TOSTER: Two One-Sided Tests (TOST) Equivalence Testing

Description

Two one-sided tests (TOST) procedure to test equivalence for t-tests, correlations, differences between proportions, and meta-analyses, including power analysis for t-tests and correlations. Allows you to specify equivalence bounds in raw scale units or in terms of effect sizes. See: Lakens (2017) doi:10.1177/1948550617697177.

Author(s)

Maintainer: Aaron Caldwell arcaldwell49@gmail.com

Authors:

• Daniel Lakens D.Lakens@tue.nl

See Also

Useful links:

• https://aaroncaldwell.us/TOSTERpkg/

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

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

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

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

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

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("simple", "cd", "c", "tnull"),
  estimates = c("raw", "SMD"),
  ci_lines,
  ci_shades,
  ...
)

describe(x, ...)

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

Arguments

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

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

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)

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

Value

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

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

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

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

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

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

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

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 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

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()

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

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)

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

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

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

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

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

Value

A results object containing:

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

Value

A results object containing:

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

library("TOSTER")

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

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

Value

A results object containing:

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

Value

A results object containing:

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

Value

A results object containing:

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

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

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

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

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

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

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

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

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()

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

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

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

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)

F-test Power

Description

Power analysis for TOST for an F-test

Usage

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

Arguments

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

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

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

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)

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

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

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

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

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

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 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

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

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.

TOST with Wilcoxon-Mann-Whitney tests

Description

A function for TOST using the non-parametric methods of the Wilcoxon-Mann-Whitney family of tests. This function uses the normal approximation and applies a continuity correction automatically.

Usage

wilcox_TOST(
  x,
  ...,
  hypothesis = "EQU",
  paired = FALSE,
  eqb,
  low_eqbound,
  high_eqbound,
  ses = "rb",
  alpha = 0.05
)

## Default S3 method:
wilcox_TOST(
  x,
  y = NULL,
  hypothesis = "EQU",
  paired = FALSE,
  eqb,
  low_eqbound,
  high_eqbound,
  ses = c("rb", "odds", "cstat"),
  alpha = 0.05,
  mu = 0,
  ...
)

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

Arguments

Details

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

If only x is given, or if both x and y are given and paired is TRUE, a Wilcoxon signed rank test of the null that the distribution of x (in the one sample case) or of x - y (in the paired two sample case) is symmetric about mu is performed.

Otherwise, if both x and y are given and paired is FALSE, a Wilcoxon rank sum test (equivalent to the Mann-Whitney test: see the Note) is carried out. In this case, the null hypothesis is that the distributions of x and y differ by a location shift.

Value

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

• "TOST": A table of class "data.frame" containing two-tailed wilcoxon signed rank 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 information on standardized effect size.

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

• "method": Type of non-parametric test.

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

References

David F. Bauer (1972). Constructing confidence sets using rank statistics. Journal of the American Statistical Association 67, 687–690. doi: 10.1080/01621459.1972.10481279.

Myles Hollander and Douglas A. Wolfe (1973). Nonparametric Statistical Methods. New York: John Wiley & Sons. Pages 27–33 (one-sample), 68–75 (two-sample). Or second edition (1999).

See Also

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

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

Examples

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

Test for Association/Correlation Between Paired Samples

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). Unlike the stats version of cor.test, this function allows users to set the null to a value other than zero.

Usage

z_cor_test(
  x,
  y,
  alternative = c("two.sided", "less", "greater", "equivalence", "minimal.effect"),
  method = c("pearson", "kendall", "spearman"),
  alpha = 0.05,
  null = 0
)

Arguments

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. See vignette("correlations") for more details.

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

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(), corsum_test(), plot_cor(), power_z_cor()

Examples

# example code
x <- c(44.4, 45.9, 41.9, 53.3, 44.7, 44.1, 50.7, 45.2, 60.1)
y <- c( 2.6,  3.1,  2.5,  5.0,  3.6,  4.0,  5.2,  2.8,  3.8)
# Sig test
z_cor_test(x, y, method = "kendall", alternative = "t", null = 0)
# MET test
z_cor_test(x, y, method = "kendall", alternative = "min", null = .2)