Bayesian Modeling via Frequentist Goodness-of-Fit

Description

A Bayesian data modeling scheme that performs four interconnected tasks: (i) characterizes the uncertainty of the elicited parametric prior; (ii) provides exploratory diagnostic for checking prior-data conflict; (iii) computes the final statistical prior density estimate; and (iv) executes macro- and micro-inference.

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Number of claims on an insurance policy

Description

The number of claims on an automobile insurance policy made by k = 9461 individuals during a single year.

Usage

data("AutoIns")

Format

A vector of length 9461.

value

number of auto insurance claims by the i^{th} person

Source

Efron, B. and Hastie, T., 2016. Computer Age Statistical Inference (Vol. 5). Cambridge University Press.

Frequency of child illness

Description

Results of a study that followed k = 602 pre-school children in north-east Thailand from June 1982 through September 1985. Researchers recorded the number of times a child became ill during every 2-week period.

Usage

data("ChildIll")

Format

A vector of length k=602.

value

number of times the i^{th} child became ill during the study

Source

Bohning, D., 2000. Computer-assisted Analysis of Mixtures and Applications: Meta-analysis, Disease Mapping, and Others (Vol. 81). CRC press.

Corbet's Butterfly data

Description

The number of times Alexander Corbet captured a species of butterfly during a two-year period in Malaysia.

Usage

data("CorbBfly")

Format

A vector of length k = 501.

value

number of times Corbet captured the i^{th} species

Source

Fisher, R.A., Corbet, A.S. and Williams, C.B., 1943. "The relation between the number of species and the number of individuals in a random sample of an animal population." The Journal of Animal Ecology, pp.42-58.

References

Efron, B. and Hastie, T., 2016. Computer Age Statistical Inference (Vol. 5). Cambridge University Press.

Conduct Finite Bayes Inference on a DS object

Description

A function that generates the finite Bayes prior and posterior distribution, along with the Bayesian credible interval for the posterior mean.

Usage

DS.Finite.Bayes(DS.GF.obj, y.0, n.0 = NULL, 
             cred.interval = 0.9, iters = 25)

Arguments

Value

Author(s)

Doug Fletcher, Subhadeep Mukhopadhyay

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Efron, B., 2018. "Bayes, Oracle Bayes, and Empirical Bayes," Technical Report.

Examples

## Not run: 
### Finite Bayes: Rat with theta_71 (y_71 = 4, n_71 = 14)
data(rat)
rat.start <- gMLE.bb(rat$y, rat$n)$estimate
rat.ds <- DS.prior(rat, max.m = 4, rat.start. family = "Binomial")
rat.FB <- DS.FiniteBayes(rat.ds, y.0 = 4, n.0 = 14)
plot(rat.FB)

## End(Not run)

Full and Excess Entropy of DS(G,m) prior

Description

A function that calculates the full entropy of a DS(G,m) prior. For DS(G,m) with m > 0, also returns the excess entropy qLP.

Usage

DS.entropy(DS.GF.obj)

Arguments

Value

Author(s)

Doug Fletcher

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Examples

data(rat)
rat.start <- gMLE.bb(rat$y, rat$n)$estimate
rat.ds <- DS.prior(rat, max.m = 4, rat.start, family = "Binomial")
DS.entropy(rat.ds)

Execute MacroInference (mean or mode) on a DS object

Description

A function that generates macro-estimates with their uncertainty (standard error).

Usage

DS.macro.inf(DS.GF.obj, num.modes = 1, 
             method = c("mean", "mode"), 
             iters = 25, exposure = NULL)

Arguments

Value

Author(s)

Doug Fletcher, Subhadeep Mukhopadhyay

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Examples

## Not run: 
### MacroInference: Mode
data(rat)
rat.start <- gMLE.bb(rat$y, rat$n)$estimate
rat.ds <- DS.prior(rat, max.m = 4, rat.start. family = "Binomial")
rat.ds.macro <- DS.macro.inf(rat.ds, num.modes = 2, method = "mode", iters = 5)
rat.ds.macro
plot(rat.ds.macro)
### MacroInference: Mean
data(ulcer)
ulcer.start <- gMLE.nn(ulcer$y, ulcer$se)$estimate
ulcer.ds <- DS.prior(ulcer, max.m = 4, ulcer.start)
ulcer.ds.macro <- DS.macro.inf(ulcer.ds, num.modes = 1, method = "mean", iters = 5)
ulcer.ds.macro
plot(ulcer.ds.macro)
## End(Not run)

MicroInference for DS Prior Objects

Description

Provides DS nonparametric adaptive Bayes and parametric estimate for a specific observation y_0.

Usage

DS.micro.inf(DS.GF.obj, y.0, n.0, e.0 = NULL)

Arguments

Details

Returns an object of class DS.GF.micro that can be used in conjunction with plot command to display the DS posterior distribution for the new study.

Value

Author(s)

Doug Fletcher, Subhadeep Mukhopadhyay

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Examples

### MicroInference for Naval Shipyard Data: sample where y = 0 and n = 5
data(ship)
ship.ds <- DS.prior(ship, max.m = 2, c(.5,.5), family = "Binomial")
ship.ds.micro <- DS.micro.inf(ship.ds, y.0 = 0, n.0 = 5)
ship.ds.micro
plot(ship.ds.micro)

Posterior Expectation and Modes of DS object

Description

A function that determines the posterior expectations E(\theta_0 | y_0) and posterior modes for a set of observed data.

Usage

DS.posterior.reduce(DS.GF.obj, exposure)

Arguments

Value

Returns k \times 4 matrix with the columns indicating PEB mean, DS mean, PEB mode, and DS modes for k observations in the data set.

Author(s)

Doug Fletcher

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Examples

data(rat)
rat.start <- gMLE.bb(rat$y, rat$n)$estimate
rat.ds <- DS.prior(rat, max.m = 4, rat.start, family = "Binomial")
DS.posterior.reduce(rat.ds)

Prior Diagnostics and Estimation

Description

A function that generates the uncertainty diagnostic function (U-function) and estimates DS(G,m) prior model.

Usage

DS.prior(input, max.m = 8, g.par, 
         family = c("Normal","Binomial", "Poisson"), 
         LP.type = c("L2", "MaxEnt"), 
         smooth.crit = "BIC", iters = 200, B = 1000,
		 max.theta = NULL)

Arguments

Details

Function can take m=0 and will return the Bayes estimate with given starting parameters. Returns an object of class DS.GF.obj; this object can be used with plot command to plot the U-function (Ufunc), Deviance Plots (mDev), and DS-G comparison (DS_G).

Value

Author(s)

Doug Fletcher, Subhadeep Mukhopadhyay

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Mukhopadhyay, S., 2017. "Large-Scale Mode Identification and Data-Driven Sciences," Electronic Journal of Statistics, 11(1), pp.215-240.

Examples

data(rat)
rat.start <- gMLE.bb(rat$y, rat$n)$estimate
rat.ds <- DS.prior(rat, max.m = 4, rat.start, family = "Binomial")
rat.ds
plot(rat.ds, plot.type = "Ufunc")
plot(rat.ds, plot.type = "DSg")
plot(rat.ds, plot.type = "mDev")

Samples data from DS(G,m) distribution.

Description

Generates samples of size k from DS(G,m) prior distribution.

Usage

DS.sampler(k, g.par, LP.par, con.prior, LP.type, B)

DS.sampler.post(k, g.par, LP.par, y.0, n.0, 
                con.prior, LP.type, B)

Arguments

Details

DS.sampler.post uses the same type of sampling as DS.sampler to generate random values from a DS posterior distribution.

Value

Vector of length k containing sampled values from DS prior or DS posterior.

Author(s)

Doug Fletcher, Subhadeep Mukhopadhyay

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Mukhopadhyay, S., 2017. "Large-Scale Mode Identification and Data-Driven Sciences," Electronic Journal of Statistics, 11(1), pp.215-240.

Examples

##Extracted parameters from rat.ds object
rat.g.par <- c(2.3, 14.1)
rat.LP.par <- c(0, 0, -0.5)
samps.prior <- DS.sampler(25, rat.g.par, rat.LP.par, con.prior = "Beta")
hist(samps.prior,15)
##Posterior for rat data
samps.post <- DS.sampler.post(25, rat.g.par, rat.LP.par, 
							y.0 = 4, n.0 = 14, con.prior = "Beta")
hist(samps.post, 15)

Norberg life insurance data

Description

The number of claims y_i on a life insurance policy for each of k=72 Norwegian occupational categories and the total number of years the workers in each category were exposed to risk (E_i).

Usage

data("NorbergIns")

Format

A data frame of the occupational group number (group), the number of deaths (deaths), and the years of exposure (exposure) for i = 1,...,72.

group

Occupational group number

deaths

The number of deaths in the occupational group resulting in a claim on a life insurance policy.

exposure

The total number of years of exposure to risk for those who passed.

Source

Norberg, R., 1989. "Experience rating in group life insurance," Scandinavian Actuarial Journal, 1989(4), pp. 194-224.

References

Koenker, R. and Gu, J., 2017. "REBayes: An R Package for Empirical Bayes Mixture Methods," Journal of Statistical Software, Articles, 82(8), pp. 1-26.

Arsenic levels in oyster tissue

Description

Results from an inter-laboratory study involving k = 28 measurements for the level of arsenic in oyster tissue. y is the mean level of arsenic from a lab and se is the standard error of the measurement.

Usage

data("arsenic")

Format

A data frame of (y_i, se_i) for i = 1,...,28.

y

mean level of arsenic in the tissue measured by the i^{th} lab

se

the standard error of the measurement by i^{th} lab

Source

Wille, S. and Berman, S., 1995. "Ninth round intercomparison for trace metals in marine sediments and biological tissues," NRC/NOAA.

Determine LP basis functions for prior distribution g

Description

Determines the LP basis for a given parametric prior distribution.

Usage

gLP.basis(x, g.par, m, con.prior, ind)

Arguments

Value

Matrix with m columns of values for the LP-Basis functions evaluated at x-values.

Author(s)

Subhadeep Mukhopadhyay, Doug Fletcher

References

Mukhopadhyay, S. and Fletcher, D., 2018. "Generalized Empirical Bayes via Frequentist Goodness of Fit," Nature Scientific Reports, 8(1), p.9983, https://www.nature.com/articles/s41598-018-28130-5.

Mukhopadhyay, S., 2017. "Large-Scale Mode Identification and Data-Driven Sciences," Electronic Journal of Statistics, 11(1), pp.215-240.

Mukhopadhyay, S. and Parzen, E., 2014. "LP Approach to Statistical Modeling," arXiv: 1405.2601.

Beta-Binomial Parameter Estimation

Description

Computes type-II Maximum likelihood estimates \hat{\alpha} and \hat{\beta} for Beta prior g\simBeta(\alpha,\beta).

Usage

gMLE.bb(success, trials, start = NULL, optim.method = "default", 
        lower = 0, upper = Inf)

Arguments

Value

Author(s)

Aleksandar Bradic

References

https://github.com/SupplyFrame/EmpiricalBayesR/blob/master/EmpiricalBayesEstimation.R

Examples

data(rat)
### MLE estimate of alpha and beta
rat.mle <- gMLE.bb(rat$y, rat$N)$estimate
rat.mle
### MoM estimate of alpha and beta
rat.mom <- gMLE.bb(rat$y, rat$N)$initial
rat.mom

Normal-Normal Parameter Estimation

Description

Computes type-II Maximum likelihood estimates \hat{\mu} and \hat{\tau}^2 for Normal prior g\simNormal(\mu, \tau^2).

Usage

gMLE.nn(value, se, fixed = FALSE, method = c("DL","SJ","REML","MoM"))

Arguments

Value

Author(s)

Doug Fletcher

References

Marin-Martinez, F. and Sanchez-Meca, J., 2010. "Weighting by inverse variance or by sample size in random-effects meta-analysis," Educational and Psychological Measurement, 70(1), pp. 56-73.

Brown, L.D., 2008. "In-season prediction of batting averages: A field test of empirical Bayes and Bayes methodologies," The Annals of Applied Statistics, pp. 113-152.

Sidik, K. and Jonkman, J.N., 2005. "Simple heterogeneity variance estimation for meta-analysis," Journal of the Royal Statistical Society: Series C (Applied Statistics), 54(2), pp. 367-384.

Examples

data(ulcer)
### MLE estimate of alpha and beta
ulcer.mle <- gMLE.nn(ulcer$y, ulcer$se, method = "DL")$estimate
ulcer.mle
ulcer.reml <- gMLE.nn(ulcer$y, ulcer$se, method = "REML")$estimate
ulcer.reml

Negative-Binomial Parameter Estimation

Description

Computes Type-II Maximum likelihood estimates \hat{\alpha} and \hat{\beta} for gamma prior g\sim Gamma(\alpha, \beta).

Usage

gMLE.pg(cnt.vec, exposure = NULL, start.par = c(1,1))

Arguments

Value

Returns a vector where the first component is \alpha and the second component is the scale parameter \beta for the gamma distribution: \frac{1}{\Gamma(\alpha)\beta^\alpha} \theta^{\alpha-1}e^{-\frac{\theta}{\beta}}.

Author(s)

Doug Fletcher

References

Koenker, R. and Gu, J., 2017. "REBayes: An R Package for Empirical Bayes Mixture Methods," Journal of Statistical Software, Articles, 82(8), pp. 1-26.

Examples

### without exposure
data(ChildIll)
ill.start <- gMLE.pg(ChildIll)
ill.start
### with exposure
data(NorbergIns)
X <- NorbergIns$deaths
E <- NorbergIns$exposure/344
norb.start <- gMLE.pg(X, exposure = E)
norb.start

Galaxy Data

Description

The observed rotation velocities and their uncertainties of Low Surface Brightness (LSB) galaxies, along with the physical radius of the galaxy.

Usage

data("galaxy")

Format

A data frame of (y_i, se_i, X_i) for i = 1,...,318.

y

actual observed (smoothed) velocity

se

uncertainty of observed velocity

X

physical radius of the galaxy

Source

De Blok, W.J.G., McGaugh, S.S., and Rubin, V. C., 2001. "High-resolution rotation curves of low surface brightness galaxies. II. Mass models," The Astronomical Journal, 122(5), p. 2396.

Rat Tumor Data

Description

Incidence of endometrial stromal polyps in k=70 studys of female rats in control group of a 1977 study on the carcinogenic effects of a diabetic drug phenformin. For each of the k groups, y represents the number of rats who developed the tumors out of n total rats in the group.

Usage

data("rat")

Format

A data frame of (y_i, n_i) for i = 1,...,70.

y

number of female rats in the i^{th} study who developed polyps/tumors

n

total number of rats in the i^{th} study

Source

National Cancer Institute (1977), "Bioassay of phenformin for possible carcinogenicity," Technical Report No. 7.

References

Gelman, A., Carlin, J.B., Stern, H.S., Dunson, D.B., Vehtari, A., and Rubin, D.B., 2014. Bayesian Data Analysis (Vol. 3). Boca Raton, FL: CRC press.

Tarone, R.E., 1982. "The use of historical control information in testing for a trend in proportions," Biometrics, pp. 215-220.

Portsmouth Navy Shipyard Data

Description

Data represents results of quality-control inspections executed by Portsmouth Naval Shipyard on lots of welding materials. The data has k=5 observations of number of defects y out of the total number of tested n=5.

Usage

data("ship")

Format

A data frame of (y_i, n_i) for i = 1,...,5.

y

number of defects found in the i^{th} inspection

n

total samples tested in the i^{th} inspection

Source

Martz, H.F. and Lian, M.G., 1974. "Empirical Bayes estimation of the binomial parameter," Biometrika, 61(3), pp. 517-523.

Nasal Steroid Data

Description

The standardized mean difference y_i and standard errors se_i for seven randomised studies on the use of topical steroids in treatment of chronic rhinosinusitis with nasal polyps.

Usage

data("steroid")

Format

A data frame of (y_i, se_i) for i = 1,...,7.

y

standard mean difference of clinical trials for topical steroids found in the i^{th} study

se

standard error of the standard mean difference for the i^{th} study

Source

IntHout, J., Ioannidis, J. P., Rovers, M. M., & Goeman, J. J., 2016. "Plea for routinely presenting prediction intervals in meta-analysis," BMJ open, 6(7), e010247.

Intestinal surgery data

Description

Data involves the number of malignant lymph nodes removed during intestinal surgery for k=844 cancer patients. For each patient, n is the total number of satellite nodes removed during surgery from a patient and y is the number of malignant nodes.

Usage

data("surg")

Format

A data frame of (y_i, n_i) for i = 1,...,844.

y

number of malignant lymph nodes removed from the i^{th} patient

n

total number of lymph nodes removed from the i^{th} patient

Source

Efron, B., 2016. "Empirical Bayes deconvolution estimates," Biometrika, 103(1), pp. 1-20.

Rolling Tacks Data

Description

An experiment that requires a common thumbtack to be "flipped" n=9 times. Out of these total number of flips, y is the total number of times that the thumbtack landed point up.

Usage

data("tacks")

Format

A data frame of (y_i, n_i) for i = 1,...,320.

y

number of times a thumbtack landed point up in the i^{th} trial

n

total number of flips for the thumbtack in the i^{th} trial

Source

Beckett, L. and Diaconis, P., 1994. "Spectral analysis for discrete longitudinal data," Advances in Mathematics, 103(1), pp. 107-128.

Terbinafine trial data

Description

During several studies of the oral antifungal agent terbinafine, a proportion of the patients in the trial terminated treatment due to some adverse effects. In the data set, y_i is the number of terminated treatments and n_i is the total number of patients in the in the i^{th} trial.

Usage

data("terb")

Format

A data frame of (y_i, n_i) for i = 1,...,41.

y

number of patients who terminated treatment early in the i^{th} trial

n

total number of patients in the i^{th} clinical trial

Source

Young-Xu, Y. and Chan, K.A., 2008. "Pooling overdispersed binomial data to estimate event rate," BMC Medical Research Methodology, 8(1), p. 58.

Recurrent Bleeding of Ulcers

Description

The data consist of k=40 randomized trials between 1980 and 1989 of a surgical treatment for stomach ulcers. Each of the trials has an estimated log-odds ratio that measures the rate of occurrence of recurrent bleeding given the surgical treatment.

Usage

data("ulcer")

Format

A data frame of (y_i, se_i) for i = 1,...,40.

y

log-odds of the occurrence of recurrent bleeding in the i^{th} study

se

standard error of the log-odds for the i^{th} study

Source

Sacks, H.S., Chalmers, T.C., Blum, A.L., Berrier, J., and Pagano, D., 1990. "Endoscopic hemostasis: an effective therapy for bleeding peptic ulcers," Journal of the American Medical Association, 264(4), pp. 494-499.

References

Efron, B., 1996. "Empirical Bayes methods for combining likelihoods," Journal of the American Statistical Association, 91(434), pp. 538-550.