Dear colleages,
Eta squared is dominantly used in reflecting the explaining power the same manner as a squared partial correlation coefficient (R2p ) from multiple linear regression: as the proportion of remaining variance in the dependent variable. In the binary case, eta equals product-moment correlation coefficient (PMC), and PMC is known to underestimate the true correlation in an obvious manner. Hence, consequently, eta squared seems to understimate the true explaining power in an obvious manner.
I have studied some options as alternatives for PMC. Goodman - Kruskal gamma (G) seems quite an appealing alternative in the binary case: it reaches correctly the extreme values 1, 0, and -1, and it is robust against many sources of systematic mechanical errors in the calculation process (such as proportions of 1s and 0s, discrepancy between the scales of the variables, and the distribution of the latent variables). Because G seems to be an appealing alternative to PMC (and eta), maybe G squared could be an option for eta squared?
My question is: how should we interprete G squared? Because G reflects strictly the probability rather than covariance, it may not be interpreted as the proportion of remaining VARIANCE in the dependent variable. But what could it reflect? Ideas?
Hello Jari,
Apologies in advance if I've misunderstood your query.
For those who don't know, the gamma statistic estimates (Pc - Pd) as a proportion of all non-tied cases (Pc + Pd), (where Pc is probability of a concordant pair of cases in the data set, and Pd is the probability of a discordant pair; concordance means that if case i is rated/ranked higher on variable X than case j, then so must case i be rated/ranked higher on variable Y than case j).
As such, a squared gamma conveys no special meaning (as would a squared correlation coefficient, such as Pearson's r). Gamma itself conveys the proportion information (relative to non-tied cases). As well, since all tied cases are ignored, there is an obvious tendency/bias towards higher absolute values of gamma as the number of ordinal categories for each variable decreases. For these reasons, I wouldn't rely on gamma squared as a useful alternative for variance accounted for indicators.
Good luck with your work.
Thanks David. Appreciate that you reacted. I guess you got the point (I mean, undestanding properly my point).
I'm a bit more positive of the interpretation of G when it comes to probability. Yes, it gives more liberal estimate than Somers' D for the probability that two randon pairs in two variables are in the same order; while D proportions the number of pairs of observations in the same direction (P - Q) with ALL possible pairs (P+Q+tied pairs), G proportions P - Q with RELEVANT pairs, that is, with the pairs were we KNOW the direction (P+Q). The logic in G is same as with Sign test and Wilcoxon signed-rank test; the latter ones also omit the tied pairs when we calculate the exact probabilities. So, I share the thinking of those colleagues who do not take the liberal character in G as a crucial matter. What we usually do not think (or even know), that G is, factually, a directional measure to the same direction as eta squared is usually used in GLM settings (that is, it corresponds only one direction of Somers' D). That may be one reason for its higher-than-we-expect value. To me, this hidden directionality came from bushes as a side result of the comparisons (if interested in this, I can share some derivations).
I'm aware that "probability squared" may not have any traditionally accepted or used interpretation. I was just pondering what COULD that interpretation BE. What would it reflect - if anything? If you would make a free-of-responsibility first-idea or to give civilized quess, what could come close to the interpretation of "probability squared"? Do we have any analogues in some other areas of statistics or practices? Do we take second powers of some other statistic that is parallel to "probability"? I may add also a crucial question, WHY we interprete the "correlation squared" as "proportion of remaining variance"?- How we came up this this interpretation? Could we interprete "probability squared" as "proportion of remaining probability"?
Hello again Jari,
Jumping to your last question, the obvious answer (to me) as to why the squared correlation is interpreted as proportion of shared variance stems from the fact that the sum of squared deviations of one variable, Y, from any X & Y pair, can be shown to be divisible into the sum of squares of regression plus the sum of squares of errors (Y -Yestimate). From there, r^2 can be shown to be the proportion ss(reg) / (ss(reg) + ss(resid)); in other words, a proportion of total variance (in Y) that is shared with X (or vice versa).
The next to the last question, what would you call a proportion/probability squared, is a bit more challenging. In probability, multiplied values are usually used as estimates of joint probability of independent events. I'll have to think about that one, as nothing obvious leaps to mind.
What might be useful, though, is if you could show that the distribution of Gsqrd is more linear, interval, or whatever goal is of interest than the distribution of G (which is why Fisher's z transform for Pearson r is so helpful). Maybe simulation of G values based on fixed marginal distributions?
Good luck with your work.
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