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6-1 Introduction To Empirical Models 6-1 Introduction To Empirical Models.

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Presentation on theme: "6-1 Introduction To Empirical Models 6-1 Introduction To Empirical Models."— Presentation transcript:

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4 6-1 Introduction To Empirical Models

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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.

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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

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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.

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A Multiple Regression Model:

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14 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

15 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.

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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

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation Notation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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6-2.1 Least Squares Estimation

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Regression Assumptions and Model Properties

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Regression Assumptions and Model Properties

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Regression and Analysis of Variance

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Regression and Analysis of Variance

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

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

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

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

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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.

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

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

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The Analysis of Variance Approach

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

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6-2.3 Confidence Intervals in Simple Linear Regression

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6-2.3 Confidence Intervals in Simple Linear Regression

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6-2.4 Prediction of New Observations

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6-2.4 Prediction of New Observations

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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

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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.

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6-2.5 Checking Model Adequacy

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6-2.5 Checking Model Adequacy

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6-2.5 Checking Model Adequacy

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6-2.5 Checking Model Adequacy

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6-2.5 Checking Model Adequacy

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6-2.6 Correlation and Regression The sample correlation coefficient between X and Y is

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

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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.

56 6-3 Multiple Regression 6-3.1 Estimation of Parameters in Multiple Regression

57 6-3 Multiple Regression 6-3.1 Estimation of Parameters in Multiple Regression The least squares function is given by The least squares estimates must satisfy

58 6-3 Multiple Regression 6-3.1 Estimation of Parameters in Multiple Regression The least squares normal equations are The solution to the normal equations are the least squares estimators of the regression coefficients.

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64 6-3 Multiple Regression 6-3.1 Estimation of Parameters in Multiple Regression

65 6-3 Multiple Regression 6-3.2 Inferences in Multiple Regression
Test for Significance of Regression

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Inference on Individual Regression Coefficients This is called a partial or marginal test

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Confidence Intervals on the Mean Response and Prediction Intervals

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Confidence Intervals on the Mean Response and Prediction Intervals

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Confidence Intervals on the Mean Response and Prediction Intervals

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A Test for the Significance of a Group of Regressors

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Residual Analysis

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Residual Analysis

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Residual Analysis

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Residual Analysis

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Residual Analysis

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Influential Observations

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Influential Observations

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Multicollinearity

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6-4.1 Polynomial Models

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6-4.1 Polynomial Models

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6-4.1 Polynomial Models

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6-4.1 Polynomial Models

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6-4.1 Polynomial Models

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6-4.2 Categorical Regressors Many problems may involve qualitative or categorical variables. The usual method for the different levels of a qualitative variable is to use indicator variables. For example, to introduce the effect of two different operators into a regression model, we could define an indicator variable as follows:

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6-4.2 Categorical Regressors

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6-4.2 Categorical Regressors

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6-4.3 Variable Selection Procedures Best Subsets Regressions

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6-4.3 Variable Selection Procedures Backward Elimination

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6-4.3 Variable Selection Procedures Forward Selection

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6-4.3 Variable Selection Procedures Stepwise Regression

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