28. SIMPLE LINEAR REGRESSION III
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1 8. SIMPLE LINEAR REGRESSION III Ftted Values and Resduals US Domestc Beers: Calores vs. % Alcohol To each observed x, there corresponds a y-value on the ftted lne, y ˆ = βˆ + βˆ x. The are called ftted values. ŷ They are the values of y whch would be predcted by the estmated lnear regresson model, at the observed values of x. Eg: For the Beer example, the ftted Calores/ oz are y ˆ = x, where x s the % Alcohol. Calores/ oz % A lcohol 6 7 Regresson Analyss: Calores/ oz versus % Alcohol Calores/ oz = % Alcohol Constant % Alcohol S =.838 R-Sq = 48.% R-Sq(adj) = 47.% O Doul s has a % Alcohol of.4. Therefore, the correspondng ftted value s y ˆ = (.)(.4) = Note that ths "predcton" s smaller than the actual Calores for ths beer, y = 7. Regresson Resdual Error Total 878
2 Ftted Lne Plot, Calores vs % Alcohol Calores/ oz = % Alcohol In general, the observed data values y wll be dfferent from the ftted values ŷ. Calores/ oz % A lcohol 6 7 S.838 R-Sq 48.% R-Sq(adj) 47.% The dfferences between the data and ftted values are called the resduals, e = y y. Eg: In the Beer example, the resdual correspondng to O Doul s s = 3.4. We can thnk of the resduals e as proxes for the underlyng errors ε. A gven resdual wll be postve f the data pont les above the ftted lne, and negatve f the pont s below the ftted lne. Because of the way the least squares estmators are determned, the resduals wll always sum to zero. ˆ 7 Resduals Versus % Alcohol (response s Calores/ oz) The resduals are very mportant, snce they tell us how well the least squares lne fts the data. To assess the qualty of the ft, we can plot the resduals aganst the x 's, or aganst the ftted values. Resdual If the model s adequate, then these plots should show no systematc patterns or structure. If they do show structure, then the model may need to be modfed. - Resdual plots should be a routne part of any regresson analyss % A lcohol 6 7 They often reveal unexpected features n the data, such as nonlnear relatonshps (parabola), non-constant varance (wedge), and outlers, whch mght never have been dscovered otherwse.
3 Eg: For the Beer example, the plot of resduals versus the %Alcohol shows the bggest outler, Sam Adams Lght, though the resdual for O Doul s does not look that bad. The actual errors for these two ponts may be much larger, though, snce they may have dragged up the ftted lne, causng the appearance of a lnear pattern wth postve slope for the other resduals. After we remove these two outlers, the ftted slope ncreases from. to 3.4, and the resdual plot seems reasonably free of patterns. In partcular, the varablty seems not to change radcally wth x. Thus, the cleaned data set does not strongly contradct any of the assumptons underlyng the lnear regresson model. Regresson Analyss: Calores/ oz versus % Alcohol WITHOUT O DOUL S, SAM ADAMS LIGHT Calores/ oz = % Alcohol Constant % Alcohol S = 7.49 R-Sq = 63.% R-Sq(adj) = 63.% Regresson Resdual Error Total Resduals Versus % Alcohol, O'DOUL'S AND SAM ADAMS LIGHT DELETED (response s Calores/ oz) Next, we assess the varablty of y for a gven x. A helpful example for ths s Salary vs. Heght. Ftted Lne Plot for Salary vs. Heght Salary = Heght 67 6 S 9.7 R-Sq 7.4% R-Sq(adj) 7.3% Resdual - Salary % A lcohol Heght
4 A numercal measure of the closeness of the least squares lne to the data s provded by the sum of squared resduals, SSE = n e = If all the data ponts le exactly on a lne, then SSE wll be zero. (Why?) SSE stands for "Sum of Squares for Error". But SSE s not equal to the sum of squares of the actual errors, n ε. (Why?) = Nevertheless, we can use SSE to estmate the varance σ of the actual errors ε. = n = ( y yˆ ) An unbased estmator of σ s gven by SSE s = = Re sd. Mean Square. () n Snce we are estmatng two parameters, we have n "degrees of freedom". We can thnk of s as an "average squared predcton error", snce SSE s the sum of the squared devatons between the actual and predcted values ( y and yˆ). You mght be tempted to use s or SSE to compare dfferent models ftted to a gven set of y's. Ths s actually a very bad way to pck a model! You need to frst adjust for the fact that dfferent models may have dfferent numbers of explanatory varables. (We wll descrbe how to do ths later f tme permts). Eg: For the Salary vs. Heght data, s = Thus, we can estmate the standard devaton of the salary for a gven heght to be s = $9.7/Month. Ths tells us about the natural varablty whch would be expected n salary, even f the heght of the graduate were held fxed. Regresson Analyss: Salary versus Heght Constant Heght S = 9.7 R-Sq = 7.4% R-Sq(adj) = 7.3% Regresson Resdual Error Total Don't confuse s wth the sample varance of a batch of numbers, n ( y y). () n = We also used the name s for (), but t's not the same as (). There s a connecton, though, snce we can thnk of () as an "average squared predcton error" based on a smpler predctor of y: Just use y to predct all the y's, and forget about the x's.
5 Testng Whether β s Zero: Is the Straght-Lne Model Useful for Predctng y from x? A basc prncple of model buldng s that we should always use the smplest model whch adequately descrbes the data. Recall the Taste Test. Why do we thnk that the Earth goes around the Sun? What s the best predctor for the number of tournaments that Serena Wllams wll wn next year? Accordngly, unless there s strong evdence of a lnear relatonshp between x and y, t s better to adhere to the null hypothess of no lnear relatonshp, H : β =. Ths produces a much smpler model, y = β + ε, n other words, the y's are a batch of data wth a normal dstrbuton (mean β, varance σ ). In fact, ths s the same model used earler n the course to make nferences about the populaton mean μ based on the sample mean and standard devaton. So f β =, the best predctor of a future y value s just the average of the observed y values. It s only f β that the values of the "explanatory varable" x can mprove our ablty to predct a future y. If β =, the use of x wll usually degrade the qualty of the forecast! Of course, we can use a scatterplot of y versus x as well as a resdual plot to help us to decde whether a lnear relatonshp exsts, but t would be nce to have a more objectve tool. Why can't we just check whether Because even f β =, the estmator βˆ s a random varable, and wll not n general be exactly zero. So βˆ has a samplng dstrbuton. Ths dstrbuton s normal, wth mean β. The standard error of βˆ can be estmated from the data. We wll call ths s. βˆ βˆ =? Mntab calls t SE Coef", that s, the standard error of the estmated coeffcent. For the Stock Market returns, s.98. Regresson Analyss: Today versus Yesterday Today = Yesterday Constant Yesterday S =.9746 R-Sq =.6% R-Sq(adj) =.6% s βˆ Regresson Resdual Error Total
6 Market Returns Today = Yesterday S.9746 R-Sq.6% R-Sq(adj).6% Now, we can test H : β = versus H A : β by convertng nto ts own t-score, βˆ t =. s βˆ βˆ The result s gven by Mntab as "T". Today For the Stock Market data,.7486 t = = If β =, then t wll have a t dstrbuton wth n degrees of freedom Yesterday The p-value correspondng to a -taled test s gven by Mntab as "P". For the Stock Market data, we get p =.. Concluson: Strong evdence of a lnear relatonshp between today's return and yesterday's return! (But the returns are stll qute hard to predct.) For the Salary vs. Heght example, the t-statstc for β s 8.3, yeldng p =. (rounded to nearest /). Apparently, heght matters. For the People Meter vs. Nelsen Ratngs, the p-value for β s agan., so there s strong evdence of a lnear relatonshp. But we mght be nterested n a dfferent null hypothess than β =. Regresson Analyss: People Meter versus Nelsen Ratng People Meter = Nelsen Ratng Constant Nelsen Ratng S =.797 R-Sq = 94.9% R-Sq(adj) = 94.3% Regresson Resdual Error Total 9 8.6
7 People Meter vs. Nelsen Ratngs People Meter = Nelsen Ratng Makng Inferences About the Slope β People Meter S.797 R-Sq 94.9% R-Sq(adj) 94.3% For any value of the true slope β, the quantty has a t-dstrbuton wth n degrees of freedom. Ths allows us to construct a confdence nterval or a test for β. A ( α)% confdence nterval for β s gven by where t α/ s based on n degrees of freedom. βˆ ± t s α βˆ, βˆ β s βˆ Nelsen Ratng. 7. For the TV example, a 9% confdence nterval for β s gven by βˆ ± t, s. βˆ where from Table 6, t. =.36 (df=8). From the Mntab output, we obtan the nterval.6 ± (.36) (.84), whch reduces to (.83,.). Note that ths nterval contans., so t s plausble that β =. (What would be the practcal mportance of ths?) To test H : β = versus H A : β at level., we can use the test statstc βˆ.6 t = = =.3, s.84 βˆ whch does not come close to the crtcal value of.36, so we do not reject H. Clearly, the p-value s very large, but we cannot compute t wthout addtonal tables. (Can get tal areas for a t-dstrbuton n Mntab, but we won t). In any case, there s lttle or no statstcal evdence to suggest that β s dfferent from. In ths example, t was more reasonable to test f β = than to test f β =. Keep n mnd, though, that Mntab bases ts "T" and "P" on a null hypothess that the true parameter values are zero. If you are nterested n a dfferent null, you must compute the test statstc manually, as above. We can get confdence nterval and hypothess test for the ntercept β n a smlar way. Use the standard errors, t-statstcs and p-values for βˆ, as gven by Mntab.
8 Eg: July Mean Temperatures n UK versus the year. Regresson Analyss: JUL versus Year UK July Mean Temperatures (Degrees C) JUL = Year S.47 R-Sq 6.% R-Sq(adj).% JUL = Year Constant Year JUL S =.47 R-Sq = 6.% R-Sq(adj) =.% Year 98 Regresson Resdual Error Total Eg: Returns on two stocks. HersheyRet.3... Hershey vs. Apple Returns, Monthly HersheyRet = AppleRet S.6696 R-Sq 4.3% R-Sq(adj) 3.4% Regresson Analyss: HersheyRet versus AppleRet HersheyRet = AppleRet Constant AppleRet S =.6696 R-Sq = 4.3% R-Sq(adj) = 3.4% AppleRet.. Regresson Resdual Error Total.88
9 Eg: Busness Week Rankngs for UG Busness Schools BW 7 Rank vs BW 6 Rank 9 8 Regresson Analyss: BW 7 Rank versus BW 6 Rank BW 7 Rank = BW 6 Rank cases used, 4 cases contan mssng values BW 7 Rank Constant BW 6 Rank S = 9.74 R-Sq = 84.8% R-Sq(adj) = 84.% 3 BW 6 Rank 4 6 Regresson Resdual Error Total 33 Eg: Medan Startng Salares for Undergrad Busness Students Resduals Versus BW 6 Rank (response s BW 7 Rank) Med. Startng Salary vs BW 7 Rank 6 Resdual - - Med. Startng Salary BW 6 Rank BW 7 Rank 7 8 9
10 Regresson Analyss: Med. Startng Salary versus BW 7 Rank Med. Startng Salary = BW 7 Rank Eg: Salary vs. Heght. Monthly Salary (Dollars) vs. Heght (Inches) 67 Constant BW 7 Rank S = 367. R-Sq = 8.7% R-Sq(adj) = 8.3% Salary Regresson Resdual Error Heght 4 3 Resduals Versus Heght (response s Salary) Do you see how the two outlers on the rght may have dragged down the lne? Resdual Heght
18. SIMPLE LINEAR REGRESSION III
8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.
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