Computes the ratio of two means (also known as the "response ratio"; RR) of
**variables on a ratio scale** (with an absolute 0). Pair with any reported
`stats::t.test()`

.

## Usage

```
means_ratio(
x,
y = NULL,
data = NULL,
paired = FALSE,
adjust = TRUE,
log = FALSE,
ci = 0.95,
alternative = "two.sided",
verbose = TRUE,
...
)
```

## Arguments

- x, y
A numeric vector, or a character name of one in

`data`

. Any missing values (`NA`

s) are dropped from the resulting vector.`x`

can also be a formula (see`stats::t.test()`

), in which case`y`

is ignored.- data
An optional data frame containing the variables.

- paired
If

`TRUE`

, the values of`x`

and`y`

are considered as paired. The correlation between these variables will affect the CIs.- adjust
Should the effect size be bias-corrected? Defaults to

`TRUE`

; Advisable for small samples.- log
Should the log-ratio be returned? Defaults to

`FALSE`

. Normally distributed and useful for meta-analysis.- ci
Confidence Interval (CI) level

- alternative
a character string specifying the alternative hypothesis; Controls the type of CI returned:

`"two.sided"`

(default, two-sided CI),`"greater"`

or`"less"`

(one-sided CI). Partial matching is allowed (e.g.,`"g"`

,`"l"`

,`"two"`

...). See*One-Sided CIs*in effectsize_CIs.- verbose
Toggle warnings and messages on or off.

- ...
Arguments passed to or from other methods. When

`x`

is a formula, these can be`subset`

and`na.action`

.

## Value

A data frame with the effect size (`Means_ratio`

or
`Means_ratio_adjusted`

) and their CIs (`CI_low`

and `CI_high`

).

## Details

The Means Ratio ranges from 0 to \(\infty\), with values smaller than 1 indicating that the second mean is larger than the first, values larger than 1 indicating that the second mean is smaller than the first, and values of 1 indicating that the means are equal.

## Note

The bias corrected response ratio reported from this function is derived from Lajeunesse (2015).

## Confidence (Compatibility) Intervals (CIs)

Confidence intervals are estimated as described by Lajeunesse (2011 & 2015) using the log-ratio standard error assuming a normal distribution. By this method, the log is taken of the ratio of means, which makes this outcome measure symmetric around 0 and yields a corresponding sampling distribution that is closer to normality.

## CIs and Significance Tests

"Confidence intervals on measures of effect size convey all the information
in a hypothesis test, and more." (Steiger, 2004). Confidence (compatibility)
intervals and p values are complementary summaries of parameter uncertainty
given the observed data. A dichotomous hypothesis test could be performed
with either a CI or a p value. The 100 (1 - \(\alpha\))% confidence
interval contains all of the parameter values for which *p* > \(\alpha\)
for the current data and model. For example, a 95% confidence interval
contains all of the values for which p > .05.

Note that a confidence interval including 0 *does not* indicate that the null
(no effect) is true. Rather, it suggests that the observed data together with
the model and its assumptions combined do not provided clear evidence against
a parameter value of 0 (same as with any other value in the interval), with
the level of this evidence defined by the chosen \(\alpha\) level (Rafi &
Greenland, 2020; Schweder & Hjort, 2016; Xie & Singh, 2013). To infer no
effect, additional judgments about what parameter values are "close enough"
to 0 to be negligible are needed ("equivalence testing"; Bauer & Kiesser,
1996).

## 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. doi:10.1890/11-0423.1

Lajeunesse, M. J. (2015). Bias and correction for the log response ratio in ecological meta-analysis. Ecology, 96(8), 2056-2063. doi:10.1890/14-2402.1

Hedges, L. V., Gurevitch, J., & Curtis, P. S. (1999). The meta-analysis of response ratios in experimental ecology. Ecology, 80(4), 1150–1156. doi:10.1890/0012-9658(1999)080[1150:TMAORR]2.0.CO;2

## See also

Other standardized differences:
`cohens_d()`

,
`mahalanobis_d()`

,
`p_superiority()`

,
`rank_biserial()`

## Examples

```
x <- c(1.83, 0.50, 1.62, 2.48, 1.68, 1.88, 1.55, 3.06, 1.30)
y <- c(0.878, 0.647, 0.598, 2.05, 1.06, 1.29, 1.06, 3.14, 1.29)
means_ratio(x, y)
#> Means Ratio (adj.) | 95% CI
#> ---------------------------------
#> 1.31 | [0.82, 2.10]
means_ratio(x, y, adjust = FALSE)
#> Means Ratio | 95% CI
#> --------------------------
#> 1.32 | [0.82, 2.13]
means_ratio(x, y, log = TRUE)
#> log(Means Ratio, adj.) | 95% CI
#> --------------------------------------
#> 0.27 | [-0.20, 0.74]
# The ratio is scale invariant, making it a standardized effect size
means_ratio(3 * x, 3 * y)
#> Means Ratio (adj.) | 95% CI
#> ---------------------------------
#> 1.31 | [0.82, 2.10]
```