Residual / explained variance TraMineR

Question

Can someone explain to me, how the explained / residual variance from the dissassoc command is computed? I tried to add up the variance within using the discrepancies of the different levels, but it didn't work. Let's take a small random sample (n=10).

data(mvad)
library(dplyr) 
set.seed(10)

mv = mvad %>% group_by(male) %>% sample_n(5)
mvad.seq <- seqdef(mv[,17:20])

# compute the dissimilarity matrix
mvad.ham <- seqdist(mvad.seq, method="HAM") 

# compute the discrepancy analysis 
d = dissassoc(mvad.ham, group = mv$male, R=10) 
d 
# Pseudo ANOVA table:
#         SS df      MSE
# Exp    1.7  1 1.700000
# Res    9.6  8 1.200000
# Total 11.3  9 1.255556

I understand how to compute the Total by hand:

sum(mvad.ham) / (2* 10) # = Total 11.3 

Questions:

1. How do you compute the Exp and Res values by hand?
2. Could you please demonstrate us how do you compute the Total Sum of Square?

I understand from (Studer et al., 2010) that the equation is:

$$ SS = \sum_{i=1}^{n} w_i(y_i - \bar{y})^2 $$

What does the $\bar{y}$ represents exactly? Could you demonstrate on this example how one can compute the $SS$ from mvad.ham manually?

Answer 1

We would need an example to see what you are making wrong. Nevertheless, from what you say I understand that you attempt to get the total within variance by adding up the variances of each group. It is the sums of squares $(SS = n \cdot var)$ that should be added up, not the variances.

The within SS of each group is computed the same way as the total SS. The sum of those within SS gives the residual SS.

The explained SS is then obtained as the difference between the total SS and the residual SS.

$SS_{exp} = SS_{tot} - SS_{res}$

I illustrate by building on your example:

(ss.tot <- sum(mvad.ham)/(nrow(mvad.seq)*2))
## [1] 11.3

ss.male <- sum(mvad.ham[mv$male=="yes",mv$male=="yes"])/(sum(mv$male=="yes")*2)
ss.female <- sum(mvad.ham[mv$male!="yes",mv$male!="yes"])/(sum(mv$male!="yes")*2)

(ss.res <- ss.female + ss.male)
## [1] 9.6

(ss.exp <- ss.tot - ss.res)
## [1] 1.7