Package 'TOSTER'

Convert to class 'htest'

Description

Convert a TOSTER result object of class 'TOSTt' or 'TOSTnp' to a list of class 'htest'.

Usage

as_htest(TOST)

Arguments

TOST	A TOSTER result object of class 'TOSTt' or 'TOSTnp'.

Value

Returns a list containing a list of class 'htest' for the result of each test with the following elements:

data.name	A character string giving the names of the data.
estimate	Estimated difference in raw units.
method	A character string indicating the performed test.
null.value	Equivalence bound.
alternative	A character string describing the alternative hypothesis.
parameter	The degrees of freedom of the distribution of the test statistic.
statistic	The value of the test statistic.
p.value	The p-value of the test.
conf.int	The confidence interval of the difference.

See Also

Other htest: htest-helpers, simple_htest()

Examples

res1 = t_TOST(formula = extra ~ group,data = sleep,eqb = .5,smd_ci = "goulet")
as_htest(res1)

---

Comparing Correlations between independent studies with bootstrapping

Description

A function to compare standardized mean differences (SMDs) between studies. This function is intended to be used to compare the compatibility of original studies with replication studies (lower p-values indicating lower compatibility).

Usage

boot_compare_cor(
  x1,
  y1,
  x2,
  y2,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  method = c("pearson", "kendall", "spearman", "winsorized", "bendpercent"),
  alpha = 0.05,
  null = 0,
  R = 1999,
  ...
)

Arguments

x1, y1	numeric vectors of data values. x and y must have the same length from study 1.
x2, y2	numeric vectors of data values. x and y must have the same length from study 2.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
method	a character string indicating which correlation coefficient is to be used for the test. One of "winsorized", "bendpercent","pearson", "kendall", or "spearman", can be abbreviated.
alpha	alpha level (default = 0.05)
null	a number indicating the null hypothesis. Default is a correlation of zero.
R	number of bootstrap replicates
...	further arguments to be passed to or from methods.

Value

A list with class "htest" containing the following components:

• "p.value": numeric scalar containing the p-value for the test under the null hypothesis.

• "estimate": difference in correlations between studies.

• "conf.int": percentile (bootstrap) confidence interval for difference in correlations.

• "null.value": the specified hypothesized value for the null hypothesis.

• "alternative": character string indicating the alternative hypothesis (the value of the input argument alternative). Possible values are "greater", "less", or "two-sided".

• "method": a character string indicating how the association was measured.

• "data.name": Names of input values.

• "boot_res": List of bootstrapped results.

• "call": the matched call.

See Also

Other compare studies: boot_compare_smd(), compare_cor(), compare_smd()

---

Comparing SMDs between ndependent studies with bootstrapping

Description

A function to compare standardized mean differences (SMDs) between studies. This function is intended to be used to compare the compatibility of original studies with replication studies (lower p-values indicating lower compatibility).

Usage

boot_compare_smd(
  x1,
  y1 = NULL,
  x2,
  y2 = NULL,
  null = 0,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater"),
  R = 1999,
  alpha = 0.05
)

Arguments

x1	a (non-empty) numeric vector of data values from study 1.
y1	an optional (non-empty) numeric vector of data values from study 1.
x2	a (non-empty) numeric vector of data values from study 2.
y2	an optional (non-empty) numeric vector of data values from study 2.
null	a number indicating the null hypothesis. For TOST, this would be equivalence bound.
paired	a logical indicating whether the SMD is from a paired or independent samples design.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.
R	number of bootstrap replicates
alpha	alpha level (default = 0.05)

Value

A list with class "htest" containing the following components:

• "statistic": z-score.

• "p.value": numeric scalar containing the p-value for the test under the null hypothesis.

• "estimate": difference in SMD between studies

• "conf.int": percentile (bootstrap) confidence interval for difference in SMDs

• "null.value": the specified hypothesized value for the null hypothesis.

• "alternative": character string indicating the alternative hypothesis (the value of the input argument alternative). Possible values are "greater", "less", or "two-sided".

• "method": Type of SMD.

• "data.name": "Boostrapped" to denote summary statistics were utilized to obtain results.

• "smd": SMDs input for the function.

• "df_ci": Data frame of confidence intervals.

• "boot_res": List of bootstrapped results.

• "call": the matched call.

See Also

Other compare studies: boot_compare_cor(), compare_cor(), compare_smd()

---

Bootstrapped correlation coefficients

Description

A function for a bootstrap, percentile, method for correlation coefficients.

Usage

boot_cor_test(
  x,
  y,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  method = c("pearson", "kendall", "spearman", "winsorized", "bendpercent"),
  alpha = 0.05,
  null = 0,
  boot_ci = c("basic", "perc"),
  R = 1999,
  ...
)

Arguments

x	a (non-empty) numeric vector of data values.
y	an optional (non-empty) numeric vector of data values.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
method	a character string indicating which correlation coefficient is to be used for the test. One of "winsorized", "bendpercent","pearson", "kendall", or "spearman", can be abbreviated.
alpha	alpha level (default = 0.05)
null	a number indicating the null hypothesis. Default is a correlation of zero.
boot_ci	type of bootstrap confidence interval. Options include studentized (stud), empirical/basic (basic) and percentile (perc) confidence intervals.
R	number of bootstrap replicates
...	further arguments to be passed to or from methods.

Details

This function uses a percentile bootstrap methods for the confidence intervals. The returned p-values are calculated from a re-sampled null distribution (similar to boot_t_TOST). See vignette("correlations") for more details.

The bootstrap correlation methods in this package offer two other correlations: a Winsorized correlation and a percentage bend correlation (see Wilcox 2017). These two can modified by adding the trim (Winsorized) or beta (percentage bend) arguments. The default for both arguments is 0.2 but can be modified at the user's discretion. These calculations are based on Rand Wilcox's R functions for his book (Wilcox, 2017), and adapted from their implementation in Guillaume Rousselet's R package "bootcorci".

Value

A list with class "htest" containing the following components:

• "p.value": the p-value of the test.

• "estimate": the estimated measure of association, with name "pb", "wincor", "cor", "tau", or "rho" corresponding to the method employed.

• "null.value": the value of the association measure under the null hypothesis.

• "alternative": character string indicating the alternative hypothesis (the value of the input argument alternative).

• "method": a character string indicating how the association was measured.

• "data.name": a character string giving the names of the data.

• "call": the matched call.

References

Wilcox, R.R. (2009) Comparing Pearson Correlations: Dealing with Heteroscedasticity and Nonnormality. Communications in Statistics - Simulation and Computation, 38, 2220–2234.

Wilcox, R.R. (2017) Introduction to Robust Estimation and Hypothesis Testing, 4th edition. Academic Press.

See Also

Other Correlations: corsum_test(), plot_cor(), power_z_cor(), z_cor_test()

---

Bootstrapped TOST with log transformed t-tests

Description

A function for a bootstrap method for TOST with all types of t-tests.

Usage

boot_log_TOST(x, ...)

## Default S3 method:
boot_log_TOST(
  x,
  y = NULL,
  hypothesis = c("EQU", "MET"),
  paired = FALSE,
  var.equal = FALSE,
  eqb = 1.25,
  alpha = 0.05,
  null = 1,
  boot_ci = c("stud", "basic", "perc"),
  R = 1999,
  ...
)

## S3 method for class 'formula'
boot_log_TOST(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
y	an optional (non-empty) numeric vector of data values.
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
paired	a logical indicating whether you want a paired t-test.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
eqb	Equivalence bound. Can provide 1 value (negative value is taken as the lower bound) or 2 specific values that represent the upper and lower equivalence bounds.
alpha	alpha level (default = 0.05)
null	Null hypothesis value for a two-tailed test (default is 1).
boot_ci	type of bootstrap confidence interval. Options include studentized (stud), empirical/basic (basic) and percentile (perc) confidence intervals.
R	number of bootstrap replicates
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

The implemented test(s) corresponds to the proposal of Chapter 16 of Efron and Tibshirani (1994), and recommended by He et al (2022). Returns TOSTt class object with bootstrapped based results. This is approximately equivalent to the percentile bootstrap method mentioned by He et al (2014).

For two-sample tests, the test is of $\bar log(x) - \bar log(y)$ (mean of x minus mean of y). For paired samples, the test is of the difference scores (z), wherein $z = log(x) - log(y) = log(x)/log(y)$, and the test is of $\bar z$ (mean of the difference/ratio scores).

For details on the calculations in this function see vignette("robustTOST").

Value

An S3 object of class "TOSTt" is returned containing the following slots:

• "TOST": A table of class "data.frame" containing two-tailed t-test and both one-tailed results.

• "eqb": A table of class "data.frame" containing equivalence bound settings.

• "effsize": table of class "data.frame" containing effect size estimates.

• "hypothesis": String stating the hypothesis being tested.

• "smd": List containing the results of the means ratio calculation.

  • Items include: d (means ratio estimate), dlow (lower CI bound), dhigh (upper CI bound), d_df (degrees of freedom for SMD), d_sigma (SE), d_lambda (non-centrality), J (bias correction), smd_label (type of SMD), d_denom (denominator calculation)

• "alpha": Alpha level set for the analysis.

• "method": Type of t-test.

• "decision": List included text regarding the decisions for statistical inference.

• "boot": List containing the bootstrap samples.

References

Efron, B., & Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press

He, Y., Deng, Y., You, C., & Zhou, X. H. (2022). Equivalence tests for ratio of means in bioequivalence studies under crossover design. Statistical Methods in Medical Research, 09622802221093721.

Food and Drug Administration (2014). Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs — General Considerations. Center for Drug Evaluation and Research. Docket: FDA-2014-D-0204. https://www.fda.gov/regulatory-information/search-fda-guidance-documents/bioavailability-and-bioequivalence-studies-submitted-ndas-or-inds-general-considerations

See Also

Other Robust tests: boot_t_TOST(), boot_t_test(), brunner_munzel(), log_TOST(), wilcox_TOST()

Other TOST: boot_t_TOST(), simple_htest(), t_TOST(), tsum_TOST(), wilcox_TOST()

---

Bootstrap SES Calculation

Description

Standardized effect size (SES), these are the effect sizes not considered SMDs.

Usage

boot_ses_calc(
  x,
  ...,
  paired = FALSE,
  ses = "rb",
  alpha = 0.05,
  boot_ci = c("basic", "stud", "perc"),
  R = 1999
)

## Default S3 method:
boot_ses_calc(
  x,
  y = NULL,
  paired = FALSE,
  ses = c("rb", "odds", "logodds", "cstat"),
  alpha = 0.05,
  boot_ci = c("basic", "stud", "perc"),
  R = 1999,
  ...
)

## S3 method for class 'formula'
boot_ses_calc(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
paired	a logical indicating whether you want a paired t-test.
ses	Standardized effect size. Default is "rb" for rank-biserial correlation. Options also include "cstat" for concordance probability, or "odds" for Wilcoxon-Mann-Whitney odds (otherwise known as Agresti's generalized odds ratio).
alpha	alpha level (default = 0.05)
boot_ci	type of bootstrap confidence interval. Options include studentized (stud), empirical/basic (basic) and percentile (perc) confidence intervals.
R	number of bootstrap replicates
y	an optional (non-empty) numeric vector of data values.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

For details on the calculations in this function see vignette("robustTOST").

Value

A data frame containing the standardized effect size.

See Also

Other effect sizes: boot_smd_calc(), ses_calc(), smd_calc()

Examples

## Not run: 
boot_ses_calc(formula = extra ~ group, data = sleep, paired = TRUE, ses = "r")

## End(Not run)

---

Bootstrapped SMD Calculation

Description

A function to only calculate standardized mean differences with bootstrap confidence intervals.

Usage

boot_smd_calc(
  x,
  ...,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  boot_ci = c("stud", "basic", "perc"),
  R = 1999
)

## Default S3 method:
boot_smd_calc(
  x,
  y = NULL,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  mu = 0,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  boot_ci = c("stud", "basic", "perc"),
  R = 1999,
  ...
)

## S3 method for class 'formula'
boot_smd_calc(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
paired	a logical indicating whether you want a paired t-test.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
alpha	alpha level (default = 0.05)
bias_correction	Apply Hedges' correction for bias (default is TRUE).
rm_correction	Repeated measures correction to make standardized mean difference Cohen's d(rm). This only applies to repeated/paired samples. Default is FALSE.
glass	A option to calculate Glass's delta as an alternative to Cohen's d type SMD. Default is NULL to not calculate Glass's delta, "glass1" will use the first group's SD as the denominator whereas "glass2" will use the 2nd group's SD.
boot_ci	type of bootstrap confidence interval. Options include studentized (stud), empirical/basic (basic) and percentile (perc) confidence intervals.
R	number of bootstrap replicates
y	an optional (non-empty) numeric vector of data values.
mu	Null value. Deviating from zero will give the x-y-mu.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

For details on the calculations in this function see vignette("SMD_calcs").

Value

A data frame containing the SMD estimates.

See Also

Other effect sizes: boot_ses_calc(), ses_calc(), smd_calc()

Examples

## Not run: 
boot_smd_calc(formula = extra ~ group,data = sleep, paired = TRUE, smd_ci = "nct")

## End(Not run)

---

Bootstrapped t-test

Description

A function for a bootstrap method for t-tests.

Usage

boot_t_test(x, ...)

## Default S3 method:
boot_t_test(
  x,
  y = NULL,
  var.equal = FALSE,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  mu = 0,
  alpha = 0.05,
  boot_ci = c("stud", "basic", "perc"),
  R = 1999,
  ...
)

## S3 method for class 'formula'
boot_t_test(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
y	an optional (non-empty) numeric vector of data values.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
paired	a logical indicating whether you want a paired t-test.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
mu	a number specifying an optional parameter used to form the null hypothesis. See ‘Details’.
alpha	alpha level (default = 0.05)
boot_ci	type of bootstrap confidence interval. Options include studentized (stud), empirical/basic (basic) and percentile (perc) confidence intervals.
R	number of bootstrap replicates
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

The implemented test(s) corresponds to the proposal of Chapter 16 of Efron and Tibshirani (1994).

For two-sample tests, the test is of $\bar x - \bar y$ (mean of x minus mean of y). For paired samples, the test is of the difference scores (z), wherein $z = x - y$, and the test is of $\bar z$ (mean of the difference scores). For one-sample tests, the test is of $\bar x$ (mean of x).

For details on the calculations in this function see vignette("robustTOST").

Value

A list with class "htest" containing the following components:

• "statistic": the value of the t-statistic.

• "parameter": the degrees of freedom for the t-statistic.

• "p.value": the p-value for the test.

• "conf.int": a confidence interval for the mean appropriate to the specified alternative hypothesis.

• "estimate": the estimated mean or difference in means depending on whether it was a one-sample test or a two-sample test.

• "null.value": the specified hypothesized value of the mean or mean difference. May be 2 values.

• "stderr": the standard error of the mean (difference), used as denominator in the t-statistic formula.

• "alternative": a character string describing the alternative hypothesis.

• "method": a character string indicating what type of t-test was performed.

• "data.name": a character string giving the name(s) of the data.

References

Efron, B., & Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press.

See Also

Other Robust tests: boot_log_TOST(), boot_t_TOST(), brunner_munzel(), log_TOST(), wilcox_TOST()

Examples

# example code

boot_t_test(extra ~ group, data = sleep)

---

Bootstrapped TOST with t-tests

Description

A function for a bootstrap method for TOST with all types of t-tests.

Usage

boot_t_TOST(x, ...)

## Default S3 method:
boot_t_TOST(
  x,
  y = NULL,
  hypothesis = "EQU",
  paired = FALSE,
  var.equal = FALSE,
  eqb,
  low_eqbound,
  high_eqbound,
  eqbound_type = "raw",
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  mu = 0,
  R = 1999,
  boot_ci = c("stud", "basic", "perc"),
  ...
)

## S3 method for class 'formula'
boot_t_TOST(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
y	an optional (non-empty) numeric vector of data values.
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
paired	a logical indicating whether you want a paired t-test.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
eqb	Equivalence bound. Can provide 1 value (negative value is taken as the lower bound) or 2 specific values that represent the upper and lower equivalence bounds.
low_eqbound	lower equivalence bounds (deprecated).
high_eqbound	upper equivalence bounds (deprecated).
eqbound_type	Type of equivalence bound. Can be set to "SMD" for standardized mean difference (i.e., Cohen's d) or "raw" for the mean difference. Default is "raw". Raw is strongly recommended as SMD bounds will produce biased results.
alpha	alpha level (default = 0.05)
bias_correction	Apply Hedges' correction for bias (default is TRUE).
rm_correction	Repeated measures correction to make standardized mean difference Cohen's d(rm). This only applies to repeated/paired samples. Default is FALSE.
glass	A option to calculate Glass's delta as an alternative to Cohen's d type SMD. Default is NULL to not calculate Glass's delta, "glass1" will use the first group's SD as the denominator whereas "glass2" will use the 2nd group's SD.
mu	a number indicating the true value of the mean for the two tailed test (or difference in means if you are performing a two sample test).
R	number of bootstrap replicates
boot_ci	type of bootstrap confidence interval. Options include studentized (stud), empirical/basic (basic) and percentile (perc) confidence intervals.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

The implemented test(s) corresponds to the proposal of Chapter 16 of Efron and Tibshirani (1994). Returns TOSTt class object with bootstrapped based results. Please note that the repeated measures "corrected" effect size is not available.

For two-sample tests, the test is of $\bar x - \bar y$ (mean of x minus mean of y). For paired samples, the test is of the difference scores (z), wherein $z = x - y$, and the test is of $\bar z$ (mean of the difference scores). For one-sample tests, the test is of $\bar x$ (mean of x).

For details on the calculations in this function see vignette("robustTOST").

Value

An S3 object of class "TOSTt" is returned containing the following slots:

• "TOST": A table of class "data.frame" containing two-tailed t-test and both one-tailed results.

• "eqb": A table of class "data.frame" containing equivalence bound settings.

• "effsize": table of class "data.frame" containing effect size estimates.

• "hypothesis": String stating the hypothesis being tested

• "smd": List containing the results of the standardized mean difference calculations (e.g., Cohen's d).

  • Items include: d (estimate), dlow (lower CI bound), dhigh (upper CI bound), d_df (degrees of freedom for SMD), d_sigma (SE), d_lambda (non-centrality), J (bias correction), smd_label (type of SMD), d_denom (denominator calculation)

• "alpha": Alpha level set for the analysis.

• "method": Type of t-test.

• "decision": List included text regarding the decisions for statistical inference.

• "boot": List containing the bootstrap samples.

References

Efron, B., & Tibshirani, R. J. (1994). An introduction to the bootstrap. CRC press.

See Also

Other Robust tests: boot_log_TOST(), boot_t_test(), brunner_munzel(), log_TOST(), wilcox_TOST()

Other TOST: boot_log_TOST(), simple_htest(), t_TOST(), tsum_TOST(), wilcox_TOST()

---

Brunner-Munzel Test

Description

This is a generic function that performs a generalized asymptotic Brunner-Munzel test in a fashion similar to t.test.

Usage

brunner_munzel(
  x,
  ...,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater"),
  mu = 0.5,
  alpha = 0.05,
  perm = FALSE,
  max_n_perm = 10000
)

## Default S3 method:
brunner_munzel(
  x,
  y,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater"),
  mu = 0.5,
  alpha = 0.05,
  perm = FALSE,
  max_n_perm = 10000,
  ...
)

## S3 method for class 'formula'
brunner_munzel(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
paired	a logical indicating whether you want a paired test.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater" or "less". You can specify just the initial letter.
mu	a number specifying an optional parameter used to form the null hypothesis (Default = 0.5). This can be thought of as the null in terms of the relative effect, p = P (X < Y ) + 0.5 * P (X = Y); See ‘Details’.
alpha	alpha level (default = 0.05)
perm	a logical indicating whether or not to perform a permutation test over approximate t-distribution based test (default is FALSE). Highly recommend to set perm = TRUE when sample size per condition is less than 15.
max_n_perm	the maximum number of permutations (default is 10000).
y	an optional (non-empty) numeric vector of data values.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

This function is made to provide a test of stochastic equality between two samples (paired or independent), and is referred to as the Brunner-Munzel test.

This tests the hypothesis that the relative effect, discussed below, is equal to the null value (default is mu = 0.5).

The estimate of the relative effect, which can be considered as value similar to the probability of superiority, refers to the following:

$\hat p = p(X<Y) + \frac{1}{2} \cdot P(X=Y)$

Note, for paired samples, this does not refer to the probability of an increase/decrease in paired sample but rather the probability that a randomly sampled value of X. This is also referred to as the "relative" effect in the literature. Therefore, the results will differ from the concordance probability provided by the ses_calc function.

The brunner_munzel function is based on the npar.t.test and npar.t.test.paired functions within the nparcomp package (Konietschke et al. 2015).

Value

A list with class "htest" containing the following components:

• "statistic": the value of the test statistic.

• "parameter": the degrees of freedom for the test statistic.

• "p.value": the p-value for the test.

• "conf.int": a confidence interval for the relative effect appropriate to the specified alternative hypothesis.

• "estimate": the estimated relative effect.

• "null.value": the specified hypothesized value of the relative effect.

• "stderr": the standard error of the relative effect.

• "alternative": a character string describing the alternative hypothesis.

• "method": a character string indicating what type of test was performed.

• "data.name": a character string giving the name(s) of the data.

References

Brunner, E., Munzel, U. (2000). The Nonparametric Behrens-Fisher Problem: Asymptotic Theory and a Small Sample Approximation. Biometrical Journal 42, 17 -25.

Neubert, K., Brunner, E., (2006). A Studentized Permutation Test for the Nonparametric Behrens-Fisher Problem. Computational Statistics and Data Analysis.

Munzel, U., Brunner, E. (2002). An Exact Paired Rank Test. Biometrical Journal 44, 584-593.

Konietschke, F., Placzek, M., Schaarschmidt, F., & Hothorn, L. A. (2015). nparcomp: an R software package for nonparametric multiple comparisons and simultaneous confidence intervals. Journal of Statistical Software 64 (2015), Nr. 9, 64(9), 1-17. http://www.jstatsoft.org/v64/i09/

See Also

Other Robust tests: boot_log_TOST(), boot_t_TOST(), boot_t_test(), log_TOST(), wilcox_TOST()

Examples

data(mtcars)
brunner_munzel(mpg ~ am, data = mtcars)

---

Comparing two independent correlation coefficients

Description

A function to compare correlations between studies. This function is intended to be used to compare the compatibility of original studies with replication studies (lower p-values indicating lower compatibility).

Usage

compare_cor(
  r1,
  df1,
  r2,
  df2,
  method = c("fisher", "kraatz"),
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  null = 0
)

Arguments

r1	Correlation study 1.
df1	Degrees of freedom from study 1 (if a simple correlation the df is N-2).
r2	Correlation study 2.
df2	Degrees of freedom from study 2 (if a simple correlation the df is N-2).
method	Method for determining differences. Default, "z", will use Fisher's transformation, while "Kraatz" will use the Kraatz method.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
null	a number indicating the null hypothesis. For TOST, this would be equivalence bound.

Details

This function tests for differences between correlations. @return A list with class "htest" containing the following components:

• "statistic": z-score

• "p.value": numeric scalar containing the p-value for the test under the null hypothesis.

• "estimate": difference in SMD between studies.

• "null.value": the specified hypothesized value for the null hypothesis.

• "alternative": character string indicating the alternative hypothesis (the value of the input argument alternative). Possible values are "greater", "less", or "two-sided".

• "method": Type of SMD.

• "data.name": "Summary Statistics" to denote summary statistics were utilized to obtain results.

• "cor": Correlation input for the function.

• "call": the matched call.

References

Counsell, A., & Cribbie, R. A. (2015). Equivalence tests for comparing correlation and regression coefficients. The British journal of mathematical and statistical psychology, 68(2), 292–309. https://doi.org/10.1111/bmsp.12045

Anderson, S., & Hauck, W. W. (1983). A new procedure for testing equivalence in comparative bioavailability and other clinical trials. Communications in Statistics-Theory and Methods, 12(23), 2663-2692.

See Also

Other compare studies: boot_compare_cor(), boot_compare_smd(), compare_smd()

---

Comparing SMDs between independent studies

Description

A function to compare standardized mean differences (SMDs) between studies. This function is intended to be used to compare the compatibility of original studies with replication studies (lower p-values indicating lower compatibility).

Usage

compare_smd(
  smd1,
  n1,
  se1 = NULL,
  smd2,
  n2,
  se2 = NULL,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  null = 0,
  TOST = FALSE
)

Arguments

smd1, smd2	SMDs from study 1 & 2, respectively.
n1, n2	sample size(s) from study 1 & 2, respectively (can be 1 number or vector of 2 numbers).
se1, se2	User supplied standard errors (SEs). This will override the internal calculations.
paired	a logical indicating whether the SMD is from a paired or independent samples design. If a one-sample design, then paired must be set to TRUE.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
null	a number indicating the null hypothesis. For TOST, this would be equivalence bound.
TOST	Defunct: use alternative argument. Logical indicator (default = FALSE) to perform two one-sided tests of equivalence (TOST).

Details

This function tests for differences between SMDs from independent studies (e.g., original vs replication).

Value

A list with class "htest" containing the following components:

• "statistic": z-score.

• "p.value": numeric scalar containing the p-value for the test under the null hypothesis.

• "estimate": difference in SMD between studies.

• "null.value": the specified hypothesized value for the null hypothesis.

• "alternative": character string indicating the alternative hypothesis (the value of the input argument alternative). Possible values are "greater", "less", or "two-sided".

• "method": Type of SMD.

• "data.name": "Summary Statistics" to denote summary statistics were utilized to obtain results.

• "smd": SMDs input for the function.

• "sample_sizes": Sample sizes input for the function.

• "call": the matched call.

See Also

Other compare studies: boot_compare_cor(), boot_compare_smd(), compare_cor()

---

Association/Correlation Test from Summary Statistics

Description

Test for association between paired samples, using one of Pearson's product moment correlation coefficient, Kendall's $\tau$ (tau) or Spearman's $\rho$ (rho). This is the updated version of the TOSTr function.

Usage

corsum_test(
  r,
  n,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  method = c("pearson", "kendall", "spearman"),
  alpha = 0.05,
  null = 0
)

Arguments

r	observed correlation
n	number of pairs of observations
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
method	a character string indicating which correlation coefficient is to be used for the test. One of "pearson", "kendall", or "spearman", can be abbreviated.
alpha	alpha level (default = 0.05)
null	a number indicating the null hypothesis. Default is a correlation of zero.

Details

This function uses Fisher's z transformation for the correlations, but uses Fieller's correction of the standard error for Spearman's $\rho$ and Kendall's $\tau$.

Value

A list with class "htest" containing the following components:

• "statistic": z-score.

• "p.value": the p-value of the test.

• "estimate": the estimated measure of association, with name "cor", "tau", or "rho" corresponding to the method employed.

• "null.value": the value of the association measure under the null hypothesis.

• "alternative": character string indicating the alternative hypothesis (the value of the input argument alternative).

• "method": a character string indicating how the association was measured.

• "data.name": a character string giving the names of the data.

• "call": the matched call.

References

Goertzen, J. R., & Cribbie, R. A. (2010). Detecting a lack of association: An equivalence testing approach. British Journal of Mathematical and Statistical Psychology, 63(3), 527-537. https://doi.org/10.1348/000711009X475853, formula page 531.

See Also

Other Correlations: boot_cor_test(), plot_cor(), power_z_cor(), z_cor_test()

---

TOST One Sample T-Test

Description

TOST One Sample T-Test in jamovi. This function is not meant to be utilized in R. See t_TOST function.

Usage

dataTOSTone(
  data,
  vars,
  mu = 0,
  hypothesis = "EQU",
  low_eqbound = -0.5,
  high_eqbound = 0.5,
  eqbound_type = "raw",
  alpha = 0.05,
  desc = FALSE,
  plots = FALSE,
  low_eqbound_d = -999999999,
  high_eqbound_d = -999999999,
  smd_type = "g"
)

Arguments

data	the data as a data frame
vars	a vector of strings naming variables of interest in data
mu	a number (default: 0) to compare against
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis;
low_eqbound	a number (default: -0.5) the lower equivalence bounds
high_eqbound	a number (default: 0.5) the upper equivalence bounds
eqbound_type	'SMD' (default) or 'raw'; whether the bounds are specified in Cohen's d or raw units respectively
alpha	alpha level (default = 0.05)
desc	TRUE or FALSE (default), provide descriptive statistics
plots	TRUE or FALSE (default), provide plots
low_eqbound_d	deprecated
high_eqbound_d	deprecated
smd_type	'd' (default) or 'g'; whether the calculated effect size is biased (d) or bias-corrected (g).

Value

A results object containing:

results$text					a html
results$tost					a table
results$eqb					a table
results$effsize					a table
results$desc					a table
results$plots					an array of images

Tables can be converted to data frames with asDF or as.data.frame. For example:

results$tost$asDF

as.data.frame(results$tost)

Examples

library("TOSTER")

dataTOSTone(data=iris, vars="Sepal.Width", mu=3, low_eqbound=-0.3, high_eqbound=0.3,
            alpha=0.05, desc=TRUE, plots=TRUE)

TOSTone(m=3.05733, mu=3, sd=0.4358663, n=150, low_eqbound_d=-0.3, high_eqbound_d=0.3, alpha=0.05)

---

TOST Paired Samples T-Test

Description

TOST Paired Samples T-Test in jamovi. This function is not meant to be utilized in R. See t_TOST function.

Usage

dataTOSTpaired(
  data,
  pair1,
  pair2,
  hypothesis = "EQU",
  low_eqbound = -0.5,
  high_eqbound = 0.5,
  eqbound_type = "raw",
  alpha = 0.05,
  desc = FALSE,
  plots = FALSE,
  low_eqbound_dz = -999999999,
  high_eqbound_dz = -999999999,
  indplot = FALSE,
  diffplot = FALSE,
  smd_type = "g"
)

Arguments

data	the data as a data frame
pair1	A string naming the first part of the pair
pair2	A string naming the second part of the pair
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
low_eqbound	a number (default: 0.5) the lower equivalence bounds
high_eqbound	a number (default: 0.5) the upper equivalence bounds
eqbound_type	'SMD' (default) or 'raw'; whether the bounds are specified in standardized mean difference (Cohen's dz) or raw units respectively
alpha	alpha level (default = 0.05)
desc	TRUE or FALSE (default), provide descriptive statistics
plots	TRUE or FALSE (default), provide plots
low_eqbound_dz	deprecated
high_eqbound_dz	deprecated
indplot	TRUE or FALSE (default), provide plot of paired data.
diffplot	TRUE or FALSE (default), provide plot of difference scores.
smd_type	'd' (default) or 'g'; whether the calculated effect size is biased (d) or bias-corrected (g).

Value

A results object containing:

results$text					a html
results$tost					a table
results$eqb					a table
results$effsize					a table
results$desc					a table
results$plots					an image
results$indplot					an image
results$diffplot					an image

Tables can be converted to data frames with asDF or as.data.frame. For example:

results$tost$asDF

as.data.frame(results$tost)

References

Mara, C. A., & Cribbie, R. A. (2012). Paired-Samples Tests of Equivalence. Communications in Statistics - Simulation and Computation, 41(10), 1928-1943. formula page 1932. Note there is a typo in the formula: n-1 should be n (personal communication, 31-08-2016)

Examples

## Not run: 
library("TOSTER")

dataTOSTpaired(data = randu, pair1 = "x", pair2="y", low_eqbound = -0.3,
               high_eqbound = 0.3, alpha = 0.05, desc = TRUE, plots = TRUE)

## End(Not run)

---

TOST Correlation

Description

TOST for correlations in jamovi. This function is not meant to be utilized in R.

Usage

dataTOSTr(
  data,
  pairs,
  cor_type = "pearson",
  hypothesis = "EQU",
  low_eqbound_r = -0.3,
  high_eqbound_r = 0.3,
  alpha = 0.05,
  desc = FALSE,
  plots = FALSE
)

Arguments

data	the data as a data frame
pairs	a list of vectors of strings naming variables to correlate from data
cor_type	a character string indicating which correlation coefficient is to be used for the test. One of "pearson", "kendall", or "spearman", can be abbreviated.
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
low_eqbound_r	lower equivalence bounds (e.g., -0.3) expressed in a correlation effect size
high_eqbound_r	upper equivalence bounds (e.g., 0.3) expressed in a correlation effect size
alpha	alpha level (default = 0.05)
desc	TRUE or FALSE (default), provide descriptive statistics
plots	TRUE or FALSE (default), provide plots

Value

A results object containing:

results$text					a preformatted
results$tost					a table
results$desc					a table
results$plots					an array of images

Tables can be converted to data frames with asDF or as.data.frame. For example:

results$tost$asDF

as.data.frame(results$tost)

---

TOST Independent Samples T-Test

Description

TOST Independent Samples T-Test for jamovi. This function is not meant to be utilized in R. See t_TOST function.

Usage

dataTOSTtwo(
  data,
  deps,
  group,
  var_equal = FALSE,
  hypothesis = "EQU",
  low_eqbound = -0.5,
  high_eqbound = 0.5,
  eqbound_type = "raw",
  alpha = 0.05,
  desc = FALSE,
  plots = FALSE,
  descplots = FALSE,
  low_eqbound_d = -999999999,
  high_eqbound_d = -999999999,
  smd_type = "g"
)

Arguments

data	the data as a data frame
deps	a vector of strings naming dependent variables in data
group	a string naming the grouping variable in data; must have two levels
var_equal	TRUE or FALSE (default), assume equal variances
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
low_eqbound	a number (default: -0.5) the lower equivalence/MET bounds
high_eqbound	a number (default: 0.5) the upper equivalence/MET bounds
eqbound_type	'SMD' (default) or 'raw'; whether the bounds are specified in Cohen's d or raw units respectively
alpha	alpha level (default = 0.05)
desc	TRUE or FALSE (default), provide descriptive statistics
plots	TRUE or FALSE (default), provide effect size plots
descplots	TRUE or FALSE (default), provide plots
low_eqbound_d	deprecated
high_eqbound_d	deprecated
smd_type	'd' (default) or 'g'; whether the calculated effect size is biased (d) or bias-corrected (g).

Value

A results object containing:

results$text					a html
results$tost					a table
results$eqb					a table
results$effsize					a table
results$desc					a table
results$plots					an array of images
results$descplots					an array of images

Tables can be converted to data frames with asDF or as.data.frame. For example:

results$tost$asDF

as.data.frame(results$tost)

References

Berger, R. L., & Hsu, J. C. (1996). Bioequivalence Trials, Intersection-Union Tests and Equivalence Confidence Sets. Statistical Science, 11(4), 283-302.

Gruman, J. A., Cribbie, R. A., & Arpin-Cribbie, C. A. (2007). The effects of heteroscedasticity on tests of equivalence. Journal of Modern Applied Statistical Methods, 6(1), 133-140, formula for Welch's t-test on page 135

Examples

library(TOSTER)

## Load iris dataset, remove one of the three groups so two are left

data<-iris[which(iris$Species!="versicolor"),]

## TOST procedure on the raw data

dataTOSTtwo(data, deps="Sepal.Width", group="Species", var_equal = TRUE, low_eqbound = -0.5,
            high_eqbound = 0.5, alpha = 0.05, desc = TRUE, plots = TRUE)

---

TOST Two Proportions

Description

TOST Two Proportions for jamovi. This function is not meant to be utilized in R.

Usage

datatosttwoprop(
  data,
  var,
  level,
  group,
  hypothesis = "EQU",
  low_eqbound = -0.1,
  high_eqbound = 0.1,
  alpha = 0.05,
  desc = FALSE,
  plot = FALSE
)

Arguments

data	.
var	.
level	.
group	.
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
low_eqbound	a number (default: -0.1) the lower equivalence bounds
high_eqbound	a number (default: 0.1) the upper equivalence bounds
alpha	alpha level (default = 0.05)
desc	TRUE or FALSE (default), provide descriptive statistics
plot	TRUE or FALSE (default), provide plot

Value

A results object containing:

results$tost					a table
results$eqb					a table
results$desc					a table
results$plot					an image

Tables can be converted to data frames with asDF or as.data.frame. For example:

results$tost$asDF

as.data.frame(results$tost)

---

Equivalence Test for ANOVA Results

Description

Performs equivalence test on the partial eta-squared (pes) value from ANOVA results.

Usage

equ_anova(object, eqbound, MET = FALSE, alpha = 0.05)

Arguments

object	an object of returned by either Anova, aov, or afex_aov
eqbound	Equivalence bound for the partial eta-squared.
MET	logical indicator to perform a minimal effect test rather than equivalence test (default is FALSE).
alpha	alpha used for the test (e.g., 0.05).

Details

For details on the calculations in this function see vignette("the_ftestTOSTER").

Value

Returns a data frame containing the ANOVA results with equivalence tests added.

The following abbreviations are used in the table:

• effect: name of the effect.

• df1: Degrees of Freedom in the numerator (i.e. DF effect).

• df2: Degrees of Freedom in the denominator (i.e., DF error).

• F: F-value.

• p.null: p-value (probability of the data given the null hypothesis).

• pes: partial Eta-Squared measure of effect size.

• eqbound: equivalence bound.

• p.equ: p-value (probability of the data given the equivalence hypothesis)

References

Campbell, H., & Lakens, D. (2021). Can we disregard the whole model? Omnibus non‐inferiority testing for R2 in multi‐variable linear regression and in ANOVA. British Journal of Mathematical and Statistical Psychology, 74(1), 64-89. doi: 10.1111/bmsp.12201

See Also

Other f-test: equ_ftest()

---

Equivalence Test using an F-test

Description

Performs equivalence test on the partial eta-squared (pes) value for using an F-test.

Usage

equ_ftest(Fstat, df1, df2, eqbound = NULL, eqb, MET = FALSE, alpha = 0.05)

Arguments

Fstat	The F-statistic from the F-test.
df1	Degrees of freedom for the numerator.
df2	Degrees of freedom for the denominator.
eqbound	Defunct argument for quivalence bound for the partial eta-squared. Default is NULL.
eqb	Defunct argument for quivalence bound for the partial eta-squared.
MET	logical indicator to perform a minimal effect test rather than equivalence test (default is FALSE).
alpha	alpha used for the test (e.g., 0.05).

Details

For details on the calculations in this function see vignette("the_ftestTOSTER").

Value

Object of class '"htest"

• "statistic": The value of the F-statistic.

• "parameter": The degrees of freedom for the F-statistic.

• "p.value": The he p-value for the test.

• "conf.int": A confidence interval for the partial eta-squared statistic.

• "estimate": Estimate of partial eta-squared.

• "null.value": The specified for the equivalence test.

• "method": A string indicating the type of F-test.

• "data.name": A required string indicating that this was calculated from summary statistics.

References

Campbell, H., & Lakens, D. (2021). Can we disregard the whole model? Omnibus non‐inferiority testing for R2 in multi‐variable linear regression and in ANOVA. British Journal of Mathematical and Statistical Psychology, 74(1), 64-89. doi: 10.1111/bmsp.12201

See Also

Other f-test: equ_anova()

---

Extract Paired Correlation

Description

A function for estimating the correlation from a paired samples t-test. Useful for when using tsum_TOST and the correlation is not available.

Usage

extract_r_paired(m1, sd1, m2, sd2 = NULL, n, tstat = NULL, pvalue = NULL)

Arguments

m1	mean of group 1.
sd1	standard deviation of group 1.
m2	mean of group 2.
sd2	standard deviation of group 2.
n	Sample size (number of pairs)
tstat	The t-value from a paired samples t-test
pvalue	The two-tailed p-value from a paired samples t-test

Value

An estimate of the correlation.

References

Lajeunesse, M. J. (2011). On the meta‐analysis of response ratios for studies with correlated and multi‐group designs. Ecology, 92(11), 2049-2055

---

Data

Description

A dataset from a study on the Hawthrone effect published by McCambridge et al. The dataset has 5 variables (participant_ID, totaldrinking.x, group, totaldrinking.y, totaldrinking.diff)

Usage

hawthorne

Format

An object of class data.frame with 5474 rows and 5 columns.

Source

McCambridge, J., Wilson, A., Attia, J., Weaver, N., & Kypri, K. (2019). Randomized trial seeking to induce the Hawthorne effect found no evidence for any effect on self-reported alcohol consumption online. Journal of Clinical Epidemiology, 108, 102–109.

---

Helpers for htest objects

Description

Functions to help interpret or display objects of the class htest

Usage

df_htest(htest, test_statistics = TRUE, show_ci = TRUE, extract_names = TRUE)

describe_htest(htest, alpha = NULL, digits = 3)

Arguments

htest	A S3 object of the class htest
test_statistics	A logical variable to display the test statistics.
show_ci	A logical variable to display the confidence interval.
extract_names	A logical variable to take the names from the S3 object (i.e., statistic for t.test would be "t")
alpha	alpha level (default = 0.05)
digits	integer indicating the number of decimal places.

See Also

Other htest: as_htest(), simple_htest()

Examples

# simple example with t-test
tres = t.test(extra ~ group, data = sleep)

# As a data frame
df_htest(tres)

# Describe t-test results
describe_htest(tres)

---

TOST with log transformed t-tests

Description

A function for TOST on the log-transformed data using parametric t-tests.

Usage

log_TOST(
  x,
  ...,
  hypothesis = "EQU",
  paired = FALSE,
  var.equal = FALSE,
  eqb = 1.25,
  alpha = 0.05,
  null = 1
)

## Default S3 method:
log_TOST(
  x,
  y = NULL,
  hypothesis = c("EQU", "MET"),
  var.equal = FALSE,
  paired = FALSE,
  eqb = 1.25,
  alpha = 0.05,
  null = 1,
  ...
)

## S3 method for class 'formula'
log_TOST(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
paired	a logical indicating whether you want a paired t-test.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
eqb	Equivalence bound; default is 1.25 (FDA guidelines). Can provide 1 value (reciprocal value is taken as the lower bound) or 2 specific values that represent the upper and lower equivalence bounds.
alpha	alpha level (default = 0.05)
null	Null hypothesis value for a two-tailed test (default is 1).
y	an optional (non-empty) numeric vector of data values.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

For details on the calculations in this function see vignette("robustTOST").

For two-sample tests, the test is of $\bar log(x) - \bar log(y)$ (mean of x minus mean of y). For paired samples, the test is of the difference scores (z), wherein $z = log(x) - log(y) = log(x)/log(y)$, and the test is of $\bar z$ (mean of the difference/ratio scores).

Value

An S3 object of class ⁠"TOSTt⁠" is returned containing the following slots:

• "TOST": A table of class "data.frame" containing two-tailed t-test and both one-tailed results.

• "eqb": A table of class "data.frame" containing equivalence bound settings.

• "effsize": table of class "data.frame" containing effect size estimates.

• "hypothesis": String stating the hypothesis being tested

• "smd": List containing the results of the means ratio calculation.

  • Items include: d (means ratio estimate), dlow (lower CI bound), dhigh (upper CI bound), d_df (degrees of freedom for SMD), d_sigma (SE), d_lambda (non-centrality), J (bias correction), smd_label (type of SMD), d_denom (denominator calculation)

• "alpha": Alpha level set for the analysis.

• "method": Type of t-test.

• "decision": List included text regarding the decisions for statistical inference.

References

He, Y., Deng, Y., You, C., & Zhou, X. H. (2022). Equivalence tests for ratio of means in bioequivalence studies under crossover design. Statistical Methods in Medical Research, 09622802221093721.

Food and Drug Administration (2014). Bioavailability and Bioequivalence Studies Submitted in NDAs or INDs — General Considerations. Center for Drug Evaluation and Research. Docket: FDA-2014-D-0204. https://www.fda.gov/regulatory-information/search-fda-guidance-documents/bioavailability-and-bioequivalence-studies-submitted-ndas-or-inds-general-considerations

See Also

Other Robust tests: boot_log_TOST(), boot_t_TOST(), boot_t_test(), brunner_munzel(), wilcox_TOST()

Examples

data(mtcars)
# Default FDA bioequivalence bounds
log_TOST(mpg ~ am,
data = mtcars)

---

Plot correlation coefficients

Description

Function to produce plots of the distribution of standard correlation coefficients.

Usage

plot_cor(
  r,
  n,
  method = c("pearson", "spearman", "kendall"),
  type = c("c", "cd"),
  levels = c(0.68, 0.9, 0.95, 0.999)
)

Arguments

r	The observed correlation coefficient.
n	Total number of observations (sample size).
method	The method by which the coefficient was calculated: pearson, spearman, or kendall (default is "pearson")
type	Choose whether to plot a "consonance" function ("c"), consonance density ("cd"), or both (c("c","cd"); default option).
levels	Numeric vector of confidence levels to display

Details

This function was created so that users could create consonance plots of Pearson's correlation coefficient. These types of plots are discussed by Schweder T, Hjort NL. (2016, ISBN:9781316445051) and Rafi Z, Greenland S. (2020) doi:10.1186/s12874-020-01105-9.

Value

Returns plot of the distribution of the correlation coefficient.

See Also

Other Correlations: boot_cor_test(), corsum_test(), power_z_cor(), z_cor_test()

Other plotting functions: plot_pes(), plot_smd()

---

Plot partial eta-squared

Description

Function to produce plots of the distribution of partial eta-squared.

Usage

plot_pes(
  Fstat,
  df1,
  df2,
  type = c("c", "cd"),
  levels = c(0.68, 0.9, 0.95, 0.999)
)

Arguments

Fstat	The F-statistic from the F-test.
df1	Degrees of freedom for the numerator.
df2	Degrees of freedom for the denominator.
type	Choose whether to plot a "consonance" function ("c"), consonance density ("cd"), or both (c("c","cd"); default option).
levels	Numeric vector of confidence levels to display

Details

This function was created so that users could create consonance plots of partial eta-squared from ANOVA-level effects. These types of plots are discussed by Schweder T, Hjort NL. (2016, ISBN:9781316445051) and Rafi Z, Greenland S. (2020) doi:10.1186/s12874-020-01105-9.

Value

Returns plot of the distribution of partial eta-squared

See Also

Other plotting functions: plot_cor(), plot_smd()

---

Plot Distribution of a SMD

Description

Function to produce plots of the distribution of the standardized mean difference

Usage

plot_smd(
  d,
  df,
  lambda = NULL,
  sigma = NULL,
  smd_ci = c("goulet", "nct", "t", "z"),
  smd_label = "SMD",
  type = c("c", "cd"),
  levels = c(0.5, 0.9, 0.95, 0.999)
)

Arguments

d	Estimate of the standardized mean difference
df	degrees of freedom for the standardized mean difference
lambda	The non-centrality parameter for the standardized mean difference
sigma	The standard error for the standardized mean difference
smd_ci	Method for calculating SMD confidence intervals. Methods include Goulet, noncentral t (nct), central t (t), and normal method (z).
smd_label	Label for the x-axis indicating the SMD measure
type	Choose whether to plot a "consonance" function ("c"), consonance density ("cd"), or both (c("c","cd"); default option).
levels	Numeric vector of confidence levels to display

Details

This function was created so that users could create plots from their own SMD calculations and were inspired by the concurve R package. The difficulty is that specific information must be past onto this function. The calculations for the standardized mean difference can be found in the vignettes of this package. These types of plots are discussed by Schweder T, Hjort NL. (2016, ISBN:9781316445051) and Rafi Z, Greenland S. (2020) doi:10.1186/s12874-020-01105-9.

Value

Returns plot of the distribution of the standardized mean difference.

See Also

Other plotting functions: plot_cor(), plot_pes()

---

F-test Power

Description

Power analysis for TOST for an F-test

Usage

power_eq_f(alpha = 0.05, df1, df2, eqbound)

Arguments

alpha	alpha used for the test (e.g., 0.05)
df1	Degrees of freedom for the numerator
df2	Degrees of freedom for the denominator
eqbound	Equivalence bound for the partial eta-squared

Value

Object of class '"power.htest"

References

Campbell, H., & Lakens, D. (2021). Can we disregard the whole model? Omnibus non‐inferiority testing for R2 in multi‐variable linear regression and in ANOVA. British Journal of Mathematical and Statistical Psychology, 74(1), 64-89. doi: 10.1111/bmsp.12201

See Also

Other power: power_t_TOST(), power_z_cor()

Examples

## Statistical power for alpha = 0.05, 3 groups, n = 80 per group, equivalence bound of
## partial eta squared = 0.01, assuming true effect = 0.
## df1 = number of groups - 1 = 3 - 1 = 2.
## df2 = Total N - number of groups = 240 - 3 = 237.
power_eq_f(alpha=0.05, df1=3, df2 = 237, eqbound = 0.01)

---

Power calculations for TOST for t-tests

Description

Calculates the exact power of two one sided t-tests (TOST) for one, two, and paired samples.

Usage

power_t_TOST(
  n = NULL,
  delta = 0,
  sd = 1,
  eqb,
  low_eqbound = NULL,
  high_eqbound = NULL,
  alpha = NULL,
  power = NULL,
  type = "two.sample"
)

Arguments

n	number of observations per group. 2 sample sizes, in a vector, can be provided for the two sample case.
delta	true difference in means (default is 0).
sd	population standard deviation. Standard deviation of the differences for paired samples.
eqb	Equivalence bound. Can provide 1 value (negative value is taken as the lower bound) or 2 specific values that represent the upper and lower equivalence bounds.
low_eqbound	Lower equivalence bounds. Deprecated use eqb.
high_eqbound	Upper equivalence bounds. Deprecated use eqb.
alpha	a priori alpha-level (i.e., significance level).
power	power of the TOST procedure (1-beta).
type	string specifying the type of t-test.

Details

The exact calculations of power are based on Owen’s Q-function or by direct integration of the bivariate non-central t-distribution (inspired by the PowerTOST package). Approximate power is implemented via the non-central t-distribution or the ‘shifted’ central t-distribution.

Note

The power function in this package is limited. Please see the PowerTOST R package for more options.

References

Phillips KF. Power of the Two One-Sided Tests Procedure in Bioequivalence. J Pharmacokin Biopharm. 1990;18(2):137–44. doi: 10.1007/BF01063556

Diletti D, Hauschke D, Steinijans VW. Sample Size Determination for Bioequivalence Assessment by Means of Confidence Intervals. Int J Clin Pharmacol Ther Toxicol. 1991;29(1):1–8.

See Also

Other power: power_eq_f(), power_z_cor()

---

TOST Power for Tests of Two Proportions

Description

Power analysis for TOST for difference between two proportions using Z-test (pooled)

Usage

powerTOSTtwo.prop(
  alpha,
  statistical_power,
  prop1,
  prop2,
  N,
  low_eqbound_prop,
  high_eqbound_prop
)

power_twoprop(
  p1,
  p2,
  n = NULL,
  null = 0,
  alpha = NULL,
  power = NULL,
  alternative = c("two.sided", "one.sided", "equivalence")
)

Arguments

alpha	a priori alpha-level (i.e., significance level).
statistical_power	Deprecated. desired power (e.g., 0.8)
prop1	Deprecated. expected proportion in group 1.
prop2	Deprecated. expected proportion in group 2.
N	Deprecated. sample size (e.g., 108)
low_eqbound_prop	Deprecated. lower equivalence bounds (e.g., -0.05) expressed in proportion
high_eqbound_prop	Deprecated. upper equivalence bounds (e.g., 0.05) expressed in proportion
p1, p2	Proportions in each respective group.
n	Sample size per group.
null	the null hypothesis value.
power	statistical power (1-beta).
alternative	equivalence, one-sided, or two-sided test. Can be abbreviated.

Value

Calculate either achieved power, equivalence bounds, or required N, assuming a true effect size of 0. Returns a string summarizing the power analysis, and a numeric variable for number of observations, equivalence bounds, or power.

References

Silva, G. T. da, Logan, B. R., & Klein, J. P. (2008). Methods for Equivalence and Noninferiority Testing. Biology of Blood and Marrow Transplantation: Journal of the American Society for Blood and Marrow Transplantation, 15(1 Suppl), 120-127. https://doi.org/10.1016/j.bbmt.2008.10.004

Julious, S. A. & Campell, M. J. (2012). Tutorial in biostatistics: sample sizes for parallel group clinical trials with binary data. Statistics in Medicine, 31:2904-2936.

Chow, S.-C., Wang, H., & Shao, J. (2007). Sample Size Calculations in Clinical Research, Second Edition (2 edition). Boca Raton: Chapman and Hall/CRC.

Examples

## Sample size for alpha = 0.05, 90% power, assuming true effect prop1 = prop 2 = 0.5,
## equivalence bounds of 0.4 and 0.6 (so low_eqbound_prop = -0.1 and high_eqbound_prop = 0.1)

#powerTOSTtwo.prop(alpha = 0.05, statistical_power = 0.9, prop1 = 0.5, prop2 = 0.5,
#    low_eqbound_prop = -0.1, high_eqbound_prop = 0.1)

   power_twoprop(alpha = 0.05, power = 0.9, p1 = 0.5, p2 = 0.5,
   null = 0.1, alternative = "e")

## Power for alpha = 0.05, N 542 , assuming true effect prop1 = prop 2 = 0.5,
## equivalence bounds of 0.4 and 0.6 (so low_eqbound_prop = -0.1 and high_eqbound_prop = 0.1)

#powerTOSTtwo.prop(alpha = 0.05, N = 542, prop1 = 0.5, prop2 = 0.5,
#    low_eqbound_prop = -0.1, high_eqbound_prop = 0.1)

power_twoprop(alpha = 0.05, n = 542, p1 = 0.5, p2 = 0.5,
   null = 0.1, alternative = "e")


#Example 4.2.4 from Chow, Wang, & Shao (2007, p. 93)
#powerTOSTtwo.prop(alpha=0.05, statistical_power=0.8, prop1 = 0.75, prop2 = 0.8,
#    low_eqbound_prop = -0.2, high_eqbound_prop = 0.2)

power_twoprop(alpha = 0.05, power = 0.8, p1 = 0.75, p2 = 0.8,
   null = 0.2, alternative = "e")

# Example 5 from Julious & Campbell (2012, p. 2932)
#powerTOSTtwo.prop(alpha=0.025, statistical_power=0.9, prop1 = 0.8, prop2 = 0.8,
#    low_eqbound_prop=-0.1, high_eqbound_prop=0.1)
 power_twoprop(alpha = 0.025, power = 0.9, p1 = 0.8, p2 = 0.8,
   null = 0.1, alternative = "e")
# From Machin, D. (Ed.). (2008). Sample size tables for clinical studies (3rd ed).

# Example 9.4b equivalence of two proportions (p. 113) #
# powerTOSTtwo.prop(alpha=0.010, statistical_power=0.8, prop1 = 0.5, prop2 = 0.5,
#    low_eqbound_prop = -0.2, high_eqbound_prop = 0.2)/2
power_twoprop(alpha = 0.01, power = 0.8, p1 = 0.5, p2 = 0.5,
   null = 0.2, alternative = "e")

---

Power Calculations for Correlations

Description

Calculates the approximate power for a z-test based on a Pearson product-moment correlation.

Usage

power_z_cor(
  n = NULL,
  rho = NULL,
  power = NULL,
  null = 0,
  alpha = NULL,
  alternative = c("two.sided", "less", "greater", "equivalence")
)

powerTOSTr(alpha, statistical_power, N, low_eqbound_r, high_eqbound_r)

Arguments

n	number of observations.
rho	true correlation value (alternative hypothesis).
power	statistical power (1-beta).
null	the null hypothesis value.
alpha	a priori alpha-level (i.e., significance level).
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", or "equivalence" (TOST). You can specify just the initial letter.
statistical_power	Deprecated. desired power (e.g., 0.8)
N	Deprecated. number of pairs (e.g., 96)
low_eqbound_r	Deprecated. lower equivalence bounds (e.g., -0.3) expressed in a correlation effect size
high_eqbound_r	Deprecated. upper equivalence bounds (e.g., 0.3) expressed in a correlation effect size

Value

An object of the class power.htest. This will include the sample size (n), power, beta (1-power), alpha (significance level), null value(s), alternative hypothesis, and a text string detailing the method.

powerTOSTr has been replaced by the power_z_cor function. The function is only retained for historical purposes.

See Also

Other Correlations: boot_cor_test(), corsum_test(), plot_cor(), z_cor_test()

Other power: power_eq_f(), power_t_TOST()

Examples

## Sample size for alpha = 0.05, 90% power, equivalence bounds of
## r = -0.1 and r = 0.1, assuming true effect = 0
#powerTOSTr(alpha=0.05, statistical_power=0.9, low_eqbound_r=-0.1, high_eqbound_r=0.1)
power_z_cor(alternative = "equivalence", alpha = .05, null = .1, power = .9, rho = 0)

## Sample size for alpha = 0.05, N=536, equivalence bounds of
## r = -0.1 and r = 0.1, assuming true effect = 0
#powerTOSTr(alpha=0.05, N=536, low_eqbound_r=-0.1, high_eqbound_r=0.1)
power_z_cor(alternative = "equivalence", alpha = .05, null = .1, n = 536, rho = 0)

## Equivalence bounds for alpha = 0.05, N=536, statistical power of
## 0.9, assuming true effect = 0
#powerTOSTr(alpha=0.05, N=536, statistical_power=0.9)

---

Power One Sample t-test

Description

Power analysis for TOST for one-sample t-test (Cohen's d). This function is no longer maintained please use power_t_TOST.

Usage

powerTOSTone(alpha, statistical_power, N, low_eqbound_d, high_eqbound_d)

powerTOSTone.raw(alpha, statistical_power, N, sd, low_eqbound, high_eqbound)

Arguments

alpha	alpha used for the test (e.g., 0.05)
statistical_power	desired power (e.g., 0.8)
N	sample size (e.g., 108)
low_eqbound_d	lower equivalence bounds (e.g., -0.5) expressed in standardized mean difference (Cohen's d)
high_eqbound_d	upper equivalence bounds (e.g., 0.5) expressed in standardized mean difference (Cohen's d)
sd	standard deviation.
low_eqbound	lower equivalence bounds (e.g., -0.5) expressed in raw scores
high_eqbound	upper equivalence bounds (e.g., 0.5) expressed in raw scores

Value

Calculate either achieved power, equivalence bounds, or required N, assuming a true effect size of 0. Returns a string summarizing the power analysis, and a numeric variable for number of observations, equivalence bounds, or power.

References

Chow, S.-C., Wang, H., & Shao, J. (2007). Sample Size Calculations in Clinical Research, Second Edition - CRC Press Book. Formula 3.1.9

Examples

## Sample size for alpha = 0.05, 90% power, equivalence bounds of
## Cohen's d = -0.3 and Cohen's d = 0.3, and assuming true effect = 0
powerTOSTone(alpha=0.05, statistical_power=0.9, low_eqbound_d=-0.3, high_eqbound_d=0.3)

## Power for sample size of 121, alpha = 0.05, equivalence bounds of
## Cohen's d = -0.3 and Cohen's d = 0.3, and assuming true effect = 0

powerTOSTone(alpha=0.05, N=121, low_eqbound_d=-0.3, high_eqbound_d=0.3)

## Equivalence bounds for sample size of 121, alpha = 0.05, statistical power of
## 0.9, and assuming true effect d = 0

powerTOSTone(alpha=0.05, N=121, statistical_power=.9)

#' ## Sample size for alpha = 0.05, 90% power, equivalence bounds of -0.3 and 0.3 in
## raw units, assuming pooled standard deviation of 1, and assuming true effect d = 0
powerTOSTone.raw(alpha=0.05, statistical_power=0.9, sd = 1, low_eqbound=-0.3, high_eqbound=0.3)

## Power for sample size of 121, alpha = 0.05, equivalence bounds of
## -0.3 and 0.3 in raw units, assuming pooled standard deviation of 1, and assuming true effect = 0

powerTOSTone.raw(alpha=0.05, N=121, sd = 1, low_eqbound=-0.3, high_eqbound=0.3)

## Power for sample size of 121, alpha = 0.05, statistical power of
## 0.9, and assuming true effect = 0

powerTOSTone.raw(alpha=0.05, N=121, statistical_power=.9, sd=1)

---

Power Paired Sample t-test

Description

Power analysis for TOST for dependent t-test (Cohen's dz). This function is no longer maintained please use power_t_TOST.

Usage

powerTOSTpaired(alpha, statistical_power, N, low_eqbound_dz, high_eqbound_dz)

powerTOSTpaired.raw(
  alpha,
  statistical_power,
  low_eqbound,
  high_eqbound,
  sdif,
  N
)

Arguments

alpha	alpha used for the test (e.g., 0.05)
statistical_power	desired power (e.g., 0.8)
N	number of pairs (e.g., 96)
low_eqbound_dz	lower equivalence bounds (e.g., -0.5) expressed in standardized mean difference (Cohen's dz)
high_eqbound_dz	upper equivalence bounds (e.g., 0.5) expressed in standardized mean difference (Cohen's dz)
low_eqbound	lower equivalence bounds (e.g., -0.5) expressed in raw mean difference
high_eqbound	upper equivalence bounds (e.g., 0.5) expressed in raw mean difference
sdif	standard deviation of the difference scores

Value

Calculate either achieved power, equivalence bounds, or required N, assuming a true effect size of 0. Returns a string summarizing the power analysis, and a numeric variable for number of observations, equivalence bounds, or power.

References

Chow, S.-C., Wang, H., & Shao, J. (2007). Sample Size Calculations in Clinical Research, Second Edition - CRC Press Book. Formula 3.1.9

Examples

## Sample size for alpha = 0.05, 80% power, equivalence bounds of
## Cohen's dz = -0.3 and Cohen's d = 0.3, and assuming true effect = 0
powerTOSTpaired(alpha=0.05,statistical_power=0.8,low_eqbound_dz=-0.3,high_eqbound_dz=0.3)

## Sample size for alpha = 0.05, N = 96 pairs, equivalence bounds of
## Cohen's dz = -0.3 and Cohen's d = 0.3, and assuming true effect = 0
powerTOSTpaired(alpha=0.05,N=96,low_eqbound_dz=-0.3,high_eqbound_dz=0.3)

## Equivalence bounds for alpha = 0.05, N = 96 pairs, statistical power of
## 0.8, and assuming true effect = 0
powerTOSTpaired(alpha=0.05,N=96,statistical_power=0.8)

## Sample size for alpha = 0.05, 80% power, equivalence bounds of -3 and 3 in raw units
## and assuming a standard deviation of the difference scores of 10, and assuming a true effect = 0
powerTOSTpaired.raw(alpha=0.05,statistical_power=0.8,low_eqbound=-3, high_eqbound=3, sdif=10)

## Sample size for alpha = 0.05, N = 96 pairs, equivalence bounds of -3 and 3 in raw units
## and assuming a standard deviation of the difference scores of 10, and assuming a true effect = 0
powerTOSTpaired.raw(alpha=0.05,N=96,low_eqbound=-3, high_eqbound=3, sdif=10)

## Equivalence bounds for alpha = 0.05, N = 96 pairs, statistical power of 0.8
## and assuming a standard deviation of the difference scores of 10, and assuming a true effect = 0

---

Power Two Sample t-test

Description

Power analysis for TOST for independent t-test (Cohen's d). This function is no longer maintained please use power_t_TOST.

Usage

powerTOSTtwo(alpha, statistical_power, N, low_eqbound_d, high_eqbound_d)

powerTOSTtwo.raw(
  alpha,
  statistical_power,
  N,
  sdpooled,
  low_eqbound,
  high_eqbound,
  delta = 0
)

Arguments

alpha	alpha used for the test (e.g., 0.05)
statistical_power	desired power (e.g., 0.8)
N	sample size per group (e.g., 108)
low_eqbound_d	lower equivalence bounds (e.g., -0.5) expressed in standardized mean difference (Cohen's d)
high_eqbound_d	upper equivalence bounds (e.g., 0.5) expressed in standardized mean difference (Cohen's d)
sdpooled	specify the pooled standard deviation
low_eqbound	lower equivalence bounds (e.g., -0.5) expressed in raw scale units (e.g., scalepoints)
high_eqbound	upper equivalence bounds (e.g., 0.5) expressed in raw scale units (e.g., scalepoints)
delta	hypothesized true value for the difference between the 2 means. Default is zero.

Value

Calculate either achieved power, equivalence bounds, or required N, assuming a true effect size of 0. Returns a string summarizing the power analysis, and a numeric variable for number of observations, equivalence bounds, or power.

References

Chow, S.-C., Wang, H., & Shao, J. (2007). Sample Size Calculations in Clinical Research, Second Edition - CRC Press Book. Formula 3.2.4 with k = 1

Examples

## Sample size for alpha = 0.05, 80% power, equivalence bounds of
## Cohen's d = -0.4 and Cohen's d = 0.4, assuming true effect = 0
powerTOSTtwo(alpha=0.05, statistical_power=0.8, low_eqbound_d=-0.4, high_eqbound_d=0.4)

## Statistical power for alpha = 0.05, N = 108 per group, equivalence bounds of
## Cohen's d = -0.4 and Cohen's d = 0.4, assuming true effect = 0
powerTOSTtwo(alpha=0.05, N=108, low_eqbound_d=-0.4, high_eqbound_d=0.4)

## Equivalence bounds for alpha = 0.05, N = 108 per group, statistical power of
## 0.8, assuming true effect = 0
powerTOSTtwo(alpha=0.05, N=108, statistical_power=0.8)

## Sample size for alpha = 0.05, 80% power, equivalence bounds of -200 and 200 in raw
## units, assuming pooled standard deviation of 350, and assuming true effect = 0
powerTOSTtwo.raw(alpha=0.05,statistical_power=0.8,low_eqbound=-200,high_eqbound=200,sdpooled=350)

## Power for alpha = 0.05, N = 53 per group, equivalence bounds of
## -200 and 200 in raw units, assuming sdpooled = 350 and true effect = 0
powerTOSTtwo.raw(alpha=0.05, N=53, low_eqbound=-200, high_eqbound=200, sdpooled=350)

## Equivalence bounds for alpha = 0.05, N = 108 per group, statistical power of
## 0.8, assuming true effect = 0
powerTOSTtwo.raw(alpha=0.05, N=53, statistical_power=0.8, sdpooled=350)

---

Non-parametric standardized effect sizes (replicates of ses_calc)

Description

Effect sizes for simple (one or two sample) non-parametric tests. Suggested to use ses_calc function instead.

Usage

rbs(x, y = NULL, mu = 0, conf.level = 0.95, paired = FALSE)

np_ses(
  x,
  y = NULL,
  mu = 0,
  conf.level = 0.95,
  paired = FALSE,
  ses = c("rb", "odds", "cstat")
)

Arguments

x	a (non-empty) numeric vector of data values.
y	an optional (non-empty) numeric vector of data values.
mu	a number indicating the value around which (a-)symmetry (for one-sample or paired samples) or shift (for independent samples) is to be estimated. See stats::wilcox.test.
conf.level	confidence level of the interval.
paired	a logical indicating whether you want to calculate a paired test.
ses	Rank-biserial (rb), odds (odds), and concordance probability (cstat).

Details

This method was adapted from the effectsize R package. The rank-biserial correlation is appropriate for non-parametric tests of differences - both for the one sample or paired samples case, that would normally be tested with Wilcoxon's Signed Rank Test (giving the matched-pairs rank-biserial correlation) and for two independent samples case, that would normally be tested with Mann-Whitney's U Test (giving Glass' rank-biserial correlation). See stats::wilcox.test. In both cases, the correlation represents the difference between the proportion of favorable and unfavorable pairs / signed ranks (Kerby, 2014). Values range from -1 indicating that all values of the second sample are smaller than the first sample, to +1 indicating that all values of the second sample are larger than the first sample.

In addition, the rank-biserial correlation can be transformed into a concordance probability (i.e., probability of superiority) or into a generalized odds (WMW odds or Agresti's generalized odds ratio).

Ties

When tied values occur, they are each given the average of the ranks that would have been given had no ties occurred. No other corrections have been implemented yet.

Value

Returns a list of results including the rank biserial correlation, logical indicator if it was a paired method, setting for mu, and confidence interval.

Confidence Intervals

Confidence intervals for the standardized effect sizes are estimated using the normal approximation (via Fisher's transformation).

References

• Cureton, E. E. (1956). Rank-biserial correlation. Psychometrika, 21(3), 287-290.

• Glass, G. V. (1965). A ranking variable analogue of biserial correlation: Implications for short-cut item analysis. Journal of Educational Measurement, 2(1), 91-95.

• Kendall, M.G. (1948) Rank correlation methods. London: Griffin.

• Kerby, D. S. (2014). The simple difference formula: An approach to teaching nonparametric correlation. Comprehensive Psychology, 3, 11-IT.

• King, B. M., & Minium, E. W. (2008). Statistical reasoning in the behavioral sciences. John Wiley & Sons Inc.

• Cliff, N. (1993). Dominance statistics: Ordinal analyses to answer ordinal questions. Psychological bulletin, 114(3), 494.

• Tomczak, M., & Tomczak, E. (2014). The need to report effect size estimates revisited. An overview of some recommended measures of effect size.

---

SES Calculation

Description

Standardized effect size (SES), these are the effect sizes not considered SMDs.

Usage

ses_calc(x, ..., paired = FALSE, ses = "rb", alpha = 0.05)

## Default S3 method:
ses_calc(
  x,
  y = NULL,
  paired = FALSE,
  ses = c("rb", "odds", "logodds", "cstat"),
  alpha = 0.05,
  mu = 0,
  ...
)

## S3 method for class 'formula'
ses_calc(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
paired	a logical indicating whether you want a paired t-test.
ses	Standardized effect size. Default is "rb" for rank-biserial correlation. Options also include "cstat" for concordance probability, or "odds" for Wilcoxon-Mann-Whitney odds (otherwise known as Agresti's generalized odds ratio).
alpha	alpha level (default = 0.05)
y	an optional (non-empty) numeric vector of data values.
mu	number indicating the value around which (a-)symmetry (for one-sample or paired samples) or shift (for independent samples) is to be estimated. See stats::wilcox.test.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

For details on the calculations in this function see vignette("robustTOST").

Value

A data frame containing the standardized effect size.

See Also

Other effect sizes: boot_ses_calc(), boot_smd_calc(), smd_calc()

Examples

## Not run: 
ses_calc(formula = extra ~ group, data = sleep, paired = TRUE, ses = "r")

## End(Not run)

---

One, two, and paired samples hypothesis tests

Description

Performs one or two sample t-tests or Wilcoxon-Mann-Whitney rank-based tests with expanded options compared to t.test, brunner_munzel, or wilcox.test.

Usage

simple_htest(
  x,
  ...,
  paired = FALSE,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  mu = NULL,
  alpha = 0.05
)

## Default S3 method:
simple_htest(
  x,
  y = NULL,
  test = c("t.test", "wilcox.test", "brunner_munzel"),
  paired = FALSE,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  mu = NULL,
  alpha = 0.05,
  ...
)

## S3 method for class 'formula'
simple_htest(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
paired	a logical indicating whether you want a paired t-test.
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
mu	a number specifying an optional parameter used to form the null hypothesis. See ‘Details’.
alpha	alpha level (default = 0.05)
y	an optional (non-empty) numeric vector of data values.
test	a character string specifying what type of hypothesis test to use. Options are limited to "wilcox.test", "t.test", or "brunner_munzel". You can specify just the initial letter.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

The type of test, t-test/Wilcoxon-Mann-Whitney/Brunner-Munzel, can be selected with the "test" argument.

alternative = "greater" is the alternative that x is larger than y (on average). If alternative = "equivalence" then the alternative is that the difference between x and y is between the two null values mu.. If alternative = "minimal.effect" then the alternative is that the difference between x and y is less than the lowest null value or greater than the highest.

For more details on each possible test (brunner_munzel, stats::t.test, or stats::wilcox.test), please read their individual documentation.

Value

A list with class "htest" containing the following components:

• statistic: the value of the t-statistic.

• parameter: the degrees of freedom for the t-statistic.

• p.value: the p-value for the test.

• conf.int: a confidence interval for the mean appropriate to the specified alternative hypothesis.

• estimate: the estimated mean or difference in means depending on whether it was a one-sample test or a two-sample test.

• null.value: the specified hypothesized value of the mean or mean difference. May be 2 values.

• stderr: the standard error of the mean (difference), used as denominator in the t-statistic formula.

• alternative: a character string describing the alternative hypothesis.

• method: a character string indicating what type of t-test was performed.

• data.name: a character string giving the name(s) of the data.

See Also

Other TOST: boot_log_TOST(), boot_t_TOST(), t_TOST(), tsum_TOST(), wilcox_TOST()

Other htest: as_htest(), htest-helpers

Examples

data(mtcars)
simple_htest(mpg ~ am,
data = mtcars,
alternative = "e",
mu = 3)

---

SMD Calculation

Description

A function to only calculate standardized mean differences.

Usage

smd_calc(
  x,
  ...,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  smd_ci = c("nct", "goulet", "t", "z")
)

## Default S3 method:
smd_calc(
  x,
  y = NULL,
  paired = FALSE,
  var.equal = FALSE,
  alpha = 0.05,
  mu = 0,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  smd_ci = c("nct", "goulet", "t", "z"),
  ...
)

## S3 method for class 'formula'
smd_calc(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
paired	a logical indicating whether you want a paired t-test.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
alpha	alpha level (default = 0.05)
bias_correction	Apply Hedges' correction for bias (default is TRUE).
rm_correction	Repeated measures correction to make standardized mean difference Cohen's d(rm). This only applies to repeated/paired samples. Default is FALSE.
glass	A option to calculate Glass's delta as an alternative to Cohen's d type SMD. Default is NULL to not calculate Glass's delta, "glass1" will use the first group's SD as the denominator whereas "glass2" will use the 2nd group's SD.
smd_ci	Method for calculating SMD confidence intervals. Methods include Goulet, noncentral t (nct), central t (t), and normal method (z).
y	an optional (non-empty) numeric vector of data values.
mu	Null value. Deviating from zero will give the x-y-mu.
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

For details on the calculations in this function see vignette("SMD_calcs").

Value

A data frame containing the SMD estimates.

See Also

Other effect sizes: boot_ses_calc(), boot_smd_calc(), ses_calc()

Examples

## Not run: 
smd_calc(formula = extra ~ group,data = sleep, paired = TRUE, smd_ci = "nct")

## End(Not run)

---

TOST with t-tests

Description

A function for TOST with all types of t-tests.

Usage

t_TOST(
  x,
  ...,
  hypothesis = "EQU",
  paired = FALSE,
  var.equal = FALSE,
  eqb,
  low_eqbound,
  high_eqbound,
  eqbound_type = "raw",
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  smd_ci = c("nct", "goulet", "t", "z")
)

## Default S3 method:
t_TOST(
  x,
  y = NULL,
  hypothesis = c("EQU", "MET"),
  paired = FALSE,
  var.equal = FALSE,
  eqb,
  low_eqbound,
  high_eqbound,
  eqbound_type = c("raw", "SMD"),
  alpha = 0.05,
  mu = 0,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  smd_ci = c("nct", "goulet", "t", "z"),
  ...
)

## S3 method for class 'formula'
t_TOST(formula, data, subset, na.action, ...)

Arguments

x	a (non-empty) numeric vector of data values.
...	further arguments to be passed to or from methods.
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
paired	a logical indicating whether you want a paired t-test.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
eqb	Equivalence bound. Can provide 1 value (negative value is taken as the lower bound) or 2 specific values that represent the upper and lower equivalence bounds.
low_eqbound	lower equivalence bounds (deprecated).
high_eqbound	upper equivalence bounds (deprecated).
eqbound_type	Type of equivalence bound. Can be set to "SMD" for standardized mean difference (i.e., Cohen's d) or "raw" for the mean difference. Default is "raw". Raw is strongly recommended as SMD bounds will produce biased results.
alpha	alpha level (default = 0.05)
bias_correction	Apply Hedges' correction for bias (default is TRUE).
rm_correction	Repeated measures correction to make standardized mean difference Cohen's d(rm). This only applies to repeated/paired samples. Default is FALSE.
glass	A option to calculate Glass's delta as an alternative to Cohen's d type SMD. Default is NULL to not calculate Glass's delta, "glass1" will use the first group's SD as the denominator whereas "glass2" will use the 2nd group's SD.
smd_ci	Method for calculating SMD confidence intervals. Methods include Goulet, noncentral t (nct), central t (t), and normal method (z).
y	an optional (non-empty) numeric vector of data values.
mu	a number indicating the true value of the mean for the two tailed test (or difference in means if you are performing a two sample test).
formula	a formula of the form lhs ~ rhs where lhs is a numeric variable giving the data values and rhs either 1 for a one-sample or paired test or a factor with two levels giving the corresponding groups. If lhs is of class "Pair" and rhs is 1, a paired test is done.
data	an optional matrix or data frame (or similar: see model.frame) containing the variables in the formula formula. By default the variables are taken from environment(formula).
subset	an optional vector specifying a subset of observations to be used.
na.action	a function which indicates what should happen when the data contain NAs. Defaults to getOption("na.action").

Details

For details on the calculations in this function see vignette("IntroTOSTt") & vignette("SMD_calcs").

For two-sample tests, the test is of $\bar x - \bar y$ (mean of x minus mean of y). For paired samples, the test is of the difference scores (z), wherein $z = x - y$, and the test is of $\bar z$ (mean of the difference scores). For one-sample tests, the test is of $\bar x$ (mean of x).

Value

An S3 object of class "TOSTt" is returned containing the following slots:

• "TOST": A table of class "data.frame" containing two-tailed t-test and both one-tailed results.

• "eqb": A table of class ⁠"data.frame⁠" containing equivalence bound settings.

• "effsize": table of class "data.frame" containing effect size estimates.

• "hypothesis": String stating the hypothesis being tested

• "smd": List containing the results of the standardized mean difference calculations (e.g., Cohen's d).

  • Items include: d (estimate), dlow (lower CI bound), dhigh (upper CI bound), d_df (degrees of freedom for SMD), d_sigma (SE), d_lambda (non-centrality), J (bias correction), smd_label (type of SMD), d_denom (denominator calculation)

• "alpha": Alpha level set for the analysis.

• "method": Type of t-test.

• "decision": List included text regarding the decisions for statistical inference.

See Also

Other TOST: boot_log_TOST(), boot_t_TOST(), simple_htest(), tsum_TOST(), wilcox_TOST()

Examples

data(mtcars)
t_TOST(mpg ~ am,
data = mtcars,
eqb = 3)

---

TOST function for meta-analysis

Description

A function for providing TOST tests of equivalence from meta-analysis results.

Usage

TOSTmeta(
  ES,
  var,
  se,
  low_eqbound_d,
  high_eqbound_d,
  alpha,
  plot = TRUE,
  verbose = TRUE
)

Arguments

ES	meta-analytic effect size
var	meta-analytic variance
se	standard error
low_eqbound_d	lower equivalence bounds (e.g., -0.5) expressed in standardized mean difference (Cohen's d)
high_eqbound_d	upper equivalence bounds (e.g., 0.5) expressed in standardized mean difference (Cohen's d)
alpha	alpha level (default = 0.05)
plot	set whether results should be plotted (plot = TRUE) or not (plot = FALSE) - defaults to TRUE
verbose	logical variable indicating whether text output should be generated (verbose = TRUE) or not (verbose = FALSE) - default to TRUE

Value

Returns TOST Z-value 1, TOST p-value 1, TOST Z-value 2, TOST p-value 2, alpha, low equivalence bound d, high equivalence bound d, Lower limit confidence interval TOST, Upper limit confidence interval TOST

References

Rogers, J. L., Howard, K. I., & Vessey, J. T. (1993). Using significance tests to evaluate equivalence between two experimental groups. Psychological Bulletin, 113(3), 553, formula page 557.

Examples

## Run TOSTmeta by specifying the standard error
TOSTmeta(ES=0.12, se=0.09, low_eqbound_d=-0.2, high_eqbound_d=0.2, alpha=0.05)
## Run TOSTmeta by specifying the variance
TOSTmeta(ES=0.12, var=0.0081, low_eqbound_d=-0.2, high_eqbound_d=0.2, alpha=0.05)
## If both variance and se are specified, TOSTmeta will use standard error and ignore variance
TOSTmeta(ES=0.12, var=9999, se = 0.09, low_eqbound_d=-0.2, high_eqbound_d=0.2, alpha=0.05)

---

Methods for TOSTnp objects

Description

Methods defined for objects returned from the wilcox_TOST function.

Usage

## S3 method for class 'TOSTnp'
print(x, digits = 4, ...)

## S3 method for class 'TOSTnp'
describe(x, digits = 3, ...)

Arguments

x	object of class TOSTnp.
digits	Number of digits to print for p-values
...	further arguments passed through, see description of return value for details. TOSTnp-methods.

Value

• print: Prints short summary of the tests.

• describe: Verbose description of results.

Examples

# example code
data(mtcars)
res1 = wilcox_TOST(mpg ~ am,data = mtcars,eqb = 3)

# PRINT
print(res1)

# DESCRIPTION
describe(res1)

---

TOST function for a one-sample t-test (Cohen's d)

Description

Development on this function is complete, and for new code we recommend switching to tsum_TOST, which is easier to use, more featureful, and still under active development.

Usage

TOSTone(
  m,
  mu,
  sd,
  n,
  low_eqbound_d,
  high_eqbound_d,
  alpha,
  plot = TRUE,
  verbose = TRUE
)

TOSTone.raw(
  m,
  mu,
  sd,
  n,
  low_eqbound,
  high_eqbound,
  alpha,
  plot = TRUE,
  verbose = TRUE
)

Arguments

m	mean
mu	value to compare against
sd	standard deviation
n	sample size
low_eqbound_d	lower equivalence bounds (e.g., -0.5) expressed in standardized mean difference (Cohen's d)
high_eqbound_d	upper equivalence bounds (e.g., 0.5) expressed in standardized mean difference (Cohen's d)
alpha	alpha level (default = 0.05)
plot	set whether results should be plotted (plot = TRUE) or not (plot = FALSE) - defaults to TRUE
verbose	logical variable indicating whether text output should be generated (verbose = TRUE) or not (verbose = FALSE) - default to TRUE
low_eqbound	lower equivalence bounds (e.g., -0.5) expressed in raw units
high_eqbound	upper equivalence bounds (e.g., 0.5) expressed in raw units

Value

Returns TOST t-value 1, TOST p-value 1, TOST t-value 2, TOST p-value 2, degrees of freedom, low equivalence bound, high equivalence bound, Lower limit confidence interval TOST, Upper limit confidence interval TOST

Examples

## Test observed mean of 0.54 and standard deviation of 1.2 in sample of 100 participants
## against 0.5 given equivalence bounds of Cohen's d = -0.3 and 0.3, with an alpha = 0.05.
TOSTone(m=0.54,mu=0.5,sd=1.2,n=100,low_eqbound_d=-0.3, high_eqbound_d=0.3, alpha=0.05)

---

TOST function for a dependent t-test (Cohen's dz)

Description

Development on this function is complete, and for new code we recommend switching to tsum_TOST, which is easier to use, more featureful, and still under active development.

Usage

TOSTpaired(
  n,
  m1,
  m2,
  sd1,
  sd2,
  r12,
  low_eqbound_dz,
  high_eqbound_dz,
  alpha,
  plot = TRUE,
  verbose = TRUE
)

TOSTpaired.raw(
  n,
  m1,
  m2,
  sd1,
  sd2,
  r12,
  low_eqbound,
  high_eqbound,
  alpha,
  plot = TRUE,
  verbose = TRUE
)

Arguments

n	sample size (pairs)
m1	mean of group 1
m2	mean of group 2
sd1	standard deviation of group 1
sd2	standard deviation of group 2
r12	correlation of dependent variable between group 1 and group 2
low_eqbound_dz	lower equivalence bounds (e.g., -0.5) expressed in standardized mean difference (Cohen's dz)
high_eqbound_dz	upper equivalence bounds (e.g., 0.5) expressed in standardized mean difference (Cohen's dz)
alpha	alpha level (default = 0.05)
plot	set whether results should be plotted (plot = TRUE) or not (plot = FALSE) - defaults to TRUE
verbose	logical variable indicating whether text output should be generated (verbose = TRUE) or not (verbose = FALSE) - default to TRUE
low_eqbound	lower equivalence bounds (e.g., -0.5) expressed in raw scores
high_eqbound	upper equivalence bounds (e.g., 0.5) expressed in raw scores

Value

Returns TOST t-value 1, TOST p-value 1, TOST t-value 2, TOST p-value 2, degrees of freedom, low equivalence bound, high equivalence bound, low equivalence bound in dz, high equivalence bound in dz, Lower limit confidence interval TOST, Upper limit confidence interval TOST

References

Mara, C. A., & Cribbie, R. A. (2012). Paired-Samples Tests of Equivalence. Communications in Statistics - Simulation and Computation, 41(10), 1928-1943. https://doi.org/10.1080/03610918.2011.626545, formula page 1932. Note there is a typo in the formula: n-1 should be n (personal communication, 31-8-2016)

Examples

## Test means of 5.83 and 5.75, standard deviations of 1.17 and 1.29 in sample of 65 pairs
## with correlation between observations of 0.75 using equivalence bounds in Cohen's dz of
## -0.4 and 0.4 (with default alpha setting of = 0.05).
TOSTpaired(n=65,m1=5.83,m2=5.75,sd1=1.17,sd2=1.29,r12=0.75,low_eqbound_dz=-0.4,high_eqbound_dz=0.4)

---

TOST function for a correlations

Description

Development on TOSTr is complete, and for new code we recommend switching to corsum_test, which is easier to use, more featureful, and still under active development.

Usage

TOSTr(n, r, low_eqbound_r, high_eqbound_r, alpha, plot = TRUE, verbose = TRUE)

Arguments

n	number of pairs of observations
r	observed correlation
low_eqbound_r	lower equivalence bounds (e.g., -0.3) expressed in a correlation effect size
high_eqbound_r	upper equivalence bounds (e.g., 0.3) expressed in a correlation effect size
alpha	alpha level (default = 0.05)
plot	set whether results should be plotted (plot = TRUE) or not (plot = FALSE) - defaults to TRUE
verbose	logical variable indicating whether text output should be generated (verbose = TRUE) or not (verbose = FALSE) - default to TRUE

Value

Returns TOST p-value 1, TOST p-value 2, alpha, low equivalence bound r, high equivalence bound r, Lower limit confidence interval TOST, Upper limit confidence interval TOST

References

Goertzen, J. R., & Cribbie, R. A. (2010). Detecting a lack of association: An equivalence testing approach. British Journal of Mathematical and Statistical Psychology, 63(3), 527-537. https://doi.org/10.1348/000711009X475853, formula page 531.

Examples

TOSTr(n=100, r = 0.02, low_eqbound_r=-0.3, high_eqbound_r=0.3, alpha=0.05)

---

Methods for TOSTt objects

Description

Methods defined for objects returned from the t_TOST and boot_t_TOST functions.

Usage

## S3 method for class 'TOSTt'
print(x, digits = 4, ...)

## S3 method for class 'TOSTt'
plot(
  x,
  type = c("cd", "c", "tnull"),
  estimates = c("raw", "SMD"),
  ci_lines,
  ci_shades,
  ...
)

describe(x, ...)

## S3 method for class 'TOSTt'
describe(x, digits = 3, ...)

Arguments

x	object of class TOSTt.
digits	Number of digits to print for p-values
...	further arguments passed through, see description of return value for details..
type	Type of plot to produce. Default is a consonance density plot "cd". Consonance plots (type = "cd") and null distribution plots (type = "tnull") can also be produced. Note: null distribution plots only available for estimates = "raw".
estimates	indicator of what estimates to plot; options include "raw" or "SMD". Default is is both: c("raw","SMD").
ci_lines	Confidence interval lines for plots. Default is 1-alpha*2 (e.g., alpha = 0.05 is 90%)
ci_shades	Confidence interval shades when plot type is "cd".

Value

• print: Prints short summary of the tests.

• plot: Returns a plot of the effects.

• describe: Verbose description of results.

Examples

# example code
# Print

res1 = t_TOST(mpg ~ am, data = mtcars, eqb = 3)

res1
# Print with more digits
print(res1, digits = 6)

# Plot with density plot - only raw values (SLOW)
#plot(res1, type = "cd", estimates = "raw")
# Plot with consonance - only raw values (SLOW)
#plot(res1, type = "c", estimates = "raw")
# Plot null distribution - only raw values
#plot(res1, type = "tnull", estimates = "raw")

# Get description of the results
describe(res1)

---

TOST function for an independent t-test (Cohen's d)

Description

Development on TOSTtwo is complete, and for new code we recommend switching to tsum_TOST, which is easier to use, more featureful, and still under active development.

Usage

TOSTtwo(
  m1,
  m2,
  sd1,
  sd2,
  n1,
  n2,
  low_eqbound_d,
  high_eqbound_d,
  alpha,
  var.equal,
  plot = TRUE,
  verbose = TRUE
)

TOSTtwo.raw(
  m1,
  m2,
  sd1,
  sd2,
  n1,
  n2,
  low_eqbound,
  high_eqbound,
  alpha,
  var.equal,
  plot = TRUE,
  verbose = TRUE
)

Arguments

m1	mean of group 1
m2	mean of group 2
sd1	standard deviation of group 1
sd2	standard deviation of group 2
n1	sample size in group 1
n2	sample size in group 2
low_eqbound_d	lower equivalence bounds (e.g., -0.5) expressed in standardized mean difference (Cohen's d)
high_eqbound_d	upper equivalence bounds (e.g., 0.5) expressed in standardized mean difference (Cohen's d)
alpha	alpha level (default = 0.05)
var.equal	logical variable indicating whether equal variances assumption is assumed to be TRUE or FALSE. Defaults to FALSE.
plot	set whether results should be plotted (plot = TRUE) or not (plot = FALSE) - defaults to TRUE
verbose	logical variable indicating whether text output should be generated (verbose = TRUE) or not (verbose = FALSE) - default to TRUE
low_eqbound	lower equivalence bounds (e.g., -0.5) expressed in raw scale units (e.g., scalepoints)
high_eqbound	upper equivalence bounds (e.g., 0.5) expressed in raw scale units (e.g., scalepoints)

Value

Returns TOST t-value 1, TOST p-value 1, TOST t-value 2, TOST p-value 2, degrees of freedom, low equivalence bound, high equivalence bound, low equivalence bound in Cohen's d, high equivalence bound in Cohen's d, Lower limit confidence interval TOST, Upper limit confidence interval TOST

References

Berger, R. L., & Hsu, J. C. (1996). Bioequivalence Trials, Intersection-Union Tests and Equivalence Confidence Sets. Statistical Science, 11(4), 283-302.

Gruman, J. A., Cribbie, R. A., & Arpin-Cribbie, C. A. (2007). The effects of heteroscedasticity on tests of equivalence. Journal of Modern Applied Statistical Methods, 6(1), 133-140, formula for Welch's t-test on page 135

Examples

## Eskine (2013) showed that participants who had been exposed to organic
## food were substantially harsher in their moral judgments relative to
## those exposed to control (d = 0.81, 95% CI: [0.19, 1.45]). A
## replication by Moery & Calin-Jageman (2016, Study 2) did not observe
## a significant effect (Control: n = 95, M = 5.25, SD = 0.95, Organic
## Food: n = 89, M = 5.22, SD = 0.83). Following Simonsohn's (2015)
## recommendation the equivalence bound was set to the effect size the
## original study had 33% power to detect (with n = 21 in each condition,
## this means the equivalence bound is d = 0.48, which equals a
## difference of 0.384 on a 7-point scale given the sample sizes and a
## pooled standard deviation of 0.894). Using a TOST equivalence test
## with default alpha = 0.05, not assuming equal variances, and equivalence
## bounds of d = -0.43 and d = 0.43 is significant, t(182) = -2.69,
## p = 0.004. We can reject effects larger than d = 0.43.

TOSTtwo(m1=5.25,m2=5.22,sd1=0.95,sd2=0.83,n1=95,n2=89,low_eqbound_d=-0.43,high_eqbound_d=0.43)

---

TOST function for two proportions (raw scores)

Description

Development on TOSTtwo.prop is complete, and for new code we recommend switching to twoprop_test, which is easier to use, more featureful, and still under active development.

Usage

TOSTtwo.prop(
  prop1,
  prop2,
  n1,
  n2,
  low_eqbound,
  high_eqbound,
  alpha,
  ci_type = "normal",
  plot = TRUE,
  verbose = TRUE
)

Arguments

prop1	proportion of group 1
prop2	proportion of group 2
n1	sample size in group 1
n2	sample size in group 2
low_eqbound	lower equivalence bounds (e.g., -0.1) expressed in proportions
high_eqbound	upper equivalence bounds (e.g., 0.1) expressed in proportions
alpha	alpha level (default = 0.05)
ci_type	confidence interval type (default = "normal"). "wilson" produces Wilson score intervals with a Yates continuity correction while "normal" calculates the simple asymptotic method with no continuity correction.
plot	set whether results should be plotted (plot = TRUE) or not (plot = FALSE) - defaults to TRUE
verbose	logical variable indicating whether text output should be generated (verbose = TRUE) or not (verbose = FALSE) - default to TRUE

Value

Returns TOST z-value 1, TOST p-value 1, TOST z-value 2, TOST p-value 2, low equivalence bound, high equivalence bound, Lower limit confidence interval TOST, Upper limit confidence interval TOST

References

Tunes da Silva, G., Logan, B. R., & Klein, J. P. (2008). Methods for Equivalence and Noninferiority Testing. Biology of Blood Marrow Transplant, 15(1 Suppl), 120-127.

Yin, G. (2012). Clinical Trial Design: Bayesian and Frequentist Adaptive Methods. Hoboken, New Jersey: John Wiley & Sons, Inc.

Examples

## Equivalence test for two independent proportions equal to .65 and .70, with 100 samples
## per group, lower equivalence bound of -0.1, higher equivalence bound of 0.1, and alpha of 0.05.

TOSTtwo.prop(prop1 = .65, prop2 = .70, n1 = 100, n2 = 100,
   low_eqbound = -0.1, high_eqbound = 0.1, alpha = .05)

---

TOST with t-tests from Summary Statistics

Description

A function for TOST with all types of t-tests from summary statistics.

Usage

tsum_TOST(
  m1,
  sd1,
  n1,
  m2 = NULL,
  sd2 = NULL,
  n2 = NULL,
  r12 = NULL,
  hypothesis = c("EQU", "MET"),
  paired = FALSE,
  var.equal = FALSE,
  eqb,
  low_eqbound,
  high_eqbound,
  mu = 0,
  eqbound_type = c("raw", "SMD"),
  alpha = 0.05,
  bias_correction = TRUE,
  rm_correction = FALSE,
  glass = NULL,
  smd_ci = c("nct", "goulet", "t", "z")
)

Arguments

m1	mean of group 1.
sd1	standard deviation of group 1.
n1	sample size in group 1.
m2	mean of group 2.
sd2	standard deviation of group 2.
n2	sample size in group 2.
r12	correlation of dependent variable between group 1 and group 2.
hypothesis	'EQU' for equivalence (default), or 'MET' for minimal effects test, the alternative hypothesis.
paired	a logical indicating whether you want a paired t-test.
var.equal	a logical variable indicating whether to treat the two variances as being equal. If TRUE then the pooled variance is used to estimate the variance otherwise the Welch (or Satterthwaite) approximation to the degrees of freedom is used.
eqb	Equivalence bound. Can provide 1 value (negative value is taken as the lower bound) or 2 specific values that represent the upper and lower equivalence bounds.
low_eqbound	lower equivalence bounds (deprecated).
high_eqbound	upper equivalence bounds (deprecated).
mu	a number indicating the true value of the mean for the two tailed test (or difference in means if you are performing a two sample test).
eqbound_type	Type of equivalence bound. Can be set to "SMD" for standardized mean difference (i.e., Cohen's d) or "raw" for the mean difference. Default is "raw". Raw is strongly recommended as SMD bounds will produce biased results.
alpha	alpha level (default = 0.05)
bias_correction	Apply Hedges' correction for bias (default is TRUE).
rm_correction	Repeated measures correction to make standardized mean difference Cohen's d(rm). This only applies to repeated/paired samples. Default is FALSE.
glass	A option to calculate Glass's delta as an alternative to Cohen's d type SMD. Default is NULL to not calculate Glass's delta, "glass1" will use the first group's SD as the denominator whereas "glass2" will use the 2nd group's SD.
smd_ci	Method for calculating SMD confidence intervals. Methods include Goulet, noncentral t (nct), central t (t), and normal method (z).

Details

For details on the calculations in this function see vignette("IntroTOSTt") & vignette("SMD_calcs").

For two-sample tests, the test is of $m1 - m2$ (mean of 1 minus mean of 2). For paired samples, the test is of the difference scores (z), wherein $z = m1 - m2$, and the test is of $\bar z$ (mean of the difference scores). For one-sample tests, the test is of $\bar m1$ (mean of group 1).

Value

An S3 object of class "TOSTt" is returned containing the following slots:

• "TOST": A table of class "data.frame" containing two-tailed t-test and both one-tailed results.

• "eqb": A table of class "data.frame" containing equivalence bound settings.

• "effsize": table of class "data.frame" containing effect size estimates.

• "hypothesis": String stating the hypothesis being tested.

• "smd": List containing the results of the standardized mean difference calculations (e.g., Cohen's d).

  • Items include: d (estimate), dlow (lower CI bound), dhigh (upper CI bound), d_df (degrees of freedom for SMD), d_sigma (SE), d_lambda (non-centrality), J (bias correction), smd_label (type of SMD), d_denom (denominator calculation)

• "alpha": Alpha level set for the analysis.

• "method": Type of t-test.

• "decision": List included text regarding the decisions for statistical inference.

See Also

Other TOST: boot_log_TOST(), boot_t_TOST(), simple_htest(), t_TOST(), wilcox_TOST()

Examples

# example code
# One sample test
tsum_TOST(m1 = 0.55, n1 = 18, sd1 = 4, eqb  = 2)

---

Test of Proportions between 2 Independent Groups

Description

This is a hypothesis testing function that mimics prop.test, but focuses only on testing differences in proportions between two groups. This function utilizes a z-test to calculate the p-values (may be inaccurate with small sample sizes).

Usage

twoprop_test(
  p1,
  p2,
  n1,
  n2,
  null = NULL,
  alpha = 0.05,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  effect_size = c("difference", "odds.ratio", "risk.ratio")
)

Arguments

p1, p2	Proportions in each respective group.
n1, n2	sample size in each respective group.
null	a number indicating the null hypothesis of the difference in proportions between two groups.
alpha	alpha level (default = 0.05)
alternative	a character string specifying the alternative hypothesis, must be one of "two.sided" (default), "greater", "less", "equivalence" (TOST), or "minimal.effect" (TOST). You can specify just the initial letter.
effect_size	the effect size estimate, and confidence intervals, to calculate. Options include the difference between both proportions ("difference"), odds ratio ("odds.ratio"), or risk ratio ("risk.ratio").

Details

The hypothesis test for differences in proportions can be made on the raw proportions scale, the odds ratio, or the risk ratio (details below). This function uses the large sample size asymptotic approximations for both the p-value and confidence interval calculations. There should be a good deal of caution when sample sizes are small. The p-values for the differences in proportions will differ from base prop.test due to the use of the unpooled standard error (see below).

Differences in Proportions

Differences in proportions test is based on the following calculation:

$d = p_1 - p_2$

The standard error of $d$ is calculated as the following:

$se(d) = \sqrt{\frac{p_1 \cdot (1-p_1)}{n_1} + \frac{p_2 \cdot (1-p_2)}{n_2}}$

The z-test, with $d_0$ being the null value, is then calculated as the following (standard normal distribution evaluated to calculate p-value):

$z = \frac{d - d_0}{se(d)}$

The confidence interval can then be calculated as the following:

$d_{lower},d_{upper} = d \pm z_{\alpha} \cdot se(d)$

Risk Ratio

The ratio between proportions test is based on the following calculation:

$\phi = p_1/p_2$

The standard error of $ln(\phi)$ is calculated as the following:

$se(ln(\phi)) = \sqrt{\frac{1-p_1}{n_1 \cdot p_1} + \frac{1-p_2}{n_2 \cdot p_2}}$

The z-test, with $\phi_0$ being the null value, is then calculated as the following (standard normal distribution evaluated to calculate p-value):

$z = \frac{ln(\phi) - ln(\phi_0)}{se(ln(\phi))}$

The confidence interval can then be calculated as the following:

$\phi_{lower} = \phi \cdot e^{-z_{\alpha} \cdot se(ln(\phi))}$

$\phi_{upper} = \phi \cdot e^{z_{\alpha} \cdot se(ln(\phi))}$

Odds Ratio

The ratio between proportions test is based on the following calculation: (p1/q1) / (p2/q2)

$OR = \frac{p_1}{1-p_1} / \frac{p_2}{1-p_2}$

The standard error of $ln(OR)$ is calculated as the following:

$se(ln(OR)) = \sqrt{\frac{1}{n_1 \cdot p_1 + 0.5} + \frac{1}{n_1 \cdot (1-p_1) + 0.5} + \frac{1}{n_2 \cdot p_2 + 0.5} + \frac{1}{n_2 \cdot (1-p_2) + 0.5} }$

The z-test, with $OR_0$ being the null value, is then calculated as the following (standard normal distribution evaluated to calculate p-value):

$z = \frac{ln(OR) - ln(OR_0)}{se(ln(OR))}$

The confidence interval can then be calculated as the following:

$OR_{lower},OR_{upper} = exp(ln(OR) \pm z_{\alpha} \cdot se(ln(OR)))$

Value

An S3 object of the class htest.

References

Gart, J. J., & Nam, J. M. (1988). Approximate interval estimation of the ratio of binomial parameters: a review and corrections for skewness. Biometrics, 323-338.

Tunes da Silva, G., Logan, B. R., & Klein, J. P. (2008). Methods for Equivalence and Noninferiority Testing. Biology of Blood Marrow Transplant, 15(1 Suppl), 120-127.

Yin, G. (2012). Clinical Trial Design: Bayesian and Frequentist Adaptive Methods. Hoboken, New Jersey: John Wiley & Sons, Inc.

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