Direct and indirect effects in dummy variable regression

Transcription

1 mercan Journal of Theoretcal and ppled Statstcs 214; 3(2): Publshed onlne February 2, 214 ( do: /j.ajtas Drect and ndrect effects n dummy varable regresson Oyeka I..., Nwankwo hke H. * Department of Statstcs, Nnamd zkwe Unversty, wka, Ngera Emal address: chkeezeoke@yahoo.com (N. hke H.) To cte ths artcle: Oyeka I..., Nwankwo hke H.. Drect and Indrect Effects n Dummy Varable Regresson. mercan Journal of Theoretcal and ppled Statstcs. Vol. 3, No. 2, 214, pp do: /j.ajtas bstract: Ths paper proposes and develops the use of the non-cummulatve dummy varables of 1 s and s to represent levels of parent ndependent varables n dummy varable multple regresson models. The regresson coeffcents obtaned usng the proposed methods are easer to nterprete and clearly understand than the use of the cummulatvely coded ordnal dummy varables of 1 s and s that could be used for the same purpose. The proposed method also enables the smultaneous estmaton of the total, absolute or overall effect of a parent ndependent varable as well as ts drect effect through ts representatve dummes and ts ndrect effect on a gven ndependent varable through the medaton of other parent ndependent varables n the model was demonstrated. The use of these procedures was llustrated wth an eample. Keywords: Dummy Varables, Total Effect, Drect Effect, Indrect Effect, Parent Varables, Medaton Model 1. Introducton umulatvely coded ordnal dummy varables of 1 s and s have been used to represent ndependent varables n a regresson model [1]. Estmates of the partal regresson effects of these dummy varables on a gven ndependent varable are provded. Interpretaton of these regresson coeffcents s often dffcult and not easly understood, ths s because the regresson coeffcent of an ordnal dummy varable representng a gven level of a parent ndependent varable s nterpreted as the effect on the dependent varable per unt change n the level of the parent ndependent varable represented by that dummy varable n comparsm or relatve to a unt ncrease n the level of the parent ndependent varable represented by an mmedately precedng ordnal dummy varable or per unt decrease n the level of the parent ndependent varable represented by an mmedately succeedng ordnal dummy varable [2]. Ths nterpretaton s rather cumbersome. 2. The Proposed Method n easer to nterprete and understand method s to perhaps use the more regular non-cumulatve dummy varables of 1 s and s to represent levels of parent ndependent varables n a regresson model. It may also be of nterest to use ths method to estmate the total, absolute or overall effect of a parent ndependent varable as well as ts drect effect through ts representatve dummes and ts ndrect effect on a gven ndependent varable through the medaton of other parent ndependent varables n the model as dscussed below. Here, the dependent varable may or may not be quanttatve. The dependent varable or the so-called parent ndependent varables may also each be ether a quanttatve or qualtatve varable. ut for the present purpose, each of these parent ndependent varables that s not already categorcal s to be parttoned nto a number of mutually eclusve categores, classes or levels. Now suppose y s the score, value, observaton on th subject on a crteron or dependent varable Y, for 1, 2,, n. Suppose further that the effects of characterstcs,,, etc, wth levels a, b, c, of each subject wth respect to the dependent varable are of nterest. To use these parent ndependent varables n a dummy varable regresson model, we would represent each of them wth one dummy varable of 1 s and s less than the number of ts levels or categores. Ths s to ensure that the resultng desgn matr s of full column rank and hence non-sngular ([2],[3], and [4]). Thus f the parent ndependent varable has a levels coded 1, 2,.., a, then these a levels are represented by a-1 dummy varables of 1 s and s n the regresson model, that s by the dummy varables 1 ;, 2;,..., a for the th subject, 1, 2,, n. Other parent ndependent varables are smlarly represented. Thus factor wth b levels s represented by b 1 dummy varables of 1 s and s and factor wth c levels s represented by c 1 dummy varables of 1 s and s,

2 mercan Journal of Theoretcal and ppled Statstcs 214; 3(2): and so on. Now a dummy varable multple regresson model epressng the dependence of scores or observatons y drawn from the crteron varable Y on the parent ndependent varables, and represented by dummy varables, 2;,..., a,..., c, s epressed as y + e (1) + + 2; 2; a a c c +... where j s are partal regresson coeffcents and e s are error terms uncorrelated wth j s wth ( ) E, for 1, 2,, n. The partal regresson coeffcent j of the dummy varable jrepresentng the jth level of a gven parent ndependent varable s nterpreted as the change n e the dependent varable per unt change n the jth level of that parent ndependent varable relatve to or n comparsm wth ts other levels holdng all other parent ndependent varables n the model at constant levels [2]. The epected value of y of equaton (1) s E ( y ) + + 2; 2; a a c c +... (2) Use of the usual least squares method wth equaton (1) gves the ftted or predcted dummy varable multple regresson model as yˆ b b (3) + b + b2; 2; ba a + b c c for 1, 2, n Note that f n equaton (1) or (3) we set some selected dummy varables equal to 1 and some equal to, several other models and estmates of the parameters of these models descrbng the dependence of the crteron varable on varous combnatons of the levels of the dfferent parent ndependent varables used n the model are obtaned, thereby further hghlghtng the tremendous versatlty and usefulness of dummy varable regresson models n statstcal modellng. Use of the usual F test enables one assess the adequacy of a hypothessed model n correctly descrbng the true pattern of relatonshps between the dependent varable and the set of parent ndependent varables used n the model. If the model fts, that s, f the model s adequate, leadng to a rejecton of the null hypothess, n whch case not all the regresson coeffcents are zero, then one may proceed to test other hypotheses and also estmate addtonal parameters of nterest, ncludng absolute drect and ndrect effects of parent ndependent varables on the crteron varable. Now to estmate the so called drect effect [5] whch s actually the partal regresson coeffcent or effect of a gven parent ndependent varable say wth z levels on a crteron or dependent varable Y, we treat the dummy varables representng the parent ndependent varable as ntermedate varables between and Y n a regresson model. Then followng the method of path analyss we obtan the requred drect effect of on Y as a weghted sum of the partal regresson coeffcents of the dummy varables used as regressor on the dependent varable. For eample, we obtan the drect effect dr; of the parent ndependent varable on each of ts representatve dummy varables j ;, 1, 2, n; j 1, 2,.., a - 1 Specfcally, to determne the partal regresson coeffcent or the so-called drect effect of the parent ndependent varable on the dependent varable Y, we take the partal dervatve of the epected value of y of equaton 1, that s of equaton 2 wth respect to, obtanng ( ) ( ) ( 2; ) ( a ) ( s; ) y d d d d d dr; 2;... a S; s s 1, 2,.. for all parent ndependent varables dfferent from. or Snce s a 1 dr; j 1 ( ) d (4) ( ) d s; S; ( ) s; because d ( ) Now the weght α, to be appled to d, the partal regresson effect of the dummy varable ; representng the jth level of the parent mpendent j varable on the dependent varable Y s obtaned by fttng a smple regresson model of regressng on j ;

3 46 Oyeka I... and Nwankwo hke H.: Drect and Indrect Effects n Dummy Varable Regresson for j 1, 2,, a-1 usng assgned numercal codes. Thus for the dummy varable j ; representng the jth level of the parent ndependent varable we obtan the weght α jby fttng the smple regresson lne α + (5) j ; ; α for 1, 2, n; j 1, 2,, a -1 and E(e ) Now takng the partal dervatve of the epected value ; of equaton 5 wth respect to, we obtan of j ( ) d α for j 1, 2,, a -1 (6) The drect effect whch s actually the partal regresson effect of the parent ndependent varable on the dependent varable Y s now obtaned by usng equaton 6 n equaton 4 as a 1 dr; j 1 α (7) whose sample estmate s obtaned as a weghted sum of the sample estmates of the partal regresson coeffcents as ˆ j ; b j ; a 1 dr; j 1 b α b (8) n advantage of usng dummy varables to represent ndependent varables n a multple regresson model s that t enables separate estmaton of the partal effect of each level or category of a parent ndependent varable on a dependent varable whch clearly provdes addtonal nformaton. It also enables the smultaneous estmaton of not only the drect effects as we have already seen, but also the total or absolute effect and the ndrect effect of a parent ndependent varable on a dependent varable through the medaton of other parent ndependent varables n the regresson model. The ndrect effect of a gven parent ndependent varable on a dependent varable s the dfference between ts total or absolute effect and ts drect effect through ts representatve dummy varables. The total or absolute effect tself s the smple regresson coeffcent or regresson effect of the parent ndependent varable usng drectly ts assgned numercal codes on the dependent varable. Thus the ndrect effect nd; of the parent ndependent varable on a dependent varable Y through the medaton of other parent ndependent varables n the model s estmated as ts total or absolute effect b t ; less ts drect effect b dr ;. That s b nd; bt; bdr; (9) where b t ; s the sample estmate of the smple regresson coeffcent or effect of the parent ndependent varable usng ts numercal codes on the dependent varable Y. The total, drect and ndrect effects of other parent ndependent varables n the model (Equaton 1) are smlarly estmated. 3. Illustratve Eample researcher s nterested n estmatng the effects of maternal age (), mother s body weght (W) and her party (P) has on brth weght ( y) of her most recent lve brth. She collected a random sample of 25 newly delvered mothers as shown n table 1 below Table 1: hld brth weght and some demographc factors of a random sample of 25 mothers S/N Mother s ge () Party (P) Mother s ody Weght (W) aby s Weght (;y) To use dummy varable multple regresson methods wth data of table 1 above we frst partton age of mother () nto three groups or levels as (1) < 25years (2) years (3) 3 years[(1), (2) and (3) beng parent ndependent varables representng the varous age ranges]; Party nto three classes or groups as (1) or 1, (2) 2 3 (3) 4 or more, [(1), (2) and (3) beng the parent ndependent varables for ndcated party]. Mother s body weght s grouped nto two classes (1) 7kg, (2) > 7kg. [Smlarly, (1) and (2) are parent ndependent varables for the two weght ntervals]. Hence maternal age () and party (P) each wth 3 levels wll each be represented by 2 dummy varables of 1 s and s whle Mother s body weght wth 2

4 mercan Journal of Theoretcal and ppled Statstcs 214; 3(2): levels wll be represented wth only 1 dummy varable of 1 s and s n the dummy varable regresson model. The resultng desgn matr s Table 2: Desgn Matr X of Dummy varables for the data on table 1 S/No X X 2; X P X 2;P X W aby s Weght Hence the ftted dummy varable multple regresson model epressng the dependence of chld brth on maternal age, body weght and party represented by dummy varables s yˆ (1).5122;.5122; P +.782; P. 69 W Now, for the drect effects of the parent ndependent varables on Y (e baby weght), we obtan the regresson coeffcents ( α j s) each of on, 2 ; on, Pon ; P, 2 Pon P wth the followng results ; 1 vs yelds ; α.7 (11) 2 vs yelds 2; ; P α.11 (12) 2; α.166 (13) P ; P 2 vs P yelds 2; P.34-.8P ; α.8 (14) 2; P 1 vs P yelds P P lso 1 W vs W yelds 1W ; W α.46 (15) W Thus, the drect effect of mothers age () on the baby s weght (y) from equaton (8) usng equaton (1) and equatons (11) and (12), s (.622.7) + (.5.11). 49 b dr ; (16) Smlarly, the drect effect of party (P) on baby s weght (y) usng equatons (8), (13) and (14) s

5 48 Oyeka I... and Nwankwo hke H.: Drect and Indrect Effects n Dummy Varable Regresson ( ) + (.78.8). 41 b (17) dr; P nd fnally, the drect effect of mother body weght (W) on baby s weght (y) usng equaton (8), (1) and (15) s b (18) dr; W For the total effects, the dependent varable, baby s weght (y), s regressed on each of the parent ndependent varables 1, 2 and 3 for the three levels of mother s age (), another 1, 2 and 3 parent ndependent varables for the three levels of party (P) and two levels of mother s body weght (W) represented by 1 and 2. These regressons produced the followng equatons and Y (for mother s ge) (19) Y P (for Party) (2) Y W (for Mother s Weght) (21) The coeffcents for, P and W respectvely from equatons (19), (2) and (21) are the total effects of age, party and mother s weght. The ndrect effects for mother s age, party and mother s weght on baby s weght are obtaned usng equaton (9), thus ndrect effect for mother s age s b b b (22) nd; t; dr; For party, t s b nd; p bt; p bdr; p (23) and lastly, the ndrect effect of mother s weght on baby s weght s 4. oncluson b (24) nd; w We have presented a method for the estmaton of total or absolute effects and the drect effect of parent ndependent varables on a dependent varable through the effects of ther set of representatve dummy varables of 1 s and s as well as ndrect effects of these parent ndependent varables through the medaton of other parent ndependent varable n a dummy varable regresson model. It s shown that an advantage of usng these dummy varables of 1 s and s s the ease of understandng and nterpretaton of the resultng estmated regresson coeffcents n comparson wth the regresson coeffcents obtaned when cummulatvely coded ordnal dummy varables of 1 s and s are used n such models. References [1] Oyeka I... and Nwankwo,. H. (212): Use of Ordnal Dummy Varables n Regresson Models. IOSR Journal of Mathematcs, Vol. 2, Issue 5 (Sep Oct 212), pp 1 7. [2] oyle, R. P. (197): Path nalyss and Ordnal Data. mercan Journal of Socology, 47, 197, [3] Oyeka I... (1993): Estmatng Effects n Ordnal Dummy Varable Regresson. STTISTI, anno L.111, n.2 pp [4] Neter, J., Wasserman, W. and Kutner, M. H. (1983): ppled Lnear Regresson Models. Rchard D. Irwn Inc, Illnos. [5] Wrght, S. (196): Path oeffcents and Path Regresson: lternatve to omplementary oncepts. ometrcs, Volume 16, pp