Lecture 14 Review of Lecture 13 What we’ll talk about today? 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 11/17/2018 ST3131, Lecture 14

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 11/17/2018 ST3131, Lecture 14

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 11/17/2018 ST3131, Lecture 14

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 11/17/2018 ST3131, Lecture 14

11/17/2018 ST3131, Lecture 14 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 11/17/2018 ST3131, Lecture 14

11/17/2018 ST3131, Lecture 14

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 11/17/2018 ST3131, Lecture 14

Y vs X1: Y=11.989+.004X1, t-test=.09, R_sq=0.0 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? 11/17/2018 ST3131, Lecture 14

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. 11/17/2018 ST3131, Lecture 14

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 intercept and slope . 11/17/2018 ST3131, Lecture 14

Function: Check linearity or homogeneity assumptions on Xj 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, 11/17/2018 ST3131, Lecture 14

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., 11/17/2018 ST3131, Lecture 14

Function: Check Independence, homogeneity of the measurement errors 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. 11/17/2018 ST3131, Lecture 14

11/17/2018 ST3131, Lecture 14

11/17/2018 ST3131, Lecture 14

11/17/2018 ST3131, Lecture 14

11/17/2018 ST3131, Lecture 14

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? 11/17/2018 ST3131, Lecture 14