Sum Mean n

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1 tatstcal Methods I (EXT 75) Page 147 ummary data Itermedate Calculatos X = 83 Y = 8 X = 51 Y = 368 Mea of X = X = Mea of Y = Y = 14.5 XY = 1348 = 16 Correcto factors ad Corrected values (ums of squares ad crossproducts) CF for X Cxx = Corrected X xx = CF for Y Cyy = 349 Corrected Y yy = 379 CF for XY Cxy = Corrected CP XY xy = ANOVA Table (values eeded): Total = 379 Regresso = / = Error = = Model Parameter Estmates lope = b 1 = 1 Y Y. X X. 1 X X. xy xx =165.5 / = Itercept = b = Y-b 1 X = * = Regresso Equato Y = b + b 1 * X + e = Y = * X + e Regresso Le um Mea ource df M F Regresso Error Total Tabular F.5; 1, 14 = 4.6 Y Tabular F.1; 1, 14 = 8.86 = b + b 1 * X = Y = * X tadard error of b 1 : b1 1 ME X X. ME xx so b = Cofdece terval o b1 where b 1 = ad t (.5/, 14df) =.145 P( * * ) =.95 P( ) =.95 Testg b1 agast a specfed value: e.g. H: 1 = 5 versus H1: 1 5

2 tatstcal Methods I (EXT 75) Page 148 where b 1 = , b1 = ad t (.5/, 14df) =.145 = ( ) / = tadard error of the regresso le (.e. Y ) : 1 X X. ME YX ˆ X X. 1 tadard error of the dvdual pots (.e. Y ): Ths s a lear combato of Y ad e, so the varaces are the sum of the varace of these two, where the varace of e s ME. The stadard error s the ME YX = YX ˆ 1 XX. 1 XX. ME ME ME 1 XX. XX. 1 1 tadard error of b s the same as the stadard error of the regresso le where X = quare Root of [ ( /9.4375)] = Cofdece terval o b, where b = ad t (.5/, 14df) =.145 P( * * ) =.95 P( ) =.95 Estmate the stadard error of a dvdual observato for umber of parastes for a te-yearold fsh: Y b b1x = *X= *1 = quare Root of [ *(1+.65+( )/9.4375)] = quare Root of [ *(1+.65+( )/9.4375)] = Cofdece terval o Y X=1 P( * Y X= * ) =.95 P( Y X= ) =.95 Calculate the coeffcet of Determato ad correlato R = or % r = ee A output Overvew of results ad fdgs from the A program

3 tatstcal Methods I (EXT 75) Page 149 I. Objectve 1 : Determe f older fsh have more parastes. (A ca provde ths) A. Ths determato would be made by examg the slope. The slope s the mea chage paraste umber for each ut crease age. The hypothess tested s H : 1 = versus H 1 : 1 1. If ths umber does ot dffer from zero, the there s o apparet relatoshp betwee age ad umber of parastes. If t dffers from zero ad s postve, the parastes crease wth age. If t dffers from zero ad s egatve, the parastes decrease wth age.. For a smple lear regresso we ca exame the F test of the model, the F test of the Type I, the F test of the Type II, the F test of the Type III or the t-test of the slope. For a smple lear regresso these all provde the same result. For multple regressos (more tha 1 depedet varable) we would exame the Type II or Type III F test (these are the same regresso) or the t-test of regresso coeffcets. [Alteratvely, a cofdece terval ca be placed o the coeffcet, ad f the terval does ot clude, the estmate of the coeffcet s sgfcatly dfferet from zero]. B. I ths case, the F tests metoed had values of 54.86, ad the probablty of ths F value wth 1 ad 14 d.f. s less tha.1. Lkewse, the t test of the slope was 7.41, whch was also sgfcat at the same level. Note that t =F, these are the same test. We ca therefore coclude that the slope does dffer from zero. ce t s postve we further coclude that older fsh have more parastes. II. Objectve : Estmate the rate of accumulato of parastes. (A ca provde ths) A. The slope for ths example s parastes per year (ote the uts). It s postve, so we expect paraste umbers to crease by 1.8 per year. B. The stadard error for the slope was Ths value s provded by A ad ca be used for hypothess testg or cofdece tervals. A provdes a t-test of H : 1 =, but hypotheses about values other tha zero must be requested (A TET statemet) or calculated by had. The cofdece terval ths case s: Ths calculato was doe prevously ad s partly repeated below. P[b 1 t /,14 d.f. b1 1 b 1 + t /,14 d.f. b1 ]=.95 P[ (.46689) (.46689)]=.95 P[ ]=.95 Note that ths cofdece terval does ot clude zero, so t dffers sgfcatly from zero. III. Estmate the tercept wth cofdece terval. A. The tercept may also requre a cofdece terval. Ths was calculated prevously ad was; P( ) =.95

4 tatstcal Methods I (EXT 75) Page 15 IV. Determe how may parastes a 1 year old fsh would have. (A ca provde ths) A. Estmatg a Y value for a partcular X smply requres solvg the equato for the le wth the Y b b1x whch for coeffcets of ad 1.87 ad for a 1-year-old fsh (X =1) s Y = (1) = = V. Place a cofdece terval o the 1 year old fsh estmate. (A ca provde ths) A. The cofdece terval for ths was estmated prevously: P( x= )=.95. B. There are may reasos why ths type of calculato may be of terest. We ca place a cofdece terval o ay value of X, cludg the tercept where X = (ths was doe prevously). The tercept s ofte the most terestg pot o the regresso le, but ot always. C. There s oe very specal characterstc of the cofdece tervals (of ether dvdual pots or meas). The cofdece terval s arrowest at the mea of X, ad gets wder to ether sde of the mea. The graph below for out example demostrates ths property. 5 Regresso wth cofdece bads P ar a s t e s D. Age (years) VI. Determe f a lear model s adequate ad assumptos met. (A ca provde most of ths) A. Idepedece : Ths s a dffcult assumpto to evaluate. There are some techques advaced statstcal methods, but these wll ot be covered here. The best guaratee for depedece s to radomze wherever ad wheever possble. B. Normalty : The ormalty of the resduals or devatos from regresso ca be evaluated wth the PROC UNIVARIATE hapro-wlks test. The W value was.96 ad the P<W was We would ot reject the ull hypothess of data s ormalty dstrbuted wth these results.

5 tatstcal Methods I (EXT 75) Page 151 Homogeety ad other cosderatos : Resdual plots are a mportat tool evaluatg possble problems regresso, some of whch we have ot see before. The ormal resdual plot, whe all s well, should reflect just radom scatter about the regresso le. A e + - Resdual Plot example s gve below. X The three resdual plots below all show possble problems. From left to rght the problems dcated are (1) the data s curved ad caot be adequately descrbed by a straght le, () the varace s ot homogeeous ad (3) there s a outler. e Resdual Plot e Resdual Plot e X Resdual Plot X + - X A outler s a observato whch appears to be too large or too small comparso to the other values. Data should be checked carefully to sure that the pot s correct. If t s correct, but s way out of le relatve to other values. t may be ecessary to omt the pot. The resdual plot for our example s gve below. Ca you detect ay potetal problems? 4 Age Resdual Plot Resduals Age

6 tatstcal Methods I (EXT 75) Page 15 VII. A old publshed artcle states that the rate of accumulato should be about 5 per year. Test our estmate agast 5.. (A ca provde ths f you ask cely) A. A automagcally test the hypothess that H : 1 =. However, ay value ca be tested. The b1 bh ME ME test s the usual oe-sample t-test, t,where 1 b as prevously xx b1 X X metoed. For ths example, t.467 VIII. Fal otes o regresso ad correlato. (A ca provde most of ths) A. The much over-rated R. The regresso accouts for a certa fracto of the total. The fracto of the total that s accouted for by the regresso s called coeffcet of determato ad s deoted R. It s calculated as R = Reg / Total. Ths value s usually multpled by 1 ad expressed as a percet. For our example the value was 79.7% of the total varato accouted for by the model. Ths s pretty good, I guess. However, for some aalyses we expect much hgher (legth - weght relatoshps for example) ad for others much lower (try to predct how may fsh you wll get a et at a partcular depth or for a partcular sze stream). Ths statstc does ot provde ay test, but may be useful for comparg betwee smlar studes o smlar materal. B. The square root of the R value s equal to the Pearso product momet correlato coeffcet, usually deoted a r. Ths value s calculated as b1 1 X X. Y Y. X X. Y Y. 1 1 xx xy yy ad s equal to.896 for our example. C. The correlato coeffcet s utless ad rages from -1 to +1. D. A perfect verse correlato gves a value of -1. Ths correspods to a egatve slope regresso, but the R value wll ot reflect the egatve because t s squared. A perfect correlato gves a value of +1 (postve slope regresso). A correlato of zero ca be represeted as radom scatter about a horzotal le (slope = regresso). Y Y Perfect verse correlato X Perfect correlato X

7 tatstcal Methods I (EXT 75) Page 153 E. The perfect correlato value of 1 (+ or ) also correspods to a perfect regresso, where the R value would dcate that 1% of the varato the total was accouted for by the Y Correlato = model. The error ths case would be zero. X About Cross products Cross products, X Y,are used a umber of related calculatos. Note from the calculatos below that whe ay of the calculatos equals zero, all of the others wll also go to zero. As a result whe the covarace s zero the slope, correlato coeffcet, R ad Regresso are also zero. As a result of ths, the commo test of hypothess of terest regresso, H: 1, ca be tested by testg ay of the statstcs below. A t-test of the slope or a F test of the MRegresso are both testg the same hypothess. Recall that we saw that from the terrelatoshps of probablty dstrbutos that a t wth d.f. = F wth 1, d.f. um of cross products = XY ( Y Y)( X X) 1 Covarace = XY ( Y Y)( X X) ( Y Y)( X X) lope = XY 1 XX 1 ( X X ) ( Y Y)( X X) XY 1 Regresso = XX 1 ( X X ) ( Y Y)( X X) Correlato coeffcet = r XY 1 XX YY ( X X) ( Y Y) 1 1 ( Y )( ) R = r Y X X XY 1 = XX YY ( X X) ( Y Y) 1 1 Regresso Total

8 tatstcal Methods I (EXT 75) Page 154 ummary Regresso s used to descrbg a relatoshp betwee two varables usg pared observatos from the varables. The tercept s the pot where the le crosses the Y axs ad the slope s the chage Y per ut X. Varace s derved from the sum of squared devatos from the regresso le. The regresso model s gve by The populato regresso model s gve by Y = 1X for observatos ad = X for the regresso le tself. yx. 1 Estmated from a sample the regresso le s Yˆ b b1x There are four assumptos usually made for a regresso, 1) Normalty (at each value of X ), ) Idepedece (1) of the observatos (Y, Y j ) from each other ad () of the devatos (e j ) from the rest of the model). 3) Homogeety of varace at each value of X. 4) The X values are measured wthout error (.e. all varato ad devatos s vertcal).

9 tatstcal Methods I (EXT 75) Page 155 Multple Regresso The objectves are the same as for smple lear regresso, the testg of hypotheses about potetal relatoshps (correlato), fttg ad documetg relatoshps, ad estmatg parameters wth cofdece tervals. The bg dfferece s that a multple regresso wll correlate a depedet varable (Y ) wth several depedet varables (X 's). The regressos equato s smlar. The sample equato s Y 1X1 X 3X3 The assumptos for the regresso are the same as for mple Lear Regresso The degrees of freedom for the error a smple lear regresso were, where the two degrees of freedom lost from the error represeted oe for the terecept ad oe for the slope. I multple regresso the degrees of freedom are p, where p s the total umber of regresso parameters ftted cludg oe for the tercept. The terpretato of the parameter estmates are the same (uts are Y uts per X uts, ad measure the chage Y for a 1 ut chage X). Dagostcs are mostly the same for smple lear regresso ad multple regresso. Resduals ca stll be examed for outlers, homogeety, curvature, etc. as wth LR. The oly dfferece s that, sce we have several X's, we would usually plot the resduals o Yhat ( Y ˆ ) stead of a sgle X varable. Normalty would be evaluated wth the PROC UNIVARIATE test of ormalty. There s oly really oe ew ssue here, ad ths s the way we estmate the parameters. If the depedet (X) varables were totally ad absolutely depedet (covarace or correlato = ), the t would't make ay dfferece f we ftted them oe at a tme or all together, they would have the same value. However, practce there wll always be some correlato betwee the X varables. If two X varables were PERFECTLY correlated, they would both accout for the AME varato Y, so whch would get the varato? If two X varables are oly partally correlated they would share part of the varato Y, so how s t parttoed? To demostrate ths we wll look at a smple example ad develop a ew otato called the Extra. For multple regresso there wll be, as wth smple lear regresso, a for the MODEL. Ths lumps together all for all varables. Ths s ot usually very formatve. We wll wat to look at the varables dvdually. To do ths there are several types of avalable A, two of whch are of partcular terest, TYPE 1 ad TYPE 3. I PROC REG these are ot provded by default. To see them you must request them. Ths ca be doe by addg the optos 1 ad/or to the model statemet. For regresso the Type II ad Type III are the same. I PROC GLM, whch wll do regressos cely, but has fewer regresso dagostcs tha PROC REG, the TYPE 1 ad TYPE 3 are provded by default.

10 tatstcal Methods I (EXT 75) Page 156 To do multple regresso A we smply specfy a model wth the varables of terest. For example, a regresso o Y wth 3 varables X 1, X ad X 3 would be specfed as PROC REG; MODEL Y = X1 X X3; To get the 1 ad we add PROC REG; MODEL Y = X1 X X3 /ss1 ss; Example wth Extra The smple example s doe wth created data set. Y X1 X X Now let s look at smple lear regressos for each varable depedetly, frst for varable X 1. If we do a smple lear regresso o X 1 we get the followg result. The Total s , ad ths wll ot chage regardless of the model sce t s adjusted oly for the tercept ad all models wll clude a tercept. If we ft a regresso of Y o X 1 the result s Model = 3.978, so the sum of squared accouted for by X 1 whe t eters aloe s If we ft X aloe, the result s Model = If we the ft both X 1 ad X together, would the resultg model be = 8.93? No, the model actually comes out to be 4.74 because of some covarace betwee the two varables. o how much would X 1 add to the model f X was ftted frst ad how much would X add f X 1 was ftted frst? We ca calculate the extra for X 1, ftted after X, ad for X ftted after X 1. The varable X aloe accouted for a sum of squares equal to ad whe X 1 was added the accouted for was 4.74, so X 1 eterg after X accouted for a addtoal = Therefore, we ca state that the accouted for by X 1, eterg the model after X, s Lkewse, we ca calculate the that X accouted for eterg after X 1. Together they accout for = 4.74 ad X 1 aloe accouted for 3.978, so X accouted for a addtoal = =.96 whe t etered after X 1. We eed a smpler otato to dcate the sum of square for each varable ad whch other varables have bee adjusted for before t eters the model. The sum of squares for X 1 ad X eterg aloe wll be X1 ad X, respectvely. Whe X 1 s adjusted for

11 tatstcal Methods I (EXT 75) Page 157 X ad vce versa the otato wll be X1 X ad X X1, respectvely. For the calculatos above the results were: X1 = 3.978, X = 4.115, X1 X = ad X X1 =.96. Fally, cosder a model ftted o all three varables. A model ftted to X ad X 3, wthout X 1, yelds Model = Whe X 1 s added to the model, so that all 3 varables are ow the model the, the accouted for s How much of ths s due to X 1 eterg after X ad X 3 are already the model? Calculate =.53. Ths sum of squares s deoted X1 X, X3. I summary, X 1 accouts for whe t eters aloe, whe t eters after X ad.53 whe t eters after both X ad X 3 together. Clearly, how much varato X 1 accouts for depeds o what varables are already the model, so we caot just talk about the sum of squares for X 1. We ca use the ew otato to descrbe the sum of squares for X 1 that dcates whch other varable are the model. Ths s the otato of the extra sum of squares. The otato s (X1) for X 1 aloe the model (adjusted for oly the tercept), (X1 X) dcatg X 1 adjusted for X oly, ad (X1, X3) dcatg that X 1 s etered after, or adjusted for, both X ad X 3. For our example; X1 = X1 X= X1 X, X3 =.53 The same procedure would be doe for each of the other two varables. We would calculate the same seres of values for the varable X ; X, X X1 or X X3 ad X X1, X3. The seres for varable X3 would be; X3, X3 X1 or X3 X ad X3 X1, X3. These values are gve the table below. Extra d.f. Error Error X X X X1 X X X X1 X X3 X X X X3 X X1 X,X X X1,X X3 X1,X All of these are prevously adjusted oly for the tercept (X, the correcto factor), ad ths wll always be the case for our examples. We could clude a otato for the tercept the extra (e.g. X1 X; X1 X, X; X1 X, X, X3; etc.), but sce X would always preset we wll omt ths from our otato.

12 tatstcal Methods I (EXT 75) Page 158 Partal sums of squares or Type II Wth so may possble sums of squares whch oes are wll be useful to us? The sums of squares ormally used for a multple regresso s called the partal sum of squares, the sum of squares where each varable s adjusted for all other varables the model. These are X1 X,X3; X X1,X3; ad X3 X1,X. Ths type of sum of squares s sometmes called the fully adjusted, or uquely attrbutable. I A they are called the TYPE II or TYPE III sum of squares sce these two types are the same for regresso aalyss. A provdes TYPE II PROC REG ad TYPE III PROC GLM by default. Testg ad evaluato of varables multple regresso s usually doe wth the TYPE II or TYPE III. ANOVA table for ths aalyss (F.5,1,8=5.3), usg the TYPE III (Partal ). ource d.f. M F value X1 X,X X X1,X X3 X1,X ERROR equetal sums of squares or Type I Whe we ft regresso, we are terested oe of two types of, ormally the partals sum of squares. There s aother type of sum of squares called the sequetally adjusted. These sum of squares are adjusted a sequetal or seral fasho. Each s adjusted for the varables prevously etered the model, but ot for varables etered after, so t s mportat to ote the order whch the varables are etered the model. For the model [Y = X 1 X X 3 ], X 1 would be frst ad adjusted for othg else (except the tercept X). X would eter secod, be adjusted for X 1, but ot for X 3. X 3 eters last ad s adjusted for both X 1 ad X. Usg our extra otato these are X1; X X1 ad X3 X1,X. These sums of squares have a umber of potetal problems. Ufortuately, the are dfferet depedg o the order the varables are etered, so dfferet researchers would get dfferet results. As a result the use of ths type s rare ad s oly used where there s a mathematcal reaso to place the varables a partcular order. Its use s restrcted pretty much to polyomal regressos whch use a seres of power terms (e.g. 3 Y 1X X 3X ) ad some other odd applcatos (e.g. some cases Aalyss of Covarace). Ivestgators sometmes feel that they kow whch varables are more mportat but ths s ot justfcato for usg sequetal sums of squares. o, we wll ot use sequetal at all, but they are provded by default by A PROC GLM. Multple Regresso wth A Ths same data set was ru wth A. The program was

13 tatstcal Methods I (EXT 75) Page 159 **********************************************; *** EXT75 Multple Regresso Example 1 ***; **********************************************; OPTION L=78 P=78 NODATE oceter oumber; DATA ONE; INFILE CARD MIOVER; TITLE1 'EXT75 MULTIPLE REGREION EXAMPLE #1'; INPUT Y X1 X X3; CARD; PROC PRINT DATA=ONE; TITLE 'Data Lstg'; RUN; ee A output Appedx 8 Note: The PROC REGREION secto PROC REG DATA=ONE LINEPRINTER; TITLE 'Aalyss wth PROC REG'; MODEL Y = X1 X X3; OUTPUT OUT=NEXT P=P R=E TUDENT=studet rstudet=rstudet lcl=lcl lclm=lclm ucl=ucl uclm=uclm; RUN; OPTION P=35; TITLE 'Resdual plot'; PLOT REIDUAL.*PREDICTED.='E'; RUN; QUIT; The overall model tatstcs for the dvdual varables The resdual plot Resduals, cofdece tervals ad uvarate aalyss proc prt data=ext; var Y X1 X X3 P E studet rstudet lcl ucl lclm uclm; ru; OPTION P=61; PROC UNIVARIATE DATA=NEXT NORMAL PLOT; VAR E; RUN; Output from proc prt, partcular the terpretato of the varables: studet, rstudet, lcl, ucl, lclm ad uclm Output from proc uvarate, especally the test of ormalty Ths same aalyss was doe wth GLM PROC GLM DATA=ONE; TITLE 'Aalyss wth PROC GLM'; MODEL Y = X1 X X3; RUN; QUIT;

14 tatstcal Methods I (EXT 75) Page 16 The results are the same, we oly wat to look at the Type I ad Type III. Evaluato of Multple Regresso If your objectve s to test the 3 varables jotly ( H: 1, ad 3 ) or dvdually ( H: ), you are doe at ths pot. Noe of the varables s sgfcatly dfferet from zero. If, however, your objectve s to develop the smplest possble, most parsmoous model, you may delete the varables oe at tme. Why oe at a tme? Because whe you remove a varable everythg chages sce they are adjusted for each other. We would remove the least sgfcat varable (the oe wth the smallest F value). I ths case that frst step would be to remove X. ANOVA table for aalyss of the varables X 1 ad X 3 aloe. (F.5,1,9 = 5.117). Note that X 1 s ow sgfcat, but X 3 s ot ad may be removed as step. ource d.f. M F value X1 X X3 X ERROR The varable X 1 s stll sgfcat. (F.5,1,1 = 4.965) ource d.f. M F value X ERROR Ths oe at a tme varable removal process s called stepwse regresso. More specfcally, t would be called backward selecto stepwse regresso. It s called backward because t starts wth a full model ad removes oe varable at at tme. There also exst a forward stepwse regresso where the best sgle varable s foud to start wth ad addtoal varables are added to the model f they meet the sgfcace requremets. Multple Regresso wth A (see A output Appedx 9) A has a program for stepwse model developmet. Ths s accomplshed wth PROC REG, wth the specfcato of a selecto opto. PROC REG DATA=ONE LINEPRINTER; TITLE 'tepwse aalyss wth PROC REG'; MODEL Y = X1 X X3 / selecto=backward; RUN; I the tal step (TEP ) the full, 3-parameter model s ftted, ad the parameter estmates are evaluated.

15 tatstcal Methods I (EXT 75) Page 161 Backward Elmato: tep All Varables Etered: R-quare =.4163 ad C(p) = 4. Aalyss of Varace um of Mea ource DF quares quare F Value Pr > F Model Error Corrected Total tep 1 s the frst removal, ths case of the varable X. The results for the remag varables are the gve.. Backward Elmato: tep 1 Varable X Removed: R-quare =.43 ad C(p) =.1784 um of Mea ource DF quares quare F Value Pr > F Model Error Corrected Total Parameter tadard Varable Estmate Error Type II F Value Pr > F Itercept X X tep s the ext removal (f eeded), ths case of the varable X 3. The result for the remag varable s the gve. Backward Elmato: tep Varable X3 Removed: R-quare =.3811 ad C(p) =.4819 um of Mea ource DF quares quare F Value Pr > F Model Error Corrected Total Parameter tadard Varable Estmate Error Type II F Value Pr > F Itercept X Fally A prts a summary of varable removals. All varables left the model are sgfcat at the.1 level. ummary of Backward Elmato Varable Number Partal Model tep Removed Vars I R-quare R-quare C(p) F Value Pr > F 1 X X Iterpretato of regresso Objectves ca vary regresso. You may be terested testg the correlatos (actually partal correlatos due to the adjustmet of oe varable for aother), or you may be

16 tatstcal Methods I (EXT 75) Page 16 terested the parameter estmates ad the resultg model (the full model or the reduced model from stepwse). Most aspects of the evaluato are smlar to what we observed wth smple lear regresso. The parameter estmates are terpreted as before, the chage Y per ut X. Of course, ow they are adjusted for other effects. tadard errors are provded for cofdece tervals, as well as a test of each regresso coeffcet agast (zero). Cofdece tervals are placed o the parameters the same as wth LR although the calculatos dffer. The d.f. for the t value s based o the ME (for the fal model) as wth smple lear regresso. The parameter ad stadard errors ca be estmated A. Resdual evaluato s very smlar to LR, but resduals are usually plotted o Yhat stead of X, sce there are several depedet varables (.e. X's). Evaluato of the resduals usg PROC UNIVARIATE for testg ormalty ad outler detecto s the same as for LR. Fully adjusted also mea fully adjusted regresso coeffcet (also partal reg. coeff.). A REG does ot gve tests of lke GLM, but the tests of the values are the same as the tests of the Type III. There are a few thgs that are dfferet. The R value s ow called the coeffcet of multple determato (stead of the coeffcet of determato). As dscussed, we ow evaluate for the dvdual varables. Note that the tests of TYPE III are detcal to the tests of the regresso coeffcets (see GLM hadout). PROC REG does oly the latter, ad wll ot do the former. There s a sute of ew dagostcs for evaluatg the multple depedet varables ad ther terrelatos. We wll ot dscuss these, except to say that f the depedet varables are hghly correlated wth each other (a correlato coeffcet, r, of aroud.9), the the parameter estmates ca fluctuate wldly ad upredctably ad may ot be useful. Also ote a curous behavor of the varables whe they occur together. Whe oe depedet varable X s adjusted for aother, sometmes t's are larger tha what t would be for that varable aloe ad sometmes athe are smaller. Ths s upredctable ad ca go ether way. For example. The X1 was whe the varable was aloe, but dropped to whe adjusted for X, ad creased to whe adjusted for X 3. It dropped to.53 whe adjusted for both. I essece the varables sometmes compete wth each other for sums of squares ad at other tmes ehace each others ablty to accout for sums of squares. Extra X X X3.37 X1 X X X1.96 X1 X X3 X X X3 3.9 X3 X. X1 X,X3.53 X X1,X3.819 X3 X1,X.116

17 tatstcal Methods I (EXT 75) Page 163 Adjusted Not oly wll the of oe varable crease or decrease as other varables are added, the regresso coeffcet values wll chage. They may eve chage sg, ad hece terpretato. Although the terpretato does ot usually chage, sometmes varables combato do ot ecessarly have the same terpretato as they mght have had whe aloe. ummary Multple regresso shares a lot terpretato ad dagostcs wth LR. Most dagostcs are the same as wth LR. The coeffcets ad sums of squares of the varables should be adjusted for each other. Ths s the sequetal sum of squares or the Type II or Type III A. Ths s the bg ad mportat dfferece from LR.