sail (version 0.1.0)

gendata: Simulation Scenario from Bhatnagar et al. (2018+) sail paper

Description

Function that generates data of the different simulation studies presented in the accompanying paper. This function requires the truncnorm package to be installed.

Usage

gendata(n, p, corr, E = truncnorm::rtruncnorm(n, a = -1, b = 1), betaE,
  SNR, parameterIndex)

Arguments

n

number of observations

p

number of main effect variables (X)

corr

correlation between predictors

E

simulated environment vector of length n. Can be continuous or integer valued. Factors must be converted to numeric. Default: truncnorm::rtruncnorm(n, a = -1, b = 1)

betaE

exposure effect size

SNR

signal to noise ratio

parameterIndex

simulation scenario index. See details for more information.

Value

A list with the following elements:

x

matrix of dimension nxp of simulated main effects

y

simulated response vector of length n

e

simulated exposure vector of length n

Y.star

linear predictor vector of length n

f1

the function f1 evaluated at x_1 (f1(X1))

f2

the function f1 evaluated at x_1 (f1(X1))

f3

the function f1 evaluated at x_1 (f1(X1))

f4

the function f1 evaluated at x_1 (f1(X1))

betaE

the value for ${\beta }_E$

f1.f

the function f1

f2.f

the function f2

f3.f

the function f3

f4.f

the function f4

X1

an n length vector of the first predictor

X2

an n length vector of the second predictor

X3

an n length vector of the third predictor

X4

an n length vector of the fourth predictor

scenario

a character representing the simulation scenario identifier as described in Bhatnagar et al. (2018+)

causal

character vector of causal variable names

not_causal

character vector of noise variables

Details

We evaluate the performance of our method on three of its defining characteristics: 1) the strong heredity property, 2) non-linearity of predictor effects and 3) interactions.

Heredity Property

Truth obeys weak hierarchy (parameterIndex = 2) $$Y\ast =f_1(X_1)+f_2(X_2)+{\beta }_E\ast X_E+X_E\ast f_3(X_3)+X_E\ast f_4(X_4)$$

Truth only has interactions (parameterIndex = 3)$$Y\ast =X_E\ast f_3(X_3)+X_E\ast f_4(X_4)$$

Non-linearity

Truth is linear (parameterIndex = 4) $$Y\ast =\sum _{j=1}^4{\beta }_j{X}_j+{\beta }_E\ast X_E+X_E\ast X_3+X_E\ast X_4$$

Interactions

Truth only has main effects (parameterIndex = 5) $$Y\ast =\sum _{j=1}^4{f}_j(X_j)+{\beta }_E\ast X_E$$

.

The functions are from the paper by Lin and Zhang (2006):

f2

f2 <- function(t) 3 * (2 * t - 1)^2

f3

f3 <- function(t) 4 * sin(2 * pi * t) / (2 - sin(2 * pi * t))

f4

f4 <- function(t) 6 * (0.1 * sin(2 * pi * t) + 0.2 * cos(2 * pi * t) + 0.3 * sin(2 * pi * t)^2 + 0.4 * cos(2 * pi * t)^3 + 0.5 * sin(2 * pi * t)^3)

The response is generated as $$Y=Y\ast +k\ast error$$ where Y* is the linear predictor, the error term is generated from a standard normal distribution, and k is chosen such that the signal-to-noise ratio is SNR = Var(Y*)/Var(error), i.e., the variance of the response variable Y due to error is 1/SNR of the variance of Y due to Y*

The covariates are simulated as follows as described in Huang et al. (2010). First, we generate $w1,\dots ,wp,u,v$ independently from $Normal(0,1)$ truncated to the interval [0,1] for $i=1,\dots ,n$. Then we set $x_j=(w_j+t\ast u)/(1+t)$ for $j=1,\dots ,4$ and $x_j=(w_j+t\ast v)/(1+t)$ for $j=5,\dots ,p$, where the parameter $t$ controls the amount of correlation among predictors. This leads to a compound symmetry correlation structure where $Corr(x_j,x_k)=t^2/(1+t^2)$, for $1\le j\le 4,1\le k\le 4$, and $Corr(x_j,x_k)=t^2/(1+t^2)$, for $5\le j\le p,5\le k\le p$, but the covariates of the nonzero and zero components are independent.

References

Lin, Y., & Zhang, H. H. (2006). Component selection and smoothing in multivariate nonparametric regression. The Annals of Statistics, 34(5), 2272-2297.

Huang J, Horowitz JL, Wei F. Variable selection in nonparametric additive models (2010). Annals of statistics. Aug 1;38(4):2282.

Bhatnagar SR, Yang Y, Greenwood CMT. Sparse additive interaction models with the strong heredity property (2018+). Preprint.

Examples

# NOT RUN {
DT <- gendata(n = 75, p = 100, corr = 0, betaE = 2, SNR = 1, parameterIndex = 1)
# }