Correlation Analysis

Transcription

1 Simple Regression

2 Correlation Analysis Correlation analysis is used to measure strength of the association (linear relationship) between two variables Correlation is only concerned with strength of the relationship No causal effect is implied with correlation Correlation was first presented in Chapter 3

3 Correlation Analysis The population correlation coefficient is denoted ρ (the Greek letter rho) The sample correlation coefficient is where r = s s x xy s y s xy (x = i x)(y n 1 i y)

4 Hypothesis Test for Correlation To test the null hypothesis of no linear association, H 0 :ρ = the test statistic follows the Student s t distribution with (n ) degrees of freedom: 0 t = r (n ) (1 r )

5 Decision Rules Hypothesis Test for Correlation Lower-tail test: H 0 : ρ 0 H 1 : ρ < 0 Upper-tail test: H 0 : ρ 0 H 1 : ρ > 0 Two-tail test: H 0 : ρ = 0 H 1 : ρ 0 α α α/ α/ -t α t α -t α/ t α/ Reject H 0 if t < -t n-, α Reject H 0 if t > t n-, α Reject H 0 if t < -t n-, α/ or t > t n-, α/ r (n ) Where t = has n - d.f. (1 r )

6 Introduction to Regression Analysis Regression analysis is used to: Predict the value of a dependent variable based on the value of at least one independent variable Explain the impact of changes in an independent variable on the dependent variable Dependent variable: the variable we wish to explain (also called the endogenous variable) Independent variable: the variable used to explain the dependent variable (also called the exogenous variable)

7 Linear Regression Model The relationship between X and Y is described by a linear function Changes in Y are assumed to be caused by changes in X Linear regression population equation model Y = β + β x + i 0 1 i ε i Where β 0 and β 1 are the population model coefficients and ε is a random error term.

8 Simple Linear Regression Model The population regression model: Dependent Variable Population Y intercept Y = β + β X + i 0 Population Slope Coefficient 1 Independent Variable i ε i Random Error term Linear component Random Error component

9 Simple Linear Regression Model (continued) Y Observed Value of Y for X i Y = β + β X + i 0 1 i ε i Predicted Value of Y for X i ε i Random Error for this X i value Slope = β 1 Intercept = β 0 X i X

10 Simple Linear Regression Equation The simple linear regression equation provides an estimate of the population regression line Estimated (or predicted) y value for observation i y ˆ = b + i Estimate of the regression intercept 0 b Estimate of the regression slope 1 x i Value of x for observation i The individual random error terms e i have a mean of zero e ˆ i = ( yi - yi) = yi -(b0 + b1xi)

11 Least Squares Estimators b 0 and b 1 are obtained by finding the values of b 0 and b 1 that minimize the sum of the squared differences between y and ŷ : min SSE = min e i = min (y i yˆ i ) = min [y i (b 0 + b 1 x i )] Differential calculus is used to obtain the coefficient estimators b 0 and b 1 that minimize SSE

12 Least Squares Estimators The slope coefficient estimator is (continued) n (x x)(y y) i i i= 1 b 1 = = n x (xi x) i= 1 r xy s s Y X And the constant or y-intercept is b0 = y b1x The regression line always goes through the mean x, y

13 Finding the Least Squares Equation The coefficients b 0 and b 1, and other regression results in this chapter, will be found using a computer Hand calculations are tedious Statistical routines are built into Excel Other statistical analysis software can be used

14 Linear Regression Model Assumptions The true relationship form is linear (Y is a linear function of X, plus random error) The error terms, ε i are independent of the x values The error terms are random variables with mean 0 and constant variance, σ (the constant variance property is called homoscedasticity) E[ε ] = 0 i and E[ε i ] = σ for (i = 1, K,n) The random error terms, ε i, are not correlated with one another, so that E[ε i ε j ] = 0 for all i j

15 Interpretation of the Slope and the Intercept b 0 is the estimated average value of y when the value of x is zero (if x = 0 is in the range of observed x values) b 1 is the estimated change in the average value of y as a result of a one-unit change in x

16 Simple Linear Regression Example A real estate agent wishes to examine the relationship between the selling price of a home and its size (measured in square feet) A random sample of 10 houses is selected Dependent variable (Y) = house price in $1000s Independent variable (X) = square feet

17 Sample Data for House Price Model House Price in $1000s (Y) Square Feet (X)

18 Graphical Presentation House Price ($1000s) House price model: scatter plot Square Feet

19 Regression Using Excel Tools / Data Analysis / Regression

20 Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 The regression equation is: house price = (square feet) ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet

21 Graphical Presentation House price model: scatter plot and regression line Intercept = House Price ($1000s) Square Feet Slope = house price = (square feet)

22 Interpretation of the Intercept, b 0 house price = (square feet) b 0 is the estimated average value of Y when the value of X is zero (if X = 0 is in the range of observed X values) Here, no houses had 0 square feet, so b 0 = just indicates that, for houses within the range of sizes observed, $98,48.33 is the portion of the house price not explained by square feet

23 Interpretation of the Slope Coefficient, b 1 house price = (square feet) b 1 measures the estimated change in the average value of Y as a result of a oneunit change in X Here, b 1 = tells us that the average value of a house increases by.10977($1000) = $109.77, on average, for each additional one square foot of size

24 Measures of Variation Total variation is made up of two parts: SST = SSR + SSE Total Sum of Squares Regression Sum of Squares Error Sum of Squares = SST (y y = SSR (yˆ y SSE = (y i ) i ) ˆ i yi) where: y = Average value of the dependent variable y i = Observed values of the dependent variable ŷ i = Predicted value of y for the given x i value

25 Measures of Variation (continued) SST = total sum of squares Measures the variation of the y i values around their mean, y SSR = regression sum of squares Explained variation attributable to the linear relationship between x and y SSE = error sum of squares Variation attributable to factors other than the linear relationship between x and y

26 y i Y _ y y SST = (y i - y) Measures of Variation _ SSE = (y i - y i ) _ SSR = (y i - y) (continued) y _ y x i X

27 Coefficient of Determination, R The coefficient of determination is the portion of the total variation in the dependent variable that is explained by variation in the independent variable The coefficient of determination is also called R-squared and is denoted as R SSR R = = SST regression sum of squares total sum of squares note: 0 R 1

28 Examples of Approximate r Values Y r = 1 Y r = 1 X Perfect linear relationship between X and Y: 100% of the variation in Y is explained by variation in X r = 1 X

29 Examples of Approximate r Values Y Y X 0 < r < 1 Weaker linear relationships between X and Y: Some but not all of the variation in Y is explained by variation in X X

30 Examples of Approximate r Values Y r = 0 No linear relationship between X and Y: r = 0 X The value of Y does not depend on X. (None of the variation in Y is explained by variation in X)

31 Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 SSR R = = = SST % of the variation in house prices is explained by variation in square feet ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet

32 Correlation and R The coefficient of determination, R, for a simple regression is equal to the simple correlation squared R = r xy

33 Estimation of Model Error Variance An estimator for the variance of the population model error is σˆ = s e = n i= 1 e n SSE n Division by n instead of n 1 is because the simple regression model uses two estimated parameters, b 0 and b 1, instead of one i = s e = s e is called the standard error of the estimate

34 Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 s e = ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet

35 Comparing Standard Errors s e is a measure of the variation of observed y values from the regression line Y Y small s e X large se X The magnitude of s e should always be judged relative to the size of the y values in the sample data i.e., s e = $41.33K is moderately small relative to house prices in the $00 - $300K range

36 Inferences About the Regression Model The variance of the regression slope coefficient (b 1 ) is estimated by s b1 se = (xi x) = (n se 1)s x where: s b1 = Estimate of the standard error of the least squares slope s e = SSE n = Standard error of the estimate

37 Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 s = b ANOVA df SS MS F Significance F Regression Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet

38 Comparing Standard Errors of the Slope S b1 is a measure of the variation in the slope of regression lines from different possible samples Y Y small S b1 X large S b1 X

39 Inference about the Slope: t Test t test for a population slope Is there a linear relationship between X and Y? Null and alternative hypotheses H 0 : β 1 = 0 H 1 : β 1 0 Test statistic t = d.f. (no linear relationship) (linear relationship does exist) b β s 1 = n b 1 1 where: b 1 = regression slope coefficient β 1 = hypothesized slope s b1 = standard error of the slope

40 Inference about the Slope: t Test (continued) House Price in $1000s (y) Square Feet (x) Estimated Regression Equation: house price = (sq.ft.) The slope of this model is Does square footage of the house affect its sales price?

41 Inferences about the Slope: t Test Example H 0 : β 1 = 0 H 1 : β 1 0 From Excel output: Coefficients Intercept b 1 Standard Error s b1 t Stat P-value Square Feet t = b β s b t = =

42 H 0 : β 1 = 0 H 1 : β 1 0 d.f. = 10- = 8 t 8,.05 =.3060 α/=.05 Inferences about the Slope: t Test Example Test Statistic: t = 3.39 From Excel output: Intercept Square Feet α/=.05 Reject H 0 Do not reject H 0 Reject H 0 -t n-,α/ 0 t n-,α/ Coefficients b 1 Standard Error sb 1 (continued) t Stat P-value Decision: Reject H 0 Conclusion: There is sufficient evidence that square footage affects house price t

43 H 0 : β 1 = 0 H 1 : β 1 0 Inferences about the Slope: t Test Example P-value = From Excel output: (continued) P-value Coefficients Standard Error t Stat P-value Intercept Square Feet This is a two-tail test, so the p-value is P(t > 3.39)+P(t < -3.39) = (for 8 d.f.) Decision: P-value < α so Reject H 0 Conclusion: There is sufficient evidence that square footage affects house price

44 Confidence Interval Estimate for the Slope Confidence Interval Estimate of the Slope: b 1 tn,α/sb < β1 < b1 + tn,α/s Excel Printout for House Prices: 1 b 1 d.f. = n - Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet At 95% level of confidence, the confidence interval for the slope is (0.0337, )

45 Confidence Interval Estimate for the Slope (continued) Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet Since the units of the house price variable is $1000s, we are 95% confident that the average impact on sales price is between $33.70 and $ per square foot of house size This 95% confidence interval does not include 0. Conclusion: There is a significant relationship between house price and square feet at the.05 level of significance

46 F-Test for Significance F Test statistic: F = MSR MSE where MSR = SSR k MSE = SSE n k 1 where F follows an F distribution with k numerator and (n k - 1) denominator degrees of freedom (k = the number of independent variables in the regression model)

47 Excel Output Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations 10 ANOVA df Regression 1 MSR F = = = MSE With 1 and 8 degrees of freedom SS MS F Significance F P-value for the F-Test Residual Total Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept Square Feet

48 0 H 0 : β 1 = 0 H 1 : β 1 0 α =.05 df 1 = 1 df = 8 Do not reject H 0 Critical Value: F α = 5.3 F-Test for Significance α =.05 Reject H 0 F.05 = 5.3 F Test Statistic: MSR F = = MSE Decision: Reject H 0 at α = 0.05 Conclusion: (continued) There is sufficient evidence that house size affects selling price

49 Prediction The regression equation can be used to predict a value for y, given a particular x For a specified value, x n+1, the predicted value is ˆ y = b + b x n n+ 1

50 Predictions Using Regression Analysis Predict the price for a house with 000 square feet: house price = (sq.ft.) = (000) = The predicted price for a house with 000 square feet is ($1,000s) = $317,850

51 Relevant Data Range When using a regression model for prediction, only predict within the relevant range of data Relevant data range House Price ($1000s) Risky to try to extrapolate far beyond the range of observed X s Square Feet

52 Estimating Mean Values and Predicting Individual Values Confidence Interval for the expected value of y, given x i Goal: Form intervals around y to express uncertainty about the value of y for a given x i Y y y = b 0 +b 1 x i Prediction Interval for an single observed y, given x i x i X

53 Confidence Interval for the Average Y, Given X Confidence interval estimate for the expected value of y given a particular x i Confidence interval for E(Y n+ 1 X n+ 1 ): yˆ n+ 1 ± t n,α/ s e 1 n + (x n + 1 (xi x) x) Notice that the formula involves the term (x x n + 1 ) so the size of interval varies according to the distance x n+1 is from the mean, x

54 Prediction Interval for an Individual Y, Given X Confidence interval estimate for an actual observed value of y given a particular x i Confidence interval for yˆ n+ 1 : yˆ n+ 1 ± t n,α/ s e 1+ 1 n + (x n + 1 (xi x) x) This extra term adds to the interval width to reflect the added uncertainty for an individual case

55 Estimation of Mean Values: Example Confidence Interval Estimate for E(Y n+1 X n+1 ) Find the 95% confidence interval for the mean price of,000 square-foot houses Predicted Price y i = ($1,000s) 1 (xi x) yˆ n + 1 ± tn-,α/se + = ± 37.1 n (x x) i The confidence interval endpoints are and , or from $80,660 to $354,900

56 Estimation of Individual Values: Example Confidence Interval Estimate for y n+1 Find the 95% confidence interval for an individual house with,000 square feet Predicted Price y i = ($1,000s) 1 (Xi X) ˆ n + 1 ± tn-1,α/se 1+ + = ± 10.8 n (X X) y i The confidence interval endpoints are and 40.07, or from $15,500 to $40,070

57 Finding Confidence and Prediction Intervals in Excel In Excel, use PHStat regression simple linear regression Check the confidence and prediction interval for x= box and enter the x-value and confidence level desired

58 Finding Confidence and Prediction Intervals in Excel (continued) Input values y Confidence Interval Estimate for E(Y n+1 X n+1 ) Confidence Interval Estimate for individual y n+1