LINEAR REGRESSION ANALYSIS. MODULE VIII Lecture Indicator Variables
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1 LINEAR REGRESSION ANALYSIS MODULE VIII Lecture - 7 Indcator Varables Dr. Shalabh Department of Maematcs and Statstcs Indan Insttute of Technology Kanpur
2 Indcator varables versus quanttatve explanatory varable The quanttatve explanatory varables can be converted nto ndcator varables. For example, f e ages of persons are grouped as follows: Group : day to years Group : years to 8 years Group : 8 years to years Group 4: years to 7 years Group 5: 7 years to 5 years en e varable age can be represented by four dfferent ndcator varables. Snce t s dffcult to collect e data on ndvdual ages, so s wll help n easy collecton of data. A dsadvantage s at some loss of nformaton occurs. For example, f e ages n years are,, 4, 5, 6, 7 and suppose e ndcator varable s defned as f age of person s > 5 years D = 0 f age of person s 5 years. Then ese values become 0, 0, 0,,,. Now lookng at e value, one can not determne f t corresponds to age 5, 6 or 7 years.
3 Moreover, f a quanttatve explanatory varable s grouped nto m categores, en (m -) parameters are requred whereas f e orgnal varable s used as such, en only one parameter s requred. Treatng a quanttatve varable as qualtatve varable ncreases e complexty of e model. The degrees of freedom for error are also reduced. Ths can effect e nferences f data set s small. In large data sets, such effect may be small. The use of ndcator varables does not requre any assumpton about e functonal form of e relatonshp between study and explanatory varables.
4 4 Regresson analyss and analyss of varance The analyss of varance s usually used n analyzng e data from e desgned experments. There s a connecton between e statstcal tools used n analyss of varance and regresson analyss. We consder e case of analyss of varance n one way classfcaton and establsh ts relaton w regresson analyss. One way classfcaton Let ere are k samples each of sze n from k normally dstrbuted populatons only n er means but ey have same varance y = µ + ε, =,,..., k; j =,,..., n j j = µ + ( µ µ ) + ε = µ + τ + ε j j σ. Ths can be expressed as N µ σ = k (, ),,,...,. The populaton dffer where y j s e j observaton for e fxed treatment effect τ = µ µ or factor level, µ s e general mean effect, j are dentcally and ndependently dstrbuted random errors followng N(0, σ ). Note at k τ = µ µ, τ = 0. = The null hypoess s H : τ = τ =... = τ = 0 H 0 : τ 0 for atleast one. k ε
5 5 µ Employng meod of least squares, we obtan e estmator of and as follows: τ S ( y ) k n k n j j = j= = j= = ε = µ τ k n S = 0 ˆ µ = yj = µ nk = j= y where y n = yj. n j = n S = 0 ˆ τ = y ˆ µ = y y τ j n j= Based on s, e correspondng test statstc s F 0 n k ( y y) k = = k n ( yj y ) = j= kn ( ) whch follows F-dstrbuton w k - and k (n - ) degrees of freedom when null hypoess s true. The decson rule s to reject H 0 whenever F0 Fα ( k, kn ( )) and t s concluded at e k treatment means are not dentcal.
6 6 Connecton w regresson To llustrate e connecton between fxed effect one way analyss of varance and regresson, suppose ere are treatments so at e model becomes y = µ + τ + ε, =,,...,, j =,,..., n. j j There are treatments whch are e ree levels of a qualtatve factor. For example, e temperature can have ree possble levels low, medum and hgh. They can be represented by two ndcator varables as f e observaton s from treatment D = 0 oerwse, D f e observaton s from treatment =. 0 oerwse. The regresson model can be rewrtten as where st D : value of D for j observaton w treatment j nd D : value of D for j observaton w treatment. j yj = β0 + βdj + βd j + εj, =,,; j =,,..., n Note at parameters n regresson model are β0, β, β. parameters n analyss of varance model are µτ,, τ, τ. We establsh a relatonshp between e two sets of parameters.
7 7 Suppose treatment s used on j observaton, so D j =, D j = 0 and y = β + β. + β.0 + ε j 0 j = β + β + ε. 0 j In case of analyss of varance model, s s represented as y = µ + τ + ε j j = µ + ε where µ = µ + τ j β + β = µ 0. If treatment s appled on j observaton, en - n regresson model set up, D = 0, D = j j and y = β + β.0 + β.+ ε j 0 j = β + β + ε 0 j. - n analyss of varance model set up, y = µ + τ + ε j j = µ + ε where µ = µ + τ j β + β = µ 0.
8 When treatment s used on j observaton, en - n regresson model set up, D = D = 0 j j y = β + β.0 + β.0 + ε j 0 = β + ε 0 j. j 8 - n analyss of varance model set up y = µ + τ + ε j j = µ + ε j where µ = µ + τ β. 0 = µ So fnally, ere are followng ree relatonshps β + β = µ 0 β + β = µ 0 β = µ 0 β = µ 0 β = µ µ β µ µ =.
9 9 In general, f ere are k treatments, en (k - ) ndcator varables are needed. The regresson model s gven by y = β + β D + β D β D + ε, =,,..., k; j =,,..., n j 0 j j k k, j j where D j f j observaton gets treatment = 0 oerwse. In s case, e relatonshp s β β = µ So always estmates e mean of k treatment and estmates e dfferences between e means of treatment 0 and k treatment. 0 k β = µ µ k, =,,..., k. β
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