June 9, Correlation and Simple Linear Regression Analysis

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1 June 9, 2004 Correlaton and Smple Lnear Regresson Analyss

2 Bvarate Data We wll see how to explore the relatonshp between two contnuous varables. X= Number of years n the job, Y=Number of unts produced X= Handspan,, Y= Heght X=Number of hours studyng, Y= Exam score Bvarate (pared) data for each unt n the sample we record the value for each of the two varables. It s useful to dentfy one varable as the dependent (response) varable and the other varable as the ndependent (explanatory) varable. Depended Varable (Y)( The varable of prmary nterest, the one to be predcted Independent Varable (X)( A varable provdng bass for predctng the depended varable; explanng the behavor of the depended varable.

3 Example 1: Heght and Handspan D a ta : H eght (n.) Span (cm ) and so on, fo r n = 167 observatons. Data shown are the frst 12 observatons of a data set that ncludes the heghts and fully stretched handspans of 167 college students.

4 Tools to descrbe Bvarate varables Scatterplot,, a two-dmensonal graph of data values. Independent Varable Depended Varable Correlaton,, a statstc that measures the strength and drecton of a lnear relatonshp. Regresson equaton,, an equaton that descrbes the average relatonshp between a depended (Y)( ) and ndependent (X)( ) varable.

5 Tools to descrbe Bvarate varables Scatterplot,, a two-dmensonal graph of data values. Regresson Lne Independent Varable Y=b 0 +b 1 X Depended Varable Coeffcent of Correlaton,, a statstc (.e. a number calculated from the data) that measures the strength and drecton of a lnear relatonshp. Regresson equaton,, an equaton that descrbes the average relatonshp between a depended (Y)( ) and ndependent (X)( ) varable.

6 Lookng for Patterns wth Scatterplots Questons to Ask about a Scatterplot What s the average pattern? Does t look lke a straght lne or s t curved? If t s a lnear pattern what s the drecton? How much do ndvdual ponts vary from the average pattern? Are there any unusual data ponts?

7 Example 1: Heght and Handspan (cont) Taller people tend to have greater handspan measurements than shorter people do. When two varables tend to ncrease together, we say that they have a postve assocaton. The handspan and heght measurements may have a lnear relatonshp.

8 Example 2: Drver Age and Maxmum Legblty Dstance of Hghway Sgns A research frm determned the maxmum dstance at whch each of 30 drvers could read a newly desgned sgn. The 30 partcpants n the study ranged n age from 18 to 82 years old. We want to examne the relatonshp between age and the sgn legblty dstance. What s X and what s Y n ths example?

9 Example 2: Drver Age and Maxmum Legblty Dstance of Hghway Sgns (cont) We see a negatve assocaton wth a lnear pattern. We can use a straght-lne equaton to model ths relatonshp.

10 Example of Non-Lnear relatonshp

11 Measurng Strength and Drecton of a LINEAR relatonshp wth Correlaton The most often used measure of lnear assocaton between two varables s the Pearson coeffcent of correlaton r. The coeffcent of correlaton r ndcates the strength and the drecton of a lnear relatonshp. The strength of the relatonshp s determned by the closeness of the ponts to a straght lne. The drecton s determned by whether one varable generally ncreases or generally decreases when the other varable ncreases.

12 Formula of Formula of r ( ) { } ( ) { } = = y x y y n x x n y x y x n s y y s x x n r where s the sample sze,.e. the number of pared observatons. To compute r by hand, you need fve quanttes n. and,,, 2 2 y x y x y x

13 Propertes and Interpretaton of r r s a measure of LINEAR ASSOCIATION r s always between 11 and +1 The magntude ndcates the strength: The closer r s to 11 or 1, the more tghtly the ponts on the scatterplot are clustered around a lne. r = 11 or +1 ndcates a perfect lnear relatonshp r = 0 the ponts are not LINEARLY assocated; ths does NOT mean there s no assocaton. The sgn of r ndcates the drecton of the relatonshp. Postve ndcates that when one varable ncreases the other s lkely to ncrease as well. Negatve ndcates that when one varable ncreases the other s lkely to decrease.

14

15 Example 1: Heght and Handspan (cont) r = => a farly strong postve lnear relatonshp. Example 2: Drver Age and Maxmum Legblty Dstance of Hghway Sgns (cont) r = -.8 => a farly strong negatve lnear relatonshp.

16 Example 3: Age and Hours of TV Vewng Relatonshp between age and hours of daly televson vewng for 1913 survey respondents. Correlaton Coeffcent r = 0.12 => a weak connecton. Note: a few clamed to watch more than 20 hours/day!

17 The Coeffcent of Determnaton By squarng the correlaton coeffcent we obtan the coeffcent of determnaton. The coeffcent of determnaton r 2, s the proporton of the total varaton n the dependent varable Y explaned by the varaton of the ndependent varable X. Example 1: 1 r = +.74 => r 2 =.55=55% Example 2: 2 r = -.8 => r 2 =.64=64% Note that 0 r 2 1. What would r 2 =1 tell us?

18 Descrbng Lnear Patterns wth a Regresson Lne When the best equaton for descrbng the relatonshp between X and Y s a straght lne, the equaton s called the regresson lne. Two purposes of the regresson lne: to estmate the average value of Y at any specfed value of X to predct the value of Y for an ndvdual, gven that ndvdual s X value

19 Example 4: Iron Is there a relatonshp between X= the concentraton of ron n the det and Y= ron n the blood? If we can determne a relatonshp (.e. an equaton) between the two varables, then we mght use ths equaton 1. To fnd the mean concentraton of ron n the blood for ndvduals wth a specfc concentraton of ron n ther det, e.g. for X=80ppm. 2. To predct a someone s concentraton of ron n the blood based on the concentraton of ron n hs/hers det.

20 The Equaton for the Regresson Lne We want to fnd an equaton to express Y n terms of X. We wll call ths functon regresson lne of Y on X. Ths equaton s of the form ŷ yˆ = b + b x 0 1 s spoken as y-hat hat, and t s also referred to ether as predcted y or estmated y. b 0 s the ntercept of the straght lne. The ntercept s the value of y when x = 0. b 1 s the slope of the straght lne. The slope tells us how much of an ncrease (or decrease) there s for the y varable when the x varable ncreases by one unt. The sgn of the slope tells us whether y ncreases or decreases when x ncreases.

21 Whch lne? There are many possble lnes that could be drawn through the cloud of ponts n the scatterplot:

22 Determne the Regresson Lne The dea behnd ths method s to choose the lne that comes as close as possble to all the data ponts smultaneously. We want the vertcal dstances from the ponts (observed) to the lne (predcted) to be as small as possble ths means our error n predctng y s small.

23 Least Squares Prncble Q : Where does ths equaton come from? A: : It s the lne that s best n the sense that t mnmzes the sum of the squared errors n the vertcal ( (Y ) drecton Y * * * * * errors X

24 Least Squares Estmates The regresson lne for obtaned from a sample s an estmate of the true (unknown) relatonshp between the two varables. Thus the values of b 0 and b 1 are referred to as the LS estmates of regresson coeffcents. By mnmzng the sum of the least squares we obtaned the followng formulas for the slope and the ntercept b b n x n y y b x ( x ) 2 1 =, 2 n x 0 = 1 n x y

25 Example 1: Heght and Handspan (cont.) Regresson equaton: Handspan = Heght Estmate the average handspan for people 60 nches tall: Average handspan = (60) = 18 cm. Predct the handspan for someone who s 60 nches tall: Predcted handspan = (60) = 18 cm.

26 Example 5: Iron (cont) Is there a relatonshp between X= the concentraton of ron n the det and Y= ron n the blood? Iron Det Iron Blood IronBlood Y = *X IronDet Regresson Summary: b 0 = 5.95 b 1 = r =.839 r 2 =.703

27 Explanaton of Terms b 1 = the sample slope. For every unt ncrease n X we expect Y to ncrease by b 1. Example: For every ncrease of 1mg of ron n the det we expect blood ron to ncrease by mg. r = the correlaton, vares between -11 and 1. A correlaton of -11 means a perfect negatve relatonshp, a correlaton of +1 means a perfect postve relatonshp. A correlaton coeffcent of zero means no relatonshp. Example: Our correlaton coeffcent of ndcates a strong postve relatonshp. r 2 = the percent of varaton n the response varable that s explaned by the predctor. Example: Our r 2 of.703 means that 70.3% of the ndvdual varaton n blood ron concentraton can be explaned by ron n the det.

28 Example 5: Expected GPA MINITAB: Ftted Lne Plot 3.8 GPA = Sleep (Hours) S = R-Sq = 8.0 % R-Sq(adj) = 7.0 % r = r 2 = 8.0% b 1 = GPA b 0 = Sleep (Hours)

29 Example 5: Expected GPA (cont) 1. Suppose a student gets 10 hours of sleep. What would ther expected GPA be? Is ths a good estmate? Explan n terms of r-sq. r Equaton s: Y = *X The predcted GPA for 10 hours of sleep s: *10 = For someone who gets 10 hours of sleep we expect them to have a GPA of Ths wll NOT be a very good predctor because the r-squared r value s only.08. Sleepng hours only explans 8% of the varaton n GPA. Most of the varaton n GPA s unaccounted for. 2. Suppose a student gets 18 hours of sleep. Can we predct the GPA of ths student usng the regresson equaton? We can not use the regresson equaton to predct the GPA of a student that sleeps 18 hours per day because 18 hours s not n the range of the values of X n the data set.

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