eplogprob.marg - Compute approximate marginal inclusion probabilities from pvalues

eplogprob.marg calculates approximate marginal posterior inclusion probabilities from p-values computed from a series of simple linear regression models using a lower bound approximation to Bayes factors. Used to order variables and if appropriate obtain initial inclusion probabilities for sampling using Bayesian Adaptive Sampling bas.lm

Usage

eplogprob.marg(Y, X, thresh = 0.5, max = 0.99, int = TRUE)

Arguments

Y

response variable

X

design matrix with a column of ones for the intercept

thresh

the value of the inclusion probability when if the p-value > 1/exp(1), where the lower bound approximation is not valid.

max

maximum value of the inclusion probability; used for the bas.lm function to keep initial inclusion probabilities away from 1.

int

If the Intercept is included in the linear model, set the marginal inclusion probability corresponding to the intercept to 1

Value

eplogprob.prob returns a vector of marginal posterior inclusion probabilities for each of the variables in the linear model. If int = TRUE, then the inclusion probability for the intercept is set to 1.

Details

Sellke, Bayarri and Berger (2001) provide a simple calibration of p-values

BF(p) = -e p log(p)

which provide a lower bound to a Bayes factor for comparing H0: beta = 0 versus H1: beta not equal to 0, when the p-value p is less than 1/e. Using equal prior odds on the hypotheses H0 and H1, the approximate marginal posterior inclusion probability

p(beta != 0 | data ) = 1/(1 + BF(p))

When p > 1/e, we set the marginal inclusion probability to 0.5 or the value given by thresh. For the eplogprob.marg the marginal p-values are obtained using statistics from the p simple linear regressions

P(F > (n-2) R2/(1 - R2)) where F ~ F(1, n-2) where R2 is the square of the correlation coefficient between y and X_j.

References

Sellke, Thomas, Bayarri, M. J., and Berger, James O. (2001), “Calibration of p-values for testing precise null hypotheses”, The American Statistician, 55, 62-71.

See also

bas

Author

Merlise Clyde clyde@stat.duke.edu

Examples

library(MASS)
data(UScrime)
UScrime[,-2] = log(UScrime[,-2])
eplogprob(lm(y ~ ., data=UScrime))
#> (Intercept)           M          So          Ed         Po1         Po2 
#>   1.0000000   0.9480823   0.5000000   0.9883193   0.5045770   0.5000000 
#>          LF         M.F         Pop          NW          U1          U2 
#>   0.5000000   0.5304659   0.5807429   0.8046414   0.5000000   0.6858642 
#>         GDP        Ineq        Prob        Time 
#>   0.6069289   0.9900000   0.9412475   0.5782754