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1 ctvty #3: Smple Lnear Regresson Resources: actgpa.sav; beer.sav; In the last actvty, we learned how to quantfy the strength of the lnear relatonshp between two varables through correlaton coeffcents. Ths s mportant nformaton, but we re generally more nterested n modelng the relatonshp between two varables. In other words, we want to come up wth a way to predct values of a dependent (outcome) varable once we re gven a set of ndependent (predctor) varables. Let s look at a couple smple examples before we begn. ) fter graduatng from SU, you re quckly hred by Toyota to help them desgn new cars. Toyota wants you to determne the gas mleage of cars based on ther specfcatons (before they are actually bult). our job, then, s to complete two tasks: () Explan what factors nfluence mleage (and determne how much each factor contrbutes to mleage) () Use values of those factors to predct the mleage of pre-producton automobles. Lst several factors that nfluence the mleage of a car. hch factor do you beleve has the bggest mpact on mleage? ) Now suppose you re only nterested n modelng the relatonshp between the weght of a car and ts mleage. ou record the weght and mleage of 38 cars and create the followng scatterplot: Mleage (MPG) hat s the general relatonshp between the varables? Could you use ths scatterplot to make future mleage predctons? eght ( lb) 3) our textbook gves an example of predctng the amount of water that flows through the Nle Rver based on prevous measurements: February Nle Inflows January Nle Inflows

2 Stuaton: Over 4 hgh school senors apply to St. mbrose each summer. The admssons offce looks at each student s hgh school transcrpts as well as ther CT scores n order to make an admssons decson. Students who are not predcted to have success at SU (based on low CT scores or poor transcrpts) are not admtted to the Unversty. e wll attempt to predct an ndvdual s frst-year GP based on that ndvdual s CT scores. 4) Suppose you only know that the average GP for SU freshmen s.9. hat would be your best predcton for the GP of an applcant who earned a score of on the CT? How about for an applcant wth an CT score of 9? 5) thout any knowledge of the relatonshp between CT scores and GP, our best guess s the overall mean GP. Suppose we were gven the GPs of 3 former SU students along wth ther CT scores. The followng table lsts the condtonal mean GPs for these 3 students. Interpret what these condtonal means represent. Based on ths knowledge, what GP would you predct for the applcant wth an CT score of? Do you beleve ths predcton wll be more accurate than your prevous predcton? CT Score verage GP (n 3).8 (n 3). (n 5). 3 (n 3).4 4 (n 4).6 5 (n )??? 6 (n 3) 3. 7 (n ) 3. 8 (n 4) (n ) (n ) 3.8 6) Usng the table of condtonal means, predct the GP of a student whose CT score was 5. Make another predcton for a student whose CT score was 7. Do you see any problems wth ths condtonal means approach?

3 7) Let s use all the avalable nformaton. The followng scatterplot dsplays the relatonshp between a student s CT scores and GP for 3 former SU students. Before we make our predcton, let s examne the relatonshp between these varables. Does t appear as though GP and CT scores have a lnear relatonshp? Estmate the value of the correlaton coeffcent Freshman GP hat would happen to r f ths outler were added? CT Score 8) hen two varables appear to have a lnear relatonshp, we can descrbe the overall relatonshp by fttng a straght lne through the ponts. ou remember that y mx + b, where m slope and b y- ntercept, s the equaton for a straght lne. Carefully draw a straght lne that best fts the data n the above scatterplot. By estmatng the slope and y-ntercept, make a guess as to the equaton of your best-ft lne. (The same scatterplot graphed n a dfferent wndow has been provded to help you) GP CT 9) e would expect that no two people n ths class drew exactly the same lne through the data. How can we tell whch lne really does gve us the best ft? Can you thnk of a way to measure how well a lne fts a scatterplot of data? : ) To llustrate how we determne the lne that best fts the data, let s move on to a smaller data set. The followng table dsplays the meda expendtures (n mllons of dollars) and the number of bottles shpped (n mllons) for 6 major brands of beer. ould you expect a relatonshp between these two varables? If so, what knd of relatonshp would you expect? Brand Meda Expendture Shpment Busch Mller Genune Draft Bud Lght Coors Lght Mller Lte. 5.9 Budweser Source: Superbrands 998; //97

4 ) The followng scatterplot dsplays the two varables for the sx brands of beer. Does t appear as though the varables have a lnear relatonshp? Estmate the correlaton coeffcent and draw a straght lne through the data. Estmate the equaton for that best-ft lne. 4 3 Bottles Shpped (n mllons) Meda Expendtures (n mllons of $) ) I had SPSS calculate the regresson lne for ths set of data. It turns out that y.x +.76 s the lne of best ft. ou can see ths least-squares regresson lne on the followng scatterplot. Does ths lne provde us wth % accuracy n our predctons? Use ths equaton and/or the regresson lne to predct the shpment for a brand that spent $45 mllon on advertsng. ou should also use the equaton to fnd the predcted shpment for a brand that spent $76.6 on advertsng (whch s exactly what Coors spent). 4 Bottles Shpped Meda Expendtures

5 Bottles Shpped 3) No regresson lne wll provde us wth perfect accuracy n predcton (there s always some amount of measurement error). In calculatng the equaton of the best-ft lne, we try to mnmze the total amount of dstance between each data pont and the predcton lne (ths dstance wll always be greater than zero, due to the fact that our predcton s never perfect). Snce we use (meda expendtures) to predct (bottle shpped), we want a lne that s as close as possble to the ponts n the vertcal drecton. The followng graph dsplays the errors we wsh to mnmze: Predcted } Our error n predcton for ths value of. ctual value Meda Expendtures 4) Because some of the dstances wll be negatve and some wll be postve, we fnd the square of each dstance (absolute values aren t used, snce they are dffcult to work wth mathematcally). The least-squares regresson lne mnmzes the sum of the squared errors. Let s calculate the squared errors and see f the regresson lne really does mnmze the vertcal dstances between the observed values and the predcted values. How do we calculate the predcted values of? Lest Squares Regresson Lne:.x +.76 Observed Predcted Predcton Error Squared Error Meda Expendtures Bottles Shpped ˆ " ˆ ( " ˆ ) Sum Expected value. hy?

6 5) To demonstrate the fact that.x +.76 s the least-squares regresson lne, suppose we thought the best fttng lne was.3x +. (an arbtrary guess). The followng table dsplays the predctons based on ths lne as well as the sum of squared errors of predcton. Is ths predcton lne more or less accurate than the least-squares regresson lne we found earler? nother Possble Predcton Lne:.3x +. Observed Predcted Predcton Error Squared Error Meda Expendtures Bottles Shpped ˆ " ˆ ( " ˆ ) Sum By defnton, the least-squares regresson lne wll mnmze the sum of squared resduals ( " ˆ ) These resduals (or errors) represent the vertcal dstance between each observed value and the value predcted by our regresson lne. 6) s you know, the formula for a straght lne s: mx + b. In lnear regresson, the equaton for the least-squares regresson lne s often wrtten as: ˆ ˆ " + " ˆ x. Interpret the coeffcents of ths equaton. The least-squares regresson lne for the beer data was found to be: Interpret the coeffcents for ths specfc regresson lne. 7) hen we studed NOV, we wrote out formal models. e can do the same wth lnear regresson. Each ndvdual score,, s modeled by: predcted value: ˆ " + " ˆ x + E ˆ E ˆ " ˆ # + ˆ # x whch remnds us that error (E) s equal to the dstance between an observed value and the ( ) ( " ˆ ) Now that we have a basc understandng of the geometrc and algebrac propertes of the least-squares regresson lne, you mght wonder how we go about calculatng the slope and y-ntercept of the best-fttng lne.

7 Dervaton of the Parameters of the Least Squares Regresson Lne Predcted ˆ b + b Error ˆ!! ( b + b ) Observed (, ) e must fnd the slope and y-ntercept for ths lne so that the sum of these squared errors s mnmzed. N Let Q represent the sum of squared errors: Q! ( " b " b ) e need to fnd values for b and b that wll mnmze Q. e know that to mnmze a functon, we must set ts frst dervatve equal to zero and solve. Because we have two varables n ths functon, we ll need to take partal dervatves of Q wth respect to b and b. Partal dervatve of Q wth respect to b : (we treat b as a varable and all other terms as constants)! #! ( Q! b N " b! b " b ) # (Chan Rule)!( " b " b ) ( " b " b ) "! ( " b " b )! b e set ths partal dervatve equal to zero:! " (! b! b ) "(! b! b )! nb + b! Partal dervatve of Q wth respect to b : (we treat b as a varable and all other terms as constants)! #! ( Q! b N " b! b " b ) # (Chan Rule)!( " b " b ) ( " b " b ) "! ( " b " b )! b Set the partal dervatve equal to zero:! " (! b! b ) Now we must solve ths system of two normal equatons " (! b! b )! b! " b! "! b +! b!

8 System of normal equatons:!! nb + b b!! + b! Ths system can be solved to get: b n! x y "! x! n! x "(! x ) y and " x x y " y " x b! b! b n n n " y! " x" " x!(" x ) e can rewrte b gven the followng nformaton: S! " "! "!! xy ( x )( y ) x y x y n S xx!! " ( x ) x " ) x (! n S!! " (! n ( y ) yy y " ) y Therefore, " x y! " x" y S xy S y r n" x!(" x ) S xx S x n b So, the lne that mnmzes the sum of squared errors has the followng slope and y-ntercept parameters: b! b and b r S S y x In our example, r.89; s y 43.57; s x.47. Usng the mean values of and, we can compute: ˆ " r s y # (.89).47 & % (. ˆ " # " ˆ # (.)(65.933).76 s x $ 43.57' Now that we can compute a regresson lne and nterpret ts coeffcents, we need to fnd some way of measurng the accuracy of our predcton lne. e already know that the least-squares regresson lne s the lne of best ft (t mnmzes the sum of squared resduals (oftentmes called SS resdual or SSE)). hat we don t know s whether or not that best-fttng lne actually does a good job of fttng the data. (For example, magne a scatterplot of two uncorrelated varables. The shape of the scatterplot would be a crcle. e could ft the least-squares lne to the data, but t stll wouldn t ft the data very well.

9 ( " ˆ ) 8) The frst ndex of accuracy we may want to evaluate s SSE, the sum of squared resduals. To evaluate how well SSE serves as an ndex of accuracy, let s calculate the maxmum and mnmum values of SSE. The mnmum value of SSE would occur when every observed value of falls upon the predcton lne. If ths s the case, there would be zero dstance between each pont and ts predcted value. Therefore, when we have a perfect predcton, SSE. The maxmum value of SSE would occur when we have uncorrelated varables (knowng the value of would not tell us anythng about the value of ). The scatterplot of uncorrelated varables would look lke a crcle: hat would the least-squares regresson lne look lke n ths case? ell, we always want to mnmze the sum of squared resduals. #( " a) Mnmze: where a represents the predcted value of. e know that ths s mnmzed when a the mean of (by defnton, the mean s the value that mnmzes the sum of squared devatons). Therefore, the maxmum value of SSE (mnmum predcton accuracy) s: #( ") whch we remember s called SS or SS TOTL n NOV. Some factors that nfluence the sze of SSE are: () Varaton around the lnear regresson lne Smaller SSE Larger SSE () Nonlnearty (approprateness of a lnear model). Larger SSE Smaller SSE Some problems wth usng SSE as an ndex of accuracy: () It vares wth n (addng observatons almost always ncreases SSE). e d lke a per-observaton ndex SSE n " #( " ˆ ) n " S Varance of the estmate (varance of gven ) SSE n " #( " ˆ ) S n " () It s expressed n squared unts. Standard error of estmate

10 9) Let s now evaluate the merts of usng the varance of the estmate as an ndex of the accuracy of our predcton. S #( " ˆ ) n " Ths represents the average squared vertcal dstance between predcted and observed values. The mnmum value of S would occur when we have % accuracy n predcton. It doesn t take too much thought to realze the value of S would be zero when each observed value les on the regresson lne. To see what the maxmum value of S would be, let s once agan look at a scatterplot of ndependent (uncorrelated) varables. Recall that our least-squares regresson lne would mnmze the sum of squared resduals. The resduals are mnmzed when: #( ) SSE " (whch represents the maxmum value of SSE) Therefore, the maxmum value of Max{ S } #( ") n " S would be: n " $ $ ' #( ") ' & ) & ) % n " ( & n " ) $ n " ' & ) S % n " ( % ( ) e already know the problem wth S -- t s expressed n squared unts. e can quckly evaluate the maxmum and mnmum values of the standard error of estmate, S. Snce the standard error of estmate s just the square root of the varance of estmate, the mnmum value wll be zero. The maxmum value, when we try to predct values of wth an uncorrelated, s: Max{ S } #( ") n " n " n " S (where S s the standard devaton of ) ) e can create other ndces of accuracy by parttonng SS. Remember that SS represents the total sums of squares (or the sum of squared dstances from the observed values to the mean of ). Frst, let s look at: SSE SS #( " ˆ ) #( ") ( " r ) hat does ths rato, whch we wll call (-r ), represent? ) hat s the value of (-r ) when we have % accuracy n predcton? hat s the value of ths ndex of accuracy when we have uncorrelated varables?

11 3) The ndex (-r ) s at a mnmum when we have perfect predcton accuracy. Its maxmum value occurs when we have no accuracy. Ths s opposte of what we would ntutvely lke to see. To fx ths, we could fnd the value of r. hat s the formula for r and what does t represent? r SS " SSE SS ( ) "#( ) #( ) # " " ˆ " SS reg SS 4) e wll see that these ndces of accuracy are mportant statstcs n lnear regresson analyses. The followng table summarzes the propertes of these ndces: S Index Hghest ccuracy Lowest ccuracy ( ) SS ( ") SSE " ˆ SSE n " #( " ˆ ) n " # n "& % ( S $ n " ' S SSE n " #( " ˆ ) n " n " n " S ( " r ) SSE SS #( " ˆ ) # ( "). r SS " SSE SS SS reg SS. 5) In ths course, we wll use a computer to calculate regresson lnes and summary statstcs. e must focus on evaluatng the assumptons of lnear regresson, testng the sgnfcance of r and the regresson coeffcents, selectng regresson models, and nterpretng regresson analyses. The assumptons for lnear regresson are smlar to the assumptons we made n NOV: () Exstence: e have a dstrbuton for each value of (ths assumpton s often gnored) () Independence: scores are statstcally ndependent (there are no lnks among subject samplng procedures) (3) Lnearty: The subpopulaton means all fall on a straght lne (see the llustraton n your textbook) (4) Homoscedastcty: Subpopulatons (or E) varances are equal for all values. (5) Normalty: (or equvalent E) scores for each value are normally dstrbuted. Some general comments about testng these assumptons and robustness: () There are few clear-cut gudelnes for determnng the serousness of assumpton volatons () Seldom do we have equal sample szes for values (balanced desgns are more robust) (3) Tests for volatons n assumptons often have low power (lke the F-max test n NOV) (4) ssumpton tests also have assumptons that may or may not be met.

12 6) e end ths actvty wth a look at some output produced by SPSS. lnear regresson analyss was conducted on the beer dataset: Descrptve Statstcs Mean Std. Devaton N Bottles Shpped Meda Expendtures Correlatons Bottles Meda Shpped Expendtures Sg. Bottles Shpped..89. Meda Expendtures.89.. Change Statstcs dj. Std Error of R-Square Model R R-Square F-Change df df Sg. F-Change R-Square Estmate Change.89 a a: Predctors Meda Expendtures. Dependent Bottles Shpped NOV Model Sum of Squares df Mean Square F Sg. Regresson Resdual Total Coeffcents Unstandardzed Stndrdzd 95% CI Model B Std Error Beta t Sg. Lower Upper (Constant) Meda Expend

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