Simple Linear Regression and Correlation
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1 Smple Lnear Regresson and Correlaton In ths chapter, you learn: How to use regresson analyss to predct the value of a dependent varable based on an ndependent varable The meanng of the regresson coeffcents b 0 and b 1 How to evaluate the assumptons of regresson analyss and know what to do f the assumptons are volated To make nferences about the slope and correlaton coeffcent To estmate mean values and predct ndvdual values CVE 475 Statstcal Technques n Hydrology 1/80
2 Correlaton vs. Regresson A scatter dagram can be used to show the relatonshp between two varables Correlaton analyss s used to measure strength of the assocaton (lnear relatonshp) between two varables Correlaton s only concerned wth strength of the relatonshp No causal effect s mpled wth correlaton CVE 475 Statstcal Technques n Hydrology 2/80
3 Introducton to Regresson Analyss Regresson analyss s used to: Predct the value of a dependent varable based on the value of at least one ndependent varable Explan the mpact of changes n an ndependent varable on the dependent varable Dependent varable: the varable we wsh to predct or explan (.e. runoff) Independent varable: the varable used to explan the dependent varable (.e. ranfall) CVE 475 Statstcal Technques n Hydrology 3/80
4 Smple Lnear Regresson Model Only one ndependent varable, X Relatonshp between X and s descrbed by a lnear functon Changes n are assumed to be caused by changes n X CVE 475 Statstcal Technques n Hydrology 4/80
5 Types of Relatonshps Lnear relatonshps Curvlnear relatonshps X X X X CVE 475 Statstcal Technques n Hydrology 5/80
6 Types of Relatonshps (contnued) Strong relatonshps Weak relatonshps X X X X CVE 475 Statstcal Technques n Hydrology 6/80
7 Types of Relatonshps No relatonshp (contnued) X X CVE 475 Statstcal Technques n Hydrology 7/80
8 Smple Lnear Regresson Model Dependent Varable Populaton ntercept = β + β X + 0 Populaton Slope Coeffcent 1 Independent Varable ε Random Error term Lnear component Random Error component CVE 475 Statstcal Technques n Hydrology 8/80
9 Smple Lnear Regresson Model (contnued) Observed Value of for X = β + β X ε Predcted Value of for X ε Random Error for ths X value Slope = β 1 Intercept = β 0 ε ~ N(0,σ 2 ) X X CVE 475 Statstcal Technques n Hydrology 9/80
10 Smple Lnear Regresson Model (contnued) ε ~ N(0,σ 2 ) CVE 475 Statstcal Technques n Hydrology 10/80
11 Smple Lnear Regresson Model (contnued) ε ~ N(0,σ 2 ) CVE 475 Statstcal Technques n Hydrology 11/80
12 Smple Lnear Regresson Equaton (Predcton Lne) The smple lnear regresson equaton provdes an estmate of the populaton regresson lne Estmated (or predcted) value for observaton Estmate of the regresson ntercept Estmate of the regresson slope Ŷ = b + 0 b 1 X Value of X for observaton The ndvdual random error terms e have a mean of zero CVE 475 Statstcal Technques n Hydrology 12/80
13 Least Squares Method b 0 and b 1 are obtaned by fndng the values of b 0 and b 1 that mnmze the sum of the squared dfferences between and Ŷ : mn ( = + Ŷ ) 2 mn ( (b b X )) CVE 475 Statstcal Technques n Hydrology 13/80
14 Fndng the Least Squares Equaton Computatonal formula for the slope b 1 : where b = 1 SS SS SS X SS X XX XX = ( X )( ) X = 1 = n n ( X ) X = 1 b 1 ˆ1 β n the text book 2 CVE 475 Statstcal Technques n Hydrology 14/80
15 Fndng the Least Squares Equaton Computatonal formula for the ntercept b 0 : b = b X 0 1 where b 0 ˆ0 β n the text book = 1 n n = 1 and X = 1 n n = 1 X CVE 475 Statstcal Technques n Hydrology 15/80
16 Fndng the Least Squares Equaton The coeffcents b 0 and b 1, and other regresson results n ths chapter, wll be found usng Excel CVE 475 Statstcal Technques n Hydrology 16/80
17 Interpretaton of the Slope and the Intercept b 0 s the estmated average value of when the value of X s zero b 1 s the estmated change n the average value of as a result of a one-unt change n X CVE 475 Statstcal Technques n Hydrology 17/80
18 Smple Lnear Regresson Example A real estate agent wshes to examne the relatonshp between the sellng prce of a home and ts sze (measured n square feet) A random sample of 10 houses s selected Dependent varable () = house prce n $1000s Independent varable (X) = square feet CVE 475 Statstcal Technques n Hydrology 18/80
19 Sample Data for House Prce Model House Prce n $1000s () Square Feet (X) CVE 475 Statstcal Technques n Hydrology 19/80
20 Graphcal Presentaton House Prce ($1000s) House prce model: scatter plot Square Feet CVE 475 Statstcal Technques n Hydrology 20/80
21 Regresson Usng Excel Tools / Data Analyss / Regresson CVE 475 Statstcal Technques n Hydrology 21/80
22 Excel Output Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 The regresson equaton s: house prce = (square feet) ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet CVE 475 Statstcal Technques n Hydrology 22/80
23 Graphcal Presentaton House prce model: scatter plot and regresson lne Intercept = House Prce ($1000s) Square Feet Slope = house prce = (square feet) CVE 475 Statstcal Technques n Hydrology 23/80
24 Interpretaton of the Intercept, b 0 house prce = (square feet) b 0 s the estmated average value of when the value of X s zero (f X = 0 s n the range of observed X values) Here, no houses had 0 square feet, so b 0 = just ndcates that, for houses wthn the range of szes observed, $98, s the porton of the house prce not explaned by square feet CVE 475 Statstcal Technques n Hydrology 24/80
25 Interpretaton of the Slope Coeffcent, b 1 house prce = (square feet) b 1 measures the estmated change n the average value of as a result of a oneunt change n X Here, b 1 = tells us that the average value of a house ncreases by.10977($1000) = $109.77, on average, for each addtonal one square foot of sze CVE 475 Statstcal Technques n Hydrology 25/80
26 Predctons usng Regresson Analyss Predct the prce for a house wth 2000 square feet: house prce = (sq.ft.) = (2000) = The predcted prce for a house wth 2000 square feet s ($1,000s) = $317,850 CVE 475 Statstcal Technques n Hydrology 26/80
27 Interpolaton vs. Extrapolaton When usng a regresson model for predcton, only predct wthn the relevant range of data Relevant range for nterpolaton House Prce ($1000s) Do not try to extrapolate beyond the range of observed X s Square Feet CVE 475 Statstcal Technques n Hydrology 27/80
28 Measures of Varaton Total varaton s made up of two parts: SST = SSR + SSE Total Sum of Squares Regresson Sum of Squares Error Sum of Squares = 2 SST ( = 2 SSR (Ŷ SSE = ( ) ) 2 Ŷ ) where: = Average value of the dependent varable = Observed values of the dependent varable Ŷ = Predcted value of for the gven X value CVE 475 Statstcal Technques n Hydrology 28/80
29 Measures of Varaton SST = total sum of squares Measures the varaton of the values around ther mean SSR = regresson sum of squares Explaned varaton attrbutable to the relatonshp between X and SSE = error sum of squares Varaton attrbutable to factors other than the relatonshp between X and (contnued) CVE 475 Statstcal Technques n Hydrology 29/80
30 Measures of Varaton _ _ SST = ( - ) 2 SSE = ( - ) 2 _ SSR = ( - ) 2 (contnued) _ X X CVE 475 Statstcal Technques n Hydrology 30/80
31 Coeffcent of Determnaton, r 2 The coeffcent of determnaton s the porton of the total varaton n the dependent varable that s explaned by varaton n the ndependent varable The coeffcent of determnaton s also called r-squared and s denoted as r 2 SSR r 2 = = SST regresson sum of squares total sum of squares note: 0 r 2 1 CVE 475 Statstcal Technques n Hydrology 31/80
32 Examples of Approxmate r 2 Values r 2 = 1 r 2 = 1 X Perfect lnear relatonshp between X and : 100% of the varaton n s explaned by varaton n X r 2 = 1 X CVE 475 Statstcal Technques n Hydrology 32/80
33 Examples of Approxmate r 2 Values X 0 < r 2 < 1 Weaker lnear relatonshps between X and : Some but not all of the varaton n s explaned by varaton n X X CVE 475 Statstcal Technques n Hydrology 33/80
34 Examples of Approxmate r 2 Values r 2 = 0 No lnear relatonshp between X and : r 2 = 0 X The value of does not depend on X. (None of the varaton n s explaned by varaton n X) CVE 475 Statstcal Technques n Hydrology 34/80
35 Excel Output Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 SSR r 2 = = = SST % of the varaton n house prces s explaned by varaton n square feet ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet CVE 475 Statstcal Technques n Hydrology 35/80
36 Standard Error of Estmate (contnued) Standard Error Varance ε ~ N(0,σ 2 ) CVE 475 Statstcal Technques n Hydrology 36/80
37 Standard Error of Estmate The standard devaton of the varaton of observatons around the regresson lne s estmated by σˆ S X = SSE n 2 = n = 1 ( n 2 Ŷ ) 2 Where SSE = error sum of squares n = sample sze CVE 475 Statstcal Technques n Hydrology 37/80
38 Excel Output Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 S X = ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet CVE 475 Statstcal Technques n Hydrology 38/80
39 Comparng Standard Errors S X s a measure of the varaton of observed values from the regresson lne small s X X large sx X The magntude of S X should always be judged relatve to the sze of the values n the sample data.e., S X = $41.33K s moderately small relatve to house prces n the $200 - $300K range CVE 475 Statstcal Technques n Hydrology 39/80
40 Assumptons of Regresson Use the acronym LINE: Lnearty The underlyng relatonshp between X and s lnear Independence of Errors Error values are statstcally ndependent Normalty of Error Error values (ε) are normally dstrbuted for any gven value of X Equal Varance (Homoscedastcty) The probablty dstrbuton of the errors has constant varance CVE 475 Statstcal Technques n Hydrology 40/80
41 Resdual Analyss e The resdual for observaton, e, s the dfference between ts observed and predcted value Check the assumptons of regresson by examnng the resduals Examne for lnearty assumpton Evaluate ndependence assumpton Evaluate normal dstrbuton assumpton Examne for constant varance for all levels of X (homoscedastcty) Graphcal Analyss of Resduals Can plot resduals vs. X = CVE 475 Statstcal Technques n Hydrology 41/80 Ŷ
42 Resdual Analyss for Lnearty x x resduals x resduals x Not Lnear Lnear CVE 475 Statstcal Technques n Hydrology 42/80
43 Resdual Analyss for Independence Not Independent Independent resduals X resduals X resduals X CVE 475 Statstcal Technques n Hydrology 43/80
44 Resdual Analyss for Normalty A normal probablty plot of the resduals can be used to check for normalty: Percent Resdual CVE 475 Statstcal Technques n Hydrology 44/80
45 Resdual Analyss for Equal Varance x x resduals x resduals x Non-constant varance Constant varance CVE 475 Statstcal Technques n Hydrology 45/80
46 Excel Resdual Output RESIDUAL OUTPUT Predcted House Prce Resduals Resduals House Prce Model Resdual Plot Square Feet Does not appear to volate any regresson assumptons CVE 475 Statstcal Technques n Hydrology 46/80
47 Inferences About the Slope The standard error of the regresson slope coeffcent (b 1 ) s estmated by Sb 1 = S X SS XX = S (X X X) 2 where: S b1 = Estmate of the standard error of the least squares slope S X = SSE n 2 = Standard error of the estmate CVE 475 Statstcal Technques n Hydrology 47/80
48 Excel Output Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 Sb 1 = ANOVA df SS MS F Sgnfcance F Regresson Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet CVE 475 Statstcal Technques n Hydrology 48/80
49 Comparng Standard Errors of the Slope S b1 s a measure of the varaton n the slope of regresson lnes from dfferent possble samples small S b1 X large S b1 X CVE 475 Statstcal Technques n Hydrology 49/80
50 Inference about the Slope: t Test t test for a populaton slope Is there a lnear relatonshp between X and? Null and alternatve hypotheses H 0 : β 1 = 0 H 1 : β 1 0 Test statstc t = d.f. (no lnear relatonshp) (lnear relatonshp does exst) b1 β S = n b 1 CVE 475 Statstcal Technques n Hydrology 50/ where: b 1 = regresson slope coeffcent β 1 = hypotheszed slope S b = standard 1 error of the slope
51 Inference about the Slope: t Test (contnued) House Prce n $1000s (y) Square Feet (x) Smple Lnear Regresson Equaton: house prce = (sq.ft.) The slope of ths model s Does square footage of the house affect ts sales prce? CVE 475 Statstcal Technques n Hydrology 51/80
52 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1 0 From Excel output: Coeffcents Intercept b 1 Standard Error S b1 t Stat P-value Square Feet t = b β S b t = = CVE 475 Statstcal Technques n Hydrology 52/80
53 Inferences about the Slope: H 0 : β 1 = 0 H 1 : β 1 0 t Test Example Test Statstc: t = From Excel output: b 1 Coeffcents Standard Error Intercept Square Feet Sb 1 (contnued) t Stat t P-value d.f. = 10-2 = 8 α/2=.025 Reject H 0 α/2=.025 Reject H 0 Do not reject H -t 0 α/2 t 0 α/ Decson: Reject H 0 Concluson: There s suffcent evdence that square footage affects house prce CVE 475 Statstcal Technques n Hydrology 53/80
54 Inferences about the Slope: H 0 : β 1 = 0 H 1 : β 1 0 t Test Example P-value = From Excel output: Coeffcents Standard Error Intercept Square Feet (contnued) P-value t Stat P-value Ths s a two-tal test, so the p-value s P(t > 3.329)+P(t < ) = (for 8 d.f.) Decson: P-value < α so Reject H 0 Concluson: There s suffcent evdence that square footage affects house prce CVE 475 Statstcal Technques n Hydrology 54/80
55 F Test for Sgnfcance F Test statstc: F = MSR MSE where MSR = SSR k MSE = SSE n k 1 where F follows an F dstrbuton wth k numerator and (n k - 1) denomnator degrees of freedom (k = the number of ndependent varables n the regresson model) CVE 475 Statstcal Technques n Hydrology 55/80
56 Excel Output Regresson Statstcs Multple R R Square Adjusted R Square Standard Error Observatons 10 ANOVA df Regresson 1 MSR F = = = MSE Wth 1 and 8 degrees of freedom SS MS F Sgnfcance F P-value for the F Test Resdual Total Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet CVE 475 Statstcal Technques n Hydrology 56/80
57 F Test for Sgnfcance 0 H 0 : β 1 = 0 H 1 : β 1 0 α =.05 df 1 = 1 df 2 = 8 Do not reject H 0 Crtcal Value: F α = 5.32 α =.05 Reject H 0 F.05 = 5.32 F Test Statstc: MSR F = = MSE Decson: Reject H 0 at α = 0.05 Concluson: (contnued) There s suffcent evdence that house sze affects sellng prce CVE 475 Statstcal Technques n Hydrology 57/80
58 Confdence Interval Estmate for the Slope Confdence Interval Estmate of the Slope: b ± t S 1 n 2 b 1 d.f. = n - 2 Excel Prntout for House Prces: Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet At 95% level of confdence, the confdence nterval for the slope s (0.0337, ) CVE 475 Statstcal Technques n Hydrology 58/80
59 Confdence Interval Estmate for the Slope (contnued) Coeffcents Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Snce the unts of the house prce varable s $1000s, we are 95% confdent that the average mpact on sales prce s between $33.70 and $ per square foot of house sze Ths 95% confdence nterval does not nclude 0. Concluson: There s a sgnfcant relatonshp between house prce and square feet at the.05 level of sgnfcance CVE 475 Statstcal Technques n Hydrology 59/80
60 The Sample Covarance The sample covarance measures the strength of the lnear relatonshp between two varables (called bvarate data) The sample covarance: cov (X,) = n = 1 (X X)( n 1 ) Only concerned wth the strength of the relatonshp No causal effect s mpled CVE 475 Statstcal Technques n Hydrology 60/80
61 Interpretng Covarance Covarance between two random varables: cov(x,) > 0 X and tend to move n the same drecton cov(x,) < 0 X and tend to move n opposte drectons cov(x,) = 0 X and are ndependent CVE 475 Statstcal Technques n Hydrology 61/80
62 Coeffcent of Correlaton Measures the relatve strength of the lnear relatonshp between two varables Sample coeffcent of correlaton: r = cov (X,) S X S where cov (X,) = n = 1 (X X)( n 1 ) S X = n = 1 (X X) n 1 2 S = n = 1 ( ) n 1 2 CVE 475 Statstcal Technques n Hydrology 62/80
63 Features of Correlaton Coeffcent, r Unt free Ranges between 1 and 1 The closer to 1, the stronger the negatve lnear relatonshp The closer to 1, the stronger the postve lnear relatonshp The closer to 0, the weaker the lnear relatonshp CVE 475 Statstcal Technques n Hydrology 63/80
64 Scatter Plots of Data wth Varous Correlaton Coeffcents X X r = -1 r = -.6 r = 0 X X X X r = +1 r = +.3 r = 0 CVE 475 Statstcal Technques n Hydrology 64/80
65 Usng Excel to Fnd the Correlaton Coeffcent Select Tools/Data Analyss Choose Correlaton from the selecton menu Clck OK... CVE 475 Statstcal Technques n Hydrology 65/80
66 Usng Excel to Fnd the Correlaton Coeffcent (contnued) Input data range and select approprate optons Clck OK to get output CVE 475 Statstcal Technques n Hydrology 66/80
67 Interpretng the Result r = Scatter Plot of Test Scores There s a relatvely strong postve lnear relatonshp between test score #1 and test score #2 Test #2 Score Test #1 Score Students who scored hgh on the frst test tended to score hgh on second test, and students who scored low on the frst test tended to score low on the second test CVE 475 Statstcal Technques n Hydrology 67/80
68 t Test for a Correlaton Coeffcent Hypotheses H 0 : ρ = 0 (no correlaton between X and ) H A : ρ 0 Test statstc t = r - ρ (correlaton exsts) (wth n 2 degrees of freedom) 2 1 r n 2 where r = + r 2 f b 1 > 0 r = r 2 f b 1 < 0 CVE 475 Statstcal Technques n Hydrology 68/80
69 Example: House Prces Is there evdence of a lnear relatonshp between square feet and house prce at the.05 level of sgnfcance? H 0 : ρ = 0 (No correlaton) H 1 : ρ 0 (correlaton exsts) α =.05, df = 10-2 = 8 t = r ρ 2 1 r n 2 = = CVE 475 Statstcal Technques n Hydrology 69/80
70 Example: Test Soluton r ρ t = 2 1 r n 2 d.f. = 10-2 = 8 α/2=.025 = = α/2=.025 Decson: Reject H 0 Concluson: There s evdence of a lnear assocaton at the 5% level of sgnfcance Reject H 0 Reject H t 0 α/2 Do not reject H -t 0 α/ CVE 475 Statstcal Technques n Hydrology 70/80
71 Estmatng Mean Values and Predctng Indvdual Values Goal: Form ntervals around to express uncertanty about the value of for a gven X Confdence Interval for the mean of, gven X = b 0 +b 1 X Predcton Interval for an ndvdual, gven X CVE 475 Statstcal Technques n Hydrology 71/80 X X
72 Confdence Interval for the Average, Gven X Confdence nterval estmate for the mean value of gven a partcular X Confdence nterval for µ X= X : Ŷ ± t n 2 S X h Sze of nterval vares accordng to dstance away from mean, X h (X X) 1 (X X) = + = + 2 n SSX n (X X) CVE 475 Statstcal Technques n Hydrology 72/80
73 Predcton Interval for an Indvdual, Gven X Confdence nterval estmate for an Indvdual value of gven a partcular X Confdence nterval for X= X : Ŷ ± tn 2 S X 1+ h Ths extra term adds to the nterval wdth to reflect the added uncertanty for an ndvdual case CVE 475 Statstcal Technques n Hydrology 73/80
74 Estmaton of Mean Values: Example Confdence Interval Estmate for µ X=X Fnd the 95% confdence nterval for the mean prce of 2,000 square-foot houses Predcted Prce = ($1,000s) 2 1 (X X) tn -2SX + = ± n (X X) Ŷ ± The confdence nterval endponts are and , or from $280,660 to $354,900 CVE 475 Statstcal Technques n Hydrology 74/80
75 Estmaton of Indvdual Values: Example Predcton Interval Estmate for X=X Fnd the 95% predcton nterval for an ndvdual house wth 2,000 square feet Predcted Prce = ($1,000s) 2 1 (X X) tn -1S X 1+ + = ± n (X X) Ŷ ± 2 The predcton nterval endponts are and , or from $215,500 to $420,070 CVE 475 Statstcal Technques n Hydrology 75/80
76 Fndng Confdence and Predcton Intervals n Excel In Excel, use PHStat regresson smple lnear regresson Check the confdence and predcton nterval for X= box and enter the X-value and confdence level desred CVE 475 Statstcal Technques n Hydrology 76/80
77 Fndng Confdence and Predcton Intervals n Excel (contnued) Input values Confdence Interval Estmate for µ X=X Predcton Interval Estmate for X=X CVE 475 Statstcal Technques n Hydrology 77/80
78 Ptfalls of Regresson Analyss Lackng an awareness of the assumptons underlyng least-squares regresson Not knowng how to evaluate the assumptons Not knowng the alternatves to least-squares regresson f a partcular assumpton s volated Usng a regresson model wthout knowledge of the subject matter Extrapolatng outsde the relevant range CVE 475 Statstcal Technques n Hydrology 78/80
79 Strateges for Avodng the Ptfalls of Regresson Start wth a scatter dagram of X vs. to observe possble relatonshp Perform resdual analyss to check the assumptons Plot the resduals vs. X to check for volatons of assumptons such as homoscedastcty Use a hstogram, stem-and-leaf dsplay, box-andwhsker plot, or normal probablty plot of the resduals to uncover possble non-normalty CVE 475 Statstcal Technques n Hydrology 79/80
80 Strateges for Avodng the Ptfalls of Regresson (contnued) If there s volaton of any assumpton, use alternatve methods or models If there s no evdence of assumpton volaton, then test for the sgnfcance of the regresson coeffcents and construct confdence ntervals and predcton ntervals Avod makng predctons or forecasts outsde the relevant range CVE 475 Statstcal Technques n Hydrology 80/80
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