6-1 Introduction To Empirical Models

6-1 Introduction To Empirical Models Based on the scatter diagram, it is probably reasonable to assume that the mean of the random variable Y is related to x by the following straight-line relationship: where the slope and intercept of the line are called regression coefficients. The simple linear regression model is given by where  is the random error term.

6-1 Introduction To Empirical Models

6-1 Introduction To Empirical Models We think of the regression model as an empirical model. Suppose that the mean and variance of  are 0 and 2, respectively, then The variance of Y given x is

6-1 Introduction To Empirical Models The true regression model is a line of mean values: where 1 can be interpreted as the change in the mean of Y for a unit change in x. Also, the variability of Y at a particular value of x is determined by the error variance, 2. This implies there is a distribution of Y-values at each x and that the variance of this distribution is the same at each x.

6-1 Introduction To Empirical Models

6-1 Introduction To Empirical Models A Multiple Linear Regression Model: 𝑌= 𝛽 0 + 𝛽 1 𝑥 1 + 𝛽 2 𝑥 2 +𝜖 where 𝛽 0 = the intercept of the plane 𝛽 1 , 𝛽 2 = partial regression coefficients

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation The case of simple linear regression considers a single regressor or predictor x and a dependent or response variable Y. The expected value of Y at each level of x is a random variable: We assume that each observation, Y, can be described by the model

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation Suppose that we have n pairs of observations (x1, y1), (x2, y2), …, (xn, yn). The method of least squares is used to estimate the parameters, 0 and 1 by minimizing the sum of the squares of the vertical deviations in Figure 6-6.

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation Using Equation 6-8, the n observations in the sample can be expressed as The sum of the squares of the deviations of the observations from the true regression line is

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

Cross-products Matrix 6-2 Simple Linear Regression Sums of Squares and Cross-products Matrix The Sums of squares and cross-products matrix is a convenient way to summarize the quantities needed to do the hand calculations in regression. It also plays a key role in the internal calculations of the computer. It is outputed from PROC REG and PROC GLM if the XPX option is included on the model statement. The elements are X’X Intercept X Y n 𝑖=1 𝑛 𝑋 𝑖 𝑖=1 𝑛 𝑌 𝑖 𝑖=1 𝑛 𝑋 𝑖 2 𝑖=1 𝑛 𝑋 𝑖 𝑌 𝑖 𝑖=1 𝑛 𝑌 𝑖 2

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression 6-2.1 Least Squares Estimation

6-2 Simple Linear Regression Regression Assumptions and Model Properties

6-2 Simple Linear Regression Regression Assumptions and Model Properties

6-2 Simple Linear Regression Regression and Analysis of Variance 𝑆𝑆𝑇=𝑆𝑦𝑦= 𝑖=1 𝑛 ( 𝑦 𝑖 − 𝑦 ) 2 = 𝑖=1 𝑛 𝑦 2 − ( 𝑖=1 𝑛 𝑦𝑖)2 𝑛

6-2 Simple Linear Regression Regression and Analysis of Variance

6-2 Simple Linear Regression Example 6-1 OPTIONS NOOVP NODATE NONUMBER LS=140; DATA ex61; INPUT salt area @@; LABEL salt='Salt Conc' area='Roadway area'; CARDS; 3.8 0.19 5.9 0.15 14.1 0.57 10.4 0.4 14.6 0.7 14.5 0.67 15.1 0.63 11.9 0.47 15.5 0.75 9.3 0.6 15.6 0.78 20.8 0.81 14.6 0.78 16.6 0.69 25.6 1.3 20.9 1.05 29.9 1.52 19.6 1.06 31.3 1.74 32.7 1.62 PROC REG DATA=EX61; MODEL SALT=AREA/XPX R; PLOT SALT*AREA; TITLE 'SCATTER PLOT'; RUN; QUIT;

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression Use of t-Tests Suppose we wish to test An appropriate test statistic would be

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression Use of t-Tests We would reject the null hypothesis if

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression Use of t-Tests Suppose we wish to test An appropriate test statistic would be

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression Use of t-Tests We would reject the null hypothesis if

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression Use of t-Tests An important special case of the hypotheses of Equation 6-23 is These hypotheses relate to the significance of regression. Failure to reject H0 is equivalent to concluding that there is no linear relationship between x and Y.

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression Use of t-Tests

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression Use of t-Tests

6-2 Simple Linear Regression

6-2 Simple Linear Regression The Analysis of Variance Approach

6-2 Simple Linear Regression 6-2.2 Testing Hypothesis in Simple Linear Regression The Analysis of Variance Approach

6-2 Simple Linear Regression 6-2.3 Confidence Intervals in Simple Linear Regression

6-2 Simple Linear Regression 6-2.3 Confidence Intervals in Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression 6-2.4 Prediction of Future Observations

6-2 Simple Linear Regression

6-2 Simple Linear Regression 6-2.4 Prediction of Future Observations

6-2 Simple Linear Regression 6-2.5 Checking Model Adequacy Fitting a regression model requires several assumptions. Errors are uncorrelated random variables with mean zero; Errors have constant variance; and, Errors be normally distributed. The analyst should always consider the validity of these assumptions to be doubtful and conduct analyses to examine the adequacy of the model

6-2 Simple Linear Regression 6-2.5 Checking Model Adequacy The residuals from a regression model are ei = yi - ŷi , where yi is an actual observation and ŷi is the corresponding fitted value from the regression model. Analysis of the residuals is frequently helpful in checking the assumption that the errors are approximately normally distributed with constant variance, and in determining whether additional terms in the model would be useful.

6-2 Simple Linear Regression 6-2.5 Checking Model Adequacy As an approximate check of normality, construct a frequency histogram or a normal probability plot of residuals. Standardize the residuals by computing 𝑑 𝑖 = 𝑒 𝑖 / 𝜎 2 , i = 1, 2, …, n. If the errors are normally distributed, approximately 95% of the standardized residuals should fall in the interval (−2, +2).

6-2 Simple Linear Regression 6-2.5 Checking Model Adequacy Plot the residuals (1) in time sequence (if known), (2) against the 𝑦 𝑖 , and (3) against the independent variable 𝑥. 𝑦 , ln 𝑦, 1 𝑦

6-2 Simple Linear Regression 6-2.5 Checking Model Adequacy

6-2 Simple Linear Regression 6-2.5 Checking Model Adequacy

6-2 Simple Linear Regression Standardized Residual 𝑑 10 = 𝑒 10 𝜎 2 = - 2.183 𝑑 12 =2.186

6-2 Simple Linear Regression 6-2.5 Checking Model Adequacy

6-2 Simple Linear Regression Example 6-1 (cont.) OPTIONS NOOVP NODATE NONUMBER LS=140; DATA ex61; INPUT salt area @@; LABEL salt='Salt Conc' area='Roadway area'; CARDS; 3.8 0.19 5.9 0.15 14.1 0.57 10.4 0.4 14.6 0.7 14.5 0.67 15.1 0.63 11.9 0.47 15.5 0.75 9.3 0.6 15.6 0.78 20.8 0.81 14.6 0.78 16.6 0.69 25.6 1.3 20.9 1.05 29.9 1.52 19.6 1.06 31.3 1.74 32.7 1.62 DATA EX61N; AREA=1.25; OUTPUT; DATA EX61N1; SET EX61 EX61N; PROC REG DATA=EX61N1; MODEL SALT=AREA/CLM CLI r; /* CLM for (100-α) % confidence limits for the expected value of the dependent variable, CLI for (100- α) % confidence limits for an individual predicted value */ TITLE 'CIs FOR MEAN RESPONSE AND FUTURE OBSERVATION'; RUN; QUIT;

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression

6-2 Simple Linear Regression 6-2.6 Correlation and Regression 𝜌 as the population correlation coefficient, which is a measure of the strength of the linear relationship between Y and X in the population or joint distribution. The sample correlation coefficient between X and Y is 𝑟= 𝑆 𝑥𝑦 𝑆 𝑥𝑥 𝑆 𝑦𝑦

6-2 Simple Linear Regression 6-2.6 Correlation and Regression The sample correlation coefficient is also closely related to the slope in a linear regression model

6-2 Simple Linear Regression Correlation Analysis Correlation is a measure of the linear relationship between two random variables, X and Y. It is a parameter of the bivariate distribution (joint distribution) of X & Y and will be denoted by the Greek letter rho, 𝜌. 𝜌 = Corr(X, Y) −1≤𝜌≤1 If 𝜌=1 then X and Y have a perfect direct linear relationship. If 𝜌=−1 then X and Y have a perfect inverse linear relationship. If 𝜌=0 then X and Y have no linear relationship. If 0≤𝜌≤1 then X and Y are directly related. The closer to 1 the more perfect the linear relationship. If −1≤𝜌≤0 then X and Y are inversely related. The closer to −1 the more perfect the linear relationship.

6-2 Simple Linear Regression

6-2 Simple Linear Regression 6-2.6 Correlation and Regression It is often useful to test the hypotheses The appropriate test statistic for these hypotheses is Reject H0 if |t0| > t/2,n-2.

6-2 Simple Linear Regression Example 6-1 (Continued) OPTIONS NOOVP NODATE NONUMBER LS=80; DATA ex61; INPUT salt area @@; LABEL salt='Salt Conc' area='Roadway area'; CARDS; 3.8 0.19 5.9 0.15 14.1 0.57 10.4 0.4 14.6 0.7 14.5 0.67 15.1 0.63 11.9 0.47 15.5 0.75 9.3 0.6 15.6 0.78 20.8 0.81 14.6 0.78 16.6 0.69 25.6 1.3 20.9 1.05 29.9 1.52 19.6 1.06 31.3 1.74 32.7 1.62 PROC CORR DATA=EX61; VAR SALT AREA; TITLE 'Correlation between SALT and AREA'; PROC REG; MODEL salt=area/XPX; TITLE 'Linear Regression of SALT vs AREA'; RUN; QUIT;

6-2 Simple Linear Regression

6-2 Simple Linear Regression