What exactly is the "proportion of variability explained"?

One often hears to say "more than 70% variability is explained by ..." What exactly is meant by this? Th proportion of the sum of squares (SSE), or mean sum of squares (MSE)? For example in the following anova table:

                                    Df Sum Sq Mean Sq F value Pr(>F)
as.factor(site)                    444   8357   18.82   163.1 <2e-16 ***
as.factor(year)                     12    569   47.43   410.9 <2e-16 ***
as.factor(month)                     5    863  172.53  1494.8 <2e-16 ***
as.factor(year):as.factor(month)    60    769   12.82   111.1 <2e-16 ***
Residuals                        34188   3946    0.12
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
7176 observations deleted due to missingness


could we say that most of the variability was explained by site? We see that site covers most of the SSE but as there is a lot of sites, the MSE for site is almost the lowest in the table.

And how would I interpret this in practice? I want to know where is the variability, whether it varies mostly accross time or space. Is the site actually the biggest source of variability, or is it a month and year? Shall I read SSE or MSE column for this purpose?

PS: please note I am not a professional statistician, so if you are about to respond with a lot of math then please make also some simple summary for dummies :-)

• possible duplicates Commented Aug 2, 2013 at 17:55
• my question is not purely technical as the linked one; it has a bit of the interpretational/philosophical level. I don't stick to particular technology like anova; I try to find proper means to answer my question. Commented Aug 2, 2013 at 19:30
• you use % in SSE but not in MSE which is what you are looking in the first part of the question. Commented Aug 2, 2013 at 19:34
• Thanks @Metrics! And what do you think is more appropriate to consider, if I want to answer the question whether my variable varies mostly in time or in space (i.e. accross sites)? Commented Aug 2, 2013 at 20:13
• I think it should be site/space since SSE is higher for that. Commented Aug 2, 2013 at 20:16

1 Answer

When you hear "more than 70% variability is explained by ...", the speaker is referring to the sums of squares (SS), not the mean squares (MS). I should note that exactly what they mean is not certain; they could be referring to either eta-squared or partial eta-squared:
\begin{align} \eta^2&=\frac{SS~IV_j}{SS~Total} \\ ~\\ ~\\ \eta^2_\text{partial}&=\frac{SS~IV_j}{SS~IV_j+ SS~Residuals} \end{align} Part of the reason why is that the SS can be partitioned (at least if you are using type I SS, see here), but the MS cannot.

You raise a good point that there is more opportunity for a given factor to contribute to the variability in the response when there are more groups in that factor (this assumes, of course, that there is real variability in the levels of the factor). Many people forget, or are ignorant of, this fact. Unfortunately, it is not possible to get around this issue. The implication of this is that the question 'which factor is most important' may not be answerable in an absolute sense, but only relative to something else.

• Thanks gung! So what (SSE or MSE) do you think is more appropriate to consider, if I want to answer the question whether my variable varies mostly in time or in space (i.e. accross sites)? You can use my example for explanation. Thanks! Commented Aug 2, 2013 at 21:04
• It is more appropriate to use SS, not MS. Commented Aug 2, 2013 at 21:13