ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable Y, producg a feld of observatos. The questo the arses: What s the best le (or curve to draw through ths feld of pots? Values of X are cotrolled by the expermeter, so the o-radom varable x s called the cotrolled varable or the depedet varable or the regressor. Values of Y are radom, but are flueced by the value of x. Thus Y s called the depedet varable or the respose varable. We wat a le of best ft so that, gve a value of x, we ca predct the value of Y for that value of x. The smple lear regresso model s that the predcted value of y s y β + β x ad that the observed value of Y s Y + β x β + ε where ε s the error. It s assumed that the errors are ormally dstrbuted as ε ~ N(, σ, wth a costat varace σ. The pot estmates of the errors ε are the resduals e y yˆ. Wth the assumptos Y β + β x + ε x x Y ~ N( β + β x, σ [ (3 V[Y] s d t of x ] place, t the follows that ˆβ ad are ubased estmators of the coeffcets β ad β. ( ote lower case E ˆ β ˆ βx + β + βx x Methods for dealg wth o-lear regresso are avalable the course text, but are beyod the scope of ths course.
ENGI 343 mple Lear Regresso Page - Examples llustratg volatos of the assumptos the smple lear regresso model:.. ( volated ot lear (3 volated varace ot costat 3. 4. OK ( & (3 volated If the assumptos are true, the the probablty dstrbuto of Y x s N( β + β x, σ.
ENGI 343 mple Lear Regresso Page -3 Example. Gve that Y.5 x + ε, where ε ~ N(,, fd the probablty that the observed value of y at x 8 wll exceed the observed value of y at x 7. Y ~ N(.5 x, Let Y 7 the observed value of y at x 7 ad Y 8 the observed value of y at x 8, the Y 7 ~ N(6.5, ad Y 8 ~ N(6, Y 8 Y 7 ~ N(6 6.5, + μ.5 σ 4 (.5 P[ Y8 Y7 > ] P Z > Φ(.5.43 Despte β <, P[Y 8 > Y 7 ] > 4%! For ay x the rage of the regresso model, more tha 95% of all Y wll le wth σ ( ether sde of the regresso le.
ENGI 343 mple Lear Regresso Page -4 Dervato of the coeffcets ˆβ ad ˆβ of the regresso le y ˆ β ˆ + β x : We eed to mmze the errors. Each error s estmated by the observed resdual e y yˆ. Mmze errors.? NO e Use the E (sum of squares due to errors e ( y ˆ β ˆ β x f ( ˆ β, ˆ β o Fd ˆβ ad ˆβ such that. ˆ β ˆ β [Note: ˆ β ˆ, β are varables, whle x, y are costats.] ˆ β ( ˆ ˆ ( ˆ ˆ β β β β ( y x + x y ad ( y ˆ ˆ β β ( ˆ ˆ ˆ x x β x + β x x y ( β x ˆ β or, equvaletly, x x ˆ β ˆ β x y ˆ β x x y x y x x y x y x (3 (4
ENGI 343 mple Lear Regresso Page -5 The soluto to the lear system of two ormal equatos ( ad ( s: from the lower row of matrx equato (4: ˆβ x y, (where x y x x y x x y ad x ( x x x ( or, equvaletly, ˆ sample covarace of x, y β ; sample varace of x ad, from equato (: ˆ β ( y ˆ β x. A form that s less susceptble to roud-off errors (but less coveet for maual computatos s ( x x( y y ( x x ˆβ ad ˆ β ˆ β x. y The regresso le of Y o x s y y ( x x ˆβ. Equato ( guaratees that all smple lear regresso les pass through the cetrod ( x, y of the data. It turs out that the smple lear regresso method remas vald eve f the values of the regressor x are also radom. However, ote that terchagg x wth y, (so that Y s the regressor ad X s the respose, results a dfferet regresso le (uless X ad Y are perfectly correlated..
ENGI 343 mple Lear Regresso Page -6 Example. (the same data set as Example.6: pared two sample t test Ne voluteers are tested before ad after a trag programme. Fd the le of best ft for the posteror (after trag scores as a fucto of the pror (before trag scores. Voluteer: 3 4 5 6 7 8 9 After trag: 75 66 69 45 54 85 58 9 6 Before trag: 7 65 64 39 5 85 5 9 58 Let Y score after trag ad X score before trag. I order to use the smple lear regresso model, the assumptos Y β + β x + ε x x Y ~ N( β + β x, σ must hold. From a plot of the data ( http://www.egr.mu.ca/~ggeorge/343/demos/regress.xls, ad http://www.egr.mu.ca/~ggeorge/343/demos/ex.mpj, oe ca see that the assumptos are reasoable. catter plot Normal probablty plot of resduals
ENGI 343 mple Lear Regresso Page -7 Calculatos: x y x x. y y 7 75 584 54 565 65 66 45 49 4356 3 64 69 496 446 476 4 39 45 5 755 5 5 5 54 6 754 96 6 85 85 75 75 75 7 5 58 74 36 3364 8 9 9 8464 837 88 9 58 6 3364 3596 3844 um: 578 65 39384 484 4397 x y x y x y 9 484 578 65 776 [Note: ( * sample covarace of (X, Y ] x ( x x x 9 39384 578 37 [Note: ( * sample varace of X ] x y 776 ˆ β.876 37 ˆ x x y ˆ 578 9 ad β ( β ( 65.876.3445 x Each predcted value ŷ of Y s the estmated usg yˆ ˆ β + ˆ β x.34 +.87 x ad the pot estmates of the ukow errors ε are the observed resduals e y yˆ. [Use u-rouded values.34... ad.87... to fd resduals.] A measure of the degree to whch the regresso le fals to expla the varato Y s the sum of squares due to error, E e ( y ˆ β ˆ β x whch s gve the adjog table. x y ŷ e e 7 75 73.98979..5 65 66 67.89898.899 3.66 64 69 67.886.97 3.8854 39 45 45.7597.76.76 5 54 55.7736.774.9493 85 85 85.33.33.98 5 58 56.58747.45.995 9 9 9.39.39.537 58 6 6.887.98.368 E 3.84
ENGI 343 mple Lear Regresso Page -8 A Alteratve Formula for E: ˆ β y βˆ x E y ( y ˆ β x ˆ x y y ˆ β β x x ( y y ˆ ( x x( y y + ˆ β β ( x x yy ˆ β ˆ + β But ˆ β ( ( ( E yy βˆ or E E yy or ( ( ( yy ( I ths example, 37 5548 7 76 E 9 37 3.84 However, ths formula s very sestve to roud-off errors: If all terms are rouded off prematurely to three sgfcat fgures, the 4 55 7 7 E 5.85 ( d.p. 9 4 ( y ˆ T ( y y E e y
ENGI 343 mple Lear Regresso Page -9 The total varato Y s the T (sum of squares - total: yy T ( y y (whch s ( the sample varace of y. I ths example, T 5 548 / 9 77.555... The total varato (T ca be parttoed to the varato that ca be explaed by the regresso le ( R ( yˆ y ad the varato that remas uexplaed by the regresso le (E. T R + E ˆ β yy The proporto of the varato Y that s explaed by the regresso le s kow as the coeffcet of determato r R T E T I ths example, r (3.8... / 77.555....994... Therefore the regresso model ths example explas 99.% of the total varato y. Note: R ˆ β ad T yy r x x x y y y The coeffcet of determato s just the square of the sample correlato coeffcet r. Thus r r.996. It s o surprse that the two sets of test scores ths example are very strogly correlated. Most of the pots o the graph are very close to the regresso le y.87 x +.34.
ENGI 343 mple Lear Regresso Page - A pot estmate of the ukow populato varace σ of the errors ε s the sample varace or mea square error s ME E / (umber of degrees of freedom. But the calculato of s cludes two parameters that are estmated from the data: ˆβ E ad ˆβ. Therefore two degrees of freedom are lost ad ME. I ths example, ME.973. A cocse method of dsplayg some of ths formato s the ANOVA table (used Chapters ad of Devore for aalyss of varace. The f value the top rght corer of the table s the square of a t value that ca be used a hypothess test o the value of the slope coeffcet β. equece of maual calculatos: {, x, y, x,, y } {,, yy } ˆ β ˆ β, R, T } { R, E } { MR, ME } f t {, ource Degrees of ums of quares Mea quares f Freedom Regresso R 73.74... MR R / MR/ME 73.74... 868.4... Error E 3.8... ME E / ( 7.973... Total T 77.555... 8 To test H o : β (o useful lear assocato agast H a : β (a useful lear assocato exsts, we compare t f to t α/, (.
ENGI 343 mple Lear Regresso Page - I ths example, t 868.4... 9.4... >> t.5, 7 (the p-value s < 7 so we reject H o favour of H a at ay reasoable level of sgfcace α. ˆ β The stadard error s b of ˆβ s s / so the t value s also equal to. ME x x Yet aother alteratve test of the sgfcace of the lear assocato s a hypothess test o the populato correlato coeffcet ρ, (H o : ρ vs. H a : ρ, usg the test statstc above. Example.3 t r r, whch s etrely equvalet to the other two t statstcs (a Fd the le of best ft to the data x 3 4 y 6. 5.3 4. 5. 4.4 3.4.6 3..8. (b Estmate the value of y whe x. (c Why ca t the regresso le be used to estmate y whe x? (d Fd the sample correlato coeffcet. (e Does a useful lear relatoshp betwee Y ad x exst? (a A plot of these data follows. The Excel spreadsheet fle for these data ca be foud at http://www.egr.mu.ca /~ggeorge/343/demos /regress3.xls.
ENGI 343 mple Lear Regresso Page - The summary statstcs are Σ x 6 Σ y 38 Σ x 4 Σ 45.6 Σ y 63.6 From whch Σ Σ x Σ y 5 Σ x (Σ x 44 yy Σ y (Σ y 86.6 ˆ β 5.5 44 ˆ y ˆ β ad β 5.48 o the regresso le s x y 5.489.56 x (3 d.p. (b x y 5.488....55... 3.38 ( d.p. (c x y 5.488....55... 5.7 <! ( d.p. Problem: x s outsde the sample rage for x. LR model may be vald. I oe word: EXTRAPOLATION. (d (e x y 5 r.977....93 44 86.6 x x y y ( ( 5 R 6.4 ( 44 T yy ( 86.6 / 8.66 ad E T R 8.66 6.4....65...
ENGI 343 mple Lear Regresso Page -3 The ANOVA table s the: ource d.f. M f R 6.4444... 6.4444... 49.7... E 8.6555....369... T 9 8.66 from whch t f 7.5 (3 d.p. But t.5,8 5.4... t > t.5, 8 Therefore reject H o : β favour of H a : β at ay reasoable level of sgfcace α. [p-value...] r.977... 8 OR t 7. 5 r.85983... reject H o : ρ favour of H a : ρ (a sgfcat lear assocato exsts. R 6.4 [Also, from the ANOVA table, r.8598 T 8.66 Therefore the regresso le explas ~86% of the varato y. r r.97, as before.] Cofdece ad Predcto Itervals The smple lear regresso model Y β + β x + ε leads to a le of best ft the least squares sese, whch provdes a expected value of Y gve a value for x : yˆ ˆ β + ˆ β x E[ Y x ] μ Y x. The ucertaty ths expected value has two compoets: the square of the stadard error of the scatter of the observed pots about the regresso le ( σ /, ad the ucertaty the posto of the regresso le tself, whch creases wth the dstace of the chose x from the cetrod of the data but decreases wth creasg ( x x spread of the full set of x values: σ. The ukow varace σ of dvdual pots about the true regresso le s estmated by the mea square error s ME.
ENGI 343 mple Lear Regresso Page -4 Thus a ( α% cofdece terval for the expected value of edpots at Y at x x o has ( ˆ ˆ ( xo x β β x ± t s + + o α /, ( The predcto error for a sgle pot s the resdual E Y ŷ, whch ca be treated as the dfferece of two depedet radom varables. The varace of the predcto error s the ( x x V[ E] V[ Y] V[ yˆ ] σ σ + + + Thus a ( α% predcto terval for a sgle future observato of Y at x x o has edpots at ( ˆ ˆ ( xo x β + β xo ± tα /, ( s + + The predcto terval s always wder tha the cofdece terval. Example.3 (cotued (f Fd the 95% cofdece terval for the expected value of Y at x ad x 5. (g Fd the 95% predcto terval for a future value of Y at x ad at x 5. (f α 5% α/.5 Usg the varous values from parts (a ad (e: t.5, 8.36... s.5779... x.6 4.4 ˆβ 5.4888... ˆβ.555... x o the 95% CI for μ Y s ( ˆ ˆ ( xo x β x ± t s + β + o α /, ( 3.3777... ±.385...... 3.3777... ±.4395....94 E[ Y ] < 3.8 (to 3 s.f.
ENGI 343 mple Lear Regresso Page -5 x o 5 the 95% CI for μ Y 5 s ( ˆ ˆ ( xo x β x ± t s + β + o α /, (... ±.385....9777...... ±.58....4 E[ Y 5 ].46 (to 3 s.f. (g x o the 95% PI for Y s ( ˆ ˆ ( xo x β x ± t s + + β + o α /, ( 3.3777... ±.385...... 3.3777... ±.3898....99 Y < 4.77 (to 3 s.f. at x x o 5 the 95% PI for Y s ( ˆ ˆ ( xo x β x ± t s + + β + o α /, (... ±.385....9777...... ±.888....6 < Y <.3 (to 3 s.f. at x 5 Note how the cofdece ad predcto tervals both become wder the further away from the cetrod the value of x o s. The two tervals at x 5 are wde eough to cross the x-axs, whch s a llustrato of the dagers of extrapolato beyod the rage of x for whch data exst. ketch of cofdece ad predcto tervals for Example 3 (f ad (g: (f 95% Cofdece Itervals (g 95% Predcto Itervals