ModeDeltaLMNPP {NPP}

Calculate Posterior Mode of the Power Parameter in Normalized Power Prior with Grid Search, Normal Linear Model

Description

The function returns the posterior mode of the power parameter \delta in normal linear model. It calculates the log of the posterior density (up to a normalizing constant), and conduct a grid search to find the approximate mode.

Usage

ModeDeltaLMNPP(y.Cur, y.Hist, x.Cur = NULL, x.Hist = NULL, npoints = 1000,
               prior = list(a = 1.5, b = 0, mu0 = 0, Rinv = matrix(1, nrow = 1),
                            delta.alpha = 1, delta.beta = 1))

Arguments

y.Cur

a vector of individual level of the response y in current data.

y.Hist

a vector of individual level of the response y in historical data.

x.Cur

a vector or matrix or data frame of covariate observed in the current data. If more than 1 covariate available, the number of rows is equal to the number of observations.

x.Hist

a vector or matrix or data frame of covariate observed in the historical data. If more than 1 covariate available, the number of rows is equal to the number of observations.

npoints

is a non-negative integer scalar indicating number of points on a regular spaced grid between [0, 1], where we calculate the log of the posterior and search for the mode.

prior

a list of the hyperparameters in the prior for model parameters (\beta, \sigma^2) and \delta. The form of the prior for model parameter (\beta, \sigma^2) is in the section "Details".

a a positive hyperparameter for prior on model parameters. It is the power a in formula (1/\sigma^2)^a; See details.

b equals 0 if a flat prior is used for \beta. Equals 1 if a normal prior is used for \beta; See details.

mu0 a vector of the mean for prior \beta|\sigma^2. Only applicable if b = 1.

Rinv inverse of the matrix R. The covariance matrix of the prior for \beta|\sigma^2 is \sigma^2 R^{-1}.

delta.alpha is the hyperparameter \alpha in the prior distribution Beta(\alpha, \beta) for \delta.

delta.beta is the hyperparameter \beta in the prior distribution Beta(\alpha, \beta) for \delta.

Details

If b = 1, prior for (\beta, \sigma) is (1/\sigma^2)^a * N(mu0, \sigma^2 R^{-1}), which includes the g-prior. If b = 0, prior for (\beta, \sigma) is (1/\sigma^2)^a. The outputs include posteriors of the model parameter(s) and power parameter, acceptance rate when sampling \delta, and the deviance information criteria.

Author(s)

Zifei Han [e-mail]

References

Ibrahim, J.G., Chen, M.-H., Gwon, Y. and Chen, F. (2015). The Power Prior: Theory and Applications. Statistics in Medicine 34:3724-3749.

Duan, Y., Ye, K. and Smith, E.P. (2006). Evaluating Water Quality: Using Power Priors to Incorporate Historical Information. Environmetrics 17:95-106.

Berger, J.O. and Bernardo, J.M. (1992). On the development of reference priors. Bayesian Statistics 4: Proceedings of the Fourth Valencia International Meeting, Bernardo, J.M, Berger, J.O., Dawid, A.P. and Smith, A.F.M. eds., 35-60, Clarendon Press:Oxford.

Jeffreys, H. (1946). An Invariant Form for the Prior Probability in Estimation Problems. Proceedings of the Royal Statistical Society of London, Series A 186:453-461.

See Also

ModeDeltaBerNPP; ModeDeltaNormalNPP; ModeDeltaMultinomialNPP; ModeDeltaNormalNPP