Good commentary, Michael. Frankly, I am not very fond of any proportion of variance effect size estimate, but the squared partial strikes as especially wicked, especially since most people who use them have no idea what they are.
Cheers, [Karl L. Wuensch]<http://core.ecu.edu/psyc/wuenschk/klw.htm> From: Michael Palij [mailto:m...@nyu.edu] Sent: Saturday, December 09, 2017 7:42 PM To: Teaching in the Psychological Sciences (TIPS) Cc: Michael Palij Subject: Fw: [tips] interpretations of partial eta squared On Fri, 08 Dec 2017 18:05:27 -0800, , Karl Louis Wuensch wrote: > Unless you can justify removing from the denominator >(total variance to be explained) that related to other effects >in the model, you should never, ever, report partial eta-squared >or partial r-squared. If you must report a proportion of variance >statistic, report semi-partial eta-squared / r- squared, known >simply as eta-squared in the context of ANOVA. Using Karls own materials on correlation, let me clarify some of the points that Karl makes above as well as pose a question: Assume we have 3 variables, AR=attitude toward animal rights, MIS=misanthrophy (a dislike of humankind), and IDEAL=Idealism, If we make AR our Y or criterion variable, and Mis = X1 and Ideal = X2, out predictor vriables, one can represent the relationship among the three variables in terms of a Venn diagram as follows: The criterion Y variable AR is subdivided into several components labelled with lower case letters: d = unexplained or error variane a = common variance or covariance of AR and MIS c = common variance or covariance of AR and Ideal b = common variance shared by AR, MIS, and IDEAL The semi-partial correlation coefficient sr identifies the correlation between a single variable and TOTAL variance of AR. In terms of an equation, sr(AR,MIS) = a/(a + b + c + d) sr(AR,MIS) = proportion of TOTAL variance explained by the common variance between AR and MIS. all of the semi-partial correlations have (a + b + c + d) in the denominator of equation used to calculate sr (given above). The semi-partial correlation is sometime referred to as a "part correlation". The semi-partial eta-squared follows a similar logic and the summ of the semi-partial eta-squard values plus the remainer error variance should sum to 1.00 because each sr has the same denominator. Partial correlations differ from semi-partial correlations in a couple of ways but the most important is what they express: semi-partial correlations (technically, its squared values) identifty the common variance between a predictor and the TOTAL variance of the criterion (in this case (AR) while the (full) partial correlations (again, technically its squared values) identify the common variance between a predictor and the UNEXPLAINED variance. The equation for (full) partial correlation for a above is pr(AR,MIS) = a / (a + d) The variance components for b and c (commoned or shared variance between these two variables and the criterion) is removed from the total variance. The question that the (full) partial correlation answers is "What proportion of the remaining unexplained variance is accounted for by the relationship between the criterion and this specific predictor after the systematic variance in criterion that is associated with other predictor is removed from the criterion's variance. The (full) partial correlations squared do NOT add up to 1.00 because they have different denominators (i.e.,[specific effect variance + error variance] and the specific effect variance is either a or b or c). Partial eta squared, following the above logic, describes how much common variance is accounted for by the independent variance of dependent variable's variance that has not been accounted for by the other independent variables. Whether one should use the semi-partial eta-squared or (full) partial eta-squared, I think, depends upon what what question one is asking or which of two reference values one can use, namely, (1) The toatl variance in the criterion or dependent variable (2) The remaining unexplained variance in the crierion or dependent variable. My question to Karl is the following: What did (full) partial correlation ever do to you to make you hold such a potent grudge against ever using them? ;-) >While SPSS does not provide this, it is easily computed as >the effect sum of squares divided by the total (corrected) >sum of squares. Tests of Between-Subjects Effects Dependent Variable: Rating Source Type III Sum of Squares df Mean Square F Sig. Partial Eta Squared Corrected Model 1318.281a 7 188.326 163.344 .000 .893 Intercept 3314.885 1 3314.885 2875.150 .000 .955 DE_Attr 1275.998 1 1275.998 1106.731 .000 .890 Gender 4.068 1 4.068 3.529 .062 .025 Gender * DE_Attr 15.894 1 15.894 13.785 .000 .091 PL_Attr .837 1 .837 .726 .396 .005 DE_Attr * PL_Attr .181 1 .181 .157 .693 .001 Gender * PL_Attr .791 1 .791 .686 .409 .005 Gender * DE_Attr * PL_Attr 4.252 1 4.252 3.688 .057 .026 Error 157.953 137 1.153 Total 4943.000 145 Corrected Total 1476.234 144 So, the plain eta-squared for DE_Atrr = 1275.988/1478.234 = .84 while the partial eta-squared = .890. >SAS will give you a confidence interval for >this estimate. People are still doing confidence intervals? I thought they moved over to credible interval. ;-) >Since your design is 2 x 2 x 2, the effect of interest is a one >degree of freedom effect. In that case, Cohen’s d is almost always a better >effect size estimate, and is easy to calculate from the marginal means and the >pooled standard deviation. Actually, given the dependence of measures of variance on aspects of the experimental design, it would seem a good idea not to use them unless one's design is VERY simple. An article by Richardson (2010) highlights some of these problem, especially the external validity of percent of variance measures as does the Olejnik & Algina (2003) article that will make one cry from all of the additional work (i.e., one eta-squared for manipulated variables, another for grouping on subject attributes like gender -- maybe we should just use Omega squared but think about that after looking at Table 2 and subsequent tables). ;-) Richardson, J. T. (2011). Eta squared and partial eta squared as measures of effect size in educational research. Educational Research Review, 6(2), 135-147. Olejnik, S., & Algina, J. (2003). Generalized eta and omega squared statistics: measures of effect size for some common research designs. Psychological methods, 8(4), 434-447. Avaialable at: https://www.researchgate.net/profile/James_Algina/publication/8968445_Generalized_Eta_and_Omega_Squared_Statistics_Measures_of_Effect_Size_for_Some_Common_Research_Designs/links/0912f51014365bef73000000/Generalized-Eta-and-Omega-Squared-Statistics-Measures-of-Effect-Size-for-Some-Common-Research-Designs.pdf I don't know if the image will make it to the Tips mail archive or the digest but it should get through to folks who receive TiPS directly. -Mike Palij New York University m...@nyu.edu<mailto:m...@nyu.edu> P.S. Thanks to Karl for providing all the stuff. ;-) --- You are currently subscribed to tips as: wuens...@ecu.edu<mailto:wuens...@ecu.edu>. 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