Topic 4: Simple Correlation and Regression Analysis

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1 Topc 4: Smple Correlato ad Regresso Aalyss Questo: How do we determe how the chages oe varable are related to chages aother varable or varables? Aswer: REGRESSION ANALYSIS Whch derves a descrpto of the fuctoal ature of the relatoshp betwee two or more varables. Examples: () Lfe-tme eargs ca be explaed by: Educatoal level Job expereces occupato Geder coutry of brth () Age of Death ca be explaed by Paret s age of death Race Healthcare accessblty weght poltcal stuato det () Attedace at hockey games ca be explaed by Rak of team Hstory of pealtes player salary volece durg the game

2 (II) Questo: How do we determe the of ths relatoshp betwee two or more varables? Aswer: ANALYSIS (whch determe the stregth of such betwee two or more varables. Ths topc wll cover the fudametal deas of ecoometrcs. We wll trasform theoretcal ecoomc relatoshps to specfc forms, by () gatherg the to estmate the parameters of these fuctos () test hypotheses about these parameters ad () do predctos. We wll brg together: & Ecoomc & Emprcal & Statstcal Wth regresso aalyss we estmate the value of oe varable ( varable) o the bass of oe or more other varables ( or explaatory varables).

3 Example: Demad Fucto Suppose the demad for Good A ca be expressed by the followg: Q A =f(p A, P B, M) ³ mult-varate relatoshp The quatty of Good A demaded s a fucto of ts ow (P A ), the prce of aother good (P B ), ad dsposable, (M). Accordg to the Law of Demad, there s a relatoshp betwee the quatty of A demaded (Q A ) ad ts prce (P A ), gve that everythg else s costat (ceters parbus). &Ths s a partal relatoshp. &Q A s referred to as the varable, whle P A, P B, ad M are or explaatory varables. ********* Ecoomc theory dctates the varable or varables whose values determe the behavour of a varable of terest..e. Ecoomc theory make clams about the varables that the value of the varable of terest. &For example, the prce of a ew wll determe the quatty of cars demaded. &Or, the terest rate o mortgages, CPI ad the GDP may partally determe the demad for ew. 3

4 But: () ecoomc theory does ot always provde the expected for all varables. Example: P B : ts the demad fucto depeds o whether the, a complemet or substtute good. () Ecoomc theory does ot always provde clear ad precse formato about the fuctoal of the fucto, f. &The fuctoal of these type of relatoshps must be specfed before estmato ca take place. Otherwse, estmato of ecoomc theory ca be. Example: A demad fucto s specfed: Q = β + β P + β P + β M A A 3 B 4 where β s are the of the model. &Parameters are usually. &We ofte wsh to estmate them. Note: Q A, P A, P B, ad M may be trasformatos of o-lear data: For example: Q = or Q = Q Q A C &These o-lear relatoshps have bee trasformed to a lear format ad hece expressed a lear regresso model. C A. 4

5 For example: Q Q Sales Sales = ( QSales ) = Q Sales ********** Oce we have determed the fuctoal form of the regresso, we ca address questos such as: If we chage the tax laws, wll there be a effect o quatty of ew demaded, (whch works through dsposable come)? Are goods A ad B substtutes or complemets (whch s determed by cross prce elastctes)? If the prce of B rses, does the quatty of A crease or decrease?,if P B creases ad t s a Complemet: Q A demaded decreases. Example: Quatty of large demaded decreases whe the prce of creases.,if P B creases ad t s a Substtute: Q A demaded creases. Example: Quatty of demaded creases whe the prce of orages creases. * ad orages are substtutes. 5

6 q Above we had examples of exact relatoshps betwee the varable ad the explaatory varables. &The fucto relatoshps were (.e. o ucertaty volved). &However, the real world, such relatoshps are ot as smple or straght forward. For Illustrato: If we look at two busesses that buld the same product, face the same producto costs ad factor prces, ther demads are ot usually the same. &There are may other dvdual factors that have bee gored. (Frm s deology, skll level of employees, marketg experece, etc.) qthere wll be some ucertaty, fucto. For Illustrato: Suppose we have several wth the same come, facg the same prces. The typcal demads from each wll dffer because other dvdual factors have bee gored by the model. Such as: q These factors project ucertaty to the fucto. 6

7 qths elemet s corporated to the fucto by cludg a stochastc error term to the model. Q = β + β P + β P + β M + ε Y Y 3 X 4 qotherwse, we could smply regard the relatoshp as explag the relatoshp betwee the varable ad a set of explaatory varables: EQ ( ) = β + β P+ β P+ β M Y Y 3 X 4 ********** For ths topc, we wll restrct our models to the followg: () Regresso:, oe varable (Y) ad oe t varable (X). (Ths ca be exteded to a mult-varate case.) () Parameters:,We assume the relatoshp betwee the depedet varable ad the depedet varable s lear terms of parameters: EY ( ) = α + βx Populato regresso le s lear terms of parameters α ad β whch depct the fuctoal relatoshp betwee X ad Y. 7

8 EY ( ) = α + βx (3) ³Meas for a gve value of X, the value of Y (average) s gve by α + βx. %We expect Y, [E(Y)], to chage as X chages. Use subscrpts to deote whch observato o X ad Y we are lookg EY ( ) = α + βx at: ³ for observato. Parameters: α, s referred to as the (populato) Y- term. β, deotes the (populato) term whch s the dervatve of Y wth respect to (w.r.t.) X: y x. (X, Y ) * Y * g E(Y )=α+βx * α * * * EY ( ) X Slope=β= E(Y ) X 8

9 Example: Let Y be the rate ad X represet the rate. The for the th observato, the expected value of s: E(Crme)= α+β(uemploymet rate) for the th observato. E(Y )=α+βx for the th observato. For each cty wth the same X, (uemploymet rate,) actual crme (Y) vares because of the other factors volved. Hece, for observato : Y = α + βx + ε ³ Populato Model where ε s the error for observato, ad hece: ε Y E Y = ( ) whch represets the dfferece betwee the observed Y (crme rate) ad the populato le, E(Y ). &β the slope measures the product of a chage uemploymet whch s mplct a chage the crme rate. 9

10 Aother example: Let Y be the umber of years of post secodary educato for dvdual. Let X be the umber of years of post secodary educato by oe of the (ether by the or, whatever f hghest,) of dvdual. Suppose we kow that α=.3 ad β=0.8. The the populato regresso le s: Y = X + ε or EY ( ) = X Notes o ths example: () Suppose dvdual s Jaso. Oe of hs parets have years of post secodary schoolg. How may years of post secodary educato do we expect Jaso to have, ceters parbus? X =4, so: EY ( ) = X EY ( ) = 3. + ( 08. )( 4) E( Y ) = 45. years. O average, we expect a dvdual wth that have years of post secodary schoolg to complete 4.5 years of post secodary schoolg. 0

11 () Other factors affect Y for =Jaso. &If the umber of years of schoolg s fact 4.5, the error term, g, equals zero: g =0. &But, f the actual umber of years of school completed s years, g =.5 for Jaso. Obvously, regresso aalyss s useful sce t allows us to the assocato or relatoshp amog varables. (.e. We ca determe how varables each other. ************ Idvdual s schoolg (Y ) E(Y)=α+βX =.3+0.8X Paret s schoolg (X )

12 The Sample Regresso Model (α ad β ) I the prevous examples we assumed that we the value of α ad β the populato parameters. Ths s urealstc, ad we usually must α ad β usg sample data. I.e. ths s aalogous to usg X as a estmator of µ, or s as a estmator of σ. Questos: : How do we decde whch ca we use for estmatg α ad β? : How ca we use formato? The regresso le takes the form: $Y = a+ bx where: () the (Ë) dcates a value or a value. () a s the of α. () b s the of β. (v) the subscrpt dcates that the th observato ad cludes all observatos the sample from to (=sample sze). As wth the populato regresso le, $Y Y typcally.

13 The dfferece betwee $Y ad Y, the error, s deoted by e : e = Y Y $ Substtutg Y$ = a+ bx e = Y a bx. Hece, we ca express the regresso model as:. Y = a+ bx + e Note: e s the sample. g s the error. e ad g are the same thg. e s. g s observable. e = Y a bx a,b are kow of αad β. sample estmates ε = Y α + βx α ad βare ukow populato parameters. 3

14 I geeral a α ad b β. } dstcto betwee a ad a populato. (Recall for all. Ths s the same dea.) X µ Notato: Let e (the sample error) be referred to as the term. Let g (the populato error) be referred to as the term. Let a be our estmator of α from a sample. Let b be our estmator of β derved from a. Sce a estmator s a or rule, we must determe a ad b from the formato. Whch method should be employed, sce there are may possble estmators? 4

15 Secto 3.3 The Method of Squares Recall we have restrcted our dscusso ths topc to a -varable (bvarate) model whch s parameters. Y = α + βx + ε Ths model s lear (Y ad X) as well as parameters. But, t s ot ecessary to our ths way. Example: & Regresso aalyss ca hadle () - models: Log( Y ) = α + βx + ε or () - models: Log( Y ) = α + βlog( X ) + ε These models are parameters, but - varables. 5

16 Y * By redefg the varables: X = Log( Y) Log( X ) * ad, the models are lear the ew varables: = * Log( Y ) = Y = α + βx + ε ad Log( Y ) = α + βlog( X ) + ε Y = α + βx + ε * * respectvely. What s mportat for the followg aalyss s that our are lear (_ ad _). Least Squares: Suppose we have a sample of sze, wth (X,Y ) par values: Plot the data pots wth Y s o the vertcal axs ad X s o the horzotal axs. Y * * * * * * * * * * * * 0 X 6

17 We ca derve a le that passes through these values by measurg α ad β by values of a ad b, such that the le passes close to the observed data. Y * * Y $ = a + bx * * * * * * slope = B * * * * 0 X How should we choose a ad b? (By the dstace betwee the observato ad the le: To llustrate, solate oe pot: (X,Y ) Y $Y = a+ bx Y } *}e e = Y Y $ X 7

18 We wat the sum of the to be as small as possble: Y Regresso le e Y e j $Y j $Y Yj X X X j The problem wth choosg a ad b so as to mmze. We could: () mmze or = e e = () mmze e Least Squares Approach =, s offsettg 8

19 Why Use The Least Squares Approach? (I) All observatos are gve weght. (II) Our estmators of α ad β, a ad b respectvely, have good statstcal propertes: () : E(a)=α ad E(b)=β () : Mmum estmators out of all lear ubased estmators (BEST). (III) smple (closed form aalytc soluto to mmzato problem.) (IV) Exteds trvally to ad o-lear relatoshps. Thus, least squares (LS) estmators of α ad β are those estmators of a ad b whch mmze the sum of the. Mathematcally, Least Squares problem s: M G = e = ( Y Y$) ( ab, ) = = where Y$ = a + bx so substtutg the above expresso to G: G = e = ( Y Y$) = ( Y a bx ). = = = Solve ths mmzato problem usg calculus.,partally G wth respect to a ad b ad equate these frst order codtos (FOC) to. Ths results two equato wth two ad we solve for a ad b: 9

20 Dervato of Least Squares Estmators: M G = ( Y a bx ) ab, () = = = ( Y a bx ) = 0 ( Y) = G ( Y a bx) = ( usg the cha rule:) a = a G = ( Y a bx )( ) = 0 ( FOC) a whch yelds the ormal equato: ( Y a bx ) = 0 e = 0 = = = = Takg the summato sg through: = = = rearragg: a = ( Y ) b X = = dvdg by : a bx = 0 a = ( Y ) b X = = a = Y bx { ( Y) a b X = 0 whch s the L.S. estmator of α. 0

21 G ( Y a bx ) = ( ) = b b Apply the cha rule: G b = ( Y a bx )( X ) = 0 = = ( XY ax bx ) = 0 = = 0 whch gves the ormal equato: = ( XY ax bx ) = = 0 = = = = = = = = = XY XY+ b( X X = XY a X b X = 0 X Y ax b X = 0 Sce a = Y - bx, we substtute that to (*) X Y ( Y-bX ) X b X = 0 X Y XY+ bx b X = 0 (*) = ) = 0

22 Rearragg such that "b" s o the LHS: b( X X ) = X Y XY = = b = = = XY X XY X Least Squares Estmator of β So, the least squares formulae for a ad b for fttg our lear regresso model to the data are: a = Y bx b = = = XY X XY X

23 Ca also wrte b as: or b b = = = ( X X)( Y Y) = ( X X) ( X X)( Y Y) ( ) = ( X X) ( ) = Note: () The deomator s the sample of X s: s X = ( ) = ( X X) () The umerator s the sample betwee X ad Y : s = ( X X)( Y Y) = XY Reflect how X ad Y are to each other. Thus, we ca also wrte: b S XY = = S X Sample cov. betwee dep. & dep. varable Sample Varace of depedet varable 3

24 () The sample regresso le always through the pot:. Recall, that the ftted regresso model s: Y$ = a+ bx ad a = Y bx. So, Y$ = Y bx + bx Y$ = Y + b( X X) So f X = X ad Y$ = Y the: Y$ = Y + b( X X) = Y. The sample regresso le (Y) $ always passes through the pot (X,Y). Y Y X X 4

25 (IV) Wth the Least Squares procedure, the of the errors (resduals) equals : e = ( Y a bx ) = = = 0 Recall the equato for determg a: = Equate to zero: = = = Y = a+ b X Y a b X = 0 ( Y a bx ) = 0 e = 0 ( Y Y$) = 0. = = = The sum of the least squares resduals s always. The least squares regresso le s derved such tha the le wll be stuated amogst the data values such that the resduals (uder- estmates of actual pot) always cacel out the resduals (over-estmates of actual pots). 5

26 Example: Phoe (Y) over a umber of years (X): Sales (Y) ( Mllos) Tme (X) (Years) =6 Populato Regresso Le: Y =α+βx +g Estmate α ad β by least squares estmators a ad b to gve sample regresso model: or equvaletly $Y = a+ bx Y = a+ bx + e. Sales (Y) Mllos Tme (X) X X Y ΣY =5.8 ΣX = Σ X = 9 ΣX Y =98. 6

27 Y = X = / 6 = 35. b XY XY = = X X = 98. ( 6)( 35. )( ) = 9 ( 6)( 35. ) = = = a = Y bx = ( )( 35. ) = = 55. Y$ = X 7

28 If you put a hat o the t varable ( Y $ ) the you do ot have to put the (e ) to the regresso le; If you do ot put the hat o Y, the you clude e. Forecastg: If tme (X) =, the Y$ = X Y$ = ( 0) Y$ = = I years, the best estmate of cellular phoe sales s. mllo. 8

29 Graph of Sales Agast Tme 9

30 Descrptve Statstcs For Both Seres 30

31 Equato Specfcato Least Squares Regresso Output 3

32 Summary: Regresso Model Terms ad Symbols Term Populato Symbol Sample Symbol Model: Y α β ε Y = a+ bx + e = + X + Error: Slope: Itercept: Equato of the le: Cocludg Remarks: ) Least squares procedure s smply a -fttg techque. It makes o about the depedet varable(s), X, the depedet varable, Y, or the error term. ) We wll eed to make about the varable(s), the varable, ad the error term f we wsh to cosder how well a ad b estmate α, the populato term ad β, the populato parameter. 3) We wll also eed to make assumptos f we wat to form estmates for predcted values of y or terval estmates for α ad β. 4) Wll also eed assumptos f we wat to test ay hypotheses about populato parameters. 3

33 Example: H 0 :β=β 0 versus Ha: β>β 0, where β 0 s some kow value. Specal test: H 0 :β=β 0 =0 versus H a :β 0 Ths tests f X s explag _. 5) We ca exted least squares prcple to - models. 6) We use least squares because t ca be show that uder certa assumptos, a ad b are estmators of α ad β, respectvely. They are also ad (.e. mmum varace).. I other words, uder certa assumptos, a ad b are the best ubased estmators of α ad β, respectvely. 33

34 Secto 3.4 Assumptos ad Estmator Propertes The populato model s : Y = α + βx + ε where =,, 3...,. The Assumptos: Gauss- Markov Theorem: Gve the fve basc assumptos, (below,) the least squares estmators the lear regresso model are best lear ubased estmators, "BLUE." Assumpto #: The radom varable g s statstcally of X. That s, E(g,X)=. Always holds f X s o- (fxed repeated samplg). Assumpto #: s dstrbuted. Assumpto #3: E(g )=_. That s, o average the dsturbace term s. Assumpto #4: Var(g )= σ ε ad X. Ths s kow as the assumpto of or costat, across all observatos. Otherwse. 34

35 Assumpto #5: Ay two errors g ad g j are statstcally of each other for j. I.e. E(g, g j )=E(g )E(g j )=_ for j. That s, zero (dsturbaces are ucorrelated). For example, dsturbace observato does ot affect observato, etc. If dsturbaces are across observatos, the there exsts or seral. Remarks: ) From assumptos through 4 we have ~N(0, σ ε ). ) Y s oly because g s radom f X s o-radom or o-. 3) EY ( ) = E( α + βx + ε ) = α + βx sce X s o - stochastc ad E( ε ) = 0. Var( Y ) = Var( α + βx + ε ) = Var( ε ) = σ ε Sce α, β ad X are o - stochastc (No varablty). Y s a lear fucto of ε ad s ormal. σ ε σ ε That s, from g ~N(0, ), t follows that Y ~N(α+βX, ). 4) As g ad g j for j are, so to are Y ad Y j for j. 35

36 Secto 3.5: Measures of Goodess of Ft Ths secto wll explore two measures of ess of : ) The Stadard of the Estmate: whch s the measure of the ft of the sample pots to the sample le. ) Coeffcet of R : whch s a measure of the ess of ft of a regresso le. Notato: Total of Y: the dfferece betwee the value of Y ad the of the y-values, Y. ( Y Y) Ths devato ca be expressed as the sum of two other devatos: ( Y Y$ ) ad (Y$ Y).,The frst devato expresso s the e : ( Y Y $ ) = e Sce the e are radom, the term devato. ( Y Y $ ) = e s referred as the,the secod devato ca be by the regresso le. ( Y$ Y) s the devato, because t s possble to. $Y Y X that dffers from because, X from 36

37 Total devato = Uexplaed devato + Explaed Devato ( ) ( ) Y Y = Y Y $ + (Y$ Y) Y Regresso Le Explaed { } Uexplaed * Y X Y Uexplaed dev. (Y- $Y ) * Total Dev. Y- Explaed Dev.( $Y - Y ) Y Y X 37

38 The two parts of the total devato are. Hece, we ca each ad sum over all observatos as follows: Y Y = Y Y $ + (Y$ Y) ( ) ( ) = = Breakg to three parts: = = ( ) Y Y The total varato ( $ ) = The total sum of squares SST Y Y The uexplaed varato The sum of squares error SSE = = e. = ( ) $Y Y The explaed varato due to the regresso. The sum of squares regresso SSR 38

39 Hece the regresso aalyss s: Total devato = Uexplaed devato + Explaed Devato ( ) ( ) Y Y = Y Y $ + (Y$ Y) SST = SSE + SSR Why are we dssectg the total to compoets? We ca ow determe the ess of of a terms of the sze of. %If the ft s perfect, =. %If the ft s ot perfect. 39

40 Calculato of SST, SSR ad SSE The best way to llustrate the calculato of these measures, s through llustrato: Example: Cellular Phoe Sales (Y) over a umber of years (X): Sales (Y) Mllos Tme (X) X X Y ΣY =5.8 ΣX = Σ X = 9 ΣX Y =98. Y = X = / 6 = 35. b = a = Y bx =55. Y$ = X 40

41 Sales (Y) Mllos ( Y Y) e = ( Y Y$ ) ( Y $ Y) ΣY =._ Σ= =6.4 Σ= =9.8 Σ= =6.33 We should fd that = +. A more coveet way to determe SST s: Total varato the depedet varable Y: SST = Y ( ) Y = = The amout of varato by the regresso equato ca be determed wthout frst calculatg. The method elmates the calculato of each estmated value ( ). varato: SSR = b ( ( X X )( Y Y ) = b( covarato of X ad Y) Oce SST ad SSR are foud, the uexplaed varato SSE s foud by subtracto: SSE= -. $Y 4

42 How Good s the Ft I Regresso Aalyss? (I) Stadard of the s = SSE ( Y Y$ ) = = e The umber of degrees of freedom ths case s because sample statstcs (_ ad ) must be calculated before the value of ca be computed. $Y The formato from ths provdes a dcato of the of the regresso model. It also dcates the of the that result. The uts of s e are always the same as those for the varable sce e s a leftover (resdual) compoet of _. To determe whether s e s large or small, we compare t to the average sze of _: Usg a coeffcet of (C.V.) type measure, we ca defe the typcal percetage error predctg _ usg the sample data ad the regresso le. 4

43 Coeffcet of Varato for regresso resduals: se C. V. resduals = * 00 Y e = ( Y Y$) =987. s = ( Y Y$) = = * =. 455 = 566. e se C. V. resduals = * 00 Y =.566 * = 084. * 00 = 8.4 For the cellular phoe example, the C.V. resduals= 8.4. I predctg the umber of cellular phoe sales each year ths sample wth the estmated model, the value wll be by 8.4%, o the average. 43

44 (II) Coeffcet of The secod measure of goodess of ft s the coeffcet of. It facltates the terpretato of the relatve amout of that has bee by the sample le. Dervg R : ( the text t s deoted r ) Recall: SST = SSE + SSR,If we dvde ths expresso by SST: SST SST SSE SSR = + SST SST SSE SSR = + SST SST These two ratos must sum to ad are M.E. Recall, SSE s the varato Y. The rato SSE SST represets the proporto of total varato that s by the regresso relato. 44

45 SSR SST ad s called the coeffcet of. The rato represets the measure of goodess of ft It measure the proporto of total varato that s explaed by the regresso le. R SSR = = SST Varato explaed Total varato If the regresso le fts all the sample pots, = ad the coeffcet of determato would acheve ts maxmum value of _: R =. Y Regresso le Perfect Ft: R =_; $ Y = Y X 45

46 As the degree of ft become accurate ad of the varato Y s explaed by ther relato wth X, R. Y Regresso le X qthe lowest value of R s, whch wll occur wheever SSR=0 ad SSE=. Ths occurs whe $Y = Y for all observatos. Y Y Regresso le X No systematc relato betwee Y ad X R =_; $Y Y = ; b=_ The regresso le s l wth a slope (b=_). Hece, X has o effect explag chages Y. 46

47 Example: Determe the coeffcet of determato for the cellular phoe example. We kow that SSR= ad SST =. Hece, SSR R = 6 33 SST =. 64. =065. The terpretato of ths result s that 6.5% of the sample varato cellular phoe sales s by the lear relato of tme years. The remag 37.5% of the varato sales s stll. Most lkely some other has bee from the regresso model that could expla some addtoal porto of the varato. ( Multple regresso model s requred.) 47

48 Secto 3.6 Aalyss The measuremet of how well two or more varables together s called aalyss. Oe measure of the populato relatoshp betwee two radom varables s the populato covarace: [ ] [(, ] = ( )( ) Cov X Y E X X Y Y Usually ths measure s a dcator of the relatve of the relatoshp betwee two varables because ts magtude depeds o the used to measure the varables. Hece, t s ecessary to stadardze the of two varables order to have a measure of ft. ( - measure.) Stadardzato s carred out by dvdg the of X ad Y by the stadard devato of X tmes the stadard devato of Y: Populato Correlato Coeffcet: ρ = Covarace of X ad Y (Std. Dev. of X) *(Std. Dev.of Y) = C ( X, Y ) σ, σ X Y 48

49 Three Specal Cases: Case #: correlato: All values of X ad Y fall o a sloped straght le: ρ= Y X Case #: Correlato: All values of X ad Y fall o a sloped straght le: ρ= Y X Case #3: Correlato: If X ad Y are learly related, sce they are radom varables, the the value of the correlato coeffcet wll be, sce C(X,Y)=0. ρ=. Thus ρ measures the stregth of the assocato betwee X ad Y., Values of ρ close to dcate a relato;, Values close to + dcate a postve relato;, Values close to - dcate a egatve correlato. 49

50 Sample Correlato Coeffcet We employ sample data to estmate the populato parameter ρ. The correlato coeffcet s deoted by the letter r. We determe the value of r the same maer as ρ, except that we substtute for each populato parameter ts best estmate based o the sample data. Sample Covarace: S : S = XY = XY ( X X )( Y Y ) Let S x ad S Y represet the sample stadard devato. Hece, the Sample Correlato Coeffcet: r S Cov of sample values of X ad Y XY = = SS X Y Sample stadard Sample stadard devato of X devato of Y = r = (X X) *(Y Y) - = (X X) * (Y Y) = = SC Sample covarato XY = SS SS Sample Varatos X Y 50

51 The Coecto Betwee Correlato ad Regresso ) The correlato coeffcet ad the coeffcet of determato measure the of the betwee X ad Y. For a model wth oly oe varable, the measure R s the of the correlato measure r, as they measure the same relatoshp. Whe there s a weak relatoshp, both measures are close to. A relato s dcated betwee the two varables X ad Y whe r approaches - or +. Ad sce the square of r s always, ad R approaches +whe there s a relato. ) The correlato coeffcet provdes formato about the of the assocato betwee X ad Y. I.e. ether postvely or egatvely related. A postve correlato must always correspod to a regresso le wth a postve (b), dcatg a drect relato. A correlato correspods to a relato. 5

52 3) I correlato aalyss the X ad Y varables are assumed to have a probablty dstrbuto. I regresso aalyss, the depedet varable X ca be cosdered as a set of gve values. Whe X s a depedet radom varable, correlato betwee X ad Y s. 4) I correlato aalyss there s o specfed depedet ad depedet varable desgato. I regresso aalyss we mpose a model, such that, Y s beg by X. If the depedet ad depedet varables are swtched, the regresso model wll be the same, although the sg o the parameter wll rema the same. The Ed 5

Simple Linear Regression

Simple Linear Regression Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato