McGraw-Hill/IrwinCopyright © 2009 by The McGraw-Hill Companies, Inc. All Rights Reserved. Simple Linear Regression Analysis Chapter 13

13-2 Simple Linear Regression 13.1The Simple Linear Regression Model and the Least Square Point Estimates 13.2Model Assumptions and the Standard Error 13.3Testing the Significance of Slope and y-Intercept 13.4Confidence and Prediction Intervals

13-3 Simple Linear Regression Continued 13.5Simple Coefficients of Determination and Correlation 13.6Testing the Significance of the Population Correlation Coefficient (Optional) 13.7An F Test for the Model 13.8Residual Analysis (Optional) 13.9Some Shortcut Formulas (Optional)

13-4 The Simple Linear Regression Model and the Least Squares Point Estimates The dependent (or response) variable is the variable we wish to understand or predict The independent (or predictor) variable is the variable we will use to understand or predict the dependent variable Regression analysis is a statistical technique that uses observed data to relate the dependent variable to one or more independent variables

13-5 Objective of Regression Analysis The objective of regression analysis is to build a regression model (or predictive equation) that can be used to describe, predict and control the dependent variable on the basis of the independent variable

13-6 Example 13.1: Fuel Consumption Case #1

13-7 Example 13.1: Fuel Consumption Case #2

13-8 Example 13.1: Fuel Consumption Case #3

13-9 Example 13.1: Fuel Consumption Case #4 The values of β 0 and β 1 determine the value of the mean weekly fuel consumption μ y|x Because we do not know the true values of β 0 and β 1, we cannot actually calculate the mean weekly fuel consumptions We will learn how to estimate β 0 and β 1 in the next section For now, when we say that μ y|x is related to x by a straight line, we mean the different mean weekly fuel consumptions and average hourly temperatures lie in a straight line

13-10 Form of The Simple Linear Regression Model y = β 0 + β 1 x + ε  y = β 0 + β 1 x + ε is the mean value of the dependent variable y when the value of the independent variable is x β 0 is the y-intercept; the mean of y when x is 0 β 1 is the slope; the change in the mean of y per unit change in x ε is an error term that describes the effect on y of all factors other than x

13-11 Regression Terms β 0 and β 1 are called regression parameters β 0 is the y-intercept and β 1 is the slope We do not know the true values of these parameters So, we must use sample data to estimate them b 0 is the estimate of β 0 and b 1 is the estimate of β 1

13-12 The Simple Linear Regression Model Illustrated

13-13 The Least Squares Estimates, and Point Estimation and Prediction The true values of β 0 and β 1 are unknown Therefore, we must use observed data to compute statistics that estimate these parameters Will compute b 0 to estimate β 0 and b 1 to estimate β 1

13-14 The Least Squares Point Estimates Estimation/prediction equation y ̂ = b 0 + b 1 x Least squares point estimate of the slope β 1

13-15 The Least Squares Point Estimates Continued Least squares point estimate of the y- intercept  0

13-16 Example 13.3: Fuel Consumption Case #1

13-17 Example 13.3: Fuel Consumption Case #2 From last slide, –Σy i = 81.7 –Σx i = –Σx 2 i = 16, –Σx i y i = 3, Once we have these values, we no longer need the raw data Calculation of b 0 and b 1 uses these totals

13-18 Example 13.3: Fuel Consumption Case #3 (Slope b 1 )

13-19 Example 13.3: Fuel Consumption Case #4 (y-Intercept b 0 )

13-20 Example 13.3: Fuel Consumption Case #5 Prediction (x = 40) y ̂ = b 0 + b 1 x = ( )(28) y ̂ = MMcf of Gas

13-21 Example 13.3: Fuel Consumption Case #6

13-22 Example 13.3: The Danger of Extrapolation Outside The Experimental Region

13-23 Model Assumptions 1.Mean of Zero At any given value of x, the population of potential error term values has a mean equal to zero 2.Constant Variance Assumption At any given value of x, the population of potential error term values has a variance that does not depend on the value of x 3.Normality Assumption At any given value of x, the population of potential error term values has a normal distribution 4.Independence Assumption Any one value of the error term ε is statistically independent of any other value of ε

13-24 Model Assumptions Illustrated

13-25 Sum of Squared Errors

13-26 Mean Square Error This is the point estimate of the residual variance σ 2 SSE is from last slide

13-27 Standard Error This is the point estimate of the residual standard deviation σ MSE is from last slide

13-28 Example 13.5: Fuel Consumption Case

13-29 Testing the Significance of the Slope A regression model is not likely to be useful unless there is a significant relationship between x and y To test significance, we use the null hypothesis: H 0 : β 1 = 0 Versus the alternative hypothesis: H a : β 1 ≠ 0

13-30 Testing the Significance of the Slope #2 If the regression assumptions hold, we can reject H 0 :  1 = 0 at the  level of significance (probability of Type I error equal to  ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p- value is less than 

13-31 Testing the Significance of the Slope #3 AlternativeReject H 0 Ifp-Value H a : β 1 > 0t > t α Area under t distribution right of t H a : β 1 < 0t < –t α Area under t distribution left of t H a : β 1 ≠ 0|t| > t α/2 * Twice area under t distribution right of |t| * That is t > t α/2 or t < –t α/2

13-32 Testing the Significance of the Slope #4 Test Statistics 100(1-α)% Confidence Interval for β 1 [b 1 ± t  /2 S b1 ] t , t  /2 and p-values are based on n–2 degrees of freedom

13-33 Example 13.6: MINITAB Output of Regression on Fuel Consumption Data

13-34 Example 13.6: Excel Output of Regression on Fuel Consumption Data

13-35 Example 13.6: Fuel Consumption Case The p-value for testing H 0 versus H a is twice the area to the right of |t|=7.33 with n-2=6 degrees of freedom In this case, the p-value is We can reject H 0 in favor of H a at level of significance 0.05, 0.01, or We therefore have strong evidence that x is significantly related to y and that the regression model is significant

13-36 A Confidence Interval for the Slope If the regression assumptions hold, a 100(1-  ) percent confidence interval for the true slope B1 is –b 1 ± t  /2 s b Here t is based on n - 2 degrees of freedom

13-37 Example 13.7: Fuel Consumption Case An earlier printout tells us: –b 1 = –s b1 = We have n-2=6 degrees of freedom –That gives us a t-value of for a 95 percent confidence interval [b 1 ± t · s b1 ] = [ ± ] = [ , ]

13-38 Testing the Significance of the y-Intercept If the regression assumptions hold, we can reject H 0 :  0 = 0 at the  level of significance (probability of Type I error equal to  ) if and only if the appropriate rejection point condition holds or, equivalently, if the corresponding p- value is less than 

13-39 Testing the Significance of the y-Intercept #2 AlternativeReject H 0 Ifp-Value H a : β 0 > 0t > t α Area under t distribution right of t H a : β 0 < 0t < –t α Area under t distribution left of t H a : β 0 ≠ 0|t| > t α/2 * Twice area under t distribution right of |t| * That is t > t α/2 or t < –t α/2

13-40 Testing the Significance of the y-Intercept #3 Test Statistics 100(1-  )% Confidence Interval for  1 t , t  /2 and p-values are based on n–2 degrees of freedom

13-41 Confidence and Prediction Intervals The point on the regression line corresponding to a particular value of x 0 of the independent variable x is y ̂ = b 0 + b 1 x 0 It is unlikely that this value will equal the mean value of y when x equals x 0 Therefore, we need to place bounds on how far the predicted value might be from the actual value We can do this by calculating a confidence interval mean for the value of y and a prediction interval for an individual value of y

13-42 Distance Value Both the confidence interval for the mean value of y and the prediction interval for an individual value of y employ a quantity called the distance value The distance value for a particular value x 0 of x is The distance value is a measure of the distance between the value x 0 of x and  Notice that the further x 0 is from , the larger the distance value

13-43 A Confidence Interval for a Mean Value of y Assume that the regression assumption holds The formula for a 100(1-  ) confidence interval for the mean value of y is as follows: This is based on n-2 degrees of freedom

13-44 Example 13.9: Fuel Consumption Case From before: –n = 8 –x 0 = 40 –  = –SS xx = 1, The distance value is

13-45 Example 13.9: Fuel Consumption Case Continued From before –x 0 = 40 is MMcf –t = –s = –Distance value is The confidence interval is

13-46 A Prediction Interval for an Individual Value of y Assume that the regression assumption holds The formula for a 100(1-  ) prediction interval for an individual value of y is as follows: This is based on n-2 degrees of freedom

13-47 Example 13.9: Fuel Consumption Case From before –x 0 = 40 is MMcf –t = –s = –Distance value is The prediction interval is

13-48 Example 13.9: MINITAB Best Fit Line for Fuel Consumption Data

13-49 Example 13.9: MINITAB Scale of Values of Fuel Consumption

13-50 Which to Use? The prediction interval is useful if it is important to predict an individual value of the dependent variable A confidence interval is useful if it is important to estimate the mean value The prediction interval will always be wider than the confidence interval

13-51 The Simple Coefficient of Determination and Correlation How useful is a particular regression model? One measure of usefulness is the simple coefficient of determination It is represented by the symbol r 2

13-52 Prediction Errors for Fuel Consumption Data When Not Using Information in X

13-53 Prediction Errors for Fuel Consumption Data When Using Information in X

13-54 Calculating The Simple Coefficient of Determination 1.Total variation is given by the formula  (y i -y ̄) 2 2.Explained variation is given by the formula  (y ̂ i -y ̄) 2 3.Unexplained variation is given by the formula  (y i -y ̂) 2 4.Total variation is the sum of explained and unexplained variation 5.r 2 is the ratio of explained variation to total variation

13-55 What Does r 2 Mean? The coefficient of determination, r 2, is the proportion of the total variation in the n observed values of the dependent variable that is explained by the simple linear regression model

13-56 Example 13.11: Fuel Consumption Case

13-57 The Simple Correlation Coefficient The simple correlation coefficient measures the strength of the linear relationship between y and x and is denoted by r Where b 1 is the slope of the least squares line

13-58 Different Values of the Correlation Coefficient

13-59 Example 13.13: Fuel Consumption Case

13-60 Two Important Points 1.The value of the simple correlation coefficient (r) is not the slope of the least square line –That value is estimated by b 1 2.High correlation does not imply that a cause-and-effect relationship exists –It simply implies that x and y tend to move together in a linear fashion –Scientific theory is required to show a cause-and-effect relationship

13-61 Testing the Significance of the Population Correlation Coefficient The simple correlation coefficient (r) measures the linear relationship between the observed values of x and y from the sample The population correlation coefficient (ρ) measures the linear relationship between all possible combinations of observed values of x and y r is an estimate of ρ

13-62 Testing ρ We can test to see if the correlation is significant using the hypotheses H 0 : ρ = 0 H a : ρ ≠ 0 The statistic is This test will give the same results as the test for significance on the slope coefficient b 1

13-63 An F Test for Model For simple regression, this is another way to test the null hypothesis H 0 : β 1 = 0 This is the only test we will use for multiple regression The F test tests the significance of the overall regression relationship between x and y

13-64 Mechanics of the F Test To test H 0 : β 1 = 0 versus Ha: β 1  0 at the  level of significance Test statistics based on F Reject H 0 if F(model) > F  or p-value <  F  is based on 1 numerator and n-2 denominator degrees of freedom

13-65 Mechanics of the F Test Graphically

13-66 Example 13.15: Fuel Consumption Case F-test at  = 0.05 level of significance Test Statistic Reject H 0 at  level of significance, since F  is based on 1 numerator and 6 denominator degrees of freedom

13-67 Residual Analysis #1 Checks of regression assumptions are performed by analyzing the regression residuals Residuals (e) are defined as the difference between the observed value of y and the predicted value of y, e = y - y ̂ Note that e is the point estimate of ε If the regression assumptions are valid, the population of potential error terms will be normally distributed with a mean of zero and a variance σ 2 Furthermore, the different error terms will be statistically independent

13-68 Residual Analysis #2 The residuals should look like they have been randomly and independently selected from normally distributed populations having mean zero and variance σ 2 With any real data, assumptions will not hold exactly Mild departures do not affect our ability to make statistical inferences In checking assumptions, we are looking for pronounced departures from the assumptions So, only require residuals to approximately fit the description above

13-69 Residual Plots 1.Residuals versus independent variable 2.Residuals versus predicted y’s 3.Residuals in time order (if the response is a time series)

13-70 Constant Variance Assumptions To check the validity of the constant variance assumption, examine residual plots against –The x values –The predicted y values –Time (when data is time series) A pattern that fans out says the variance is increasing rather than staying constant A pattern that funnels in says the variance is decreasing rather than staying constant A pattern that is evenly spread within a band says the assumption has been met

13-71 Constant Variance Visually

13-72 Example 13.16: The QHIC Case Residuals Versus Value

13-73 Example 13.16: The QHIC Case Residuals Versus the Fitted Values

13-74 Assumption of Correct Functional Form If the relationship between x and y is something other than a linear one, the residual plot will often suggest a form more appropriate for the model For example, if there is a curved relationship between x and y, a plot of residuals will often show a curved relationship

13-75 Normality Assumption If the normality assumption holds, a histogram or stem-and-leaf display of residuals should look bell-shaped and symmetric Another way to check is a normal plot of residuals –Order residuals from smallest to largest –Plot e (i) on vertical axis against z (i) Z (i) is the point on the horizontal axis under the z curve so the area under this curve to the left is (3i-1)/(3n+1) If the normality assumption holds, the plot should have a straight-line appearance

13-76 Example 13.18: The QHIC Case Normal Probability Plot of the Residuals

13-77 Independence Assumption Independence assumption is most likely to be violated when the data are time-series data –If the data is not time series, then it can be reordered without affecting the data –Changing the order would change the interdependence of the data For time-series data, the time-ordered error terms can be autocorrelated –Positive autocorrelation is when a positive error term in time period i tends to be followed by another positive value in i+k –Negative autocorrelation is when a positive error term in time period i tends to be followed by a negative value in i+k Either one will cause a cyclical error term over time

13-78 Independence Assumption Visually Positive Autocorrelation

13-79 Independence Assumption Visually Negative Autocorrelation

13-80 Some Shortcut Formulas where