Analysis of Bivariate Data

Transcription

1 Analysis of Bivariate Data

2 Data Two Quantitative variables GPA and GAES Interest rates and indices Tax and fund allocation Population size and prison population Bivariate data (x,y) Case corr&reg 2

3 Independent variable Explanatory variable Dependent Variable Response variable Goal Study the relationship, if any, between the two variables Basic Questions corr&reg 3

4 Goal Understand the direction of association Positive or negative Type of association Linear or non-linear Strength of association odd data corr&reg 4

5 1. Descriptive measure Visual tool General form, pattern of relationship Exceptions Scatter plot Explanatory variable - x axis Response Variable - y axis 2. Summary measure Strength and direction of association Correlation Coefficient corr&reg 5

6 Cause and Effect Relation between two variables Example: GAES (Graduate Exam Score) and GPA: GAES = Response (dependent) variable (y) GPA = Explanatory (independent) variable (x) DATA: GPA : GAES: corr&reg 6

7 GAES Is there a linear pattern? Look at the SCATTER PLOT (Graph--> ScatterPlot) GPA corr&reg 7

8 Linear pattern ==> linear association between GPA and GAES Changes in GAES may be explained by a linear function of GPA such as GAES = β 0 + β 1 *GPA This line summarizes the information in the scatter plot. Use it to estimate the GAES for a given GPA β 1 = slope of the line; measures by how much GAES changes for one unit change in GPA. β 0 = intercept of the line; How do we get the values of β 1 and β 0? corr&reg 8

9 Principle of least squares Fitted value ^ y a bx i i Residual = differences between actual values and estimates. Find the values of a and b that give us a line with the smallest error Error = sum of squares of residuals y i ^ y i corr&reg 9

10 xy b For our data, 1 x a y x 2 ( x)( y) n ( x) n 2 S S xy xx ( x x)( y y) ( x x) 2 X 39.1 Y 6690 Y XY 22289;So y x X corr&reg 10

11 Row GPA(x) GAES(y) x^2 xy y^ Ŷ SUM X Y Y 6690 X XY 22289; So ŷ x PREDICTED VALUE OF Y corr&reg 11

12 GAES Scatter plot of GAES vs GPA 750 Scatter plot of GAES vs GPA GPA corr&reg 12

13 GAES Scatter plot with the Least Squares line Fitted Line Plot GAES = GPA S R-Sq 82.9% R-Sq(adj) 81.2% GPA corr&reg 13

14 ŷ x Row GPA GAES FITS1 RESI1 SS_RESID corr&reg 14

15 Sum of Squares of Residuals Consider a different line, say GAES= *GPA The Sum of Squares of Residuals is much higher than SSR of the BEST line! corr&reg 15

16 Best Line Row GPA GAES FITS2 RESI2 SSR2 RESI Sum of squares of RESI2 = corr&reg 16

17 How good is the regression model? 1.Residual plot.. Plot residuals against the explanatory variable Residual = observed y - predicted (fitted) We know that the sum of residuals = 0 So, the average residuals = 0. If the model is good, then we expect to see a random distribution of residuals about the mean of 0 Any deviation from this random pattern means that the relationship between the variables is not adequately represented by our regression line. corr&reg 17

18 Resid GPA corr&reg 18

19 2.Coefficient of Determination: Measures the percentage of variation in y that is explained by its regression on x. What percentage of variability in y is due to x? - R-Squared Of all the variation in GAES, 82.9% is due to its linear association with GPA. The regression equation is GAES = GPA Predictor Coef Stdev t-ratio p Constant GPA s = R-sq = 82.9% R-sq(adj) = 81.2% GPA is useful in explaining 82.9% of the variability in GAES corr&reg 19

20 Analysis of Variance Source DF SS MS F P Regression Residual Error Total Note that Also, Total SS 2 SS Re sid R SSTotal SS Re sid s se MS Re sid n 2 s = standard error of residuals; Typical deviation of data around the best line SSResidual corr&reg 20

21 Graph Scatterplot Simple corr&reg 21

22 Cost 980 Scatterplot of Cost vs Pollutio Pollutio corr&reg 22

23 Stat Regression Regression corr&reg 23

24 corr&reg 24

25 Cost Stat Regression Fitted Line Plot Fitted Line Plot Cost = Pollutio S R-Sq 33.8% R-Sq(adj) 17.2% Pollutio corr&reg 25

26 Outliers in regression Earlier, outlier = data that is away from the rest of the data. Since there are two related data sets now, we need to fine-tune our definition. Away in which direction? Outlier = a data that is away from the regression line ==> it produces a large residual In our example, it would be obsn. # 5 with a residual of corr&reg 26

27 What if a data had an extreme x value? Then that data is called an influential observation. The data influences the regression line. This might not show up as one with large residual as the point might be close to the regression line. However, removing the point changes the slope of the line. corr&reg 27

28 Another Example corr&reg 28