UMPBT.alt {MSPRT}

UMPBT alternative

Description

Given a sample size and prespecified Type I & II error probabilities, this function obtains the objective alternative in the Uniformly Most Powerful Bayesian Test (UMPBT).

Usage

UMPBT.alt(test.type, side = "right", theta0, 
          N, N1, N2, Type1 = 0.005, 
          sigma = 1, sigma1 = 1, sigma2 = 1, 
          obs, sd.obs, obs1, obs2, pooled.sd)

Arguments

test.type

Character. Type of test. Currently, the package only allows

• oneProp for one-sample proportion tests

• oneZ for one-sample z tests

• oneT for one-sample t tests

• twoZ for two-sample z tests

• twoT for two-sample t tests.

side

Character. Direction of the composite alternative hypothesis. right for H_1 : \theta > \theta_0 (default), and left for H_1 : \theta < \theta_0.

theta0

Numeric. Hypothesized value of effect size (\theta_0) under H_0. Default: 0.5 in one-sample proportion tests, and 0 for others.

N

Positive integer. Sample size in one-sample tests.

N1

Positive integer. Sample size from Group-1 in two-sample tests.

N2

Positive integer. Sample size from Group-2 in two-sample tests.

Type1

Numeric in [0,1]. Prespecified Type I error probability. Default: 0.005.

sigma

Positive numeric. Known standard deviation in one-sample z tests. Default: 1.

sigma1

Positive numeric. Known standard deviation for Group-1 in two-sample z tests. Default: 1.

sigma2

Positive numeric. Known standard deviation for Group-2 in two-sample z tests. Default: 1.

obs

Numeric vector. The vector of observations based on which the UMPBT alternative in one-sample t test is determined. Either obs or sd.obs is required.

sd.obs

Positive numeric. The standard deviation (with divisor n-1) of observations based on which the UMPBT alternative in one-sample t test is determined. Either obs or sd.obs is required.

obs1

Numeric vector. The vector of observations from Group-1 based on which the UMPBT alternative in two-sample t test is determined. Either both obs1 and obs2, or pooled.sd is required.

obs2

Numeric vector. The vector of observations from Group-2 based on which the UMPBT alternative in two-sample t test is determined. Either both obs1 and obs2, or pooled.sd is required.

pooled.sd

Positive numeric. The pooled standard deviation of observations from Group-1 and 2 based on which the UMPBT alternative in two-sample t test is determined. Either both obs1 and obs2, or pooled.sd is required.

Value

List with two named components theta and mix.prob in one-sample proportion test. In this case, the UMPBT alternative is a mixture distribution of two points. theta contains the two points (effect sizes) and mix.prob contains their respective mixing probabilities.

Numeric in case of all the other tests. It is the UMPBT alternative effect size.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

Johnson, V. E. (2013a). Revised standards for statistical evidence.Proceed-ings of the National Academy of Sciences, 110(48):19313-19317. [Article]

Johnson, V. E. (2013b). Uniformly most powerful Bayesian tests.TheAnnals of Statistics, 41(4):1716-1741. [Article]

Pramanik S., Johnson V. E. and Bhattacharya A. (2020+). A Modified Sequential Probability Ratio Test. [Arxiv]

Examples

##### one-sample proportion test #####

## right-sided
UMPBT.alt(test.type = "oneProp", N = 30)

## left-sided
UMPBT.alt(side = "left", test.type = "oneProp", N = 30)


##### one-sample z test #####

## right-sided
UMPBT.alt(test.type = "oneZ", N = 30)

## left-sided
UMPBT.alt(side = "left", test.type = "oneZ", N = 30)


##### one-sample t test #####

## observed data

set.seed(1)
x = rnorm(n = 30, mean = 0, sd = 1.5)
	
## right-sided
UMPBT.alt(test.type = "oneT", N = 30, obs = x)

## left-sided
UMPBT.alt(side = "left", test.type = "oneT", N = 30, obs = x)


##### two-sample z test #####

## right-sided
UMPBT.alt(test.type = "twoZ", N1 = 30, N2 = 30)

## left-sided
UMPBT.alt(side = "left", test.type = "twoZ", N1 = 30, N2 = 30)


##### two-sample t test #####

## observed data

set.seed(1)
x1 = rnorm(n = 30, mean = 0, sd = 1.5)
x2 = rnorm(n = 30, mean = 0, sd = 1.5)

## right-sided
UMPBT.alt(test.type = "twoT", N1 = 30, N2 = 30, 
          obs1 = x1, obs2 = x2)

## left-sided
UMPBT.alt(side = "left", test.type = "twoT", N1 = 30, N2 = 30, 
          obs1 = x1, obs2 = x2)