Alternatives To Pearson s and Spearman s Correlation Coefficients


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1 Alteratives To Pearso s ad Spearma s Correlatio Coefficiets Floreti Smaradache Chair of Math & Scieces Departmet Uiversity of New Mexico Gallup, NM 8730, USA Abstract. This article presets several alteratives to Pearso s correlatio coefficiet ad may examples. I the samples where the rak i a discrete variable couts more tha the variable values, the mixture of Pearso s ad Spearma s gives a better result. Itroductio Let s cosider a bivariate sample, which cosists of 2 pairs (x,y). We deote these pairs by: (x, y ), (x 2, y 2 ),, (x,y ), where x i = the value of x for the ith observatio, ad y i = the value of y for the ith observatio, for ay < i <. We ca costruct a scatter plot i order to detect ay relatioship betwee variables x ad y, drawig a horizotal xaxis ad a vertical yaxis, ad plottig poits of coordiates (x i, y i ) for all i {, 2,, }. We use the stadard statistics otatios, mostly used i regressio aalysis: = i, 2 = i 2, y = i, xy = ( xy i i), 2 = i 2, () xi X = yi Y = = the mea of sample variable x, = the mea of sample variable y. Let s itroduce a otatio for the media:
2 X M = the media of sample variable x, (2) Y M = the media of sample variable y. Correlatio Coefficiets. Correlatio coefficiet of variables x ad y shows how strogly the values of these variables are related to oe aother. It is deoted by r ad r [, ]. If the correlatio coefficiet is positive, the both variables are simultaeously icreasig (or simultaeously decreasig). If the correlatio coefficiet is egative, the whe oe variable icreases while the other decreases, ad reciprocally. Therefore, the correlatio coefficiet measures the degree of lie associatio betwee two variables. We have strog relatioship if r [0.8, ] or r [, 0.8]; moderate relatioship if r (0.5, 0.8) or r (0.8, 0.5); (3) Ad weak relatioship if r [0.5, 0.5]. Correlatio coefficiet does ot deped o the measuremet uit, either o the order of variables: (x, y) or (y, x). If r = or , the there is a perfectly liear relatioship betwee x ad y. If r = 0, or close to zero, the there is ot a strog liear relatioship, but there might be a strog oliear relatioship that ca be checked o the scatter plot. The coefficiet of determiatio, deoted by r 2, represets the proportio of variatio i y due to a liear relatioship betwee x ad y i the sample: r 2 SSTo SS Re sid = SSTo =  SSRe sid SSTo (4) where SSTo = total sum of squares = ( y y) 2 = ( yi y) 2 (5) ad SSResid = residual sum of squares = ( y yˆ ) 2 = ( yi yˆ i) (6) with y ˆi = the ith predicted value = a + bx i for i {,2,,} resultig from substitutig each sample x value ito the equatio for the leastsquares lie
3 ŷ = a + bx where b = xy [( x)( y) / ] x ^2 [( x)^2/ ] (7) ad a = Y b X. (8) Obviously: coefficiet of determiatio = (correlatio coefficiet) 2. Two sample correlatio coefficiets are wellkow: ) Pearso s sample correlatio coefficiet, let s deote it by r p r p = xy [( x)( y) / ] x x y y ^ 2 [( )^2/ ] i ^2 [( )^2/ ] (9) which is the most popular; ad 2) Spearma s rak correlatio coefficiet, let s deote it by r 5, which is obtaied from the previous oe by replacig, for each i {, 2,, }, x i by its rak i the variable x, ad similarly for y i. * We propose more alterative sample correlatio coefficiets i the followig ways, replacig i Pearso s formula (9): 3.. Each x i by its deviatio from the x mea: x i x, ad each y i by its deviatio from the y mea: y i  y Each x i by its deviatio from the x miimum: x i x mi, ad each y i by its deviatio from the y miimum: y i y mi Each x i by its deviatio from the x maximum: x max x i, ad each y i by its deviatio from the y maximum: y max y i 3.4. Each x i by its deviatio from a give x k (for k {, 2,, }): x i x k ad each y i by its deviatio from the correspodig give y k : y i y k
4 Not surprisigly, all these four alterative sample correlatio coefficiets are equal to Pearso s sice they are simply related to traslatios of Cartesia axes, whose origi (0,0) is moved to ( x, y ), (x mi, y mi ), (x max, y max ), or (x k, y k ) respectively. Example: Let the variables x, y be give below: x y Table ad their scatter plot: y Graph x ) Calculatig Pearso s correlatio coefficiet: = 357; x = 35.7; = 24.3; y = 2.43; 2 = 8,989; 2 = 2,634.; y = 6,96.8;
5 r p = ) Calculatig Spearma s rak correlatio coefficiet: x y Table 2 ( + 0) i0 x = =.5 = 5.5; 2 = 55; 2 = 385; 2 = 385; y = 377; r s = ) Replacig x i by x i x ad y i by y i y for all i (deviatios from the mea): x y Table 3 Similarly: = 0, 0 because = ( xi x) = x  x + x 2 x + +x 0 x = (x + x x 0 ) 0 x x+ x x = (x + x x 0 ) 0i = 0; 0 = 0; 2 = 6,244.0; 2 =,089.06;
6 y = 2,479.29; r mea = ) Replacig x i, y i by their deviatios from the smaller x: = xx small ad y: = yy small we have a traslatio of axes agai. x y Table 4 = 297; = 3.3; 2 = 5,065; 2 = 2,372.75; y = 5,844.30; r (small) = ) Replacig x i, y i by their deviatios from the maximum: x y Table 5 = 363; = 205.7; 2 = 9,42; 2 = 5,320.3; y = 9,946.20; r (max) = ) Replacig x i by x i x 4 ad y i by y i y 4 (i this case k = 4), (x 4, y 4 ) = (4, 2.): x y Table 6 = 27; = 03.3;
7 2 = 0,953; 2 = 2,56.5; y = 4,720.9; r 4 = r i = for ay i {, 2,, 0}. Similarly if we replace i Pearso s formula (9) ad also gettig the same result equals to r p : 3.5) Each x i by its deviatio from x s media, ad each y i by its deviatio from y s media. 3.6) Each x i by its deviatio from x s stadard deviatio, ad each y i by its deviatio from y s stadard deviatio. 3.7) Each x i by x i ± a (where a is ay umber), ad each y i by y i ± b (where b is ay umber). 3.8) Each x i by x i * a (where a is ay ozero umber ad * is either divisio or multiplicatio), ad each y i by y i * b (similarly for b ad * ). Sice the cases are similar to , let s cosider two examples for the case 3.8: 3.8.) Suppose each x i i the origial example, Table, is divided by 5, while each y i is divided by 2. The: = 7.4; = 62.5; 2 = ; 2 = ; y = 69.68; r (divisio, divisio) = ) Now, let s still divide each x i i Table by 5, but this time multiply each y i with 2. The: = 7.4; = 248.6;
8 2 = ; 2 = 0,536.4; y = 2,766.72; r (divisio, multiplicatio) = So, agai these results coicide with Pearso s. More iterestig alterative correlatio coefficiets [ad give differet results from Pearso s ad Spearma s] are obtaied by doig: A mixture of Pearso s ad Spearma s correlatio coefficiets. 4. We oly replace x i by its rak amog x s, while y i remais uchaged: x rak y Table 7 = 55; = 24.3; 2 = 385; 2 = 2,634.; y = 958.4; r s,p = [ , ] Similarly, as above, let s oly replace y i by its rak amog y s, while x i remais uchaged. x y rak Table 8 = 357; = 55; 2 = 8,989; 2 = 385; y = 2,636; r p,s = [ , ].
9 Both mixture correlatio coefficiets give differet results from Pearso s ad Spearma s, actually they are i betwee. Coclusio: I the samples where the rak i a discrete variable couts more tha the variable values, this mixture of correlatio coefficiets brigs better results tha Pearso s or Spearma s. Referece: Jay Devore, Roxy Peck, Itroductory Statistics, secod editio, West Publ. Co., 994.
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