This function performs a \(\chi^2\) test for contingency
tables or tests for given probabilities. The returned effects sizes are
Cramer's V for tables with more than two rows or columns, Phi (\(\phi\))
for 2x2 tables, and Fei (פ) for tests against
given probabilities (see *Ben-Shachar et al. 2023*).

```
chi_squared_test(
data,
select = NULL,
by = NULL,
probabilities = NULL,
weights = NULL,
paired = FALSE,
...
)
```

- data
A data frame.

- select
Name(s) of the continuous variable(s) (as character vector) to be used as samples for the test.

`select`

can be one of the following:`select`

can be used in combination with`by`

, in which case`select`

is the name of the continous variable (and`by`

indicates a grouping factor).`select`

can also be a character vector of length two or more (more than two names only apply to`kruskal_wallis_test()`

), in which case the two continuous variables are treated as samples to be compared.`by`

must be`NULL`

in this case.If

`select`

select is of length**two**and`paired = TRUE`

, the two samples are considered as*dependent*and a paired test is carried out.If

`select`

specifies**one**variable and`by = NULL`

, a one-sample test is carried out (only applicable for`t_test()`

and`wilcoxon_test()`

)For

`chi_squared_test()`

, if`select`

specifies**one**variable and both`by`

and`probabilities`

are`NULL`

, a one-sample test against given probabilities is automatically conducted, with equal probabilities for each level of`select`

.

- by
Name of the variable indicating the groups. Required if

`select`

specifies only one variable that contains all samples to be compared in the test. If`by`

is not a factor, it will be coerced to a factor. For`chi_squared_test()`

, if`probabilities`

is provided,`by`

must be`NULL`

.- probabilities
A numeric vector of probabilities for each cell in the contingency table. The length of the vector must match the number of cells in the table, i.e. the number of unique levels of the variable specified in

`select`

. If`probabilities`

is provided, a chi-squared test for given probabilities is conducted. Furthermore, if`probabilities`

is given,`by`

must be`NULL`

. The probabilities must sum to 1.- weights
Name of an (optional) weighting variable to be used for the test.

- paired
Logical, if

`TRUE`

, a McNemar test is conducted for 2x2 tables. Note that`paired`

only works for 2x2 tables.- ...
Additional arguments passed down to

`chisq.test()`

.

A data frame with test results. The returned effects sizes are Cramer's V for tables with more than two rows or columns, Phi (\(\phi\)) for 2x2 tables, and Fei (פ) for tests against given probabilities.

The function is a wrapper around `chisq.test()`

and
`fisher.test()`

(for small expected values) for contingency tables, and
`chisq.test()`

for given probabilities. When `probabilities`

are provided,
these are rescaled to sum to 1 (i.e. `rescale.p = TRUE`

). When `fisher.test()`

is called, simulated p-values are returned (i.e. `simulate.p.value = TRUE`

,
see `?fisher.test`

). If `paired = TRUE`

and a 2x2 table is provided,
a McNemar test (see `mcnemar.test()`

) is conducted.

The weighted version of the chi-squared test is based on the a weighted
table, using `xtabs()`

as input for `chisq.test()`

.

Interpretation of effect sizes are based on rules described in
`effectsize::interpret_phi()`

, `effectsize::interpret_cramers_v()`

,
and `effectsize::interpret_fei()`

. Use these function directly to get other
interpretations, by providing the returned effect size as argument, e.g.
`interpret_phi(0.35, rules = "gignac2016")`

.

The following table provides an overview of which test to use for different types of data. The choice of test depends on the scale of the outcome variable and the number of samples to compare.

Samples | Scale of Outcome | Significance Test |

1 | binary / nominal | `chi_squared_test()` |

1 | continuous, not normal | `wilcoxon_test()` |

1 | continuous, normal | `t_test()` |

2, independent | binary / nominal | `chi_squared_test()` |

2, independent | continuous, not normal | `mann_whitney_test()` |

2, independent | continuous, normal | `t_test()` |

2, dependent | binary (only 2x2) | `chi_squared_test(paired=TRUE)` |

2, dependent | continuous, not normal | `wilcoxon_test()` |

2, dependent | continuous, normal | `t_test(paired=TRUE)` |

>2, independent | continuous, not normal | `kruskal_wallis_test()` |

>2, independent | continuous, normal | `datawizard::means_by_group()` |

>2, dependent | continuous, not normal | not yet implemented (1) |

>2, dependent | continuous, normal | not yet implemented (2) |

(1) More than two dependent samples are considered as *repeated measurements*.
For ordinal or not-normally distributed outcomes, these samples are
usually tested using a `friedman.test()`

, which requires the samples
in one variable, the groups to compare in another variable, and a third
variable indicating the repeated measurements (subject IDs).

(2) More than two dependent samples are considered as *repeated measurements*.
For normally distributed outcomes, these samples are usually tested using
a ANOVA for repeated measurements. A more sophisticated approach would
be using a linear mixed model.

Ben-Shachar, M.S., Patil, I., Thériault, R., Wiernik, B.M., Lüdecke, D. (2023). Phi, Fei, Fo, Fum: Effect Sizes for Categorical Data That Use the Chi‑Squared Statistic. Mathematics, 11, 1982. doi:10.3390/math11091982

Bender, R., Lange, S., Ziegler, A. Wichtige Signifikanztests. Dtsch Med Wochenschr 2007; 132: e24–e25

du Prel, J.B., Röhrig, B., Hommel, G., Blettner, M. Auswahl statistischer Testverfahren. Dtsch Arztebl Int 2010; 107(19): 343–8

`t_test()`

for parametric t-tests of dependent and independent samples.`mann_whitney_test()`

for non-parametric tests of unpaired (independent) samples.`wilcoxon_test()`

for Wilcoxon rank sum tests for non-parametric tests of paired (dependent) samples.`kruskal_wallis_test()`

for non-parametric tests with more than two independent samples.`chi_squared_test()`

for chi-squared tests (two categorical variables, dependent and independent).

```
data(efc)
efc$weight <- abs(rnorm(nrow(efc), 1, 0.3))
# Chi-squared test
chi_squared_test(efc, "c161sex", by = "e16sex")
#> # Chi-squared test for contingency tables
#>
#> Data: c161sex by e16sex (n = 900)
#>
#> χ² = 2.233, ϕ = 0.053 (very small effect), df = 1, p = 0.135
#>
# weighted Chi-squared test
chi_squared_test(efc, "c161sex", by = "e16sex", weights = "weight")
#> # Chi-squared test for contingency tables (weighted)
#>
#> Data: c161sex by e16sex (n = 910)
#>
#> χ² = 2.044, ϕ = 0.050 (very small effect), df = 1, p = 0.153
#>
# Chi-squared test for given probabilities
chi_squared_test(efc, "c161sex", probabilities = c(0.3, 0.7))
#> # Chi-squared test for given probabilities
#>
#> Data: c161sex against probabilities 30% and 70% (n = 901)
#>
#> χ² = 16.162, פ = 0.088 (very small effect), df = 1, p < .001
#>
```