Response Variable denoted by y it is the variable that is to be predicted measure of the outcome of an experiment also called the dependent variable
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1 Statistics Chapter 4 Correlatio ad Regressio If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Associatio betwee Variables Two variables are associated whe they are measured o the same idividuals whe some values of oe variable ted to occur more ofte with some values of the secod variable tha the other values of the secod variable. Example (a.) Height ad weight are associated ( positively) (b.) High school GPA ad college GPA are associated (positively) (c.) weight of car ad MPG ratig (egative) (d.) age ad whether a perso has had a heart attack (positively) Whe lookig at the relatioship betwee variables we ca (1) just look at the associatio (-or-) (2) try to explai variatio i oe variable i terms of the other variable (that is, try to predict oe variable from the other) Respose Variable deoted by y it is the variable that is to be predicted measure of the outcome of a experimet also called the depedet variable Explaatory Variable deoted by x explais the chage i the respose variable also called idepedet variable Wat to explai of predict why based o the explaatory variable x. I some cases there is o clear distictio betwee variables as far as which the respose is ad which the explaatory variable is. Method of Uderstadig Relatioship (1) Costruct umerical ad graphical display of each variable (2) costruct a scatter plot of y versus x (3) look for patters ad deviatios from patter (ie: outliers) (4) Whe the overall patter is quite regular, use a mathematical model to describe the patter. 1 P a g e
2 Scatter Diagrams A graph i which pairs of poits, (x, y), are plotted with x o the horizotal axis ad y o the vertical axis. The explaatory variable is x. The respose variable is y. Oe goal of plottig paired data is to determie if there is a liear relatioship betwee x ad y. If we have two (or more) variables we are usually iterested i the relatioship betwee the variables. Paired Data (x, y) Importat Questios How strog is the liear correlatio betwee x ad y? What lie best represets the data? Let x ad y be our variables of iterest. Suppose we have observatios (x 1, y 1 ), (x 2, y 2 ) (x, y ) Scatter plot is a plot of poits (x i, y i ) Look for: patter deviatios from patter describe the patter i terms of stregth, directio, ad form Some Patters liear patters = data poits approximately follow a straight lie Curviliear = data poits approximately follows some curve approximatio 2 P a g e
3 How Strog Is the Liear Correlatio? Not all relatioships are liearly-correlated. Statisticias eed a quatitative measure of the stregth of the liear associatio. How to do this? The Sample Correlatio Coefficiet r - measure of the stregth of liear associatio give data (x 1, y 1 ), (x 2, y 2 ) (x, y ) Statisticias use the sample correlatio coefficiet r to measure the stregth of the liear correlatio betwee paired data. Properties of Correlatio 1) r has o uits. 2) 1 r 1 3) r > 0 idicates a positive relatioship betwee x ad y, r < 0 idicates a egative relatioship. 4) r = 0 idicates o liear relatioship. 5) Switchig the explaatory variable ad respose variable does ot chage r. 6) Chagig the uits of the variables does ot chage r. 7) Correlatio coefficiet, r, measures the stregth of liear associatio. Correlatio is ot a measure of stregth of curved associatio. 8) Correlatio requires that both variables are quatitative. 9) Correlatio is ot resistat to outliers 10) Correlatio does ot provide a complete descriptio of the relatioship betwee variables. A Computatioal Formula for r 3 P a g e
4 The correlatio coefficiet is 1 i 1 i where x x, y y i 1 i 1 1 xi x yi y r 1 i1 sx sy 1, s x xi x 1 2, ad s 2 y yi y 1 i1 1 i1 The computatioal formula is r i1 x y i i xy xi x yi y i1 i1 NOTE: r = sample correlatio coefficiet computer from a radom sample of (x, y) data pairs = populatio correlatio coefficiet computed from all populatio data pairs (x, y) Illistratio Caribou (x, i hudreds) ad wolf (y) populatios 4 P a g e
5 Iterpretig the Value of r r = 0 : There is o liear relatio for the poits of the scatter diagram. r = 1 or r = 1 : There is a perfect liear relatio betwee x ad y; all poits lie o a straight lie. 0 < r < 1 : The x ad y values has a positive correlatio. As x icreases, y teds to icrease. 0 :The x ad y values have a egative correlatio. As x icreases, y teds to decrease. 1 < r < 5 P a g e
6 Example - Which of the followig shows a strog egative correlatio? Critical Thikig Expect r to vary from sample to sample. So, cosider the sigificace of r as well as its value whe assessig the stregth of a liear correlatio. (Sectio 11.4) r 1 oly implies a liear relatioship betwee x ad y. It does ot imply a cause ad effect relatioship betwee x ad y. The values of x ad y may both deped liearly o some third lurkig variable. Example - Over the past few years, there has bee a strog positive relatioship betwee the aual cosumptio of coffee ad the umber of computers sold per year. Which coclusio is the best oe to draw from this strog correlatio? a). Coffee cosumptio stimulates computer sales. b). Computer users are sophisticated ad thus are iclied to drikig coffee. c). The correlatio is purely accidetal. d). The resposes of both variables probably reflect the icreasig wealth of the citizery. Liear Regressio Liear Regressio - a mathematical techique for creatig a liear model for paired data. Based o the least-squares criterio of best fit 6 P a g e
7 Example - Caribou ad wolf populatios i Deali Natioal Park Questios Do the data poits have a liear relatioship? How do we fid a equatio for the best fittig lie? Ca we predict the value of the respose variable for a ew value of the predictor variable? What fractioal part of the variability i y is associated with the variability i x? Least-Squares Criterio - Method of Least Squares to fit Equatio s to Data Let y=respose variable ad x=explaatory variable Data: (x 1, y 1 ), (x 2, y 2 ) (x, y ) Suppose the plot of y vs. x shows a straight lie (liear) relatioship. We wat to determie the lie of best fit. A regressio lie is a straight lie that describes how the average respose value varies as the explaatory variable (x) chages. We ca use the regressio lie to predict the value of y at a give x. Equatio of a straight lie y a bx where a= the y-itercept (values of y whe x=0) ad b=slope of the lie (chage i y for a evet chage i x) If we kow a ad b we ca predict y for a give value of x. How accurate the predictio is depeds o how much scatter there is i the data about the lie. 7 P a g e
8 Calculator Liear Regressio: 2 d zero (Catalog) scroll dow to Diagostic O the press Eter, the Eter Stat Edit the eter your data ito L 1, ad L 2 Stat Calc LiReg(a+bx) Optio #8. Methods of Fittig a Straight Lie (1.) Eye Ball method (2.) Least Squares (1805) 8 P a g e
9 The method of Least Squares fids the straight lie that miimizes the sums of the squares of the vertical distaces betwee the poits ad the lie. If lie is y a bx 2 i i the we wat to miimize y a bx squares straight lie is give by i1 ŷ a bx with respect to a ad b. The least s x where b r where r = corr(x, y) sx is the stadard deviatio x ad s s y is the stadard deviatio of y y. Ad a y bx Properties of the Regressio Equatio The poit is always o the least-squares lie. The slope tells us the amout that y chages whe x icreases by oe uit. 9 P a g e
10 Example - Caribou (x, i hudreds) ad wolf (y) populatios Least-squares liear relatioship betwee caribou ad wolf populatios: yˆ x Critical Thikig: Makig Predictios We ca simply plug i x values ito the regressio equatio to calculate y values. Extrapolatio may produce urealistic forecasts 10 P a g e
11 Coefficiet of Determiatio Aother way to gauge the fit of the regressio equatio is to calculate the coefficiet of determiatio, r 2. 1). Compute r. Simply square this value to get r 2. 2). r 2 is the fractioal amout of total variatio i y that ca be explaied usig the liear model. 3). 1 r 2 is the fractioal amout of total variatio i y that is due to radom chace (or possibly due to lurkig variables). Example - The liear correlatio coefficiet for a set of paired data is r = What fractioal amout of the total variatio i y is due to radom chace ad/or to lurkig variables? Commets about Least Squares Fit s x Sice b r ad sice b is the chage i y for a uit chage i x, if r is close to 0 there will s y be little chage i y as x chages ( r 0 - gives a approximately horizotal regressio lie). The larger r is i absolute values, the greater the chage i y for a uit chage i x. The least squares regressio lie passes through the poit x, y The regressio lie of y regressed o x is differet from the regressio lie of x regressed o y. So selectig a regressio variable is importat i least squares regressio lies. ( Recall correlatio coefficiet is ot affected by iterchagig x ad y) 2 The square of the correlatio coefficiet, r, (called the coefficiet of determiatio) is the proportio of variatio i the values of y that ca be accouted for (explaied by) the least 2 squares regressio lie of y o x. Furthermore, 1 r is the proportio of total variatio i y that is due to radom chace or to the possibility of lurkig variables that ifluece y. 11 P a g e
12 Example Suppose that five idividuals with the same iitial systolic blood pressure are give a blood pressure medicatio, each with a differet dosage. After a period of time o the medicatio each idividual has their systolic blood pressure measured. The data are: Dosage (x) Systolic Blood Pressure (y) 10 mg mg mg mg mg 130 a) Make a scatter plot of the data b) Fid the correlatio betwee Dosage ad Blood Pressure c) Fid the Least squares regressio lie of the data d) Fid the coefficiet of determiatio (the proportio of variatio i the values of y that ca be accouted for (explaied by) the least squares regressio lie of y o x.) 12 P a g e
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