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Review of Lecture 13 Standard Regression Assumptions:

Lecture 14. Review of Lecture 13 Standard Regression Assumptions: a). about the form of the model b). about the measurement errors c). about the predictor variables d). about the observations

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Review of Lecture 13 Standard Regression Assumptions:

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  1. Lecture 14 • Review of Lecture 13 • Standard Regression Assumptions: • a). about the form of the model • b). about the measurement errors • c). about the predictor variables • d). about the observations • II Examples of the Anscombe’s Quartet Data show that • a). Gross Violations of assumptions will lead to serious problems • b). Summary statistics may miss or overlook the features of the data. • III Types of Residuals • a). Ordinary b). Standardized c). Studentized • What we’ll talk about today? • I Graphical Methods for Exploring Data Structures • a) Graphs before fitting b) Graphs after fitting ST3131, Lecture 14

  2. Graphical Methods Graphical methods play an important role in data analysis, especially in linear regression analysis. It can reveal some important features that summary statistics may miss, e.g., the scatter plots of the Anscombe’s data. a). Graphs before fitting a model Functions: 1) Detect outliers 2) Suggest a model b). Graphs after fitting a model Functions: 1) Checking assumption violations 2). Detecting outliers ST3131, Lecture 14

  3. Graphs before fitting a model • Functions: 1) Detect outliers, high leverage point or influential points • 2) Recognize the patterns • 3) Explore the relationship between variables • Types : 1). One-dimensional • 2). Two-dimensional • 3). Rotating plot • 4). Dynamic graphs ST3131, Lecture 14

  4. One-dimensional graphs • Histogram • Stem-and-leaf display • Dot plot • Box plot • Functions: (1) Distribution of a single variable • (2) Detect outliers, high leverage points, or influential points • Two-dimensional graphs • Matrix plot: pair-wise scatter plot • Purpose: explore patterns of pair-wise variables ST3131, Lecture 14

  5. Stem-and-leaf of Y N = 15 Leaf Unit = 0.10 2 10 88 3 11 0 4 11 2 6 11 45 6 11 7 11 9 (1) 12 0 7 12 3 6 12 4 5 12 66 3 12 8 2 13 01 ST3131, Lecture 14

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  7. Drawback when p>1, the scatter plots of Y vs Xj may or may not show linear patterns even when Y and X1, X2, …,Xp have a good or perfect linear relationship. Hamiltan’s Data ST3131, Lecture 14

  8. Hamiltan’s Data : Y, X1, X2 Fitted Results: Y vs X1: Y=11.989+.004X1, t-test=.09, R_sq=0.0 Question: Y is uncorrelated with X1? Y vs X2: Y=10.632+.195 X2, t-test=1.74, R_sq=.188 Question: Y is uncorrelated with X2? Y vs X1, X2: Y=-4.515+3.097X1+1.032X2, F-test=39222, R_sq=1.0 Question: Y is almost perfectly linearly correlated with X1 and X2? Question: What assumption is violated by the Hamiltan’s Data? ST3131, Lecture 14

  9. Rotating Plots: 3-dimensional plot Rotate the points in different directions s o that three-dimensional structure becomes apparent. Dynamic Graphs: p>3 Graphs are in a dynamic status instead of a static status. Good for exploring the structural and relationship in more than 3-dimensions. ST3131, Lecture 14

  10. b) Graphs after fitting a model • Functions: 1) Checking assumptions, • 2) Detection of outliers, high leverage points, influential points • 3). Diagnostic plots for the effect of variables • Standardized Residuals-based Plots • Normal Probability Plot of standardized residuals: • ordered standardized residuals vs normal scores • Function: • Main Idea : If the residuals are normally distributed, the ordered standardized residuals should be approximately the same as the ordered normal scores. In this case, the plot should resemble a (nearly) straight-line with interceptand slope . ST3131, Lecture 14

  11. 2. Scatter Plots of standardized residuals against each of the predictor variables Function: Check linearity or homogeneity assumptions on Xj Main Idea: Under the standard assumptions, the standardized residuals are nearly uncorrelated with each of the predictor variables. In this case, the residual points should be randomly scattered in the range. For example, ST3131, Lecture 14

  12. 3. Scatter Plots of standardized residuals against the fitted values. Function: Check Independence, homogeneity of the measurement errors Linearity of the data Main Idea: Under the Independence, homogeneity of the measurement errors Linearity of the data assumptions, the standardized residuals are nearly uncorrelated with the fitted values. In this case, the residual points should be randomly scattered in the range, e.g., ST3131, Lecture 14

  13. 4. Index Plot of standardized residuals i. e. the scatter plot of standardized residuals against the indices of observations. Function: Check Independence, homogeneity of the measurement errors Linearity of the data Main Idea: Under the assumption of independence errors, the standardized residuals should be randomly scattered within a horizontal band around 0. ST3131, Lecture 14

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  18. After-class Questions: • Is graphics before fitting a model model-based? • Is graphics after fitting a model model-based? • Why is graphics sometimes more useful than a statistic? ST3131, Lecture 14

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