SSDRegression: Sample size determination for the Bayesian Multiple Linear...

In Qianrao-Fu/SSDbain: Sample Size Determination using AAFBF for Bayesian Hypothesis Testing Implemented in bain

Sample size determination for the Bayesian Multiple Linear Regression with bain

Description

The function SSDRegression in the R package SSDbain computes the sample size for multiple linear regression using Bayes factors implemented in bain with informative hypothesis. SSDbain can be found at https://github.com/Qianrao-Fu/SSDbain. In Fu (unpublished), it is elaborated how to use SSDRegression. Users are well advised to read this paper before using SSDRegression. Users are required to install package bain from CRAN before using SSDbain package.

Usage

SSDRegression(Hyp1,Hyp2,k,rho,R_square1,R_square2,T_sim = 1000,BFthresh,eta,seed = 10,standardize = TRUE, ratio)

Arguments

Hyp1	strings that specify the former hypothesis of a pair of interested hypotheses. For example, if H0:"beta1=beta2=beta3=0" vs H1:"beta1>0&beta2>0&beta3>0", then Hyp1:"beta1=beta2=beta3=0".
Hyp2	strings that specify the latter hypothesis of a pair of interested hypotheses. For example, as the previous example, then Hyp2:"beta1>0&beta2>0&beta3>0". Note: when Hyp2 is an unconstrained hypothesis, then Hyp2="Ha"; when hyp2 is the complement hypothesis of Hyp1, then Hyp2="Hc".
k	a positive integer that specifies the number of predictors in the multiple linear regression. It should be noted that the number of predictors in the hypotheses is equal to the number of predictors in the linear regression. For example, if the model is yi=beta1 xi1+beta2 xi2+beta3 xi3+\epsilon i, k=3.
rho	a symmetric matrix that specifies the correlation between each pair of predictors. The diagonal of this matrix are ones.
R_square1	parameter used to specify the coefficient of determination under hyp1.
R_square2	parameter used to specify the coefficient of determination under hyp2.
BFthresh	a numeric value not less than 1 that specifies the required size of the Bayes factor for the true hypothesis. For example, if BFthresh=3, and H0 is compared with H1, BF01\ge3 and BF10\ge3 should be satisfied.
eta	a numeric value that specifies the probability that the Bayes factor is larger than BFthresh if either of the competing hypotheses is true.
T_sim	a positive integer that specifies the number of data sets sampled from the populations corresponding to the two competing hypotheses. It should be noted that at least T_sim=10000 is required to guarantee the stability of the results.
seed	a positive integer that specifies the seed of R's random numer of generator.
standardize	a logical value that specifies whether hypotheses regarding standardized or unstandardized regression coefficients are evaluated. With standardize=TRUEhypothese with respect to standardized regression coefficients are evaluated. With standardize=FALSE hypothese with respect to unstandardized regression coefficients are evaluated.
ratio	an optional vector that specifies the ratio between each pair of regression coefficients for each population in which the corresponding hypothesis is true.

Value

The output resulting from analyses with SSDRegression contains:

1) required sample size N

2) the probability P(BFsv>BFthresh|Hs)

3) the probability P(BFvs>BFthresh|Hv)

Note: To perform the sensitive analysis, the results are provided for three different fractions b, 2b, and 3b, where b corresponds to fraction = 1 in the call to bain.

Note

The following competing hypothese can be executed:

1) Hyp1='beta1=beta2=...=betaK=0' vs Hyp2='Ha', where Ha is the unconstrianed hypothesis. The default ratio of the regression coefficienst for Ha is 1,1,...,1.

2) Hyp1='beta1=beta2=...=betaK=0' vs Hyp2='beta1>0,beta2>0,...,betaK>0', where each of the regression coefficient may also be smaller than zero. That is, both "<" and ">" can exist in a hypothesis. The default ratio of the regression coefficienst for Hyp2 is 1,1,...,1.

3) Hyp1='beta1=beta2=...=betaK' vs Hyp2='beta1*>beta2*>...>betaK*', where 1*, 2*, ..., K* are a re-ordering of the numbers 1, 2, ..., K. The ratio of the regression coefficients 1,1, ... ,1 is used for Hyp1 and the regression coefficients are computed such that R^2 = 0.13, and the ratio of the regression coefficients Kd, (K-1)d, ..., 3d, 2d, d is used for Hyp2.

4) Hyp1='beta1>0, beta2>0, ..., betaK>0' vs Hyp2='Hc', where Hc is the complement hypothesis of Hyp1. The ratio for Hyp1 is consistent with Hyp2 in 2), and the ratio is reordered using the representative hypothesis (see Appendix B in Fu, unpublished) for Hyp2. It should be noted that in this situation only ">" or "<" is allowed.

5) Hyp1='beta1*>beta2*>...>betaK*' vs Hyp2='Hc', where Hc is the complement hypothesis of Hyp1. The ratio for Hyp1 is consistent with Hyp2 in 3), and the ratio is reordered using the representative hypothesis (see Appendix B in Fu, unpublished) for Hyp2.

The standardized regression coefficients are used in 3) and 5) to ensure the regression coefficients are comparable.

Author(s)

Qianrao Fu

References

Fu, Q.(unpublished). Sample size determination for Bayesian testing of informative hypotheses in linear regression. Gu, X., Mulder, J., and Hoijtink, H. (2017). Approximated adjusted fractional Bayes factors: A general method for testing informative hypotheses. British Journal of Mathematical and Statistical Psychology. British Journal of Mathematical and Statistical Psychology, 71(2), 229-261. doi:10.1111/bmsp.12110

Hoijtink, H., Gu, X., and Mulder, J. (2018). Bayesian evaluation of informative hypotheses for multiple populations. British Journal of Mathematical and Statistical Psychology. doi:10.1111/bmsp.12145

Examples

#Example 1: using the coefficient of determination R^2=0.13 to determine the sample size when H0 versus Ha.


Res<-SSDRegression(Hyp1='beta1=beta2=beta3=0',Hyp2='Ha',k=3,rho=matrix(c(1,0.2,0.2,0.2,1,0.2,0.2,0.2,1),nrow=3),R_square1=0,R_square2=0.13,T_sim=10000,BFthresh=3,eta=0.8,seed=10,standardize=FALSE,ratio=c(1,1,1))


#Example 2: using the coefficient of determination R^2=0.13 to determine the sample size when H0 versus H1.


Res<-SSDRegression(Hyp1='beta1=beta2=beta3=0',Hyp2='beta1>0&beta2>0&beta3>0',k=3,rho=matrix(c(1,0.2,0.2,0.2,1,0.2,0.2,0.2,1),nrow=3),R_square1=0,R_square2=0.13,T_sim=10000,BFthresh=3,eta=0.8,seed=10,standardize=FALSE,ratio=c(1,1,1))


#Example 3: using the coefficient of determination R^2=0.13 to determine the sample size when H0 versus H2.


Res<-SSDRegression(Hyp1='beta1=beta2=beta3',Hyp2='beta1>beta2>beta3',k=3,rho=matrix(c(1,0.2,0.2,0.2,1,0.2,0.2,0.2,1),nrow=3),R_square1=0,R_square2=0.13,T_sim=10000,BFthresh=3,eta=0.8,seed=10,standardize=TRUE,ratio=c(3,2,1))


#Example 4: using the coefficient of determination R^2=0.13 to determine the sample size when H1 versus H1c.


Res<-SSDRegression(Hyp1='beta1>0&beta2>0&beta3>0',Hyp2='Hc',k=3,rho=matrix(c(1,0.2,0.2,0.2,1,0.2,0.2,0.2,1),nrow=3),R_square1=0.13,R_square2=0.13,T_sim=10000,BFthresh=3,eta=0.8,seed=10,standardize=FALSE,ratio=c(1,1,1))


#Example 5: using the coefficient of determination R^2=0.13 to determine the sample size when H2 versus H2c.


Res<-SSDRegression(Hyp1='beta1>beta2>beta3',Hyp2='Hc',k=3,rho=matrix(c(1,0.2,0.2,0.2,1,0.2,0.2,0.2,1),nrow=3),R_square1=0.13,R_square2=0.13,T_sim=10000,BFthresh=3,eta=0.8,seed=10,standardize=TRUE,ratio=c(3,2,1))

Qianrao-Fu/SSDbain documentation built on Oct. 23, 2023, 10:30 p.m.