','norm','logreg')
data.combined\$B2 = factor(data.combined\$B2)
imputes = mice::mice(data.combined,
m=M,
predictorMatrix=pred_matrix,
method=imp_method,
printFlag=F)
``````

### Step 3: Stack M imputed datasets

``````stack = mice::complete(imputes, action="long", include = FALSE)
stack\$B2 = as.numeric(as.character(stack\$B2))
``````

### Step 4: Calculate population-specific weights

``````##### Internal data only
fit.gamma.I = glm(Y ~ X1 + X2 + B1 + B2, data = datan, family=gaussian())
gamma.I = coef(fit.gamma.I)

######## calculate the initial gamma for population S=1 and S=2
gamma.S1.origin = Initial.estimates(datan=datan,
gamma.I=gamma.I,
beta=betaHatExt_list[['Ext1']],
X='X1',
B=c('X2','B1','B2'),
Btype=c('continuous','continuous','binary'))
gamma.S2.origin = Initial.estimates(datan=datan,
gamma.I=gamma.I,
beta=betaHatExt_list[['Ext2']],
X=c('X1','X2'),
B=c('B1','B2'),
Btype=c('continuous','binary'))

############ calculate weights for each observation by population
stack\$wt = 0
stack[stack\$S == 0, 'wt'] = dnorm(stack[stack\$S == 0, 'Y'],
data.matrix(stack[stack\$S==0, c('Int','X1','X2','B1','B2')])%*%
matrix(gamma.I, q, 1))
stack[stack\$S == 1, 'wt'] = dnorm(stack[stack\$S == 1, 'Y'],
data.matrix(stack[stack\$S==1, c('Int','X1','X2','B1','B2')])%*%
matrix(gamma.S1.origin, q, 1))
stack[stack\$S == 2, 'wt'] = dnorm(stack[stack\$S == 2, 'Y'],
data.matrix(stack[stack\$S==2, c('Int','X1','X2','B1','B2')])%*%
matrix(gamma.S2.origin, q, 1))
## weights need to be re-scaled to 1 within individuals
stack = as.data.frame(stack %>% group_by(.id) %>% mutate(wt = wt / sum(wt)))
if(sum(is.na(stack\$wt))>0){
stack[is.na(stack\$wt)==TRUE,]\$wt = 0
}
``````

### Step 5: Estimation

``````#### point estimation
fit = glm(Y~X1+X2+B1+B2+factor(S), data=stack, family=gaussian(),
weights = stack\$wt)
coef(fit)
#> (Intercept)          X1          X2          B1          B2  factor(S)1
#>  -0.9246228  -1.0649288  -0.9831681  -0.9198052  -0.8887366   2.5444291
#>  factor(S)2
#>   4.1798795

##### Lauren's variance estimation
Louiscovar = StackImpute::Louis_Information(fit, stack, M = M)
diag(solve(Louiscovar))
#> (Intercept)          X1          X2          B1          B2  factor(S)1
#> 0.024572936 0.003252623 0.003724367 0.006970366 0.043269481 0.020332938
#>  factor(S)2
#> 0.017627224

##### Bootstrap variance
#***readers need to modify the existing function Resample.gamma.binaryY()
# to match their own Steps 1-5
# results.boot = boot(data=datan,
#                     statistic=Resample.gamma.continuousY,
#                     R=2) ##R=2 is for illustration purpose to save running time.
# Typically a larger R number, e.g.R=500 is used for reliable estimates***
# it may take hours to finish running
#broom::tidy(results.boot)\$std.error^2
``````