I am interested in calculating the standardised difference for a multinomial variable. In R there is a package called tableone that produces such calculations (https://cran.r-project.org/web/packages/tableone/vignettes/smd.html). In this vignette there is a calculation for race, with a smd of 0.036.
race (%) 0.036
black 585 (16.5) 335 (15.3)
other 213 ( 6.0) 142 ( 6.5)
white 2753 (77.5) 1707 (78.2)
to work through this calculation I used the "building block" code here https://rpubs.com/kaz_yos/smd1
> T <- c(213.0/3551.0, 2753.0/3551.0)
> C <- c(142.0/2184.0, 1707.0/2184.0)
> T
[1] 0.0599831 0.7752746
> C
[1] 0.06501832 0.78159341
> meanDiffVector<-(T-C)
> vcovT <- -1 * outer(T, T)
> diag(vcovT) <- T * (1-T)
> vcovT
[,1] [,2]
[1,] 0.05638513 -0.04650337
[2,] -0.04650337 0.17422391
> vcovC <- -1 * outer(C, C)
> diag(vcovC) <- C * (1-C)
> vcovC
[,1] [,2]
[1,] 0.06079093 -0.05081789
[2,] -0.05081789 0.17070515
> S<-(vcovC+vcovT)/2
> S
[,1] [,2]
[1,] 0.05858803 -0.04866063
[2,] -0.04866063 0.17246453
> Sinv <- solve(S)
> Sinv
[,1] [,2]
[1,] 22.292318 6.289747
[2,] 6.289747 7.572937
> smdCat <- drop(t(meanDiffVector) %*% Sinv %*% t(t(meanDiffVector)))
> smdCat
[1] 0.001267793
with a value of 0.00127 not the 0.036.
Both methods claim to follow the method in Yang and Dalton (2012) "... extension to multinomival variables is suggested in Yang et al 2012. This multinomial extension treats a single multinomial variable as multiple non-redundant dichotomous variables and use the Mahalanobis distance. The off diagonal elements of the covariance matrix on page 3 have an error, and need negation." https://www.rdocumentation.org/packages/tableone/versions/0.12.0/topics/CreateTableOne
Yang, D. and Dalton, JE. (2012). A unified approach to measuring the effect size between two groups using SAS. SAS Global Forum 2012, Paper 335-2012. http://support.sas.com/resources/papers/proceedings12/335-2012.pdf
