12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model

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Gerard Alban Rich
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1 1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed values of y wll vary(ad ot fall exactly o the le) because Y s a RV depedet upo the RV ε, the error. Y = β + β x+ ε ad E( Y) = β + β x VY ( ) ) β o = σ 0, β1, σ are ot kow to the vestgator o observed ordered pars ( x, y ),( x, y ),...,( x, y ) wll be avalable. 1 1 o The observatos are depedet The goal s to fd the true regresso le, the le for whch y s a lear fucto of x. Accordg to the model, the observed pots wll be dstrbuted about the true regresso le a radom maer. Ths says bascally that the regresso le s thele that s the "best ft" for the observed data pots.aga some pots wll be hgher, some wll be lower. Ths s the dea behd the prcple of least squares. A le provdes a good ft to data f the vertcal dstaces(stadard devatos) from the le are szmall. The measure of how well ths le estmates the values of y (called goodess of ft of the model)s the sum of the squares of the devatos. The best-ft le s the oe havg the smallest possble sum of squared devatos. 1. part1 1

2 The prcple of Least Squares: The vertcal devato of the pot ( x, y ) from the le y = b + b x s heght of pot - heght of the le = y (b + b x ) The sum of squared vertcal devatos from the pots ( x, y ),( x, y ),..., ( x, y )to the le s the: [ ] f( b, b) = y (b + b x ) = The pot estmates β ad β deoted by β ad β ad called the least squares estmates, are those values that mmze f( b, b). The estmated lear regresso le (or least squares le) s the le whose equato s y = β + β x. To mmze b ad b, we wll eed to take partal dervatves of f( b, b) 1. part1

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4 Ex 1: The followg table gves the rafall ad ruoff volume a partcular rego Aust, TX. a. Does the scatter plot of the data support the use of the smple lear regresso model? b. Calculate the pot estmates of the slope ad tercept of the populato regresso le. rafall volume (x) sum of x squared ruoff volume (y) x*y rafall volume vs ruoff volume (y) ruoff volume rafall volume ruoff volume (y) 1. part1 4

5 Example 1 cotued: c. Calculate a pot estmate of the true average ruoff volume whe rafall volume s 50 Estmatg σ ad σ. The parameter σ determes the amout of varablty of the regresso model. whe σ s large, the (x,y ) pars wll be far from the regresso le, ad whe σ s small, (x,y wll be close to the regresso le. ) pars The complcato ths s that the true le s ot kow, we are makg a guess, so our best guess of σ wll be based o how far our sample data rages from the estmated le. Defto: 1 The ftted or predcted values y 1, y,..., y are obtaed by successvely substtutg x, x,...,x to the estmated regresso le: y = β + β x: The resduals are the vertcal devatos y 1 - y 1,..., y - y from the estmated le. Warg: You caot use a least squares le to extrapolate! You caot use t to fd y values for x values outsde the gve rage. Back to Estmatg the varace: Well, wheever we estmate the varace t has come from fdg the sum of dfferece betwee each observed value ad the mea value. The same thg apples here: 1. part1 5

6 σ = ( y y ) ( y y) s called the error sum of squares(sse) SSE so you ca wrte σ =. We dvde by -(remember these are the d.f.) ad they are determed by how much s ukow about our sample. Sce we are estmatg β ad β, we have lost two degrees of freedom. A computatg formula for SSE: SSE = y β0 y β1 xy Example 1 cotued d. Calculate a pot estmate for the stadard devato. 1. part1 6

7 The Coeffcet of Determato Now we are lookg for a way to express how much varato s left uexplaed by the model. We use the total sum of squares (SST) as aother measure of varato whch leads to determg how much varato s uexplaed: yy ( y y) SST = S = = 1. part1 7

8 The quattatve measure of how much s left uexplaed by the lear regresso model s: The coeffcet of determato: SSE r = 1 SST s terpreted as the proporto of observed y varato that ca be explaed by the smple lear regresso model (attrbuted to a approxmate learrelatoshp betwee y ad x). The coeffcet of determato ca also be wrtte usg aother term, The regreso sum of squares: SSE SST SSE SST SSE SSR r = 1 = = = SST SST SST SST SST the Regresso Sum of Squares (SSR) = SST- SSE. Example 1 cotued e. What proporto of the observed varato ruoff volume ca b e attrbuted to the smple lear regresso relatoshp betwee ruoff ad rafall? 1. part1 8

9 Example : The followg summary statstcs were obtaed from a study that used regresso aalyss to vestgate the relatoshp betwee pavemet deflectol ad surface temperature of the pavemet at varous locatos o a state hghway. Here x = the temperature (degrees faraheght) ad y = deflecto adjustmet factor (y >=0): = 15 x = 145 y = x y = 139, x y = = a. Compute β, β ad the equato of the estmated 1 0 regresso le. Graph the estmated le. b. What s the estmate of the expected chage the deflecto adjustmet factor whe the temperature s creased by 1 degree? c. If a 00 degree surface temperature were wth the realm of possblty, would you use th eestmated le of part (a) to predct deflecto factor for ths temperature. Why or why ot? 1. part1 9

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