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UF STA 6126 - Linear Regression Correlation

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Linear Regression Correlation Quantitative Explanatory and Response Variables Goal Test whether the level of the response variable is associated with depends on the level of the explanatory variable Goal Measure the strength of the association between the two variables Goal Use the level of the explanatory to predict the level of the response variable Linear Relationships Notation Y Response dependent outcome variable X Explanatory independent predictor variable Linear Function Straight Line Relation Y X Plot Y on vertical axis X horizontal Slope The amount Y changes when X increases by 1 0 Line slopes upward Positive Relation 0 Line is flat No linear Relation 0 Line slopes downward Negative Relation Y intercept Y level when X 0 Example Service Pricing Internet History Resources New South Wales Family History Document Service Membership fee 20A 20 0 20A per image viewed Y Total cost of service X Number of images viewed Cost when no images viewed Incremental Cost per image viewed Y X 20 0 20X Example Service Pricing Total Cost vs Images Viewed www ihr com au 60 cost 20 00 0 20 im ages R Square 1 00 50 40 30 20 0 50 100 images 150 200 Linear Regression Probabilistic Models In practice the relationship between Y and X is not perfect Other sources of variation exist We decompose Y into 2 components Systematic Relationship with X X Random Error Random respones can be written as the sum of the systematic also thought of as the mean and random components Y X The conditional on X mean response is E Y X Least Squares Estimation Problem are unknown parameters and must be estimated and tested based on sample data Procedure Sample n individuals observing X and Y on each one Plot the pairs Y vertical axis versus X horizontal Choose the line that best fits the data Criteria Choose line that minimizes sum of squared vertical distances from observed data points to line Least Squares Prediction Equation Y a bX X X Y Y b X X 2 a Y b X Example Pharmacodynamics of LSD Response Y Math score mean among 5 volunteers Predictor X LSD tissue concentration mean of 5 volunteers Raw Data and scatterplot of Score vs LSD concentration 80 70 60 50 SCORE 40 30 20 1 2 LSD CONC Source Wagner et al 1968 3 4 5 6 7 Example Pharmacodynamics of LSD Score y 78 93 58 20 67 47 37 47 45 65 32 92 29 97 350 61 LSD Conc x 1 17 2 97 3 26 4 69 5 83 6 00 6 41 30 33 x xbar 3 163 1 363 1 073 0 357 1 497 1 667 2 077 0 001 y ybar 28 843 8 113 17 383 12 617 4 437 17 167 20 117 0 001 Sxx 10 004569 1 857769 1 151329 0 127449 2 241009 2 778889 4 313929 22 474943 Sxy 91 230409 11 058019 18 651959 4 504269 6 642189 28 617389 41 783009 202 487243 Syy 831 918649 65 820769 302 168689 159 188689 19 686969 294 705889 404 693689 2078 183343 Column totals given in bottom row of table 350 61 30 33 Y 50 087 X 4 333 7 7 202 4872 b 9 01 a Y b X 50 09 9 01 4 33 89 10 22 4749 Y 89 10 9 01X SPSS Output and Plot of Equation Coefficientsa Model 1 Unstandardized Coefficients B Std Error 89 124 7 048 9 009 1 503 Constant LSD CONC Standardized Coefficients Beta 937 t 12 646 5 994 a Dependent Variable SCORE Math Score vs LSD Concentration SPSS 80 00 Linear Regression 70 00 score 60 00 50 00 40 00 30 00 1 00 2 00 score 89 12 9 01 lsd conc R Square4 00 0 88 5 00 3 00 6 00 lsd conc Sig 000 002 Example Retail Sales U S SMSA s Y Per Capita Retail Sales X Females per 100 Males Per Capita Retail Sales vs Females per 100 Males Linear Regression 40 00 Coefficientsa 30 00 Model 1 20 00 0 00 50 00 75 00 291 t 5 096 8 829 100 00 f100m Standardized Coefficients Beta a Dependent Variable PCSALES 0 16 f100m pcsales 9 85 0 08 R Square 10 00 Constant F100M Unstandardized Coefficients B Std Error 9 851 1 933 163 018 125 00 Y 9 851 0 163 X Sig 000 000 Residuals Residuals aka Errors Difference between observed values and predicted values e Y Y Error sum of squares SSE Y Y 2 Estimate of conditional standard deviation of Y SSE n 2 2 Y Y n 2 Linear Regression Model Data Y X Mean E Y X Conditional Standard Deviation Error terms are assumed to be independent and normally distributed Parameter Estimator X X Y Y b X X X a Y b X a bX 2 2 Y Y n 2 Example Pharmacodynamics of LSD Y 89 10 9 01X Y X 78 93 58 20 67 47 37 47 45 65 32 92 29 97 1 17 2 97 3 26 4 69 5 83 6 00 6 41 Yhat e Y Yhat 78 5583 0 3717 62 3403 4 1403 59 7274 7 7426 46 8431 9 3731 36 5717 9 0783 35 04 2 12 31 3459 1 3759 SSE 253 8861 e 2 0 138161 17 14208 59 94785 87 855 82 41553 4 4944 1 893101 253 8861 253 8861 7 13 7 2 Correlation Coefficient Slope of the regression describes the direction of association if any between the explanatory X and response Y Problems The magnitude of the slope depends on the units of the variables The slope is unbounded doesn t measure strength of association Some situations arise where interest is in association between variables but no clear definition of X and Y Population Correlation Coefficient Sample Correlation Coefficient r Correlation Coefficient Pearson Correlation Measure of strength of linear association Does not delineate between explanatory and response variables Is invariant to linear transformations of Y and X Is bounded between 1 and 1 higher values in absolute value imply stronger relation Same sign positive negative as slope r X X Y Y X X Y Y 2 2 sX sY b Example Pharmacodynamics of LSD Using formulas for standard deviation from beginning of course sX 1 935 and sY 18 611 From previous calculations b 9 01 1 935 r 9 01 0 94 18 611 This represents a strong negative association between math scores and LSD tissue concentration Coefficient of Determination Measure of the variation in Y that is explained by X Step 1 Ignoring X measure the total variation in Y around its mean 2 TSS Y Y Step 2 Fit regression relating Y to X and measure the unexplained variation in Y around its predicted values SSE Y Y 2 Step 3 Take the difference variation in Y explained TSS SSE by X and divide by total 2 r TSS Example Pharmacodynamics of LSD TSS Y Y 2 2078 183 SSE Y Y 2 253 89 r2 2078 183 253 89 …


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