I have a set of 16 human heights, half of which are from males, and half from females.
| Males | 164 | 182 | 179 | 170 | 166 | 178 | 169 | 180 |
| Females | 168 | 154 | 166 | 159 | 152 | 161 | 169 | 171 |
If I code sex as 0 for female and 1 for male, the interclass correlation between height and sex is 0.64. This means r² = 0.41. I understand this to mean that the variance of height in the full sample (79 cm²) can be broken down into
- between-sex variance 0.41 × 79 = 32 cm², and
- within-sex variance (1-0.41) × 79 = 47 cm².
I am trying to get a better intuition for analysis of variance, so I am reading Fisher's 1925 Statistical Methods for Research Workers. In it, he shows how to break down variation into components more directly through an identity involving the sum of squares of deviations:
| Deg. of Freedom | Sum of sq. | Mean sq. | |
|---|---|---|---|
| Within-sex | 14 | 698 | 50 |
| Between-sex | 1 | 484 | 484 |
| Total | 15 | 1182 | 79 |
Then, as far as I understand, Fisher claims that given the between-sex SSE and the total mean square s², we can solve r out of
484 = ns²(1 + (k-1)r)
which is
484 = 2×79×(1 + 7r)
which gives us r = 0.29. (I also get something similar to this when I ask R to produce pairs of measurements and compute the correlation from those.)
This seems to imply that the interclass correlation is a whopping 0.64, while the intraclass correlation is a lower 0.29.
I'm okay with this, because Fisher explains the intraclass correlation as equivalent to the correlation between all pairs of subjects within each group, i.e. all males paired up with all other males, and all females paired up with all other females. It would make sense that the intraclass correlation is lower, because knowing the height of another member of the same class tells us less than knowing the class, because we could get an unlucky member which is not indicative of the main distribution of the class itself.
But, to my question: why is there this disparity between interclass correlation and intraclass correlation in how they split up the variance? From the interclass correlation we had
- between-sex variance 0.41 × 79 = 32 cm², and
- within-sex variance (1-0.41) × 79 = 47 cm².
but the intraclass correlation, which Fisher emphasises better represents analysis of variance, we get
- within-sex variance 0.3 × 79 = 23 cm², and
- between-sex variance (1-0.3) × 79 = 56 cm².
These are somewhat numbers, and it seems like they aren't quite representing the same thing. So what are these numbers representing, and why are they different from each other?
I have talked to an acquaintance about this but they can only say that interclass and intraclass correlations measure different things, they are unable to point to the incorrect mental leap I'm making when I assume both tell apart between-sex variance and within-sex variance. I'm turning to you in the hopes of getting more assistance with that.
