---
title: "Standardized Mean Differences"
subtitle: "The calculation of Cohen's d type effect sizes"
author: "Aaron R. Caldwell"
date: "`r Sys.Date()`"
output: 
  rmarkdown::html_vignette:
    toc: True
vignette: >
  %\VignetteIndexEntry{Standardized Mean Differences}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
editor_options: 
  chunk_output_type: console
  markdown: 
    wrap: 72
bibliography: references.bib
---

```{r setup, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
library(TOSTER)
library(ggplot2)
library(ggdist)

```

The calculation of standardized mean differences (SMDs) can be helpful
in interpreting data and are essential for meta-analysis. In psychology,
effect sizes are very often reported as an SMD rather than raw units
(though either is fine: see @Caldwell2020). In most papers the SMD is
reported as Cohen's d^[I'd argue it is more appropriate to label it as a SMD since many times researchers are not reporting Jacob Cohen's original formulation. Therefore it is more accurate descriptor to label any SMD as SMD]. The simplest form involves reporting the mean
difference (or mean in the case of a one-sample test) divided by the
standard deviation.

$$
Cohen's \space d = \frac{Mean}{SD}
$$

However, two major problems arise: bias and the calculation of the
denominator. First, the Cohen's d calculation is biased (meaning the
effect is inflated), and a bias correction (often referred to as Hedges'
g) is applied to provide an unbiased estimate. Second, the denominator
can influence the estimate of the SMD, and there are a multitude of
choices for how to calculate the denominator. To make matters worse, the
calculation (in most cases an approximation) of the confidence intervals
involves the noncentral *t* distribution. This requires calculating a
non-centrality parameter (lambda: $\lambda$), degrees of freedom ($df$),
or even the standard error (sigma: $\sigma$) for the SMD. None of these
are easy to determine and these calculations are hotly debated in the
statistics literature [@Cousineau2021].

In this package we originally opted to make the default SMD confidence intervals as the formulation
outlined by @Goulet_2018. We found that that these calculations were
simple to implement and provided fairly accurate coverage for the
confidence intervals for any type of SMD (independent, paired, or one
sample). However, even the authors have outlined some issues with the
method in a newer publication [@Cousineau2021]. 
Other packages, such as `MOTE` [@MOTE] or `effectsize` [@effectsize], 
use a simpler formulation of the noncentral t-distribution (nct). 
The default option in the package is the nct type of confidence intervals.
We have created an argument for all TOST functions 
(`tsum_TOST` and `t_TOST`) named `smd_ci` 
which allow the user to specify "goulet" (for the @Cousineau2021 method), 
"nct" (this will approximately match the results of @MOTE and @effectsize), 
"t" (central t method), or "z" (normal method).
We would strongly recommend using "nct" or "goulet" for any analysis.
It is important to remember that all of these methods are only *approximations*
of confidence intervals (of varying degrees of quality)
and therefore should be interpreted with caution.

It is my belief that SMDs provide another interesting description of the sample, 
and have very limited inferential utility (though exceptions apply). 
You may disagree, and if you are basing your inferences on the SMD, and the
associated confidence intervals, we recommend you go with a
bootstrapping approach (see `boot_t_TOST`) [@Kirby2013].

In this section we will detail on the calculations that are involved in
calculating the SMD, their associated degrees of freedom, non-centrality
parameter, and variance. If these SMDs are being reported in a
scientific manuscript, we **strongly** recommend that the formulas for
the SMDs you report be included in the methods section.

# Bias Correction (Hedges)

For all SMD calculative approaches the bias correction was calculated as
the following:

$$
J = \frac{\Gamma(\frac{df}{2})}{\sqrt{\frac{df}{2}} \cdot \Gamma(\frac{df-1}{2})} 
$$

The correction factor^[This calculation was derived from the supplementary material of
@Cousineau2021.] is calculated in R as the following:

    J <- exp ( lgamma(df/2) - log(sqrt(df/2)) - lgamma((df-1)/2) )

Hedges g (bias corrected Cohen's d) can then be calculated by
multiplying d by J

$$
g = d \cdot J
$$
When the bias correction is not applied, J is equal to 1.

# Independent Samples

For independent samples there are three calculative approaches supported
by `TOSTER`. One the denominator is the pooled standard deviation
(Cohen's d), the average standard deviation (Cohen's
d(av)), and the standard deviation of the control group (Glass's $\Delta$). 
Currently, the d or d(av) is selected by whether or not variances are assumed to
be equal. If the variances are not
assumed to be equal then Cohen's d(av) will be returned, and if
variances are assumed to be equal then Cohen's d is returned. Glass's delta can
be selected by setting the `glass` argument to "glass1" or "glass2".

## Variances Assumed Unequal: Cohen's d(av)

For this calculation, the denominator is simply the square root of the
average variance.

$$
s_{av} = \sqrt \frac {s_{1}^2 + s_{2}^2}{2}
$$

The SMD, Cohen's d(av), is then calculated as the following:

$$
d_{av} = \frac {\bar{x}_1 - \bar{x}_2} {s_{av}}
$$ 

Note: the x with the bar above it (pronounced as "x-bar") refers the
the means of group 1 and 2 respectively.

The degrees of freedom for Cohen's d(av), derived from @delacre2021, is
the following:

$$
df = \frac{(n_1-1)(n_2-1)(s_1^2+s_2^2)^2}{(n_2-1) \cdot s_1^4+(n_1-1) \cdot s_2^4}
$$

The non-centrality parameter ($\lambda$) is calculated as the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\lambda =  d_{av} \times \sqrt{\frac{n_1 \cdot n_2(\sigma^2_1+\sigma^2_2)}{2 \cdot (n_2 \cdot \sigma^2_1+n_1 \cdot \sigma^2_2)}}
$$

2. Under all other methods (nct, t, or z):

$$
\lambda = \frac{2 \cdot (n_2 \cdot \sigma_1^2 + n_1 \cdot \sigma_2^2)} {n_1 \cdot n_2 \cdot (\sigma_1^2 + \sigma_2^2)}
$$
The standard error ($\sigma$) of Cohen's d(av) is calculated as the
following from @bonett2009meta^[This changed from the original formulation, I believe the current version is a better estimate]:

$$
\sigma_{SMD} = \sqrt{d^2 \cdot (\frac{s_1^4}{n_1-1}+\frac{s_2^4}{n_2-1}) \div 8 \cdot s_{av}^4 + (\frac{s_1^2}{n_1-1}+\frac{s_2^2}{n_2-1}) \div s_{av}^2}
$$ 




## Variances Assumed Equal: Cohen's d

For this calculation, the denominator is simply the pooled standard
deviation.

$$
s_{p} = \sqrt \frac {(n_{1} - 1)s_{1}^2 + (n_{2} - 1)s_{2}^2}{n_{1} + n_{2} - 2}
$$

$$
d = \frac {\bar{x}_1 - \bar{x}_2} {s_{p}}
$$

The degrees of freedom for Cohen's d is the following:

$$
df = n_1 + n_2 - 2
$$

The non-centrality parameter ($\lambda$) is calculated as the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\lambda = d \cdot \sqrt \frac{\tilde n}{2}
$$ 

wherein, $\tilde n$ is the harmonic mean of the 2 sample sizes which
is calculated as the following:

$$
\tilde n = \frac{2 \cdot n_1 \cdot n_2}{n_1 + n_2}
$$

2. Under all other methods (nct, t, or z):

$$
\lambda = \frac{1}{n_1} +\frac{1}{n_2}
$$ 

The standard error ($\sigma$) of Cohen's d is calculated as the
following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\sigma_{SMD} = \sqrt{\frac{df}{df-2} \cdot \frac{2}{\tilde n} (1+d^2 \cdot \frac{\tilde n}{2}) -\frac{d^2}{J}}
$$ 


2. Under all other methods (nct, t, or z) it is calculated using the "unbiased" estimate from @viechtbauer2007approximate (table 1; equation 26)^[This assumes degrees of freedom of N-2]:

$$
\sigma_{SMD} = \sqrt{\frac{1}{n_1} + \frac{1}{n_2} + (1 - \frac{df-2}{df \cdot J^2}) \cdot d_s^2}
$$

## Glass's Delta

For this calculation, the denominator is simply the standard
deviation of one of the groups (`x` for `glass = "glass1"`, or `y` for 
`glass = "glass2"`.

$$
s_{c} = SD_{control \space group}
$$

$$
d = \frac {\bar{x}_1 - \bar{x}_2} {s_{c}}
$$

The degrees of freedom for Glass's delta is the following:

$$
df = n_c - 1
$$

The non-centrality parameter ($\lambda$) is calculated as the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\lambda = d \cdot \sqrt \frac{\tilde n}{2}
$$ 

wherein, $\tilde n$ is the harmonic mean of the 2 sample sizes which
is calculated as the following:

$$
\tilde n = \frac{2 \cdot n_1 \cdot n_2}{n_1 + n_2}
$$

2. Under all other methods (nct, t, or z):

$$
\lambda = \frac{1}{n_T} +  \frac{s_c^2}{n_c \cdot s_c^2}
$$ 

The standard error ($\sigma$) of Glass's delta is calculated as the
following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\sigma_{SMD} = \sqrt{\frac{df}{df-2} \cdot \frac{2}{\tilde n} (1+d^2 \cdot \frac{\tilde n}{2}) -\frac{d^2}{J}}
$$ 


2. Under all other methods^[Originally, I used an estimate of the standard error from Morris and DeShon. However, it was pointed out by colleague that this wasn't very accurate. We now use what is used in the metafor R package for the "SMD1H" which is the use of one sample SD as the denominator under the assumption of heterogeneous variances] (nct, t, or z) derived from @bonett2008:

$$
\sigma_{SMD} = \sqrt{\frac{s_e^2 \div s_c^2}{n_e-1}+ \frac{1}{n_c-1}+ \frac{d^2}{2 \cdot (n_c-1)}}
$$



# Paired Samples

For paired samples there are two calculative approaches supported by
`TOSTER`. One the denominator is the standard deviation of the change
score (Cohen's d(z)), the correlation corrected effect
size (Cohen's d(rm)), and the standard deviation of the control condition 
(Glass's $\Delta$). Currently, the choice is made by the function
based on whether or not the user sets `rm_correction` to TRUE. If
`rm_correction` is set to t TRUE then Cohen's d(rm) will be returned,
and otherwise Cohen's d(z) is returned. This can be overridden and Glass's delta
is returned if the `glass` argument is set to "glass1" or "glass2".

## Cohen's d(z): Change Scores

For this calculation, the denominator is the standard deviation of the
difference scores which can be calculated from the standard deviations
of the samples and the correlation between the paired samples.

$$
s_{diff} =  \sqrt{sd_1^2 + sd_2^2 - 2 \cdot r_{12} \cdot sd_1 \cdot sd_2}
$$

The SMD, Cohen's d(z), is then calculated as the following:

$$
d_{z} = \frac {\bar{x}_1 - \bar{x}_2} {s_{diff}}
$$

The degrees of freedom for Cohen's d(z) is the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
df = 2 \cdot (N_{pairs}-1)
$$

2. Under all other methods (nct, t, or z):

$$
df = N - 1
$$

The non-centrality parameter ($\lambda$) is calculated as the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\lambda = d_{z} \cdot \sqrt \frac{N_{pairs}}{2 \cdot (1-r_{12})}
$$

2. Under all other methods (nct, t, or z):

$$
\lambda = \frac{1}{n}
$$

The standard error ($\sigma$) of Cohen's d(z) is calculated as the
following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\sigma_{SMD} = \sqrt{\frac{df}{df-2} \cdot \frac{2 \cdot (1-r_{12})}{n} \cdot (1+d^2 \cdot \frac{n}{2 \cdot (1-r_{12})}) -\frac{d^2}{J^2}} \space \times \space \sqrt {2 \cdot (1-r_{12})}
$$

2. Under all other methods (nct, t, or z) it calculated using the "unbiased" estimate from @viechtbauer2007approximate (table 1; equation 26)^[This assumes degrees of freedom of N-1]:

$$
\sigma_{SMD} = \sqrt{\frac{1}{N} + (1 - \frac{df-2}{df \cdot J^2}) \cdot d_z^2}
$$

## Cohen's d(rm): Correlation Corrected

For this calculation, the same values for the same calculations above is
adjusted for the correlation between measures. As @Goulet_2018 mention,
this is useful for when effect sizes are being compared for studies that
involve between and within subjects designs.

First, the standard deviation of the difference scores are calculated

$$
s_{diff} =  \sqrt{sd_1^2 + sd_2^2 - 2 \cdot r_{12} \cdot sd_1 \cdot sd_2}
$$

The SMD, Cohen's d(rm), is then calculated with a small change to the
denominator[^1]:

[^1]: This is incorrectly stated in the article by @Goulet_2018; the
    correct notation is provided by @lakens2013

$$
d_{rm} = \frac {\bar{x}_1 - \bar{x}_2}{s_{diff}} \cdot \sqrt {2 \cdot (1-r_{12})} 
$$

The degrees of freedom for Cohen's d(rm) is the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
df = 2 \cdot (N_{pairs}-1)
$$

2. Under all other methods (nct, t, or z):

$$
df = N - 1
$$

The non-centrality parameter ($\lambda$) is calculated as the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\lambda = d_{rm} \cdot \sqrt \frac{N_{pairs}}{2 \cdot (1-r_{12})}
$$

2. Under all other methods (nct, t, or z):

$$
\lambda = \frac{1}{n}
$$

The standard error ($\sigma$) of Cohen's d(rm) is calculated as the
following:

$$
\sigma_{SMD} = \sqrt{\frac{df}{df-2} \cdot \frac{2 \cdot (1-r_{12})}{n} \cdot (1+d_{rm}^2 \cdot \frac{n}{2 \cdot (1-r_{12})}) -\frac{d_{rm}^2}{J^2}}
$$

## Glass's Delta

For this calculation, the denominator is simply the standard
deviation of one of the groups (`x` for `glass = "glass1"`, or `y` for 
`glass = "glass2"`.

$$
s_{c} = SD_{control \space condition}
$$

$$
d = \frac {\bar{x}_1 - \bar{x}_2} {s_{c}}
$$

The degrees of freedom for Glass's delta is the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
df = 2 \cdot N - 1
$$

2. Under all other methods (nct, t, or z):

$$
df = N - 1
$$

The non-centrality parameter ($\lambda$) is calculated as the following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\lambda = d \cdot \sqrt{\frac{N}{2 \cdot (1 - r_{12})}}
$$ 

2. Under all other methods (nct, t, or z):

$$
\lambda = \frac{1}{N}
$$ 

The standard error ($\sigma$) of Glass's delta is calculated as the
following derived^[Originally, @borenstein was utilized for this SE calculation. However, upon finding the estimate from @bonett2008, which is utilized in the `metafor` R package for heterogenous variance estimates, we have opted to use this estimate instead.] from @bonett2008:

$$
\sigma_{SMD} =     \sqrt{\frac{s_{diff}^2}{s_c^2 \cdot df} + \frac{d^2}{2 \cdot df}}
$$

# One Sample

For a one-sample situation, the calculations are very straight forward

For this calculation, the denominator is simply the standard deviation
of the sample.

$$
s={\sqrt {{\frac {1}{N-1}}\sum _{i=1}^{N}\left(x_{i}-{\bar {x}}\right)^{2}}}
$$

The SMD is then the mean of X divided by the standard deviation.

$$
d = \frac {\bar{x}} {s}
$$

The degrees of freedom for Cohen's d is the following:

$$
df = N - 1
$$

The non-centrality parameter ($\lambda$) is calculated as the following:

$$
\lambda = d \cdot \sqrt N
$$

The standard error ($\sigma$) of Cohen's d is calculated as the
following:

1. Under the @Cousineau2021 method (`smd_ci = "goulet"`):

$$
\sigma_{SMD} = \sqrt{\frac{df}{df-2} \cdot \frac{1}{N} (1+d^2 \cdot N) -\frac{d^2}{J^2}}
$$ 

2. Under all other methods (nct, t, or z):

$$
\sigma_{SMD} = \sqrt{\frac{1}{n} + \frac{d^2}{(2 \cdot n)}}
$$


# Confidence Intervals

For the SMDs calculated in this package we use the non-central *t*
method, those outlined by @Goulet_2018, and approximate methods using the *t* and normal distributions. These calculations are only
approximations and newer formulations may provide better coverage
[e.g., @Cousineau2021]. In any case, if the calculation of confidence
intervals for SMDs is of the utmost importance then I would strongly
recommend using bootstrapping techniques rather than any of these approximations whenever possible [@Kirby2013].

The calculations of the confidence intervals for the `goulet` and `nct` methods involve a
two step process: 1) using the noncentral *t*-distribution to calculate
the lower and upper bounds of $\lambda$, and 2) transforming this back
to the effect size estimate. These noncentrality methods are not without their detractors [@spanos2021]. The normal and *t* methods are much simpler and less computationally intensive, but may have varying levels of accuracy [see @viechtbauer2007approximate for more details].

## @Cousineau2021 Method (smd_ci = "goulet")

Calculate confidence intervals around $\lambda$.

$$
t_L = t_{(1/2-(1-\alpha)/2,\space df, \space \lambda)} \\
t_U = t_{(1/2+(1-\alpha)/2,\space df, \space \lambda)}
$$

Then transform back to the SMD.

$$
d_L = \frac{t_L}{\lambda} \cdot d \\
d_U = \frac{t_U}{\lambda} \cdot d 
$$

## The non-central t-method (smd_ci = "nct")

This method is commonly reported in the literature and in other R packages (e.g., `effectsize`). However, it is not without criticism [@spanos2021; @viechtbauer2007approximate].

In thise method, you calculate the non-centrality parameters necessary to form confidence intervals wherein the observed t-statistic ($t_{obs}$)is utilized.

$$
t_L = t_{(1-alpha,\space df, \space t_{obs})} \\
t_U = t_{(alpha,\space df, \space  t_{obs})}
$$
The confidence intervals can then be constructed using the non-centrality parameter and the bias correction.

$$
d_L = t_L \cdot \sqrt{\lambda} \cdot J \\
d_U = t_U \cdot \sqrt{\lambda} \cdot J 
$$



## Central t-method

This is less commonly used though it might perform alright in some scenarios, and as @bonett2008 and @viechtbauer2007approximate report it might perform adequately under with an adequately large sample size.

The limits of the t-distribution at the given alpha-level and degrees of freedom (`qt(1-alpha,df)`) are multiplied by the standard error of the calculated SMD.

$$
CI = SMD \space \pm \space t_{(1-\alpha,df)} \cdot \sigma_{SMD}
$$

## Normal method

This method will often be the most liberal estimate for the confidence interval. But as @bonett2008 and @viechtbauer2007approximate report, it might perform adequately under with an adequately large sample size. The use of the normal distribution can be justified by the fact that SMDs are asympototically normal.

The limits of the z-distribution at the given alpha-level (`qnorm(1-alpha)`) are multiplied by the standard error of the calculated SMD.

$$
CI = SMD \space \pm \space z_{(1-\alpha)} \cdot \sigma_{SMD}
$$

# Just Estimate the SMD

It was requested that a function be provided that only calculates the SMD.
Therefore, I created the `smd_calc` and `boot_smd_calc` functions^[The results differ greatly because the bootstrap CI method, studentized, is more conservative than the parametric method. This difference is more apparent with extremely small samples like that in the `sleep` dataset.].
The interface is almost the same as `t_TOST` but you don't set an equivalence bound.

```{r}
smd_calc(formula = extra ~ group,
         data = sleep,
         paired = TRUE,
         smd_ci = "nct",
         bias_correction = F)

# Setting bootstrap replications low to
## reduce compiling time of vignette
boot_smd_calc(formula = extra ~ group,
         data = sleep,
         R = 199,
         paired = TRUE,
         boot_ci = "stud",
         bias_correction = F)
```


# Plotting SMDs

Sometimes you may take a different approach to calculating the SMD, or
you may only have the summary statistics from another study. For this
reason, I have included a way to plot the SMD based on just three
values: the estimate of the SMD, the degrees of freedom, and the
non-centrality parameter/standard error. So long as all three are reported, or can be
estimated, then a plot of the SMD can be produced. The non-centrality parameter is needed if you want to plot the estimates using the `"goulet"` method while the standard error is needed if you want to plot the `z` (normal) or `t` (t-distribution) methods.

Two types of plots can be produced: consonance (`type = "c"`),
consonance density (`type = "cd"`), or both (the default option;
(`type = c("c","cd")`))

```{r fig.width=6, fig.height=6}
plot_smd(d = .43,
         df = 58,
         sigma = .33,
         smd_label = "Cohen's d",
         smd_ci = "z"
         )
```

# Comparing SMDs

In some cases, the SMDs between original and replication studies want to be
compared. Rather than looking at whether or not a replication attempt is
significant, a researcher could compare to see how compatible the SMDs
are between the two studies.

For example, say there is original study reports an effect of Cohen's dz
= 0.95 in a paired samples design with 25 subjects. However, a
replication doubled the sample size, found a non-significant effect at
an SMD of 0.2. Are these two studies compatible? Or, to put it another
way, should the replication be considered a failure to replicate?

We can use the `compare_smd` function to at least measure how often we
would expect a discrepancy between the original and replication study if
the same underlying effect was being measured (also assuming no
publication bias or differences in protocol).

We can see from the results below that, if the null hypothesis were
true, we would only expect to see a discrepancy in SMDs between studies, at
least this large, \~1% of the time.

```{r}
compare_smd(smd1 = 0.95,
            n1 = 25,
            smd2 = 0.23,
            n2 = 50,
            paired = TRUE)
```

## Raw Data

The above results are only based on an approximating the differences
between the SMDs. If the raw data is available, then the optimal
solution is the bootstrap the results. This can be accomplished with the
`boot_compare_smd` function.

For this example, we will simulate some data.

```{r}
set.seed(4522)
diff_study1 = rnorm(25,.95)
diff_study2 = rnorm(50)
boot_test = boot_compare_smd(x1 = diff_study1,
                             x2 = diff_study2,
                             paired = TRUE)

boot_test

# Table of bootstrapped CIs
knitr::kable(boot_test$df_ci, digits = 4)
```

### Visualize

The results of the bootstrapping are stored in the results. So we can
even visualize the differences in SMDs.

```{r}
library(ggplot2)

list_res = boot_test$boot_res
df1 = data.frame(study = c(rep("original",length(list_res$smd1)),
                           rep("replication",length(list_res$smd2))),
                 smd = c(list_res$smd1,list_res$smd2))
ggplot(df1,
            aes(fill = study, color =smd, x = smd))+ 
 geom_histogram(aes(y=..density..), alpha=0.5, 
                position="identity")+
 geom_density(alpha=.2) +
  labs(y = "", x = "SMD (bootstrapped estimates)") +
  theme_classic()

df2 = data.frame(diff = list_res$d_diff)
ggplot(df2,
            aes(x = diff))+ 
 geom_histogram(aes(y=..density..), alpha=0.5, 
                position="identity")+
 geom_density(alpha=.2) +
  labs(y = "", x = "Difference in SMDs (bootstrapped estimates)") +
  theme_classic()
```

# References
