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Type:

Package

Title:

A Modified Sequential Probability Ratio Test (MSPRT)

Version:

3.0

Date:

2020-11-11

Author:

Sandipan Pramanik [aut, cre], Valen E. Johnson [aut], Anirban Bhattacharya [aut]

Maintainer:

Sandipan Pramanik <sandy.pramanik@gmail.com>

Description:

Given the maximum available sample size (N) for an experiment, and the target levels of Type I and II error probabilities, this package designs a modified SPRT (MSPRT). For any designed MSPRT the package can also obtain its operating characteristics and implement the test for a given sequentially observed data. The MSPRT is defined in a manner very similar to Wald's initial proposal. The proposed test has shown evidence of reducing the average sample size required to perform statistical hypothesis tests at specified levels of significance and power. Currently, the package implements one-sample proportion tests, one and two-sample z tests, and one and two-sample t tests. A brief user guidance for this package is provided below. One can also refer to the supplemental information for the same.

Imports:

nleqslv, ggplot2, ggpubr, foreach, iterators, parallel, doParallel, datasets, graphics, grDevices, methods, stats, utils

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

NeedsCompilation:

Packaged:

2020-11-13 01:08:33 UTC; sandipanpramanik

Repository:

CRAN

Date/Publication:

2020-11-13 10:20:05 UTC

A Modified Sequential Probability Ratio Test (MSPRT)

Description

Details

Package:	MSPRT
Type:	Package
Version:	3.0
Date:	11-11-2020
License:	GPL>=2

Author(s)

Sandipan Pramanik [aut, cre], Valen E. Johnson [aut], Anirban Bhattacharya [aut]

Maintainer: Sandipan Pramanik <sandy.pramanik@gmail.com>

Sample size required to achieve higher significance

Description

Given the sample size that is available at a lower level of significance, this function calculates the sample size that is required for achieving a higher level of significance so that a desired level of Type II error probability is maintained at a desired effect size.

Usage

Nstar(test.type, N, N1, N2, 
      N.increment = 1, N1.increment = 1, N2.increment = 1, 
      lower.signif = 0.05, higher.signif = 0.005, theta0, 
      side = "right", Type2.target = 0.2, theta, 
      sigma = 1, sigma1 = 1, sigma2 = 1, plot.it = T)

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.

N

Positive integer. Sample size available at the lower level of significance in one-sample tests.

N1

Positive integer. Sample size available from Group-1 at the lower level of significance in two-sample tests.

N2

Positive integer. Sample size available from Group-2 at the lower level of significance in two-sample tests.

N.increment

Positive integer. Increment in sample size allowed while searching for the sample size that is required for achieving the higher level of significance.

N1.increment

Positive integer. Increment in sample size from Group-1 allowed while searching for the sample size that is required for achieving the higher level of significance.

N2.increment

Positive integer. Increment in sample size from Group-2 allowed while searching for the sample size that is required for achieving the higher level of significance.

lower.signif

Numeric within [0,1]. Lower level of significance. Default 0.05.

higher.signif

Numeric within [0,1]. Higher level of significance. Default: 0.005.

theta0

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

side

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

Type2.target

Numeric within [0,1]. Prespecified level of Type 2 error probability.

Default: 0.2.

theta

Numeric. Effect size value where Type2.target Type II error probability is desired at both levels of significance. Default: Fixed-design alternative (\theta_a) at the lower level of significance; that is, the effect size where the fixed design test with N samples and level of significance lower.signif has the Type II error probability Type2.target.

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.

plot.it

Logical. If TRUE (default), returns a plot. Otherwise it doesn't.

Value

One-sample tests: Numeric. The required sample size.
Two-sample tests: Numeric vector of length 2. The first and second components store the sample sizes required respectively from Group 1 and 2 for achieving the higher level of significance.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

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

Examples


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

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

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


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

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

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


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

## right-sided
Nstar(test.type = "oneT", N = 30)

## left-sided
Nstar(test.type = "oneT", side = "left", N = 30)


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

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

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


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

## right-sided
Nstar(test.type = "twoT", N1 = 30, N2 = 30)

## left-sided
Nstar(test.type = "twoT", side = "left", N1 = 30, N2 = 30)

Operating characteristics (OC) and Average Sample Number (ASN) of a designed MSPRT

Description

This function obtains the operating characteristics, that is the probability of accepting H_0 and the sample size required on average for reaching a decision, for a designed MSPRT at the specified effect size(s).

Usage

OCandASN.MSPRT(theta, design.MSPRT.object, 
               termination.threshold, test.type, side = "right", 
               theta0, Type1.target = 0.005, Type2.target = 0.2,
               N.max, N1.max, N2.max,
               sigma = 1, sigma1 = 1, sigma2 = 1, 
               batch.size, batch1.size, batch2.size, 
               nReplicate = 1e+06, nCore = max(1, detectCores() - 1),
               verbose = T, seed = 1)

Arguments

theta

Numeric vector. Vector of effect size(s) where the operating characteristics of the MSPRT is desired.

design.MSPRT.object

List. The output returned from design.MSPRT corresponding to the MSPRT for which the operating characteristics are desired.

termination.threshold

Positive numeric. Termination threshold of the designed MSPRT.

test.type

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

side

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

theta0

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

Type1.target

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

Type2.target

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

N.max

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

N1.max

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

N2.max

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

sigma

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

sigma1

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

sigma2

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

batch.size

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

batch1.size

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

batch2.size

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

nReplicate

Positive integer. Total number of replications to be used in Monte Carlo simulation for calculating the termination threshold and the operating characteristics of the MSPRT.

Default: 1,000,000.

verbose

Logical. If TRUE (default), returns messages of the current proceedings. Otherwise it doesn't.

nCore

Positive integer. Total number of cores available for computation. Can be anything \ge 1.

Default: detectCores() - 1. That is, 1 less than the total number of available cores.

seed

Integer. Random number generating seed.

Default: 1.

Details

If design.MSPRT.object is provided, one can only additionally provide nReplicate, nCore, verbose and seed (Easier option). Otherwise, just like in design.MSPRT, all the other arguments together with termination.threshold (obtained from design.MSPRT) needs to be provided adequately.

Value

Data frame.

One-sample tests: The data frame has 3 columns named theta, acceptH0.prob and EN, and the number of rows equals to the number of effect sizes (length of theta) where the operating characteristics are evaluated. Each row corresponds to a particular value of theta (effect size). The columns respectively contain the value of a particular theta (effect size), and the probability of accepting the $H_0$ and the average sample size required by the MSPRT for reaching a decision thereat.
Two-sample tests: The data frame has 4 columns named theta, acceptH0.prob, EN1 and EN2, and the number of rows equals to the number of effect sizes (length of theta) where the operating characteristics are evaluated. Each row corresponds to a particular value of theta (effect size). The columns respectively contain the value of a particular theta (effect size), and the probability of accepting the H_0 at that effect size, and the average sample size from Group-1 & 2 that is required by the MSPRT for reaching a decision thereat.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

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

Examples


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

#### right sided ####
### design
#design.oneprop.right <- design.MSPRT(test.type = 'oneProp', side = 'right',
#                                     N.max = 20)

### OC and ASN
#OC.oneprop.right <- OCandASN.MSPRT(theta = seq(design.oneprop.right$theta0, 1,
#                                               length.out = 3),
#                                   design.MSPRT.object = design.oneprop.right)

#### left sided ####
### design
#design.oneprop.left = design.MSPRT(test.type = 'oneProp', side = 'left',
#                                   N.max = 20)

### OC and ASN
#OC.oneprop.left = OCandASN.MSPRT(theta = seq(0, design.oneprop.right$theta0,
#                                             length.out = 3),
#                                 design.MSPRT.object = design.oneprop.left)

#### both sided ####
### design
#design.oneprop.both = design.MSPRT(test.type = 'oneProp', side = 'both',
#                                   N.max = 20)

### OC and ASN
#OC.oneprop.both = OCandASN.MSPRT(theta = seq(0, 1, length.out = 3),
#                                 design.MSPRT.object = design.oneprop.both)


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

#### right sided ####
### design
#design.onez.right = design.MSPRT(test.type = 'oneZ', side = 'right',
#                                 N.max = 20)

### OC and ASN
#OC.onez.right = OCandASN.MSPRT(theta = seq(design.onez.right$theta0,
#                                           design.onez.right$theta0 + 3*design.onez.right$sigma,
#                                           length.out = 3),
#                               design.MSPRT.object = design.onez.right)

#### left sided ####
### design
#design.onez.left = design.MSPRT(test.type = 'oneZ', side = 'left',
#                                N.max = 20)

### OC and ASN
#OC.onez.left = OCandASN.MSPRT(theta = seq(design.onez.left$theta0 - 3*design.onez.left$sigma,
#                                          design.onez.left$theta0,
#                                          length.out = 3),
#                              design.MSPRT.object = design.onez.left)

#### both sided ####
### design
#design.onez.both = design.MSPRT(test.type = 'oneZ', side = 'both',
#                                N.max = 20)

### OC and ASN
#OC.onez.both = OCandASN.MSPRT(theta = seq(design.onez.both$theta0 - 3*design.onez.both$sigma,
#                                          design.onez.both$theta0 + 3*design.onez.both$sigma,
#                                          length.out = 3),
#                              design.MSPRT.object = design.onez.both)


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

#### right sided ####
### design
#design.onet.right = design.MSPRT(test.type = 'oneT', side = 'right',
#                                 N.max = 20)

### OC and ASN
#OC.onet.right = OCandASN.MSPRT(theta = seq(design.onet.right$theta0, 1,
#                                           length.out = 3),
#                               design.MSPRT.object = design.onet.right)

#### left sided ####
### design
#design.onet.left = design.MSPRT(test.type = 'oneT', side = 'left',
#                                N.max = 20)

### OC and ASN
#OC.onet.left = OCandASN.MSPRT(theta = seq(-1, design.onet.left$theta0,
#                                          length.out = 3),
#                              design.MSPRT.object = design.onet.left)

#### both sided ####
### design
#design.onet.both = design.MSPRT(test.type = 'oneT', side = 'both',
#                                N.max = 20)

### OC and ASN
#OC.onet.both = OCandASN.MSPRT(theta = seq(-1, 1, length.out = 3),
#                              design.MSPRT.object = design.onet.both)


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

#### right sided ####
### design
#design.twoz.right = design.MSPRT(test.type = 'twoZ', side = 'right',
#                                 N1.max = 20, N2.max = 20)

### OC and ASN
#OC.twoz.right = OCandASN.MSPRT(theta = seq(design.twoz.right$theta0, 
#                                           design.twoz.right$theta0 + 2,
#                                           length.out = 3),
#                               design.MSPRT.object = design.twoz.right)

#### left sided ####
### design
#design.twoz.left = design.MSPRT(test.type = 'twoZ', side = 'left',
#                                N1.max = 20, N2.max = 20)

### OC and ASN
#OC.twoz.left = OCandASN.MSPRT(theta = seq(design.twoz.left$theta0 - 2,
#                                          design.twoz.left$theta0,
#                                          length.out = 3),
#                              design.MSPRT.object = design.twoz.left)

#### both sided ####
### design
#design.twoz.both = design.MSPRT(test.type = 'twoZ', side = 'both',
#                                N1.max = 20, N2.max = 20)

### OC and ASN
#OC.twoz.both = OCandASN.MSPRT(theta = seq(design.twoz.both$theta0 - 2,
#                                          design.twoz.both$theta0 + 2,
#                                          length.out = 3),
#                             design.MSPRT.object = design.twoz.both)


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

#### right sided ####
### design
#design.twot.right = design.MSPRT(test.type = 'twoT', side = 'right',
#                                 N1.max = 20, N2.max = 20)

### OC and ASN
#OC.twot.right = OCandASN.MSPRT(theta = seq(design.twot.right$theta0, 
#                                           design.twot.right$theta0 + 2,
#                                           length.out = 3),
#                               design.MSPRT.object = design.twot.right)

#### left sided ####
### design
#design.twot.left = design.MSPRT(test.type = 'twoT', side = 'left',
#                                N1.max = 20, N2.max = 20)

### OC and ASN
#OC.twot.left = OCandASN.MSPRT(theta = seq(design.twot.left$theta0 - 2,
#                                          design.twot.left$theta0,
#                                          length.out = 3),
#                              design.MSPRT.object = design.twot.left)

#### both sided ####
### design
#design.twot.both = design.MSPRT(test.type = 'twoT', side = 'both',
#                                N1.max = 20, N2.max = 20)

### OC and ASN
#OC.twot.both = OCandASN.MSPRT(theta = seq(design.twot.both$theta0 - 2,
#                                          design.twot.both$theta0 + 2,
#                                          length.out = 3),
#                              design.MSPRT.object = design.twot.both)

Internal MSPRT function: OC and ASN of a designed MSPRT in one-sample proportion tests

Description

OCandASN.MSPRT calls this function for obtaining the OC and ASN of a designed MSPRT for one-sample proportion tests. Users please refer to OCandASN.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: OC and ASN of a designed MSPRT in one-sample t tests

Description

OCandASN.MSPRT calls this function for obtaining the OC and ASN of a designed MSPRT for one-sample t tests. Users please refer to OCandASN.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: OC and ASN of a designed MSPRT in one-sample z tests

Description

OCandASN.MSPRT calls this function for obtaining the OC and ASN of a designed MSPRT for one-sample z tests. Users please refer to OCandASN.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: OC and ASN of a designed MSPRT in two-sample t tests

Description

OCandASN.MSPRT calls this function for obtaining the OC and ASN of a designed MSPRT for two-sample t tests. Users please refer to OCandASN.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: OC and ASN of a designed MSPRT in two-sample z tests

Description

OCandASN.MSPRT calls this function for obtaining the OC and ASN of a designed MSPRT for two-sample z tests. Users please refer to OCandASN.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Type II error probability of fixed design tests

Description

Obtains the Type II error probability of fixed-design tests for testing the point null hypothesis H_0 : \theta = \theta_0.

Usage

Type2.fixed_design(theta, test.type, side = "right", theta0, 
                   N, N1, N2, Type1 = 0.005, 
                   sigma = 1, sigma1 = 1, sigma2 = 1)

Arguments

theta

Numeric. Effect size where the Type II error probability is desired.

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.

Value

Numeric in [0,1]. The Type II error probability of the fixed-design test at the specified effect size value theta.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

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

Examples


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

## right-sided
Type2.fixed_design(theta = seq(0, 1, length.out = 10), 
                   test.type = "oneProp", N = 30)

## left-sided
Type2.fixed_design(theta = seq(0, 1, length.out = 10), side = "left", 
                   test.type = "oneProp", N = 30)


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

## right-sided
Type2.fixed_design(theta = seq(0, 1, length.out = 10), 
                   test.type = "oneZ", N = 30)

## left-sided
Type2.fixed_design(theta = seq(-1, 0, length.out = 10), side = "left", 
                   test.type = "oneZ", N = 30)


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

## right-sided
Type2.fixed_design(theta = seq(0, 1, length.out = 10), 
                   test.type = "oneT", N = 30)

## left-sided
Type2.fixed_design(theta = seq(-1, 0, length.out = 10), side = "left", 
                   test.type = "oneT", N = 30)


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

## right-sided
Type2.fixed_design(theta = seq(0, 1, length.out = 10), 
                   test.type = "twoZ", N1 = 30, N2 = 30)

## left-sided
Type2.fixed_design(theta = seq(-1, 0, length.out = 10), side = "left",
                   test.type = "twoZ", N1 = 30, N2 = 30)


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

## right-sided
Type2.fixed_design(theta = seq(0, 1, length.out = 10), 
                   test.type = "twoT", N1 = 30, N2 = 30)

## left-sided
Type2.fixed_design(theta = seq(-1, 0, length.out = 10), side = "left", 
                   test.type = "twoT", N1 = 30, N2 = 30)

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)

Designing the MSPRT

Description

Given the maximum available sample size and prespecified Type I & II error probabilities, this function designs/obtains the corresponding MSPRT.

Usage

design.MSPRT(test.type, side = "right", theta0, theta1 = T, 
             Type1.target = 0.005, Type2.target = 0.2, 
             N.max, N1.max, N2.max, 
             sigma = 1, sigma1 = 1, sigma2 = 1, 
             batch.size, batch1.size, batch2.size, 
             nReplicate = 1e+06, verbose = T, seed = 1)

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.

theta1

Logical, numeric or list (two components with names 'right' and 'left').

If FALSE, no comparison is done under the alternative hypothesis.
If TRUE (Default), comparison is done at the fixed-design alternative effect size (\theta_a).
If numeric, this can only be in case of one-sided tests (that is, side = "right" or "left"). The comparison is done at the specified numeric value of the alternative effect size.
If list, this can only be in case of two-sided tests (that is, side = "both"). The list has to be of the form list("right" = \theta_1, "left" = \theta_2). Then the comparison is done at alternative effect sizes \theta_1 and \theta_2.

Note: In case of two-sided tests at a given level of significance, there are two effect sizes under H_1 (one on the right of H_0 and one on the left) that corresponds to the same Type II error probability (or power). This list provides users with the ability where he/she can replace \theta_1 and \theta_2 by any effect sizes from each side in the form of a list as mentioned above, and can get the designed MSPRT together with its operating characteristics at those effect sizes.

Type1.target

Numeric within [0,1]. Prespecified level of Type I error probability.

Default: 0.005.

The MSPRT exactly maintains its Type I error probability at this value.

Type2.target

Numeric within [0,1]. Prespecified level of Type 2 error probability.

Default: 0.2.

The MSPRT approximately maintains its Type II error probability at this value at the corresponding fixed-design alternative (\theta_a).

N.max

Positive integer. Maximum available sample size in one-sample tests.

N1.max

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

N2.max

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

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.

batch.size

Integer vector. A vector denoting the number of observations that are planned to be observed at each sequential step in one-sample tests.

Default:

Proportion and z tests: rep(1, N.max).
t tests: c(2, rep(1, N.max-1)).

Default values mean the sequential analysis is performed after observing each observation. This corresponds to a sequential MSPRT. If any batch size is more than 1 (or more than 2 in the 1st step for t test) it corresponds to a group sequential MSPRT.

Note: First batch size for t tests needs to be at least 2. The length of batch.size equals to the maximum number of planned sequential analyses.

batch1.size

Integer vector. A vector denoting the number of observations that are planned to be observed from Group-1 at each sequential step in two-sample tests.

Default:

z tests: rep(1, N1.max).
t tests: c(2, rep(1, N1.max-1)).

Default values mean the sequential analysis is performed after observing each observation from Group-1.

batch2.size

Integer vector. A vector denoting the number of observations that are planned to be observed from Group-2 at each sequential step in two-sample tests.

Default:

z tests: rep(1, N2.max).
t tests: c(2, rep(1, N2.max-1)).

Default values mean the sequential analysis is performed after observing each observation from Group-2.

nReplicate

Positive integer. Total number of replications to be used in Monte Carlo simulation for calculating the termination threshold and the operating characteristics of the MSPRT.

Default: 1,000,000.

verbose

Logical. If TRUE (default), returns messages of the current proceedings. Otherwise it doesn't.

seed

Integer. Random number generating seed.

Default: 1.

Value

List. The list has the following named components in case of one-sided one-sample tests:

TypeI.attained

Numeric in [0,1]. Type I error probability attained by the designed MSPRT.

Type2.attained

Numeric in [0,1]. Type II error probability attained by the designed MSPRT at the specified alternative effect size theta1. Returned only if theta1 is TRUE or numeric.

N

List.

If theta1 = FALSE, the list has one component named H0. It stores an integer vector of length nReplicate. This is the vector of sample size required by the MSPRT for each of nReplicate Monte Carlo simulations under H_0.
If theta1 is TRUE or numeric, the list has two components named H0 and H1. Each of these stores an integer vector of length nReplicate. The stored vector under H0 is the same as in theta1 = FALSE. The H1 component stored the vector of sample size required by the MSPRT for each of nReplicate Monte Carlo simulations under the specified alternative effect size.

EN

Numeric vector.

If theta1 = FALSE, the vector is of length 1. It is the number of samples required on average by the MSPRT under H_0.
If theta1 is TRUE or numeric, the vector is of length 2. They are the number of samples required on average by the MSPRT under H_0 (first component) and the specified alternative effect size (second component), respectively.

UMPBT or theta.UMPBT

The UMPBT alternative. UMPBT in case of one-sample proportion test and theta.UMPBT in case of all the other tests. Their types are the same as their output from UMPBT.alt function.

Note: Not returned in t tests as it depends on the data.

theta1

Returned only if theta1 is anything but FALSE. Stores the effect size under H_1 where the operating characteristic of the MSPRT is obtained. Of the same type as the argument theta1.

Type2.fixed.design

Numeric in [0,1]. Type II error probability attained by the fixed design test with sample size N.max and Type I error probability Type1.target at the alternative effect size theta1.

RejectH0.threshold

Positive numeric. Threshold for rejecting H_0 in the MSPRT.

RejectH1.threshold

Positive numeric. Threshold for accepting H_1 in the MSPRT.

termination.threshold

Positive numeric. Termination threshold of the MSPRT.

In case of one-sided two-sample tests the above components are returned with following modifications:

N:

List.

If theta1 = FALSE the list has one component named H0.
theta1 is TRUE or numeric, the list has two components named H0 and H1.

Each of the named components H0 and H1 contains a list with two components named Group1 and Group2. Each of these contains the same vector corresponding to Group-1 and Group-2. In each of these, it contains the sample size required by the MSPRT in each of nReplicate Monte Carlo simulations under the respective effect size for the respective group.

EN

List.

If theta1 = FALSE the list has one component named H0.
If theta1 is TRUE or numeric, the list has two components named H0 and H1.

Each of the named components H0 or H1 contains a list with two components named Group1 and Group2. In each of these, it contains the sample size required on average by the MSPRT under the respective effect size for the respective group.

In case of two-sided tests the above components are returned with following modifications:

Type2.attained

Numeric vector of length 2 with both elements in [0,1]. The first and second component is the Type II error probability of the MSPRT at the specified alternative effect sizes theta1$right and theta1$left, respectively.

N

This is the same as in one-sided tests if theta1 is FALSE. If theta1 is TRUE or a two-component list with names right and left, this is a list with three components with names H0, right and left instead of a two-component list with names H0 and H1. Quantities stored under these components are the same as in one-sided tests except the quantities under right and left are the same performance of the designed MSPRT at the specified alternative effect sizes theta1$right and theta1$left, respectively.

EN

Numeric vector. The same as in one-sided tests if theta1 is FALSE. If theta1 is TRUE or a two-component list with names right and left, this is a numeric vector of length 3, where the first, second and third components are the average required sample size under H_0, and at the specified alternative effect sizes theta1$right and theta1$left, respectively.

Additionally, the output list also contains the provided arguments of design.MSPRT, and

nAnalyses

Positive integer. This is the maximum number of sequential analyses that is planned. This equals to the length(batch.size) in one-sample tests, and to the length(batch1.size) and length(batch2.size) in two-sample tests.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

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

Examples


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

## right-sided
#design.MSPRT(test.type = 'oneProp', side = 'right',
#             N.max = 20)

## left-sided
#design.MSPRT(test.type = 'oneProp', side = 'right',
#             N.max = 20)

## two-sided
#design.MSPRT(test.type = 'oneProp', side = 'both',
#             N.max = 20)


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

## right-sided
#design.MSPRT(test.type = 'oneZ', side = 'right',
#             N.max = 20)

## left-sided
#design.MSPRT(test.type = 'oneZ', side = 'right',
#             N.max = 20)

## two-sided
#design.MSPRT(test.type = 'oneZ', side = 'both',
#              N.max = 20)


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

## right-sided
#design.MSPRT(test.type = 'oneT', side = 'right',
#             N.max = 20)

## left-sided
#design.MSPRT(test.type = 'oneT', side = 'right',
#             N.max = 20)

## two-sided
#design.MSPRT(test.type = 'oneT', side = 'both',
#             N.max = 20)


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

## right-sided
#design.MSPRT(test.type = 'twoZ', side = 'right',
#             N1.max = 20, N2.max = 20)

## left-sided
#design.MSPRT(test.type = 'twoZ', side = 'left',
#             N1.max = 20, N2.max = 20)

## two-sided
#design.MSPRT(test.type = 'twoZ', side = 'both',
#             N1.max = 20, N2.max = 20)


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

## right-sided
#design.MSPRT(test.type = 'twoT', side = 'right',
#             N1.max = 20, N2.max = 20)

## left-sided
#design.MSPRT(test.type = 'twoT', side = 'left',
#             N1.max = 20, N2.max = 20)

## two-sided
#design.MSPRT(test.type = 'twoT', side = 'both',
#             N1.max = 20, N2.max = 20)

Internal MSPRT function: Designing the MSPRT for one-sample proportion tests

Description

design.MSPRT calls this function for designing the MSPRT in one-sample proportion tests. Users please refer to design.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Designing the MSPRT for one-sample t tests

Description

design.MSPRT calls this function for designing the MSPRT in one-sample t tests. Users please refer to design.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Designing the MSPRT for one-sample z tests

Description

design.MSPRT calls this function for designing the MSPRT in one-sample z tests. Users please refer to design.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Designing the MSPRT for two-sample t tests

Description

design.MSPRT calls this function for designing the MSPRT in two-sample t tests. Users please refer to design.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Designing the MSPRT for two-sample z tests

Description

design.MSPRT calls this function for designing the MSPRT in two-sample z tests. Users please refer to design.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Calculating effective maximum sample size to be used in designing the MSPRT in one-sample proportion test

Description

Given a maximum sample size that is planned to use, this function obtains the maximum sample size (N) that is suggested to use in designing the MSPRT for one-sample proportion tests.

Usage

effectiveN.oneProp(N, side = "right", Type1 = 0.005, theta0 = 0.5, 
                   plot.it = T)

Arguments

N

Positive integer. Maximum sample that is intended to use.

side

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

Type1

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

theta0

Numeric. Hypothesized value of effect size (\theta_0) under H_0. Default: 0.5.

plot.it

Logical. If TRUE (default), returns a plot. Otherwise it doesn't.

Value

Positive integer. This is suggested to use in OCandASN.MSPRT as the maximum availeble sample size (N) to design the MSPRT for one-sample proportion tests.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

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

Examples

effectiveN.oneProp(N = 30)

Fixed-design alternative

Description

Given a sample size and prespecified Type I & II error probabilities, this function obtains the fixed-design alternative (\theta_a) for testing the point null hypothesis H_0 : \theta = \theta_0.

Usage

fixed_design.alt(test.type, side = "right", theta0, 
                 N, N1, N2, Type1 = 0.005, Type2 = 0.2, 
                 sigma = 1, sigma1 = 1, sigma2 = 1)

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.

Type2

Numeric in [0,1]. Prespecified Type II error probability. Default: 0.2.

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.

Value

Numeric. The fixed-design alternative effect size (\theta_a).

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

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

Examples


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

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

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


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

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

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


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

## right-sided
fixed_design.alt(test.type = "oneT", N = 30)

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


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

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

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


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

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

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

Implementing the MSPRT

Description

This function implements the MSPRT for a sequentially observed data.

Usage

implement.MSPRT(obs, obs1, obs2, design.MSPRT.object, 
                termination.threshold, test.type, side = "right",
                theta0, Type1.target = 0.005, Type2.target = 0.2,
                N.max, N1.max, N2.max, 
                sigma = 1, sigma1 = 1, sigma2 = 1,
                batch.size, batch1.size, batch2.size,
                verbose = T, plot.it = 2)

Arguments

obs

Numeric vector. The vector of data in the order they are sequentially observed for one-sample tests. Note: Its length can't exceed the length of batch.size.

obs1

Numeric vector. The vector of data in the order they are sequentially observed from Group-1 for two-sample tests. Note: Its length can't exceed the length of batch1.size.

obs2

Numeric vector. The vector of data in the order they are sequentially observed from Group-2 for two-sample tests. Note: Its length can't exceed the length of batch2.size.

design.MSPRT.object

List. The output returned from design.MSPRT corresponding to the MSPRT for which the operating characteristics are desired.

termination.threshold

Positive numeric. Termination threshold of the designed MSPRT.

test.type

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

side

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

theta0

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

Type1.target

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

Type2.target

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

N.max

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

N1.max

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

N2.max

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

sigma

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

sigma1

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

sigma2

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

batch.size

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

batch1.size

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

batch2.size

Same as in design.MSPRT. Not required if design.MSPRT.object is provided.

verbose

Logical. If TRUE (default), returns messages of the current proceedings. Otherwise it doesn't.

plot.it

0, 1 or 2 (default).

if plot.it=0, no plot is returned.
if plot.it=1, only the ggplot object required to get a comparison plot is returned, but it's not plotted.
if plot.it=2, a comparison plot and the corresponding ggplot object is returned.

Details

Value

List. The list has the following named components in case of one-sided one-sample tests:

n

Positive integer. Number of samples required to reach the decision.

decision

Character. The decision reached. The possibilities are 'accept', 'reject' and 'continue'. They respectively correspond to accepting H_0, rejecting H_0 and continue sampling.

RejectH0.threshold

Positive numeric. Threshold for rejecting H_0 in the MSPRT.

RejectH1.threshold

Positive numeric. Threshold for accepting H_1 in the MSPRT.

LR

Numeric vector. Vector of weighted likelihood ratios (proportion tests) or likelihood ratios (z tests) or Bayes factor (t tests) that are computed at each step of sequential analysis until either a decision is reached or the maximum available number of samples (N.max in one-sample tests, or N1.max and N2.max in two-sample tests) has been used.

UMPBT alternative

This stores the UMPBT alternative(s) as

UMPBT for proportion tests. Of the same type as it is returned by UMPBT.alt in these tests.
theta.UMPBT for z and t tests. This is a numeric in case of z tests and a numeric vector in case of t tests. For t tests the UMPBT alternative depends on the data. So the numeric vector returned in this case contains the UMPBT alternative computed at step of sequential analysis and is based on all data observed until that step.

In case of two-sample tests, the n output above is replaced by n1 and n2. They are positive integers and refer to the number of samples from Group-1 and 2 required to reach the decision.

In case of two-sided tests at level of significance \alpha, the MSPRT carries out a right and a left sided test simultaneously at level of significance \alpha/2. In this case the outputs are same as above with following changes in components in the returned list:

LR

List. It has two components named right and left corresponding to the right and left sided tests of size \alpha/2. Each of these components stores the vector of weighted likelihood ratios (proportion tests) or likelihood ratios (z tests) or Bayes factor (t tests) that are computed at each step of sequential analysis until either a decision is reached or the maximum available number of samples (N.max in one-sample tests, or N1.max and N2.max in two-sample tests) has been used for that sided test.

UMPBT or theta.UMPBT

List with two components named right and left corresponding to the right and left sided tests of size \alpha/2. Each of these contains the UMPBT alternative (of the same type as the output from UMPBT.alt for the test with respective sides.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

References

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

Examples


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

#### right sided ####
### design
#design.oneprop.right = design.MSPRT(test.type = 'oneProp', side = 'right',
#                                    N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0.5  # change effect size to experiment
#y = rbinom(20, 1, theta.gen)
#implement.oneprop.right = implement.MSPRT(obs = y, 
#                                          design.MSPRT.object = design.oneprop.right)

#### left sided ####
### design
#design.oneprop.left = design.MSPRT(test.type = 'oneProp', side = 'left',
#                                   N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0.5  # change effect size to experiment
#y = rbinom(20, 1, theta.gen)
#implement.oneprop.left = implement.MSPRT(obs = y, 
#                                          design.MSPRT.object = design.oneprop.left)

#### both sided ####
### design
#design.oneprop.both = design.MSPRT(test.type = 'oneProp', side = 'both',
#                                   N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0.5  # change effect size to experiment
#y = rbinom(20, 1, theta.gen)
#implement.oneprop.both = implement.MSPRT(obs = y, 
#                                         design.MSPRT.object = design.oneprop.both)


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

#### right sided ####
### design
#design.onez.right = design.MSPRT(test.type = 'oneZ', side = 'right',
#                                 N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y = rnorm(20, theta.gen, design.onez.right$sigma)
#implement.onez.right = implement.MSPRT(obs = y, 
#                                       design.MSPRT.object = design.onez.right)

#### left sided ####
### design
#design.onez.left = design.MSPRT(test.type = 'oneZ', side = 'left',
#                                N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y = rnorm(20, theta.gen, design.onez.left$sigma)
#implement.onez.left = implement.MSPRT(obs = y, 
#                                      design.MSPRT.object = design.onez.left)

#### both sided ####
### design
#design.onez.both = design.MSPRT(test.type = 'oneZ', side = 'both',
#                                N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y = rnorm(20, theta.gen, design.onez.both$sigma)
#implement.onez.both = implement.MSPRT(obs = y, 
#                                      design.MSPRT.object = design.onez.both)


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

#### right sided ####
### design
#design.onet.right = design.MSPRT(test.type = 'oneT', side = 'right',
#                                 N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y = rnorm(20, theta.gen, 1)
#implement.onet.right = implement.MSPRT(obs = y, 
#                                       design.MSPRT.object = design.onet.right)

#### left sided ####
### design
#design.onet.left = design.MSPRT(test.type = 'oneT', side = 'left',
#                                N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y = rnorm(20, theta.gen, 1)
#implement.onet.left = implement.MSPRT(obs = y, 
#                                      design.MSPRT.object = design.onet.left)

#### both sided ####
### design
#design.onet.both = design.MSPRT(test.type = 'oneT', side = 'both',
#                                N.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y = rnorm(20, theta.gen, 1)
#implement.onet.both = implement.MSPRT(obs = y, 
#                                      design.MSPRT.object = design.onet.both)


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

#### right sided ####
### design
#design.twoz.right = design.MSPRT(test.type = 'twoZ', side = 'right',
#                                 N1.max = 20, N2.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y1 = rnorm(20, theta.gen/2, design.twoz.right$sigma1)
#y2 = rnorm(20, -theta.gen/2, design.twoz.right$sigma2)
#implement.twoz.right = implement.MSPRT(obs1 = y1, obs2 = y2,
#                                       design.MSPRT.object = design.twoz.right)

#### left sided ####
### design
#design.twoz.left = design.MSPRT(test.type = 'twoZ', side = 'left',
#                                N1.max = 20, N2.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y1 = rnorm(20, theta.gen/2, design.twoz.left$sigma1)
#y2 = rnorm(20, -theta.gen/2, design.twoz.left$sigma2)
#implement.twoz.left = implement.MSPRT(obs1 = y1, obs2 = y2,
#                                      design.MSPRT.object = design.twoz.left)

#### both sided ####
### design
#design.twoz.both = design.MSPRT(test.type = 'twoZ', side = 'both',
#                                N1.max = 20, N2.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y1 = rnorm(20, theta.gen/2, design.twoz.both$sigma1)
#y2 = rnorm(20, -theta.gen/2, design.twoz.both$sigma2)
#implement.twoz.both = implement.MSPRT(obs1 = y1, obs2 = y2,
#                                      design.MSPRT.object = design.twoz.both)


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

#### right sided ####
### design
#design.twot.right = design.MSPRT(test.type = 'twoT', side = 'right',
#                                 N1.max = 20, N2.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y1 = rnorm(20, theta.gen/2, 1)
#y2 = rnorm(20, -theta.gen/2, 1)
#implement.twot.right = implement.MSPRT(obs1 = y1, obs2 = y2,
#                                       design.MSPRT.object = design.twot.right)

#### left sided ####
### design
#design.twot.left = design.MSPRT(test.type = 'twoT', side = 'left',
#                                N1.max = 20, N2.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y1 = rnorm(20, theta.gen/2, 1)
#y2 = rnorm(20, -theta.gen/2, 1)
#implement.twot.left = implement.MSPRT(obs1 = y1, obs2 = y2,
#                                      design.MSPRT.object = design.twot.left)

#### both sided ####
### design
#design.twot.both = design.MSPRT(test.type = 'twoT', side = 'both',
#                                N1.max = 20, N2.max = 20)

### implementation
#set.seed(1)
#theta.gen = 0  # change effect size to experiment
#y1 = rnorm(20, theta.gen/2, 1)
#y2 = rnorm(20, -theta.gen/2, 1)
#implement.twot.both = implement.MSPRT(obs1 = y1, obs2 = y2,
#                                      design.MSPRT.object = design.twot.both)

Internal MSPRT function: Implementing the MSPRT in one-sample proportion tests

Description

implement.MSPRT calls this function for implementing the MSPRT in one-sample proportion tests. Users please refer to implement.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Implementing the MSPRT in one-sample t tests

Description

implement.MSPRT calls this function for implementing the MSPRT in one-sample t tests. Users please refer to implement.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Implementing the MSPRT in one-sample z tests

Description

implement.MSPRT calls this function for implementing the MSPRT in one-sample z tests. Users please refer to implement.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Implementing the MSPRT in two-sample t tests

Description

implement.MSPRT calls this function for implementing the MSPRT in two-sample t tests. Users please refer to implement.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

Internal MSPRT function: Implementing the MSPRT in two-sample z tests

Description

implement.MSPRT calls this function for implementing the MSPRT in two-sample z tests. Users please refer to implement.MSPRT.

Author(s)

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya