# Using smd

#### 2020-10-13

The smd package provides the smd method to compute standardized mean differences between two groups for continuous values (numeric and integer data types) and categorical values (factor, character, and logical). The method also works on matrix, list, and data.frame data types by applying smd() over the columns of the matrix or data.frame and each item of the list. The package is based on Yang and Dalton (2012).

The smd function computes the standardized mean difference for each level $$k$$ of a grouping variable compared to a reference $$r$$ level:

$d_k = \sqrt{(\bar{x}_r - \bar{x}_{k})^{\intercal}S_{rk}^{-1}(\bar{x}_r - \bar{x}_{k})}$

where $$\bar{x}_{\cdot}$$ and $$S_{rk}$$ are the sample mean and covariances for reference group $$r$$ and group $$k$$, respectively. In the case that $$x$$ is categorical, $$\bar{x}$$ is the vector of proportions of each category level within a group, and $$S_{rk}$$ is the multinomial covariance matrix.

Standard errors are computed using the formula described in Hedges and Olkin (1985):

$\sqrt{ \frac{n_r + n_k}{n_rn_k} + \frac{d_k^2}{2(n_r + n_k)} }$

# Examples

library(smd)

## Numeric

set.seed(123)
xn <- rnorm(90)
gg2 <- rep(LETTERS[1:2], each = 45)
gg3 <- rep(LETTERS[1:3], each = 30)

smd(x = xn, g = gg2)
#>   term   estimate
#> 1    B 0.03413269
smd(x = xn, g = gg3)
#>   term    estimate
#> 1    B -0.25169577
#> 2    C -0.07846864
smd(x = xn, g = gg2, std.error = TRUE)
#>   term   estimate std.error
#> 1    B 0.03413269 0.2108339
smd(x = xn, g = gg3, std.error = TRUE)
#>   term    estimate std.error
#> 1    B -0.25169577 0.2592192
#> 2    C -0.07846864 0.2582982

## Integers

xi <- sample(1:20, 90, replace = TRUE)
smd(x = xi, g = gg2)
#>   term  estimate
#> 1    B 0.1687339

## Character

xc <- unlist(replicate(2, sort(sample(letters[1:3], 45, replace = TRUE)), simplify = FALSE))
smd(x = xc, g = gg2)
#>   term  estimate
#> 1    B 0.1946887

## Factors

xf <- factor(xc)
smd(x = xf, g = gg2)
#>   term  estimate
#> 1    B 0.1946887

## Logical

xl <- as.logical(rbinom(90, 1, prob = 0.5))
smd(x = xl, g = gg2)
#>   term estimate
#> 1    B        0

## Matrices

mm <- cbind(xl, xl, xl, xl)
smd(x = mm, g = gg3, std.error = FALSE)
#>               xl          xl          xl          xl
#> [1,] -0.06765101 -0.06765101 -0.06765101 -0.06765101
#> [2,] -0.20203051 -0.20203051 -0.20203051 -0.20203051

## Lists

ll <- list(xn = xn, xi = xi, xf = xf, xl = xl)
smd(x = ll, g = gg3)
#>   variable term    estimate
#> 1       xn    B -0.25169577
#> 2       xn    C -0.07846864
#> 3       xi    B  0.30325301
#> 4       xi    C  0.36089675
#> 5       xf    B  1.50232594
#> 6       xf    C  2.23606798
#> 7       xl    B -0.06765101
#> 8       xl    C -0.20203051

## data.frames

df <- data.frame(xn, xi, xc, xf, xl)
smd(x = df, g = gg3)
#>    variable term    estimate
#> 1        xn    B -0.25169577
#> 2        xn    C -0.07846864
#> 3        xi    B  0.30325301
#> 4        xi    C  0.36089675
#> 5        xc    B  1.50232594
#> 6        xc    C  2.23606798
#> 7        xf    B  1.50232594
#> 8        xf    C  2.23606798
#> 9        xl    B -0.06765101
#> 10       xl    C -0.20203051

## Using smd with dplyr

library(dplyr, verbose = FALSE)
#> Warning: package 'dplyr' was built under R version 3.6.2
#>
#> Attaching package: 'dplyr'
#> The following objects are masked from 'package:stats':
#>
#>     filter, lag
#> The following objects are masked from 'package:base':
#>
#>     intersect, setdiff, setequal, union
df$g <- gg2 df %>% summarize_at( .vars = vars(dplyr::matches("^x")), .funs = list(smd = ~ smd(., g = g)$estimate))
#>       xn_smd    xi_smd    xc_smd    xf_smd xl_smd
#> 1 0.03413269 0.1687339 0.1946887 0.1946887      0

See: