Package 'SynDI'

Example data for Create.Synthetic()

Description

Example data set for Create.Synthetic()

Format

a list with

• nrep when generating the synthetic data, replicate the observed X nrep times

• datan simulated internal data set

• betaHatExt_list list of external model estimates

---

Create the synthetic data

Description

Creates a synthetic data set from internal data and external models.

Usage

Create.Synthetic(
  datan,
  nrep,
  Y,
  XB,
  Ytype = "binary",
  parametric,
  betaHatExt_list,
  sigmaHatExt_list = NULL
)

Arguments

datan	internal data only
nrep	number of replication when creating the synthetic data
Y	outcome name, e.g. Y='Y'
XB	all covariate names for both X and B in the target model, e.g. XB=c('X1','X2','X3','X4','B1','B2')
Ytype	the type of outcome Y, either 'binary' or 'continuous'.
parametric	choice of "Yes" or "No" for each external model. Specify whether the external model is paramtric or not, e.g. parametric=c('Yes','No')
betaHatExt_list	a list of parameter estimates of the external models. The order needs to be the same as listed in XB, and variable name is required. See example for details.
sigmaHatExt_list	a list of sigma^2 for continuous outcome fitted from linear regression. If not available or the outcome type is binary, set sigmaHatExt_list=NULL

Value

a data.frame. The combined dataset of the internal data (of size n) and the synthetic data for the given external model (of size n * nrep). This combined dataset contains a total of n*(1+nrep) rows, one intercept column (Int), one outcome column (Y), one indicator column (S), and all the predictors in the internal data. S is the indicator variable, where the internal data is indicated as S=0, and the synthetic data is indicated as S=1. The internal data part is a complete dataset without any missingness. The synthetic data part may contain missingness for certain predictors that were not used in the external model.

References

Reference: Gu, T., Taylor, J.M.G. and Mukherjee, B. (2021) Regression inference for multiple populations by integrating summary-level data using stacked imputations https://arxiv.org/abs/2106.06835.

Examples

data(create_synthetic_example)

nrep = create_synthetic_example$nrep
datan = create_synthetic_example$datan
betaHatExt_list = create_synthetic_example$betaHatExt_list

data.combined = Create.Synthetic(nrep = nrep, datan = datan, Y = 'Y', 
    XB = c('X1', 'X2', 'X3', 'X4', 'B1', 'B2'), Ytype = 'binary', 
    parametric = c('Yes', 'No'), betaHatExt_list = betaHatExt_list, 
    sigmaHatExt_list = NULL)

---

Expit function

Description

Expit function

Usage

expit(x)

Arguments

x	vector to expit

Value

numeric vector with the value of the expit function y = expit(x) = exp(x)/(1+exp(x)).

Expit helper function.

---

Example data for Initial.estimates()

Description

Example data set for Initial.estimates()

Format

a list with

• datan simulated internal data set

• gamma.I internal gamma coefficients

• beta beta estimates from external model 1

---

Internal estimation

Description

Calculate the initial estimates for external populations.

Usage

Initial.estimates(datan, gamma.I, X, B, beta, Btype)

Arguments

datan	internal data only
gamma.I	regression estimates using internal data only (datan)
X	a vector of predictor that were used in the external study, e.g. X = c('X1','X2','X3')
B	a vector of covariates that were not used in the external study, e.g. B=c('X4','B1','B2')
beta	a vector of external model estimates, the vector order should be the same as listed in X, e.g. names(beta) = c("int", "X1", "X2", "X3")
Btype	a vector of type of B, either continuous or binary. If "continuous", linear regression will be used; if "binary", logistic regression will be used. More types can be implemented manually.

Value

a numeric vector of estimated coefficients of the target model for the given external population. Assume the internal data contains p predictors. The vector is of dimension (p+1), including the estimates of the intercept.

References

Neuhaus, J. and Jewell, N. (1993). A geometric approach to assess bias due to omitted covariates in generalized linear models. Biometrika 80,807–815.

Gu, T., Taylor, J.M.G. and Mukherjee, B. (2021) Regression inference for multiple populations by integrating summary-level data using stacked imputations https://arxiv.org/abs/2106.06835.

Examples

#' data(initial_estimates_example)

datan = initial_estimates_example$datan
gamma.I = initial_estimates_example$gamma.I
beta = initial_estimates_example$beta

# calculate the initial gamma for population S=1
gamma.S1.origin = Initial.estimates(datan = datan, gamma.I = gamma.I, 
    X = c('X1', 'X2', 'X3'), B = c('X4', 'B1', 'B2'), 
    beta = beta, Btype = c('continuous', 'continuous', 'binary'))

---

Resample for bootstrap variance for binary Y

Description

Resampling function to get bootstrap variance for binary Y. Note that readers need to modify the existing function Resample.gamma.binaryY() to match their own Steps 1-5. It was only included in the package for the purpose of providing an example.

Usage

Resample.gamma.binaryY(data, indices)

Arguments

data	synthetic data
indices	row indices to replicate

Value

numeric vector of regression coefficients

References

Reference: Gu, T., Taylor, J.M.G. and Mukherjee, B. (2021) Regression inference for multiple populations by integrating summary-level data using stacked imputations https://arxiv.org/abs/2106.06835.

---

Resample for bootstrap variance continuous Y

Description

Resampling function to get bootstrap variance for continuous Y. Note that readers need to modify the existing function Resample.gamma.continuousY() to match their own Steps 1-5. It was only included in the package for the purpose of providing an example.

Usage

Resample.gamma.continuousY(data, indices)

Arguments

data	synthetic data
indices	row indices to replicate

Value

numeric vector of regression coefficients

References

Reference: Gu, T., Taylor, J.M.G. and Mukherjee, B. (2021) Regression inference for multiple populations by integrating summary-level data using stacked imputations https://arxiv.org/abs/2106.06835.

---