ࡱ> B` A=bjbj o A5"""""""tzzzz{T2:~LLLL'Հ< $h'`2"F''FF2""LLG~~~F."L"L nF ~~D""įL.~ 0-CKzt4&, ]0ZjjLį"į1hJ\<'11122R 111FFFFDvz$z"""""" Simple Linear Regression ________________________________________ 1) Discuss conceptual differences between ANOVA and regression. 2) Identify the components of the simple regression equation ((1, (0) and explain their interpretation. 3) Demonstrate the Least-Squares method for calculating (1 and (0. 4) Develop a measure for error in the regression model and demonstrate a method for comparing the variance due to error with the variance due to our model. 5) Define and explain the correlation coefficient and the coefficient of determination. 6) Discuss the relationship between correlation and causation. CI vs. ANOVA vs. Regression __________________________________________ Key word for CI: Key word for ANOVA: t-test Key word(s) for regression: Trivia Wars ______________________________________ Lets say Amherst declares war on Northampton because Northampton tries to lure Judie's into moving out of Amherst. No one actually wants to kill anyone, so we decide to settle our differences with a rousing game of Jeopardy! You are elected the Captain of Amhersts team (as if you would be selected instead of me). How are you going to choose the team? Multiple criteria: 1) Knowledge 2) Performance under pressure EX: Cindy Brady 3) Speed Historical roots in WW II Who would be a good ball turret gunner? Regression ______________________________________ What is the relationship between Grades or Money or Relationship or Health Status and Life Satisfaction? ______________________________________ How well can I predict a persons Life Satisfaction if I know their Grades or Money or Relationship or Health Status ______________________________________ How are we going to do this?   General form of Probabilistic (Regression) Models ________________________________________ y = + or y = regression line + error or y = + _______________________________________ E(y) - Regression line connects Simple Regression First-Order Single-Predictor ___________________________________________ y = (0 + (1x + ( y = x = E(y) = ( = ______________________________________ (0 y = mx + b (1 Interpretation of y-intercept and slope ________________________________________ Intercept Intercept only makes sense if x Regression equation only applies ________________________________________ Slope Change in y for a unit change in x. + implies relationship implies relationship ________________________________________ Most important point: Give me a value for x and the regression equation and I can Steps to completing a regression analysis (both simple and multiple) ________________________________________ Step 1Hypothesize the deterministic component of the model.  Step 2Use sample data to  Step 3Specify the probability distribution of the Step 4Evaluate the usefulness of  Step 5Use the model for  Fitting a model to our data (Step 2) ________________________________________ Least-Squares method Sum of the vertical distance between each point Square of the vertical distance is When in doubt, think Bribery!! ____________________________________________ You want to determine the relationship between monetary gifts and "BONUS POINTS FOR SPECIAL CONTRIBUTIONS TO CLASS" added to your final average so that you can decide how large a check to write at the end of the semester (though I do prefer cash for tax purposes). Let's say x represents the amount of money contributed by past students, and y represents the number of "Bonus Points" awarded to them.  EMBED Excel.Chart.8 \s  Fishing for a regression line ________________________________________  EMBED Excel.Chart.8 \s  XYDistanceSquared-DistanceGiftBPy=5Y=x+1y=5y=x+141-4-416168940160250204650-2040-43224 Which regression line is better? Is that the best regression line? Formulae for Least Squares Method ________________________________________ (1 = SP / SSx (0 = My ((1* Mx) ______________________________________________ SSx =  EMBED Equation.3  SP =  EMBED Equation.3  Finding the best-fit regression line ________________________________________ xYx2Xy4116489647225410653630(x = 20(y = 20((x2) = 120((xy) = 116 SSx = ((x2) [((x)2 / n] = 120 [(20)2 / 4] = 120 (400 / 4) = 120 100 = 20 SP = ((xy) [((x)((y)] / n = 116 [(20)(20) / 4] = 116 (400 /4) = 116 100 = 16 ________________________________________ (1 = SP / SSx = 16 / 20 = 0.8 (0 = My ((1* Mx) = 5 (.8)(5) = 1.0 ________________________________________ The Least-Squares Regression Line ________________________________________  EMBED Excel.Chart.8 \s  xyE(y)DistanceSquared-Distance414.2-3.210.24897.4 1.6 2.56252.6 2.4 5.76655.8-0.8 0.64019.20 Testing Example __________________________________________ Unbeknownst to you, Biff is the heir to his familys Widget fortune. For his summer job, Biff was asked to evaluate a group of employees widget making ability using a standardized widget-making test. Biffs boss (Uncle Buck) asks Biff to determine the regression equation that one would use to predict performance on the test from years of service with the company. The data appear below. x (years)y (score)x2y2xy355930251654781660843124721651842882584336411658925792144536393969189473165329292584257056420375956252252484230496(x = 35(y = 695((x2) = 133((y2) = 49,861((xy) = 2,548 Calculations ______________________________________________ SSx = ((x2) [((x)2 / n] SP = ((xy) [((x)((y)] / n ______________________________________________ (1 = SP / SSx (0 = My ((1* Mx) Widget Test Scatter Plot ________________________________________  EMBED Excel.Chart.8 \s  Assumptions regarding Error (() ________________________________________ (: essentially vertical distance from regression line _________________________________________ 1) The mean of the probability distribution = 2) The variance of the probability distribution of ( is 3) Distribution of ( is 4) Values of ( are of one another. Factors that contribute to Error ________________________________________ Two types of Error 1) Measurement Error - EX: incorrect reading of beaker 2) Chance factors EX: unusually non/reactive chemical Estimation of Variability due to Error (Step 3) ________________________________________ s2 is analogous to MSE s2 = SSE / dferror = SSE / n 2 SSE = SSy - (1(SP) SSy = (y2 [((y)2 / n] ________________________________________  (s2 = (SSE / (n-2) = (MSE s = Estimated Standard Error of the Regression Model or = Root MSE Calculate the error ______________________________________ SSy = (y2 [((y)2 / n] = 49,861 [(695)2 / 10] = 49,861 (483,025 / 10) = 49,861 48,302.5 = 1558.5 SSE = SSy - (1(SP) = 1558.5 11.0(115.5) = 1558.5 1270.5 = 288 s2 = SSE / (n-2) = 288 / (10-2) = 36 (a/k/a MSE) s = (36 = 6 (a/k/a Root MSE) Important points about error or ( ________________________________________ The smaller (, the better we can The smaller (, the more the individual data points will be around the regression line. A smaller ( implies that x is a predictor of y. Why? Also, can use this information to develop a sense of how far points should fall off the line. We can calculate a CI around the regression line. 95% of our points should fall within about 2 RMSEs of the regression line. If not, HMMMM Evaluate the usefulness of the model (Step 4) ________________________________________ Step 1: Specify the null and alternative hypotheses. Ho: (1 = 0 Ha: (1 ( 0 Step 2: Designate the rejection region by selecting (. Step 3: Obtain the critical value for your test statistic t df = n-2 Collect your data Step 5: Use your sample data to calculate: (1 SP / SSx s(1 = SE = s / (SSx Step 6: Use your parameter estimates to calculate the observed value of your test statistic t = (1 0 / s(1 Step 7: Compare tobs with tcrit: If the test statistic falls in the RR, reject the null. Otherwise, we fail to reject the null. Calculating whether (1 (slope) ( 0 ________________________________________ Ho: (1 = 0 Ha: (1 ( 0 tcrit 2.306 (df = 8; ( = .05) RR |tobs| > 2.306 Observed t = (1 0 / (s / ( SSx) = 11 0 / (6 / (10.5) = 11 / 1.85 = 5.94 We would reject the null hypothesis because tobs exceeds the tcrit. In other words, tobs falls in the rejection region. Implication:   Correlation Coefficient ________________________________________ Pearsons product moment coefficient of correlation a measure of the strength of the linear relationship between two variables. Terminology / notation: r Pearsons r correlation coefficient ________________________________________ r =  EMBED Equation.3  Interpretation: +1 perfect positive relationship (strong positive relationship) 0 no relationship (strong negative relationship) -1 perfect negative relationship  r for the Widget Example ____________________________________________ r =  EMBED Equation.3  Experience in Years =  EMBED Equation.3  =  EMBED Equation.3  = 115.5 / 127.92 = .90 Experience in Months =  EMBED Equation.3  =  EMBED Equation.3  = 1386 / 1535.074 = .90 Stress and Health ____________________________________________ There is a strong negative correlation between stress and health. Generally, the more stressed a person is, the worse their health is. But, does that mean that stress causes poor health? No...Yes...  Coefficient of Determination ________________________________________ r2 represents the proportion of the total sample variability For simple, linear regression, r2 = r2. ________________________________________ More general formula is as follows: r2 = (SSy SSE) / SSy = 1 (SSE / SSy) ________________________________________ SPSS will give us everything we need! Questions about Regression output __________________________________________ 1) What is r? 2) Is this correlation significant? 3) How much of the variance in # of colds per winter can be explained by weekend bedtime? 4) What is the y-intercept? 5) Is it significantly different from zero? 6) What is E(y) if x = 10:00 PM (10)? 7) What is E(y) if x = 2:00 AM (14)? 8) Are your answers to questions 6 and 7 meaningful? SPSS output ______________________________________________ Model Summary ModelRR2Adj R2SE1.204.041.0341.20 ANOVA Model Sum of SquaresdfMean SquareFSig.1Regression7.6817.685.32.023 Residual177.581231.44   Total185.27124    Coefficients Model Unstand CoeffStand CoefftSig.  BSEBeta  1(Constant)5.7111.69 3.38.001 bed_we-.266.12-.20-2.31.023 I just dont get it ____________________________________________ I know Im old, but I just dont get the tattoo thing. I gotta figure that people regret their decision as time passes. The data below represent 100 subjects who had tattoos etched into their skin between 1 and 5 years ago. They rated their satisfaction with their lifetime scar on a scale of 1-10 (10 = extremely satisfied). Is there a relationship between tattoo age and tattoo satisfaction? ((x)((y)((x2)((y2)((x)(y)300600110039541660 Regression Equation SP = ((xi)(yi) [((xi)((yi)] / n SSx = (xi2 [((xi)2 / n] (1 = SP / SSx (0 = My ((1* Mx) Hypothesis Test SSy = (yi2 [((yi)2 / n] SSE = SSy - (1(SP) s2 (MSE) = SSE / (n-2) t = (1 - 0 / (s / (SSxx) Correlation Coefficient r =  EMBED Equation.3  Calculating the regression parameters ______________________________________________ SP = ((xi)(yi) [((xi)((yi)] / n SSx = (xi2 [((xi)2 / n] (1 = SP / SSx (0 = My ((1* Mx) Let's do a t-test ______________________________________________ SSy = (yi2 [((yi)2 / n] SSE = SSy - (1* (SP) s2 = MSE s = t = (1 0 / (s / (SSx) We reject the null and conclude that there is a significant negative relationship between tattoo age and tattoo satisfaction. Let's calculate the correlation coefficient ____________________________________________ SSy = (yi2 [((yi)2 / n] r =  EMBED Equation.3  r2 = Although there is a significant negative relationship between tattoo age and tattoo satisfaction, age only explains about 25% of the variance in satisfaction. Clearly, other factors are involved. Skipping Class __________________________________________ In a perfect world, the correlation between the number of classes skipped and the percentage of classes skipped should be 1.00. Let's see how well the percentage of classes skipped (x) predicts the number of hours of classes skipped (y). Please calculate the regression line, the correlation coefficient, and the coefficient of determination. 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NM  !!(4E4  3Q y = 5Q ; Q ; Q3_ " NM  ""(4E4D$% M=3O&Q4$% M=3O&Q4FAtu 3OGi A 3*$@??#M&43*#M&4% = M3O]&Q Donation'4% P,MZ3Op&Q  Bonus Points'4523   43d" Z13O[% M@ 3O&Q443_ M NM  MM<4444 e@ @??@@@@@@@@@@@@@@ @ @ "@ "@ $@e??@"@@@@@@@@@@@@@@@ @@"@@ $@ @ @e>  u'  dMbP?_*+%"??U#    @?? @"@?@@@@@@@@@ @@ +?eeA;@@ +?eeA7@@ @ @  @"@  "@$@  $@&@ @  ?@ @@ @@ @@@@@@@@ @@"@@$@@~ (? DDD~ ?(? DDD~ @(ffffff@ DDD~ @(ffffff@ DDD~ @(333333@ DDD~ @(333333@ DDD~ @(333333@ DDD~ @(333333@ DDD@4D""""$SS$$$$$$$$::::::: !"~  @( !@ DDD ~ !"@(!#@ DDD!~ "$@("%@ DDD" (::(  p  6NMM?PK ]`d8  u'MXhp deskjet 960c series݀f dںں@RLdͫ"d??3` a@` a@` M` M 53d?23 M NM4 3QQ ;Q ;Q3_ M NM   4E4  3Q  y = x + 1Q ; Q ; Q3_ ! 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Line8882C788!82H87888---'--- LC&---'---  ---'---  - -   !!-- -' ' ' !FMicrosoft Excel ChartBiff8Excel.Chart.89qOh+'0@Hd| Mathew SchulkindeMatt SchulkinddMicrosoft Excel@@q՜.+,0 P u'\pMatt Schulkind Ba=S=F<X@"1Arial1Arial1Arial1Arial1cArial1cArial1Arial1^Arial1^Arial1Arial1Arial1Arial1Arial1TArial1Arial1Arial1TArial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *  `Chart36Sheet14Sheet2MSheet3TbZ  3  @@  Slope y-intercept   u'MBhp deskjet 5100 series!@d߀ odBeRLdhpztsb04ArialHBeںںhp deskjet 5100 serieskjet 5100 series,LocalOnly,DrvConvertVWK*XX"d??3` a@` 2` M` a@,3d23 M NM4 3QQ ;Q ;Q3_ M NM   4E4  3Q  y = x + 1Q ; Q ; Q3_ ! NM  !!(4E4  3Q y = 5Q ; Q ; Q3_ " NM  ""(4E4  3Q 2Least Squares Reg. LineQ ;"Q ;"Q3_ # NM  #Nd4E4D$% M=3O&Q4$% M=3O&Q4FA+m 3O f 3*$@??#M&43*#M&4% WM3O)&Q Donation'4% KMZ3OT&Q  Bonus Points'4523   43d" H;3OGAJ% M@ 3O& Q423 M NM443_ M NM  MM<4444 e@ @???@@@@@@@@@@@@@@@@@@@@ @ @ @ "@ "@ "@ $@ $@e??@ZZZZZZ?"@@@MKKKKK?@@@@@@@ @@@kiiiii@@@@@@{xxxxx@ @@@"@@@ $@ @ !@ @ NKKKKK#@e> NKKK u'  dMbP?_*+%"??U#    @?? @"@?@@@@@@@@@ @@ +?eeA;@@ +?eeA7@@ @ @  @"@  "@$@  $@&@ @  ?@ @@ @@ @@@@@@@@ @@"@@$@@~ (? DDD~ ?(? DDD~ @(ffffff@ DDD~ @(ffffff@ DDD~ @(333333@ DDD~ @(333333@ DDD~ @(333333@ DDD~ @(333333@ DDD@4D""""$SS$$$$$$$$::::::: !"~  @( !@ DDD ~ !"@(!#@ DDD!~ "$@("%@ DDD" (::(  p  6NMM?PK ]`/  u'MXhp deskjet 960c series݀f dںں@RLdͫ"d??3` a@` a@` M` M ;3d?23 M NM4 3QQ ;Q ;Q3_ M NM   4E4  3Q  y = x + 1Q ; Q ; Q3_ ! NM  !!(4E4  3Q y = 5Q ; Q ; Q3_ " NM  ""(4E4D$% M=3O&Q4$% M=3O&Q4FAZ 3OC]5 3*$@??#M&43*#M&4% ! M3ON Q Donation'4% PCjMZ3OY&Q  Bonus Points'4523   43d" "l3O#p% M@ 3O& Q443_ M NM  MM<4444 eee >@7 u'  dMbP?_*+%"??U#    @@?"@$@?@?@@@@@@@ @@ +ZZZZZZ? eeA;@@ +@<<<<@  7 u'  dMbP?_*+%" ??sU>@7 SummaryInformation(#%+DocumentSummaryInformation8/,_12269114388L(!FʫCKʫCKOle 4Xd lt| 3 Sheet1Sheet2Sheet3Chart3  WorksheetsCharts !FMicrosoft Office Excel ChartBiff8Excel.Chart.89qPRINT'*tCompObj5iObjInfo),7Workbook*cC 9 k   {   '' dArial-"System 5--------dArial-------------'- z -̙- Z !!---'--- z - $GG  -G G - -    oo))  G     ,  ,  G G---'--- Y !!---'---  }---'---  z-- , +GA-- $CAsA---'---   z},  $, K^ },  }, K---'---   zx,  $, F^ x,  x, F---'---   z $(---'---   zG $G}yGG}---'---   z $C$---'---   zN,  $, ^ N,  N, ---'---   zG $GOyGGO---'---   z $C,---'---   zg  $5 (g  g 5 ---'---   z--  ~---'---  z  ---'---  z,  4 $ ---'---  z+G 3O#?---'---  z---'---  }---'--- z ---'--- z ---'--- z  2 ^ ^  40WW2 i 50WW2 60WW2 tt"70WW2 80WW2 --90WW2 {{8100WWW---'--- z ---'--- z   2 = = " 2W 2 = = = 3W 2 =   = X 4W 2 = = s 5W---'--- z --------'---   h  2 o~6 2 o~6 2 ; ~ ExperiencehW_W=+W_WW-----'--- z -------'--- qe'd  2 o=D[ 2 o=D[dArial- 2 [ Test Score_WW4+hW_=W- -----'--- z ---'--- z - - Z !!-- -' ' '%$ \pBigDog Ba=!=[<X@"1JArial1JArial1JArial1JArial1JArial1wJArial1JArial1JArial1wJArial1wJArial"$"#,##0_);\("$"#,##0\)!"$"#,##0_);[Red]\("$"#,##0\)""$"#,##0.00_);\("$"#,##0.00\)'""$"#,##0.00_);[Red]\("$"#,##0.00\)7*2_("$"* #,##0_);_("$"* \(#,##0\);_("$"* "-"_);_(@_).))_(* #,##0_);_(* \(#,##0\);_(* "-"_);_(@_)?,:_("$"* #,##0.00_);_("$"* \(#,##0.00\);_("$"* "-"??_);_(@_)6+1_(* #,##0.00_);_(* \(#,##0.00\);_(* "-"??_);_(@_)                + ) , *  `Chart1bSheet1)Sheet2 *Sheet3"ZR3  @@  slope intercept   M Brother HL-5140 series S odXXLetterPRIV ''''1 $B dArialMDSCHULKIND-L3ENGLISHBRSP104A.EXEBRLHL04A.DLLBRBH204A.DLLBO5140.INI )  Z47 000Ix'RSP104A.EXEdArialdmdschulkindBRLHL04A.DLLEOSC"d??3` `  `  3d23 M ̙/4  3Q SkillQ ; Q ; Q3_ M NM   4E4 3Q "Regression lineQ ;Q ;Q3_  NM  !!(4E4D $% M3O&Q4$% M3O& Q4FA d 3O]O 3*@@?#M43*D@#M4% G;M3Ok!& Q  Experience'4% S MZ3O!f& Q  Test Score'4523  ̙/43d" 3_ M NM  MM<4444 e@@@@@@@@@@@@@ @eK@*kJ@S@ D2P@R@zR@M@n+U@@V@O@@R@U@R@ H@e>    dMbP?_*+%"??U    @K@@0*kJ@ ?@D:u<%@?@:u<%@@S@@0 D2P@ ?@D:u<%@?@:u<%@@R@@0zR@ ?@D:u<%@?@:u<%@@M@@0n+U@ ?@D:u<%@?@:u<%@@@V@@O@@@R@@U@@R@ @H@  + ?@ e e A7  + &@ e e A; Ntttt((((((O(  p  6NMM? ]`!  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