# Test and effect size details

#### Indrajeet Patil

#### 2021-04-13

This vignette provides a go-to summary for which test is carried out for each function included in the package and what effect size it returns. Additionally, there are also recommendations on how to interpret those effect sizes.

# Summary of tests and effect sizes

## `two_sample_test` + `oneway_anova`

No. of groups: `2` => `two_sample_test` No. of groups: `> 2` => `oneway_anova`

### between-subjects

Following (between-subjects) tests are carried out for each type of analyses-

Type | No. of groups | Test | Function used
--- | --- | --- | ---
Parametric | > 2 | Fisher’s or Welch’s one-way ANOVA | `stats::oneway.test`
Non-parametric | > 2 | Kruskal–Wallis one-way ANOVA | `stats::kruskal.test`
Robust | > 2 | Heteroscedastic one-way ANOVA for trimmed means | `WRS2::t1way`
Bayes Factor | > 2 | Fisher’s ANOVA | `BayesFactor::anovaBF`
Parametric | 2 | Student’s or Welch’s t-test | `stats::t.test`
Non-parametric | 2 | Mann–Whitney U test | `stats::wilcox.test`
Robust | 2 | Yuen’s test for trimmed means | `WRS2::yuen`
Bayesian | 2 | Student’s t-test | `BayesFactor::ttestBF`

Following effect sizes (and confidence intervals/CI) are available for each type of test-

Type | No. of groups | Effect size | CI? | Function used
--- | --- | --- | --- | ---
Parametric | > 2 | \(\eta_{p}^2\), \(\omega_{p}^2\) | Yes | `effectsize::omega_squared`, `effectsize::eta_squared`
Non-parametric | > 2 | \(\epsilon_{ordinal}^2\) | Yes | `effectsize::rank_epsilon_squared`
Robust | > 2 | \(\xi\) (Explanatory measure of effect size) | Yes | `WRS2::t1way`
Bayes Factor | > 2 | \(R_{posterior}^2\) | Yes | `performance::r2_bayes`
Parametric | 2 | Cohen’s d, Hedge’s g | Yes | `effectsize::cohens_d`, `effectsize::hedges_g`
Non-parametric | 2 | r (rank-biserial correlation) | Yes | `effectsize::rank_biserial`
Robust | 2 | \(\xi\) (Explanatory measure of effect size) | Yes | `WRS2::yuen.effect.ci`
Bayesian | 2 | \(\delta_{posterior}\) | Yes | `bayestestR::describe_posterior`

### within-subjects

Following (within-subjects) tests are carried out for each type of analyses-

Type | No. of groups | Test | Function used
--- | --- | --- | ---
Parametric | > 2 | One-way repeated measures ANOVA | `afex::aov_ez`
Non-parametric | > 2 | Friedman rank sum test | `stats::friedman.test`
Robust | > 2 | Heteroscedastic one-way repeated measures ANOVA for trimmed means | `WRS2::rmanova`
Bayes Factor | > 2 | One-way repeated measures ANOVA | `BayesFactor::anovaBF`
Parametric | 2 | Student’s t-test | `stats::t.test`
Non-parametric | 2 | Wilcoxon signed-rank test | `stats::wilcox.test`
Robust | 2 | Yuen’s test on trimmed means for dependent samples | `WRS2::yuend`
Bayesian | 2 | Student’s t-test | `BayesFactor::ttestBF`

Following effect sizes (and confidence intervals/CI) are available for each type of test-

Type | No. of groups | Effect size | CI? | Function used
--- | --- | --- | --- | ---
Parametric | > 2 | \(\eta_{p}^2\), \(\omega_{p}^2\) | Yes | `effectsize::omega_squared`, `effectsize::eta_squared`
Non-parametric | > 2 | \(W_{Kendall}\) (Kendall’s coefficient of concordance) | Yes | `effectsize::kendalls_w`
Robust | > 2 | \(\delta_{R-avg}^{AKP}\) Yes | Algina-Keselman-Penfield robust standardized difference average | `WRS2::wmcpAKP`
Bayes Factor | > 2 | \(R_{posterior}^2\) | Yes | `performance::r2_bayes`
Parametric | 2 | Cohen’s d, Hedge’s g | Yes | `effectsize::cohens_d`, `effectsize::hedges_g`
Non-parametric | 2 | r (rank-biserial correlation) | Yes | `effectsize::rank_biserial`
Robust | 2 | \(\delta_{R}^{AKP}\) (Algina-Keselman-Penfield robust standardized difference) | Yes | `WRS2::dep.effect`
Bayesian | 2 | \(\delta_{posterior}\) | Yes | `bayestestR::describe_posterior`

## `one_sample_test`

Following tests are carried out for each type of analyses-

Type | Test | Function used
--- | --- | ---
Parametric | One-sample Student’s t-test | `stats::t.test`
Non-parametric | One-sample Wilcoxon test | `stats::wilcox.test`
Robust | Bootstrap-t method for one-sample test | `trimcibt` (custom)
Bayesian | One-sample Student’s t-test | `BayesFactor::ttestBF`

Following effect sizes (and confidence intervals/CI) are available for each type of test-

Type | Effect size | CI? | Function used
--- | --- | --- | ---
Parametric | Cohen’s d, Hedge’s g | Yes | `effectsize::cohens_d`, `effectsize::hedges_g`
Non-parametric | r (rank-biserial correlation) | Yes | `effectsize::rank_biserial`
Robust | trimmed mean | Yes | `trimcibt` (custom)
Bayes Factor | \(\delta_{posterior}\) | Yes | `bayestestR::describe_posterior`

## `corr_test`

Following tests are carried out for each type of analyses. Additionally, the correlation coefficients (and their confidence intervals) are used as effect sizes-

Type | Test | CI? | Function used
--- | --- | --- | ---
Parametric | Pearson’s correlation coefficient | Yes | `correlation::correlation`
Non-parametric | Spearman’s rank correlation coefficient | Yes | `correlation::correlation`
Robust | Winsorized Pearson correlation coefficient | Yes | `correlation::correlation`
Bayesian | Pearson’s correlation coefficient | Yes | `correlation::correlation`

## `contingency_table`

Following tests are carried out for each type of analyses-

Type of data | Design | Test | Function used
--- | --- | --- | ---
Unpaired | \(n \times p\) contingency table | Pearson’s \(\chi^2\) test | `stats::chisq.test`
Paired | \(n \times p\) contingency table | McNemar’s \(\chi^2\) test | `stats::mcnemar.test`
Frequency | \(n \times 1\) contingency table | Goodness of fit (\(\chi^2\) test) | `stats::chisq.test`

Following effect sizes (and confidence intervals/CI) are available for each type of test-

Test | Effect size | CI? | Function used
--- | --- | --- | ---
Pearson’s \(\chi^2\) test | Cramer’s \(V\) | Yes | `effectsize::cramers_v`
McNemar’s test | Cohen’s \(g\) | Yes | `effectsize::cohens_g`
Goodness of fit | Cramer’s \(V\) | Yes | `effectsize::cramers_v`

## `meta_analysis`

Type | Test | Effect size | 95% CI available? | Function used
--- | --- | --- | --- | ---
Parametric | Meta-analysis via random-effects models | \(\beta\) | Yes | `metafor::metafor`
Robust | Meta-analysis via robust random-effects models | \(\beta\) | Yes | `metaplus::metaplus`
Bayes | Meta-analysis via Bayesian random-effects models | \(\beta\) | Yes | `metaBMA::meta_random`

# Effect size interpretation

See `effectsize`’s interpretation functions to check different rules/conventions to interpret effect sizes:

https://easystats.github.io/effectsize/reference/index.html#section-interpretation

# Dataframe as output

Although the primary focus of this package is to get expressions containing statistical results, one can also use it to extract dataframes containing these details.

For a more detailed summary of these dataframe: https://indrajeetpatil.github.io/statsExpressions//articles/web_only/dataframe_outputs.html

# References

- For parametric and non-parametric effect sizes: https://easystats.github.io/effectsize/articles/simple_htests.html

- For robust effect sizes: https://CRAN.R-project.org/package=WRS2/vignettes/WRS2.pdf

- For Bayesian posterior estimates: https://easystats.github.io/bayestestR/articles/bayes_factors.html

# Suggestions

If you find any bugs or have any suggestions/remarks, please file an issue on GitHub: https://github.com/IndrajeetPatil/ggstatsplot/issues

# Session Information

For details, see- https://indrajeetpatil.github.io/ggstatsplot/articles/web_only/session_info.html