Regression Quantitative A Aptitude & & Business S - PowerPoint PPT Presentation

Regression Quantitative A Aptitude & & Business S Statistics

Regr gress ession on  Regr Regression on is t the he meas measure of of average r ge relat ations onshi hip b p betwe ween t en two or or mor more e var ariab ables i in n ter erms ms of of or origi ginal uni units of of the he dat data. a. Quantitative Aptitude & Business Statistics: 2 Regression

 Regre ression analysis is i is a a statistic ical tool ool to s o stud udy the nat he natur ure e and ex and extent of functional onal r relations onshi hip b p between een two or wo or mor more e var ariab ables and t and to o es estimate t the he unk unknown n val alues of of independ pendent ent v variabl ble. e. Quantitative Aptitude & Business Statistics: 3 Regression

 Depend endent ent v variabl ble e :The Var The Variab able Whi Which i is pr predi edicted on on the he bas basis of of another er v variabl able i e is c called ed Depe Depende dent v var ariab able or or ex expl plained d variable . .  Indepe ndependen ent v var ariable : e :The Var The Variabl ble Which i Whi is us used ed t to o pr predi edict anot another variabl able is c called i ed indep epende endent nt var ariable or e or ex expl plan anatory v var ariab able. Quantitative Aptitude & Business Statistics: 4 Regression

es of Regr gres ession A sion Anal alysis ysis Uses 1.Regr Regress ession on line li fa facili lita tate tes to to pre redic ict the he valu lues of of a dep dependen endent varia riable le fro rom th the gi given ven valu lue of of independ pendent ent variable. 2.Throu ough gh St Stan anda dard Erro rror facili lita tate tes to to obt obtai ain a measu measure of of the he er error or in involv lved in in us using ng the he reg egress ession on line as lin as bas basis fo for estimat mation on. Quantitative Aptitude & Business Statistics: 5 Regression

3.Regre ressio ion c coeffi ficients ts (b xy xy and and b yx yx ) f ) facili litate tes t to calc lcula late te coef oefficien ent of of det deter erminat ation ( n (r 2 ) ) and c and coef oefficient of of cor orrelation on. 4. 4.Re Regr gression Anal n Analysis i is hi highl ghly us usef eful tool ool i in n ec econ onomics and and business. Quantitative Aptitude & Business Statistics: 6 Regression

Distinc inction ion betwee ween C n Correlat latio ion a n and d Regre ressi ssion Correl elation on Regressi ession 1. Correlation 1. Regression measures degree and measures nature and direction of extent of average relationship between relationship between variables. two or more variables. 2.It is a relat ative ve meas asur ure e 2.It is an absolute show owing a ng assoc sociat ation on measure relationship. between v een variabl ables es. Quantitative Aptitude & Business Statistics: 7 Regression

Correl elation on Regressi ession 3. Correlation 3. Regression Coefficient Coefficient is is independent of origin independent of both but not scale. origin and scale. . Correlation 4.Regression Coefficient 4. Coefficient is is not independent of independent of units of measurement. units of measurement. Quantitative Aptitude & Business Statistics: 8 Regression

Correl elation on Regressi ession 5.Correlation 5. Regression equation Coefficient is may be linear or non- linear . lies between -1 and +1. . It is not 6.It is a forecasting 6. forecasting device. device. Quantitative Aptitude & Business Statistics: 9 Regression

Regression lines  Regr egres ession on line ne X on Y on Y = + X a bY  Wher here X= D Depe epend nden ent Var ariable e Y =Inde ndepe pend nden ent var ariab able a= a=intercept pt and and b= b= slope ope Quantitative Aptitude & Business Statistics: 10 Regression

 Anot nother her w way of ay of regr egres ession on l line X e X on on Y ( ) − = − X X b Y Y xy ( ) σ − = − x X X r Y Y σ y Quantitative Aptitude & Business Statistics: 11 Regression

Regre ression c coeffic icie ients  There are two regression coefficients byx and bxy  The regression coefficient Y on X is σ = y b r . yx σ x  The regression coefficient X on Y is σ = x b r . xy σ y Quantitative Aptitude & Business Statistics: 12 Regression

Regre ression c coeffic icie ients  The regression coefficient X on Y is σ = x b r . σ xy y Quantitative Aptitude & Business Statistics: 13 Regression

 Regr egres ession on line ne Y on X on X = + Y a bX  Wher here Y= D Depe epend nden ent Var ariable e  X =Inde ndepe pend nden ent var ariab able  a= a=intercept pt and and b= b= slop ope e Quantitative Aptitude & Business Statistics: 14 Regression

 Another way of regression line Y on X ( ) − = − Y Y byx X X σ ( ) − = − y Y Y r X X σ x Quantitative Aptitude & Business Statistics: 15 Regression

Properties of Linear Regression  Two Regr Two Regression Equa Equations.  Pro roduct ct of f re regre ression c coeffi fficient. t.  Si Sign gns of of Regr Regression Coef n Coefficient and c correlation c coeff ffic icie ient. t.  In Inte ters rsecti tion o of f means.  Slopes . pes . Quantitative Aptitude & Business Statistics: 16 Regression

 Angl ngle e bet between een R Regr egression l n line nes Value of r Angle between Regression Lines a) If r=0 Regression lines are perpendicular to each other. Regression lines are b) If r=+1 or -1 coincide to become identical . Quantitative Aptitude & Business Statistics: 17 Regression

Properties of regression coefficients 1. 1.Same S ame Sign. gn. 2. 2.Bot oth c h cannot annot gr great eater er t than han one one . 3. 3.Independent ndependent of of or origi gin n but but not not of of scal ale e . 4. 4.Arithm hmet etic mean mean of of regr egressi ssion c on coef oeffici cient ents ar are e gr great eater er t than han Cor orrel elat ation c on coef oeffici cien ent. 5. 5.r,bxy bxy and and by byx hav x have e same s ame sign. gn. 6 6 .Cor orrel elat ation c on coef oeffici cient ent i is the he Geomet eometric c Mean Mean (GM) b/ b/w regr egress ssion c n coef oefficient ents. Quantitative Aptitude & Business Statistics: 18 Regression

Independent of origin but not of scale.  Thi his pr proper operty st stat ates t es that hat if the he or original pai pairs of of var variab ables es i is ( s (x, x,y) y) and and if t they hey ar are e change changed t d to o the he pai pair (u, u,v) v), w wher here x=a + x=a + p p u u and and y=c +q y=c +q v or or − q x a = × = b b and u and uv xy p p − q y c = × = b b v vu yx p q Quantitative Aptitude & Business Statistics: 19 Regression

Nor Normal Equa Equation ons  Regr egres ession on line ne Y on X on X = + Y a bX  The two normal Equations are ∑ ∑ = + Y Na b X ∑ ∑ ∑ = + 2 XY a X b X Quantitative Aptitude & Business Statistics: 20 Regression

Calcul culat ate b e byx ∑ ∑ X Y ∑ − XY N = b ( ) ∑ yx 2 X ∑ − 2 X N = − a Y b X Quantitative Aptitude & Business Statistics: 21 Regression

Nor Normal Equa Equation ons  Regression line X on Y = + X a bY  The t The two o nor normal Equat quations ns ar are e ∑ ∑ = + X Na b Y ∑ ∑ ∑ = + 2 XY a Y b Y Quantitative Aptitude & Business Statistics: 22 Regression

Calcul culat ate b e b xy xy ∑ ∑ X Y ∑ − XY N = b ( ) ∑ xy 2 Y ∑ − 2 Y N = − a X b Y Quantitative Aptitude & Business Statistics: 23 Regression

Population Linear Regression Model  Relationship between variables is described by a linear function  The change of the independent variable causes the change in the dependent variable Random andom Slope ope Y-Intercept ercept Error rror + ε + Y = a bx i Dependen ependent Indep ndepend nden ent (Respo pons nse) e) (Expl planat nator ory) Varia Va iable le Varia Va iable le Quantitative Aptitude & Business Statistics: 24 Regression

Sam ample Li e Linea near Regr egression  Using Ordi dinary Leas Least Squa quares (OLS LS), we can find the values of a and b that minimize the sum of the squared residuals: ( ) n n ∑ ∑ 2 − = ˆ 2 Y Y e i i i = = i i 1 1  Partial Differentiate w.r.t parameters a and b then ,we will get the two normal equations ∑ ∑ ∑ ∑ ∑ = + = + 2 Y Na b X XY a X b X Quantitative Aptitude & Business Statistics: 25 Regression

Fr From om the he fol ollowing ng Dat ata C Cal alculate Coeffici cient o of correlation X 1 2 3 4 5 Adver dvertisem emen ent Exp. xp. ( (Rs. l lakh akhs) Y 10 10 20 20 30 30 50 50 40 40 Sal ales es (Rs.lakh akhs) s) Quantitative Aptitude & Business Statistics: 26 Regression