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Correlation & Regression

Correlation & Regression. Correlation. Measure the strength of linear relation between 2 random variables (X & Y)  = Corr(X,Y) = Cov(X,Y)/ δ x δ y = E[(X- μ x )(Y- μ y )]/[(X- μ x ) 2 (Y- μ y ) 2 ] 1/2 Standardized Cov(X,Y) so -1   1. Strength of .

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Correlation & Regression

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  1. Correlation & Regression

  2. Correlation • Measure the strength of linear relation between 2 random variables (X & Y) •  = Corr(X,Y) = Cov(X,Y)/δxδy • = E[(X-μx)(Y-μy)]/[(X- μx)2(Y- μy)2]1/2 • Standardized Cov(X,Y) so -1   1

  3. Strength of  •  = -1  Perfect Negative linear relation •  = 1  Perfect Positive linear relation •  = 0  No linear relation • As || increases so does the strength of the relationship

  4. Sample • Cov(X,Y) = 1/(n-1) (xi -x)(yi -y) • Corr(X,Y) = r = • (xi -x)(yi -y)/[(xi -x)2 (yi -y)2]1/2

  5. Hypothesis Test • Null: H0:  = 0 • Alternative: HA:   0; reject H0 if t>tn-2,/2 • Alternative: HA:  > 0; reject H0 if t > tn-2, • Alternative: HA:  < 0; reject H0 if t < -tn-2,

  6. Rank Correlation (Spearman’s) • Sample Correlation (r) can be affected by extreme observations • Spearman’s Rank • 1st rank xi and yi then calculate sample correlation of these ranks • rs = 1- [6(d2)/n(n2-1)] • Where di = the differences of the ranked pairs

  7. Linear Regression • Find/Define relationship between dependent variable and independent variable • Use independent variable to explain the behavior of the dependent variable • Separate variation in the data into explained variation and unexplained variation (noise) • Predict the value of the dependent variable given a value for the independent variable

  8. Linear Regression Model • Predict Y given X • E(Y|X=x) = 0 + 1x • Y = 0 + 1xi + i • Assumptions: • I are random variables • E[i] = 0 • E[i i] = δ2 • E[i k] = 0 ik; they are uncorrelated

  9. Sum of Squares • Total Sum of Squares = Regression sum of squares + Error sum of squares • SST = SSR + SSE • (yi -y)2 = (yi -y)2 + e2i

  10. Coefficient of Determination (R2) • Measures how well x explain the variation in Y • R2 = SSR/SST = 1- SSE/SST = r2 • R2 measures the explained variation in the data

  11. Confidence Interval • Error Variance: S2e = e2i/(n-2) = SSE/(n-2) • Unbiased Estimate of δ2b: S2b = S2e/(xi -x)2 • t = (b-)/Sb • C.I. for Regression Slope = b-tn-2,/2Sb <  < b+tn-2,/2Sb

  12. Regression Slope Tests • H0:  = 0 or H0:   0 vs. H1:  > 0 • Reject H0 if (b-)/Sb > tn-2, • H0:  = 0 or H0:   0 vs. H1:  < 0 • Reject H0 if (b-)/Sb < -tn-2, • H0:  = 0 vs. H1:   0 • Reject H0 if (b-)/Sb > tn-2, or (b-)/Sb < -tn-2,

  13. SAS: Inches-Centimeter • Data Height; • Input inches centimeter; • Datalines; • 1 2.54 • 2 5.08 • 24 60.96 • 4 10.16 • 5 12.7 • 16 40.64 • 7 17.78 • 8 20.32 • 19 48.26 • 10 25.4 • 20 50.8 • 25 63.5 • ; • ProcPlot Data=Height; • Plot inches*centimeter; • ProcCorr Data=Height; • Title 'Correlation Matrix of Inches vs. Centimeter'; • Var inches centimeter; • ProcReg Data=Height; • Title 'Regression Line for Inches-Centimeter Data'; • Model inches=centimeter; • Plot Predicted.*centimeter = 'P' • U95M.*centimeter = '-' L95M.*centimeter = '_' • inches*centimeter = '*' / overlay; • Plot Residual.*centimeter = 'o'; • Quit;

  14. SAS: GRE – GPA Data • Data GRE_GPA; • Input GRE GPA; • Datalines; • 2100 4 • 1920 3.8 • 2290 3.8 • 1580 3.9 • 1400 3.77 • 1300 3.95 • 2020 3.8 • 1060 3.54 • 1500 3 • 1900 4 • 1900 3.7 • 1800 3.5 • 2200 4 • 1990 3.51 • 2000 4 • 1650 3.8 • 1640 3.75 • 1800 3.9 • 2300 3.91 • 2000 3.75 • 2000 3.9 • ; • ProcPlot Data=GRE_GPA; • Plot GRE*GPA; • ProcCorr Data=GRE_GPA; • Title 'Correlation Matrix of GRE vs. GPA'; • Var GRE GPA; • ProcReg Data=GRE_GPA; • Title 'Regression Line for GRE-GPA Data'; • Model GPA=GRE; • Plot Predicted.*GRE = 'P' • U95M.*GRE = '-' L95M.*GRE = '_' • GPA*GRE = '*' / overlay; • Plot Residual.*GRE = 'o'; • Quit;

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