Pardon my intrusion but a quotation from Glass & Hopkins (3rd ed, 1996) may help clarify some issues:
| A word or two about the range of possible values for r.bis [biserial r] is in |order. Unlike almost any other commonly used correlation coefficient, r.bis |can sometimes take on values below -1.00 and above +1.00. The only |significance of these extreme values of r.bis is that they reflect either incorrectness |of the assumption that the Y-scores are normally distributed, or sampling |error, when n is small, that produces the markedly platykurtic distribution |(Section 6.10) of Y-values in the sample. The unpleasant fact remains that |values of r.bis exceeding 1.00 are occasionally obtained, especially with |small samples. When n is small, r.bis is a very crude approximation of r; |indeed, it is probably best not to use r.bis when n is small, especially when |p1 differs considerably from .5 [NOTE: when the dichotomous variable |does not have equal numbers of the two values]. | | Another caution pertaining to r.bis results from the fact that if the variable |underlying X [NOTE: the dichotomy] is not normally distributed, or the |relationship is not linear, r.bis and r can differ by as much as .2 (Nunnaly & |Bernstein, 1994, p127). For these reasons, r.bis should not be viewed as an |adequate substitute for r but only as a crude approximation, unless the |underlying assumptions are known to be met and n is large, at least 100. (p136) David Howell holds a comparable but more negative view: | Two other coefficients, to which you are likely to see reference, but are most |unlikely to use, are the biserial correlation and the tetrachoric correlation. In |earlier editions of this book I showed how to calculate those coefficients, but |there does not seem to be much point in doing so anymore. I will simply |explain how they differ from the coefficients I have discussed. | [...] | The biserial is a the direct analog of the point-biserial correlation, except that |the biserial assumes underlying normality in the dichotomous variable. The |tetrachoric correlation is the direct analog of [the phi coefficient - Pearson r |based on two true dichotomies], where we assume underlying normality on |both variables. That is all you really need to know about these two coefficients. (p303, D. Howell, 2010, "Statistical Methods for Psychology", 7th ed). Bottom line: avoid using the biserial r and tetrachoric correlation unless you know that the assumptions for them are met -- even then they are likely to inaccurate estimates of the Pearson r. For those with structural equation modeling (SEM) experience, one might try some of the SEM techniques to estimate the correlations between variables but (a) one really needs to know what one is doing, and (b) one is probably going to need a large sample size for a valid analysis. If one comes across a biserial r that is greater than +1.00 or less than -1.00, then I think one should treat it the same way one might treat a negative Cronbach's alpha or a negative R-squareed: it's an indication that something is seriously wrong and you need to review the validity of your assumptions, the nature of your data, and the suitability of your analysis for the situation. -Mike Palij New York University [email protected] On Tue, 20 Apr 2010 23:31:30 -0500, Jim Clark wrote: > Hi > Following SPSS simulation generates 1000 samples of 100 x y pairs with known > population rho (#r = .9 here), then dichotomizes x to create categorical > predictor c, which is then used to calculate rb, the biserial r (I had to > track down various algorithms for this, but it seems correct ... mean rb, for > example, is very close to rho). Anyway, it illustrates that for extreme > values of rho, rb can in fact exceed 1 (presumably same at other tail). 12 > of 1000 rbs were > 1 in one simulation I ran. Perhaps there are other > factors that also influence likelihood of getting values beyond normal range > for rs (e.g., size of categories). > > input program. > comp #r = .9. > loop samp = 1 to 1000. > leave samp. > loop obs = 1 to 100. > comp x = rv.norm(0,1). > comp y = rv.norm(0,1)*SQRT(1-#r**2) + x*#r. > end case. > end loop. > end loop. > end file. > end input program. > comp c = 0. > if x > -.2 c = 1. > if c = 0 y0 = y. > if c = 1 y1 = y. > > aggre /outfile = * /presort /break = samp > /m0 = mean(y0) /m1 = mean(y1) /p = fgt(c, 0) /q = flt(c, 1) /sy = > sd(y). > > compute z = idf.normal(q, 0, 1). > compute ord = .3989*2.71828**-((z**2)/2). > compute rb = (m1 - m0)*((p*q/ord)/sy). > freq rb /forma = notable /hist. > comp rbx = (rb<-1) or (rb>+1). > freq rbx. > > It is perhaps worth noting that there are other widely used statistics that > produce "impossible" values. The Bonferroni test, for example, can produce > ps > 1 if one computes LSD p x # comparisons (as reported in SPSS, for > example). SPSS rounds these to 1. Perhaps similar convention is adopted for > rb? > > I'm hard-pressed to decide whether to thank Karl for raising this interesting > question, or berate him for taking me away from my marking to do this > exercise! Or perhaps the latter should be a thanks as well? > > Take care > Jim --- You are currently subscribed to tips as: [email protected]. To unsubscribe click here: http://fsulist.frostburg.edu/u?id=13090.68da6e6e5325aa33287ff385b70df5d5&n=T&l=tips&o=2128 or send a blank email to leave-2128-13090.68da6e6e5325aa33287ff385b70df...@fsulist.frostburg.edu
