On the Meaning of Regression Coefficients for Categorical and Continuous Variables: Model I and Model II; Effect Coding and Dummy Coding

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1 Dt_nlysisclm On the Mening of Regression for tegoricl nd ontinuous Vribles: I nd II; Effect oding nd Dummy oding R Grdner Deprtment of Psychology This describes the simple cse where there is one ctegoricl vrible (with levels) nd one continuous vrible, though the observtions cn be generlized to ny number of vribles nd levels In this sitution, we could consider the groups nd the results of bivrite regressions computed on ech one This would yield the following: Gp Gp Gp Y b + b Y b + b Y b + b Using multiple regression with Effect or Dummy oding, we could generte regression equtions for I or II I is the unique sum of squres pproch where ll vribles re entered into the regression eqution simultneously II is hierrchicl pproch where the min effects re entered first nd the two wy interction is entered on the next step This could be extended for more vribles with no loss in generlity Exmples of the coding would be s follows In this exmple, two vectors (A, nd A) re needed to define the three groups, is the continuous vrible (centered or not), nd A nd A would represent the products of the A nd vectors Only observtion is shown in ech group (nd no vlue of is shown becuse this could chnge for ech observtion in ech group), but obviously there would be more The generliztions tht follow pply to both equl nd unequl smple sizes Group Effect oding Dummy coding A A A A A A A A I For I, ll terms re entered t once, nd the regression eqution is: Y b + ba+ ba + b + b4a + b5a The following tble illustrtes the precise mening of the regression coefficients (in terms of the univrite regression coefficients shown for ech group in the first tble presented) when both Effect coding nd Dummy coding is used This is done for explntory purposes In either

2 Dt_nlysisclm cse, the multiple correltion will be identicl but the regression coefficients will tke on different vlues, nd the tests of significnce of the squred multiple semi-prtil correltions for the A vectors nd the continuous vrible will lso differ In fct, Dummy coding should not be used for I becuse it produces incorrect tests of significnce for the min effects of the ctegoricl fctor (ie, the F-rtio for the squred multiple semi-prtil correltion) oefficient Effect coding Dummy oding b b + b + b b b b b - b b - b b b - b b - b b b + b + b b b b 4 b - b b - b b 5 b - b b - b It will be noted tht the regression coefficients for Effect coding describe effects, while those for Dummy coding describe contrsts with the group coded with ll 's (group in this cse) II For II, the effects re entered hierrchiclly, beginning with the min effects only Thus, the first step in II, would produce the following regression eqution: Y b + ba+ ba + b Note in this cse, there is no product term involving A nd, hence only the menings for the regression coefficients for b b nd b given in the bove tble pply Tht is, the sme definitions re pplicble but the mening of the terms b, b, nd b chnge, nd the vlue of b is the within cells regression coefficient (b w ) for the groups This vlue is: bw n ( i )( Yi Y) n ( i )² bs ²( n ) S²( n ) This stge of the nlysis is simply n nlysis of covrince where group is the tretment vrible nd is the covrite Thus:

3 Dt_nlysisclm b the djusted men of Y for group b the djusted men of Y for group b the djusted men of Y for group As consequence, the vlues of the regression coefficients will chnge between Effect coding nd Dummy coding, but the tests of significnce will be the sme The second step in II dds the product terms, so tht the definition of the regression coefficients t this stge re identicl to those given erlier for I For obvious resons, the vlues of the regression coefficients for b, b, b, nd b re different from wht they were in the eqution generted by step Numericl Illustrtion - I The following exmple is bsed on dt from groups with dependent vrible, Y, nd centered continuous vrible, A Bivrite regression nlysis for ech of the groups yields the following sttistics Group Y men A men Intercept Slope n Stndrd Devition The following tbles present the regression coefficients for I for both Effect coding nd Dummy coding These results re used to illustrte the formule given previously I Effect oding (onstnt) b b b b Dependent Vrible: x Unstndrdized Stndrdized B Std Error Bet t Sig

4 Dt_nlysisclm 4 I Dummy oding (onstnt) d d d d Dependent Vrible: x Unstndrdized Stndrdized B Std Error Bet t Sig lose inspection of these tbles will demonstrte tht lthough both nlyses yield the sme squred multiple correltion (674), the regression sttistics generted by Effect coding re different from those for Dummy coding Following illustrtes the precise mening of these sttistics in terms of the regression coefficients obtined in the bivrite solutions oefficient Effect coding Dummy oding b b b b b b In the regression tble for effect coding presented erlier, significnt vlues were obtined for the constnt (ie, b ), the continuous vrible,, (ie, b), the first ctegoricl vrible, b (ie, b ), nd the product, b (ie, b 5 ) Thus, these vlues suggest tht: the intercept, 7986 (the vlue predicted when ) is significntly different from, the difference between the intercept for group nd the men of the intercepts (ie, b ) is significntly different from, the men slope (ie, b 5) is significntly different from, nd 4 the slope for group differs significntly from the men of the slopes (ie,

5 Dt_nlysisclm 5 The results for Dummy coding lso indicte significnt effects for the constnt,, d, nd d, but the vlues re different nd their menings re very different These results suggest tht: the intercept for group, 9 is significntly different from, the difference between the intercept for group nd the intercept for group (ie, b ) is significntly different from, the slope for group (ie, b 59) is significntly greter thn, nd 4 the slope for group is significntly less thn the slope for group (ie, II The following tbles present the regression coefficients for II for both Effect coding nd Dummy coding II Effect oding (onstnt) b b (onstnt) b b b b II Dummy oding Dependent Vrible: x Unstndrdized Stndrdized B Std Error Bet t Sig

6 Dt_nlysisclm 6 (onstnt) d d (onstnt) d d d d Dependent Vrible: x Unstndrdized Stndrdized B Std Error Bet t Sig For II, the regression coefficients for the first step re different thn those in I, even though the menings re the sme s demonstrted bove The difference, of course, is becuse the initil estimtes re different Tht is, the regression coefficient for the continuous independent vrible () is the within cells regression coefficient The vlue is: ( 8)( 5)( 6 469²) + ( 69)( 7)( ²) + ( 59)( 4)( 4 645²) ()( ²) + ()( ²) + ()( ²) nd the intercepts for ech group re the djusted mens for tht group Tht is: b the djusted men of Y for group 6 - (-667-) 668 b the djusted men of Y for group (875 - ) 844 b the djusted men of Y for group 8 - (- - ) 855 Using these vlues in the defining formul given erlier for b, b, b, nd b yields the following tble oefficient Effect coding Dummy oding b b b b In this cse, the tests of significnce for Effect coding t the first step indicte tht:

7 Dt_nlysisclm 7 the men intercept (ie, the men of the djusted mens (7787) differs significntly from, the intercept for group differs significntly from the men of the intercepts (ie, b ), the within cells regression coefficient (ie, b ) differs significntly from result On the other hnd, the results for Dummy coding t the first step indicte tht: the intercept (ie, the djusted men for group 855) differs significntly from, nd the within cells regression coefficient (ie, b ) differs significntly from It will be noted tht this is the only plce where the two nlyses describe the sme Finlly, it will be noted tht the results t step hve vlues identicl to those obtined with I

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3.

Treatment Spring Late Summer Fall 0.10 5.56 3.85 0.61 6.97 3.01 1.91 3.01 2.13 2.99 5.33 2.50 1.06 3.53 6.10 Mean = 1.33 Mean = 4.88 Mean = 3. The nlysis of vrince (ANOVA) Although the t-test is one of the most commonly used sttisticl hypothesis tests, it hs limittions. The mjor limittion is tht the t-test cn be used to compre the mens of only

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