1. 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
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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
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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
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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
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5. 11/17/2018 ST3131, Lecture 14 Stem-and-leaf of Y N = 15
Leaf Unit = 0.10
6 11
(1) 12 0
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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
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8. 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= X1, t-test=.09, R_sq=0.0
Question: Y is uncorrelated with X1?
Y vs X2: Y= X2, t-test=1.74, R_sq=.188
Question: Y is uncorrelated with X2?
Y vs X1, X2: Y= X X2, 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?
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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.
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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 intercept and slope
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11. 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,
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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.,
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13. 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.
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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?
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