Britto, Anthony

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Letter to the editor: Discussion of proposed t-statistic in“ppcor: An R Package for a fast calculation to semi-partialcorrelation coefficients,” CSAM 2015; 22:665–674Anthony Britto1,aaChair of Energy Economics, Karlsruhe Institute of Technology, Germany1. LetterDear Prof. Seongjoo Song,This letter concerns the article ppcor: An R Package for a fast calculation to semi-partial correla-tion coefficients by Prof. Seongho Kim, which appears in the November 2015 issue of Communica-tions for Statistical Applications and Methods Kim (2015). I believe that there is a discrepancy whichmerits a detailed discussion between the t-statistic of the semi-partial correlation coefficient proposedby Prof. Kim, and the one proposed by Cohen et al. (2002).Equation (2.8) in Prof. Kim’s article lists the t-statistics of the partial and semi-partial correlationcoefficients respectively asti j|S = ri j|S√n − 2 − g1 − r2i j|S, (1.1)ti( j|S ) = ri( j|S )√n − 2 − g1 − r2i( j|S ), (1.2)where ri j|S (resp. ri( j|S )) is the partial (resp. semi-partial) correlation coefficient of the random vari-ables xi and x j with g covariates. Each of these variables are in the vector of random variablesX = (x1, x2, . . . , xn)T ; XS subsequently denotes the random sub-vector of X that results from deletingxi and x j.An immediate consequence of the above formulae, which are identical as functions of ri j|S andri( j|S ), is that since ri j|S , ri( j|S ) in general, the respective t-statistics will also not agree: ti j|S , ti( j|S )(We have ri j|S = ri( j|S ) only in the trivial case where each of the g covariates has zero correlationwith xi; cf. equations (2.1) and (2.2) in Kim (2015)). However, Cohen et al. (2002) argue thatsince the partial and semi-partial correlation coefficients are different scalings of the same statistical1 Chair of Energy Economics, Karlsruhe Institute of Technology, Hertzstr. 16 Building 06.33, Karlsruhe 76187, Germany.E-mail: anthony.britto@kit.eduphenomenon, they must yield an identical t-statistic. In fact, they claim further that the t-statisticof the partial and semi-partial correlation must also equal the one yielded by the βi resulting from amultiple linear regression of xi on x j together with the other g covariates.Their argument is as follows. Consider the case of a random variable y regressed on two predictorsx1 and x2, where all variables have been standardised; the three effect sizes in the previous paragraphfor the predictor x1 for instance are then given byβ1 =ry1 − ry2r121 − r212, (1.3)ry1|2 =ry1 − ry2r12√1 − r2y2√1 − r212, (1.4)ry(1|2) =ry1 − ry2r12√1 − r212. (1.5)Since these effects differ only with regard to their denominators, “none can equal zero unless theothers are also zero, so it is also not surprising that they must yield the same ti value t-statistic for thestatistical significance of their departure from zero” (Cohen et al., 2002).The respective formulae for the t-statistics of these quantities are again found in Cohen et al.(2002) and elsewhere in the literature, for instance in the review article by Aloe and Thompson (2013).I reproduce here the formulae for the partial and semi-partial correlations (the formula for the t-statistic of the βi is presented in the appendix), reverting now to the setup and notation of Prof. Kimabove:ti j|S = ri j|S√df1 − r2i j|S, (1.6)ti( j|S ) = ri( j|S )√df1 − R2i| jS, (1.7)where df is the degrees of freedom, and R2i| jS the coefficient of determination of the linear model wherexi is regressed on x j and other covariates,x̃i = β j x̃ j +∑xk∈Xgβk x̃k (1.8)for Xg ⊂ X the set of g covariates. The tilde indicates that the variables have been standardised; sincethis means that the intercept vanishes, we havedf = n − g − 2. (1.9)Comparing equations (1.6) and (1.7) with equations (1.1) and (1.2), we see that the sole difference toProf. Kim’s article is in equation (1.2), namely, that r2i( j|S ) would need to be replaced with R2i| jS .Table A.1: Description of variables in Duncan’s datasetincome Percentage of occupational incumbents in the 1950 US Census who earned $3,500 or more per year (about$36,000 in 2017 US dollars).education Percentage of occupational incumbents in 1950 who Ire high school graduates.prestige Percentage of respondents in a social survey who rated the occupation as “good” or better in prestige.A trivial consequence of this, but one worth noting, is that such a change would automatically sim-plify equation (2.9) in Prof. Kim’s article: since ti j|S = ti( j|S ), the partial and semi-partial correlationcoefficients would then have identical p-values:pi j|S = 2Φ(−∣∣∣ti j|S ∣∣∣ , df) = 2Φ (− ∣∣∣ti( j|S )∣∣∣ , df) = pi( j|S ) , (1.10)where Φ(·) is the cumulative density function of a Student’s t distribution with degrees of freedom dfas in equation (1.9) above.I present an example calculation within the framework of Cohen et al. (2002) in the appendix,demonstrating that their formulae do in fact work as claimed. I extend my gratitude to Prof. Kim forhis R package, and look forward to a productive discussion on this discrepancy.Sincerely,Anthony BrittoAppendix: Example calculation: Duncan’s occupational prestige dataI demonstrate here that the formulae of Cohen et al. (2002) work as claimed; I employ a standard,freely-available data set, Duncan’s Occupational Prestige Data, available in the carData package inR Fox et al (2020). The data quantifies the prestige and other characteristics of 45 U.S. occupations in1950; descriptions of the three columns used in the analysis are found in Table A.1.The correlation matrix of the three variables is computed as follows:income education prestigeincome 1.0000 0.7245 0.8378education 0.7245 1.0000 0.8519prestige 0.8378 0.8519 1.0000From this matrix, it is possible to directly compute the β’s and the partial and semi-partial correlationsusing equations (1.3) through (1.5) above, where I set income to be the y variable and educationand prestige to be x1 and x2 respectively (see table below).The t-statistics of the partial and semi-partial correlations can then be computed from equations(1.6) and (1.7); for equation (1.7), we need the coefficient of determination of the model where y isregressed on x1 and x2. Using the method of least squares, this is found to be R2y = 0.7023 (df =45 − 2 = 43). As claimed by Cohen et al. (2002), the t-statistics of the partial and semi-partialcorrelation coefficients agree; for the variable prestige for instance, we see thatty2|1 = ry2|1√df1 − r2y2|1= 0.6111√431 − 0.61112= 5.0625≈ 0.4212√431 − 0.7023= ry(2|1)√df1 − R2y= ty(2|1). (A.1)This should be contrasted with what occurs if the Prof. Kim’s expression, equation (1.2), is employed:ty(2|1) = ri( j|S )√n − 2 − g1 − r2i( j|S )= 0.4212√431 − 0.4212= 3.6304. (A.2)Finally, for the t-statistic of the βi we have the usual formulat(βi) =βis.e.(βi), (A.3)where s.e.(βi) is the standard error of the estimate Cohen et al. (2002),s.e.(βi) =√1 − R2ydf(1 − R2i|J). (A.4)Here, R2y is as above, and R2i|J is the coefficient of determination of the model where the ith predictoris regressed on all other predictors. In the case of a model with just two predictors, R2i| j = r2i j, and thestandard errors of the coefficients are identical: 0.1589 in this case. Hence, from equations A.3, andthen 1.10, I finally arrive at the following coefficient table, which neatly summarises that these effectsizes exist on different scales, but nevertheless share their statistical significance.βi ryi| j ry(i| j) t–stat. p–valueeducation 0.0393 0.0377 0.0206 0.2473 0.8058prestige 0.8043 0.6111 0.4212 5.0625 0.0000ReferencesAloe A and Thompson C (2013). The synthesis of partial effect sizes, Journal of the Society for SocialWork and Research, 4, University of Chicago Press.Cohen J, Cohen P, West S, and Aiken L (2002). Applied Multiple Regression/Correlation Analysis forthe Behavioral Sciences (3rd ed.)(pp64–101), Lawrence Erlbaum Associates, Inc.Fox J, Weisberg S, and Price B (2020). carData: Companion to applied regression data sets, R packageversion 3.0-4.Kim SH (2015). ppcor: An R Package for a fast calculation to semi-partial correlation coefficients,Communications for Statistical Applications and Methods, 22, 665-674.Vallat R (2018). Pingouin: Statistics in Python, Journal of Open Source Software, 3.

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Letter to the editor: Discussion of proposed t -statistic in "ppcor: An R Package for a fast calculation to semi-partial correlation coefficients," CSAM 2015; 22:665-674

Letter to the editor: Discussion of proposed t -statistic in "ppcor: An R Package for a fast calculation to semi-partial correlation coefficients," CSAM 2015; 22:665-674

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