A DETAILED DESCRIPTION OF THE DISCREPANCY IN FORMULAS FOR THE STANDARD ERROR OF THE DIFFERENCE BETWEEN A RAW AND PARTIAL CORRELATION: A TYPOGRAPHICAL
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1 Olkin and Finn Discepancy A DETAILED DESCRIPTION OF THE DISCREPANCY IN FORMULAS FOR THE STANDARD ERROR OF THE DIFFERENCE BETWEEN A RAW AND PARTIAL CORRELATION: A TYPOGRAPHICAL ERROR IN OLKIN AND FINN (995 by Chonda M. Lockwood and David P. MacKinnon Contact: David MacKinnon David.MacKinnon@asu.edu Apil 4, 000 Technical Repot pepaed in conjunction with the manuscipt: MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., Sheets, V., & West, S. G. (000 A Compaison of Methods to Test the Significance of the Mediated Effect. Copy of olkintech.cml.wpd
2 Olkin and Finn Discepancy INTRODUCTION This epot documents the deivation of the vaiance of the diffeence between a aw (o simple and patial coelation. This function is used as a measue of the mediated effect in the manuscipt entitled A Compaison of Methods to Test the Significance, by David P. MacKinnon, Chonda M. Lockwood, Jeanne M. Hoffman, Vigil Sheets, and Stephen G. West. A discussion follows of the discepancies between deivations conducted by the authos of this manuscipt and Olkin and Finn (995 fom which this method was dawn. Olkin and Finn (995 Olkin and Finn (995 pesent a solution fo the vaiance of the diffeence between a simple coelation and the same coelation patialled fo a thid vaiable. This appoach can povide a test of mediation. To the extent that the elationship between an independent vaiable (X and a dependent vaiable (Y is caied though a mediato (M, the coelation between X and Y will be educed when patialled fo M. This function is shown in Equation, whee xy is the coelation between X and Y, my is the coelation between M and Y, and xm is the coelation between X and M. xy xmmy olkin xy xy. m xy ( ( my xm THE MULTIVARIATE DELTA METHOD The multivaiate delta method fo deiving the vaiance of a function equies a covaiance matix of the elements in the function as well as a vecto of patial deivatives of the function with espect to each element. The vaiance estimate is the covaiance matix pe- and post-multiplied by the vecto of deivatives. In the cases descibed in this epot, we have thee elements: xy, my, and xm. Fo any function f of these elements, the multivaiate delta vaiance fomula is in Equation, whee Φ is the vaiance-covaiance matix and a is the vecto of patial deivatives. σ f aφa xy my xm σ σ σ σ σ σ σ σ σ xy xy my xy xm xy my my my xm xy xm my xm xm Vaiances and Covaiances among Coelations Olkin and Siotani (976 pesented the fomulas fo asptotic vaiances of and covaiances among coelations. The fomulas to complete the covaiance matix Φ fom Equation ae given in Equations 3 though 8. xy my xm ( (
3 σ xy σ my σ xm ( xy N ( my N ( N xm ( ( 3 xm xy my xm xy my xm σ xy my + N ( ( 3 my xm xy my xm xy my σ xy xm + N ( ( 3 xy xm my xm xy my xy σ my xm + N Olkin and Finn Discepancy 3 Patial Deivatives - Olkin and Finn The patial deivatives of the Olkin and Finn function (Equation ae listed in Equations 9 though. These will be the elements of a vecto of patial deivatives called a olkin. See Appendix A fo moe detailed explanation of the deivations. olkin aolkin ( ( a a3 xy my xm olkin xm xy ( ( olkin my olkin 3 xm my olkin my xy ( ( xm 3 my xm (3 (4 (5 (6 (7 (8 (9 (0 ( Mathematica Deivations As a check on the deivations, the deivations pesented in Equations 9 though wee also conducted using Mathematica (Wolfam Reseach, 996. The esults of this pogam ae shown in Appendix B and ae identical to Equations 9 though. TYPOGRAPHICAL ERROR IN THE OLKIN AND FINN DERIVATIONS In Olkin and Finn (995, the vaiance fomula is pesented in a simplified manne, athe
4 Olkin and Finn Discepancy 4 than the multivaiate delta pesentation descibed above. The fomula in thei text is epoduced in Equation. A photocopy of the fomula fom thei text (p. 60 is pesented in Appendix C. Note that in Equation, the notation has been changed to be consistent with this epot, i.e. in tems of xy, my, and xm athe than ρ 0, ρ 0, and ρ. Olkin and Finn (995 use 0, and to efe to the vaiables Y, X and M, espectively. olkin text:va( xy xy. m aφa ( my( xm The denominato of Equation was emoved fom the patial deivatives in vecto a pesented in the text of Olkin and Finn (995. The vecto pesented in the Olkin and Finn aticle is epoduced in Equation 3 and in Appendix C, whee page 60 fom the aticle is shown. xy my xm xy my olkin text: a ( my( xm,, my my Note that the numeato of Olkin and Finn s (995 vaiance fomula (Equation looks equivalent to Equation. Because Φ is the same in the two equations, the diffeence is in the vecto a. It appeas that each element in a in the Olkin and Finn text has been divided by the quantity ( ( my xm. Howeve, afte each patial deivative fom Equation 3 was divided by this quantity, the esults do not coespond exactly to the to the multivaiate delta method pesented ealie. The patial deivatives divided by this quantity ae pesented in Equations 4 though 6. atext ( my( xm ( my ( xm a text xy my ( my( xm ( my ( my a3text xy my xm 3 ( my( xm ( my ( xm my 3 Equations 4 and 5 ae equivalent to Equations 9 and 0, espectively, when these quantities ae squaed as they ae in Equations. The discepancy is fo the thid patial deivative. Equation 6 is not equivalent to Equation. Appendix D contains a desciption of whee these methods ae equivalent and whee they ae not. In sum, the discepancy stems fom the thid element of a. We popose that this discepancy stems fom a typogaphical eo in the thid element of a in Olkin and Finn s (995 text on page 60 of the aticle. my ( (3 (4 (5 (6
5 METHOD OF FINITE DIFFERENCES APPROACH Olkin and Finn Discepancy 5 The method of finite diffeences fo the two possible patial deivative solutions was pogammed in SAS (see Appendix E.The pogam clealy showed that the patial deivative in equation is coect.
6 Olkin and Finn Discepancy 6 REFERENCES MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., Sheets, V., & West, S. G. (000. A Compaison of Methods to Test the Significance of the Mediated Effect. Manuscipt submitted to Psychological Methods. Olkin, I. & Finn, J. D. (995. Coelations Redux. Psychological Bulletin, 8 (, Olkin, I. & Siotani, M. (976. Asptotic distibution of functions of a coelation matix. In S. Ikeda (ed. Essays in Pobability and Statistics (pp Shinko Tsusho: Tokyo. SAS Institute, Inc. (996. SAS (Vesion 6. [Compute softwae]. Cay, NC: Autho. Wolfam Reseach, Inc. (996. Mathematica (Vesion 3.0 [Compute softwae]. Champaign, IL: Autho.
7 Olkin and Finn Discepancy 7 APPENDIX A DETAILED DERIVATIONS OF OLKIN AND FINN FUNCTION
8 Olkin and Finn Discepancy 8 f ( ( let a, b, c f a let u bc, let v ( c, let x ( b a u xv a a u a + + xv. Summation ule a+ u xv xv( a+ u ( a + u( xv ( xv. Quotient ule a a + xv a u a + u xv ( + ( ( ( xv 3. Substitution of esults step into step. a xv xv( u ( a u( xv ( xv 4. Deivative of a with espect to itself equals ( 5. Deivative of a u bc 0 constant equals 0 a ( xv 0 b c + ( ( (( b ( c 6. Deivative of a constant equals 0 7. Substitution of steps 5 and 6 into step 4 a b c algebaically 8. Equation educes ( (
9 ( ( Olkin and Finn Discepancy 9 9. Replace oiginal elements. b a u a + + xv. Summation ule a+ u xv xv( a+ u ( a + u( xv ( xv. Quotient ule b a + xv a u a + u xv ( + ( ( ( xv 3. Substitution of esults step into step. b xv( 0 + u ( a + u( xv 0 + ( xv xv( u ( a + u( xv ( xv 4. Deivative of a constant equals 0 Equation educes algebaically ( 5. Deivative of u bc c xf i ixf i- ( xv ( c (( b( b b( c ( b 6. Chain ule Equation educes algebaically b ( ( ( ( b ( c c a + bc b ( c ( b 7. Substitution of steps 5 and 6 into step 4. (( b ( c
10 b ( ( ( b ( c c ab cb ( c ( b ( b ( c ( b ( c [ c( b ( ab cb ] ( b ( c c ab 3 ( b ( c Olkin and Finn Discepancy 0 8. Reduce equation algebaically 9. Replace oiginal 3 elements. ( ( c a u a + + xv. Summation ule a+ u xv xv( a+ u ( a + u( xv ( xv. Quotient ule c a + xv a u a + u xv ( + ( ( ( xv 3. Substitution of esults step into step. c ( ( ( xv 0 + u a + u xv 0 + ( xv xv( u ( a + u( xv ( xv 4. Deivative of a constant equals 0 Equation educes algebaically ( u bc b 5. Deivative of xf i ixf i-
11 Olkin and Finn Discepancy ( xv ( b (( c( c c( b ( c 6. Chain ule Equation educes algebaically c ( ( ( ( b ( c b a + bc c ( b ( c 7. Substitution of steps 5 and 6 into step 4. (( b ( c c ( ( ( b ( c b ac bc ( b ( c ( b ( c [ ( ( ] ( b ( c b c ac bc ( b ( c b ac ( b ( c 3 8. Reduce equation algebaically 9. Replace oiginal elements. 3 ( (
12 Olkin and Finn Discepancy APPENDIX B MATHEMATICA DERIVATIONS
13 Olkin and Finn Discepancy 3
14 Olkin and Finn Discepancy 4 APPENDIX C PAGE 60 FROM OLKIN AND FINN (995
15 Olkin and Finn Discepancy 5
16 Olkin and Finn Discepancy 6 APPENDIX D TYPOGRAPHICAL ERROR IN OLKIN AND FINN (995
17 f ( ( FIRST PARTIAL DERIVATIVE ( ( Olkin and Finn Discepancy 7 - diffeence between the aw and patial coelation Olkin and Finn a ( ( How the quantities ae elated : ( a ( ( f ( a ( ( ( ( ( ( ( ( ( ( ( ( + ( ( ( ( ( ( ( ( ( ( + + ( ( ( ( ( ( + + ( (
18 Olkin and Finn Discepancy 8 f ( ( - diffeence between the aw and patial coelation SECOND PARTIAL DERIVATIVE Olkin and Finn 3 a ( ( How the quantities ae elated : ( a ( ( f ( a ( ( ( ( ( ( ( 3 3 ( ( ( ( 3 ( ( ( ( 3
19 Olkin and Finn Discepancy 9 f ( ( - diffeence between the aw and patial coelation THIRD PARTIAL DERIVATIVE Olkin and Finn 3 a ( ( 3 How the quantities ae elated : ( a3 ( ( f ( a3 ( ( ( ( ( ( ( 3 3 ( ( ( ( 3 3 ( ( ( (
20 Olkin and Finn Discepancy 0 f ( ( - diffeence between the aw and patial coelation ADDENDUM - TYPOGRAPHICAL ERROR IN OLKIN AND FINN A3 If Olkin and Finn a3*, note that two elements have been eplaced with Assuming the typogaphical eo : ( a3* ( ( f ( a3* 3 ( ( ( ( ( 3 ( ( 3 ( ( ( ( 3 3 ( ( ( (
21 Olkin and Finn Discepancy APPENDIX E SAS PROGRAM FOR THE METHOD OF FINITE DIFFERENCES
22 title 'method of finite diffeences veification of deivatives'; data a; input xy my xm; /*Function fo the diffeence between aw and patial coelation;*/ diffxy-((xy-xm*my/((sqt(-my*my*(sqt(-xm*xm; Olkin and Finn Discepancy /*Ou deivative; */ us(my-xm*xy /((sqt(-my*my*((-xm*xm**(3/; /*deivative fom Olkin and Finn 995 page 60';*/ olkin(xy*my-my/((sqt(-xm*xm*((-my*my**(3/; do i.0000 to.000 by.0000; xm.8+i; fnxy-((xy-xm*my/((sqt(-my*my*(sqt(-xm*xm; fdiff(fn-diff/i; output; end; cads;...8 ; poc pint; un; method of finite diffeences veification of deivatives 6 OBS RXY RMY RXM DIFFR US OLKIN I FN FDIFF method of finite diffeences veification of deivatives 7 OBS RXY RMY RXM DIFFR US OLKIN I FN FDIFF
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