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1 12er12 Chapte Bivariate i Regression (Part 1) Bivariate Regression Visual Displays Begin the analysis of bivariate data (i.e., two variables) with a scatter plot. A scatter plot - displays each observed data pair (x i, y i ) as a dot on an X/Y grid - indicates visually the strength of the relationship between the two variables Visual Displays

2 The sample correlation coefficient (r) measures the degree of linearity in the relationship between X and Y. -1 < r < +1 Strong negative relationship Strong positive relationship r = 0 indicates no linear relationship In Excel, use =CORREL(array1,array2), where array1 is the range for X and array2 is the range for Y. Weak Positive Correlation Strong Negative Correlation Strong Positive Correlation o Weak Negative Correlation

3 Nonlinear Relation r is an estimate of the population correlation coefficient ρ (rho). To test the hypothesis H 0 : ρ = 0, the test statistic is: No Correlation The critical value t α is obtained from Appendix D using ν = n 2 degrees of freedom for any α. Find the p-value using Excel s function =TDIST(t,deg_freedom,tails) or MINITAB. Equivalently, you can calculate the critical value for the correlation coefficient using This method gives a benchmark for the correlation coefficient. However, there is no p-value and is inflexible if you change your mind about α. Steps in Testing if ρ = 0 Step 1: State the Hypotheses Determine whether you are using a one or two- tailed test and the level of significance (α). H 0 : ρ = 0 H 1 : ρ 0 Step 2: Calculate the Critical Value For degrees of freedom ν = n -2, look up the critical value tα in Appendix D, then calculate

4 Steps in Testing if ρ = 0 Step 3: Make the Decision If the sample correlation coefficient r exceeds the critical value r α, then reject H 0. If using the t statistic method, reject H 0 if t > t α or if the p-value < α. Quick Rule for Significance A quick test for significance of a correlation at α =.05 is r >2/ n Role of Sample Size As sample size increases, the critical value of r becomes smaller. This makes it easier for smaller values of the sample correlation coefficient to be considered significant. A larger sample does not mean that the correlation is stronger nor does its significance imply importance. Bivariate Regression What is Bivariate Regression? Bivariate Regression analyzes the relationship between two variables. It specifies one dependent (response)) variable and one independent (predictor predictor) )variable variable. This hypothesized relationship may be linear, quadratic, or whatever.

5 Model Form Bivariate Regression Models and Parameters Unknown parameters are β 0 Intercept β 1 Slope The assumed model for a linear relationship is y i = β 0 + β 1 x i + ε i for all observations (i = 1, 2,,,n) The error term is not observable, is assumed normally distributed with mean of 0 and standard deviation σ. Models and Parameters The fitted model used to predict the expected value of Y for a given value of X is y^ i = b 0 + b 1 x i The fitted coefficients are b 0 the estimated intercept b 1 the estimated t slope Residual is e i = y i - y^ i. Residuals may be used to estimate t σ,, the standard deviation of the errors. Fitting a Regression on a Scatter Plot in Excel Step 1: - Highlight the data columns. - Click on the Chart Wizard and choose Scatter Plot - In the completed graph, click once on the points in the scatter plot to select the data - Right-click and choose Add Trendline - Choose Options and check Display Equation

6 Fitting a Regression on a Scatter Plot in Excel Slope and Intercept The ordinary least squares method (OLS) estimates the slope and intercept of the regression line so that the residuals are small. The sum of the residuals = 0 Slope and Intercept The OLS estimator for the slope is: or The sum of the squared residuals is SSE The OLS estimator t for the intercept t is:

7 Assessing Fit We want to explain the total variation in Y around its mean (SST for Total Sums of Squares) Assessing Fit The error sum of squares (SSE)) is the unexplained variation in Y The regression sum of squares (SSR)) is the explained variation in Y If the fit is good, SSE will be relatively small compared to SST. A perfect fit is indicated by an SSE = 0. The magnitude of SSE depends on n and on the units of measurement. Coefficient of Determination R 2 is a measure of relative fit based on a comparison of SSR and SST. Standard Error of Regression The standard error (s yx ) is an overall measure of model fit. 0 < R 2 < 1 Often expressed as a percent, an R 2 = 1 (i.e., 100%) indicates perfect fit. In a bivariate regression, R 2 = (r) 2 If the fitted model s predictions are perfect (SSE = 0), then s yx = 0. Thus, a small s yx indicates a better fit. Used to construct confidence intervals. Magnitude of s yx depends on the units of measurement of Y and on data magnitude.

8 Confidence Intervals for Slope and Intercept Standard error of the slope: Confidence Intervals for Slope and Intercept Confidence interval for the true slope: Confidence interval for the true intercept: Standard error of the intercept: Hypothesis Tests If β 1 = 0, then X cannot influence Y and the regression model collapses to a constant β 0 plus random error. The hypotheses to be tested are: Hypothesis Tests A t test is used with ν = n 2 degrees of freedom The test statistics for the slope and intercept are: Slope: Intercept: t n-2 is obtained from Appendix D or Excel for a given α. Reject H 0 if t > t α or if p-value < α.

9 Using Excel Using MegaStat Using MINITAB Applied Statistics in Business and Economics End of Chapter 12 Part 1 of 2