design.MSPRT: Designing the MSPRT
In MSPRT: A Modified Sequential Probability Ratio Test (MSPRT)

Description Usage Arguments Value Author(s) References Examples

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

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)

`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 : θ > θ_0 (default), and `left` for H_1 : θ < θ_0.
`theta0`	Numeric. Hypothesized value of effect size (θ_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 (θ_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" =` θ_1`, "left" =` θ_2). Then the comparison is done at alternative effect sizes θ_1 and θ_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 θ_1 and θ_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 (θ_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.

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.

Sandipan Pramanik, Valen E. Johnson and Anirban Bhattacharya

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

##### 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)