# pvalue.TOST: p-value(s) of the TOST procedure In PowerTOST: Power and Sample Size for (Bio)Equivalence Studies

## Description

Calculates the p-value(s) of the TOST procedure via students t-distribution given pe, CV and n.

## Usage

 ```1 2 3 4``` ```pvalue.TOST(pe, CV, n, logscale = TRUE, theta1, theta2, design = "2x2", robust = FALSE, both = FALSE) pvalues.TOST(pe, CV, n, logscale = TRUE, theta1, theta2, design = "2x2", robust = FALSE, both = TRUE) ```

## Arguments

 `pe` Observed point estimate of the T/R ratio or difference. In case of `logscale=TRUE` it must be given as ratio T/R. If `logscale=FALSE`, the observed difference in means. In this case, the difference may be expressed in two ways: relative to the same (underlying) reference mean, i.e. as (T-R)/R = T/R - 1; or as difference in means T-R. Note that in the former case the units of `CV`, `theta1` and `theta2` need also be given relative to the reference mean (specified as ratio). `CV` In case of `logscale=TRUE` the observed (geometric) coefficient of variation given as ratio. If `logscale=FALSE` the argument refers to the observed (residual) standard deviation of the response. In this case, standard deviation may be expressed two ways: relative to a reference mean (specified as ratio sigma/muR), i.e. again as a coefficient of variation; or untransformed, i.e. as standard deviation of the response. Note that in the former case the units of `pe`, `theta1` and `theta2` need also be given relative to the reference mean (specified as ratio). In case of cross-over studies this is the within-subject CV, in case of a parallel-group design the CV of the total variability. `n` Total number of subjects if given as scalar. Number of subjects in (sequence) groups if given as vector. `logscale` Should the data be used after log-transformation or on original scale? `TRUE` or `FALSE`. Defaults to `TRUE`. `theta1` Lower (bio-)equivalence limit. In case of `logscale=TRUE` it is given as ratio. If `logscale=FALSE`, the limit may be expressed in two ways: difference of means relative to the same (underlying) reference mean or in units of the difference of means. Note that in the former case the units of `CV`, `pe` and `theta2` need also be given relative to the reference mean (specified as ratio). Defaults to 0.8 if `logscale=TRUE` or to -0.2 if `logscale=FALSE`. `theta2` Upper (bio-)equivalence limit. In case of `logscale=TRUE` it is given as ratio. If `logscale=FALSE`, the limit may be expressed in two ways: difference of means relative to the same (underlying) reference mean or in units of the difference of means. Note that in the former case the units of `CV`, `theta0` and `theta1` need also be given relative to the reference mean (specified as ratio). If not given, `theta2` will be calculated as `1/theta1` if `logscale=TRUE` or as `-theta1` if `logscale=FALSE`. `design` Character string describing the study design. See `known.designs()` for designs covered in this package. `robust` If set to `TRUE` triggers the use of degrees of freedom according to the ‘robust’ evaluation (aka Senn’s basic estimator). These degrees of freedom are calculated as `n-seq`. See `known.designs()\$df2`. Has only effect for higher-order crossover designs. Defaults to `FALSE`. With that value the usual degrees of freedom will be used. `both` Indicates if both p-values (t-tests of pe >= theta1 and pe <= theta2) shall be given back or only the maximum. Defaults to `FALSE` for the function `pvalue.TOST()` and to `TRUE` for the function `pvalue`s`.TOST()`.

## Value

Returns the p-value(s).
Returns a vector with named elements `p.left`, `p.right` if arguments `pe` and `CV` are scalars, else a matrix with columns `p.left`, `p.right`.
`p.left` gives the p-value of testing
` HA1: theta >= theta1`
and `p.right` the p-value of testing
` HA2: theta <= theta2`
against their respective Nulls.

## Note

The formulas implemented cover balanced and unbalanced designs.

In case of argument `n` given as n(total) and is not divisible by the number of (sequence) groups the total sample size is partitioned to the (sequence) groups to have small imbalance only. A message is given in such cases.

SAS procedure TTEST with the TOST option names p.left = Upper, p.right= Lower according to the tail of the t-distribution to be used for obtaining the p-values.

## Author(s)

B. Lang, man page by D. Labes

## References

Schuirmann DJ. A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability. J Pharmacokin Biopharm. 1987;15:657–80. doi: 10.1007/BF01068419

Hauschke D, Steinijans V, Pigeot I. Bioequivalence Studies in Drug Development. Chichester: Wiley; 2007.

`CI.BE`

## Examples

 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16``` ```# Defaults: 2x2 crossover, log-transformation # BE acceptance limits 0.8 ... 1.25, usual dfs # interested in both p-values pvalues.TOST(pe = 0.95, CV = 0.3, n = 12) # gives the vector (named elements) # p.left p.right # 0.09105601 0.02250985 # i.e. 'left' hypothesis H01: theta<=theta1 can't be rejected # 'right' hypothesis H02: theta>=theta2 can be rejected # max. p-value only as 'overall' pvalue, preferred by Benjamin pvalue.TOST(pe = 0.912, CV = 0.333, n = 24) # should give 0.08777621, i.e., inequivalence can't be rejected # this is operationally identical to CI.BE(pe = 0.912, CV = .333, n = 24) # lower limit = 0.7766 outside 0.8 ... 1.25, i.e., inequivalence can't be rejected ```

### Example output

```    p.left    p.right
0.09105601 0.02250985
[1] 0.08777621
lower     upper
0.7765749 1.0710416
```

PowerTOST documentation built on Jan. 18, 2021, 5:07 p.m.