Using JMP with a Specific

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1 1 Using JMP with a Specific Example of Regression Ying Liu 10/21/ 2009 Objectives 2 Exploratory data analysis Simple liner regression Polynomial regression How to fit a multiple regression model How to fit a multiple regression model with interactions How to generate and compare candidate models Regression diagnostics Evaluating the assumptions of regression Recommendations 1

2 Brief Information about JMP October was the 20th anniversary of JMP's first release. A product of SAS. JMP grows fast because of its interactive, comprehensive, highly visual. 3 JMP discovery statistical software and JMP Genomics The functionality of JMP is built around four data-driven tasks: Data Access Data Management Data Analysis Data Presentation JMP files (.jmp.jsl,.jrn,.jrp,.jmpprj, jmpmenu) 4 JMP data ----.jmp, (data file : open all kinds of formats) JMP script---.jsl (can be saved with.txt) JMP journal---.jrn JMP report---.jrp JMP project---.jmpprj JMP menu archives---.jmpmenu 2

3 JMP 5 JMP 6 3

4 Exploratory Data Analysis 7 Correlation analysis is used to examine and describe the linear relationship between two continuous variables. Pearson correlation coefficient, r STRONG WEAK STRONG Exploratory Data Analysis 8 r=0 r=

5 Exploratory Data Analysis

6 Misundersatnding the Correlation Coefficient 11 A common error is that using correlation between two variables to conclude a cause-and effect relationship. An example: Crime rate # of policeman City A 2 20 City B 5 25 Q: Is it true that the more policemen cause high crime rate? Correlation does not necessarily imply causation. Exploring data with Scatterplots and Correlations Look for relationship using scatterplots. Add correlation coefficients to further explore the possible relationships. Data: Fitness.JMP 12 The Variables of interest are listed below: Name Name of runner Gender gender of the runner Oxy oxygen consumption (higher Oxy is associated with better aerobic fitness) Runtime time to run 1.5 miles in mins Age age of runner Weight weight of the runner (kg) RunPulse pulse rate at the end of the run RstPulse resting pulse rate MaxPulse max. pulse rate during the run. 6

7 Exploring data with Scatterplots and Correlations 13 Exploring data with Scatterplots and Correlations 14 Note: These calculations omit observations that are missing data for any variable; when this occurs, JMP indicates the number of omitted observations at the bottom of the table. 7

8 Exploring data with Scatterplots and Correlations 15 Exploring data with Scatterplots and Correlations 16 8

9 Exploring data with Scatterplots and Correlations 17 Simple Linear Regression 18 Objectives Distinguish between one_way ANOVA and simple linear regression. Understand the principles of simple linear regression. Fit a simple linear regression model. Interpret simple linear regression result from JMP. 9

10 ANOVA versus Regression 19 Continuous response Categorical Predictor Continuous Predictor One-way ANOVA Simple Linear Regression Simple Linear Regression Model The equation: Y= β 0 + β 1 X+ ε 20 Where Y is the response variable. X is the predictor variable. β 0 is the intercept parameter, which corresponding to the value of the response variable when the predictor is 0. β 1 is the slope parameter, which corresponding to the change in the response variable given one-unit change in the predictor variable. ε is an error term representing deviation of Y about the line defined by β 0 + β 1 X. 10

11 Simple Linear Regression Mode-----Variability 21 The estimates of the unknown population parameters β 0 and β 1, are obtained by the method of least squares Unexplained variability: the error sum of squares SSE ( Y ˆ i Yi ) Explained variability: The model sum of squares SSM 2 ( ˆ Y ) Y i Total variability : The corrected total sum of squares 2 ( Y ) Y i 2 The Baseline Model 22 The baseline model: Yˆ = ˆ β 0 The intercept is the sample mean of the response. To determine whether the predictor variable explains a significant amount of variability in the response variable, the simple linear model is compared to the baseline model. 11

12 Model Hypothesis Test Null hypothesis 23 H 0 : β 1 =0 Alternative hypothesis H a : β 1 0 Fitting a Simple Linear Regression 24 12

13 Fitting a Simple Linear Regression 25 Fitting a Simple Linear Regression Linear Fit Oxy = *Runtime 26 13

14 Fitting a Simple Linear Regression 27 Polynomial Regression A simple linear regression model might inadequately characterize the relationship between predictor and response variables if the association between these variables does not follow a straight. 28 Consider a polynomial regression when the residuals from a simple linear regression model exhibit curvature. In general, building higher-order polynomials is not recommended e except as it might be suggested ed by the pattern of the residuals. 14

15 Hierarchical Models In polynomial regression, if a higher-order term is statistically significant, all lower-order terms must be included in the final model. This structure maintains model hierarchy. 29 The predictor x has a calculated p-value< The p-value for x 2 is The p-value for x 3 is The p-value for x 4 is If alpha=0.05, which parameters should be included in your model? A. y=x 2, x 3, x 4 B. y=x, x 3 C. y= x, x 2, x 3 D. y= x, x 2, x 3, x 4 Fitting a Polynomial Regression Model Data set: Cars.jmp Response: MidPrice Average price of a base model and model with all the extra options. 30 Predictor: HwyMPG Average highway miles per gallon (EPA rating). 15

16 Fitting a Polynomial Regression Model 31 Steps to obtain the scatterplots

17 Fitting a Polynomial Regression Model 33 Fitting a Polynomial Regression Model 34 Can we claim the model as below? MidPrice = *HwyMPG 17

18 Res Residual Plot 35 HwyMPG Lack of Fit Test 36 H 0 : The model is adequate (no lack of fit). H a : The model is not adequate (lack of fit). Reject null hypothesis 18

19 Fitting Higher-Order Model 37 Fitting Higher-Order Model quadratic 38 19

20 Fitting Higher-Order Model without vans 39 Fitting Higher-Order Model without vans 40 20

21 Fitting Higher-Order Model without vans 41 The adjusted R-square has a value of about 56%, which is a bit larger than that for the quadratic model fit the entire data set (31%) Multiple Regression Model Model: Y=β 0 + β 1 X 1 + β k X k +ε 42 Model Hypothesis Test H 0 : β 0 = β 1 =β k H a : Not all βs equal zero. 21

22 Multiple Linear Regression vs. Simple Linear Regression Main Advantage Enables an investigation of the relationship between Y and several independent variables simultaneously. 43 Main Disadvantages Increased complexity makes it more difficult to. Ascertain which model is best Interpret epe the models. odes Model Comparison Statistics JMP software provides several ways to compare competing regression models including 44 Mean square Error (MSE)--- Smaller is better. AdjustedR bigger is better Akaike s Information Criterion----smaller is better. Mallow s C p look for models with C p <p, where equals Mallow s C p look for models with C p p, where equals the # of parameters in the model, including the intercept. 22

23 Fitting Higher-Order Model without vans 45 Backward Stepwise Regression 46 23

24 Backward Stepwise Regression 47 Backward Stepwise Regression 48 24

25 Backward Stepwise Regression 49 Backward Stepwise Regression 50 25

26 Forward Stepwise Regression 51 Backward Stepwise Regression 52 26

27 Backward and Froward Stepwise Regression 53 SSE DF MSE R 2 Adj R 2 Cp AICc Backward Forward Assumptions for Linear Rgression 54 The variables are related linearly. The errors are normally distributed with a mean zero. The errors have a constant variance. The errors are independent. 27

28 Two Types of Plots Are Useful to VerifyAassumptions 55 A scatterplot of the residuals versus the predicted values is useful for identifying problematic patterns in the residuals. A histogram and normal quantile plot are useful for identifying potential problems with non-normality. For multiple regression, leverage plots are recommended. Two Types of Plots Are Useful to VerifyAassumptions 56 A scatterplot of the residuals versus the predicted values is useful for identifying problematic patterns in the residuals. A histogram and normal quantile plot are useful for identifying potential problems with non-normality. For multiple regression, leverage plots are recommended. 28

29 A recommendation when Assumptions Have Not Been Met Data : Cars.jmp Response: MidPrice Predictor: CityMPG 57 Two Types of Plots Are Useful to VerifyAassumptions 58 29

30 Two Types of Plots Are Useful to VerifyAassumptions 59 Residual vs. City MPG Two Types of Plots Are Useful to Verify Assumptions 60 30

31 Transforming Variables 61 Variable transformations can be useful when one or more of the assumptions of linear regression are violated. They are typically used to Linearize the relationship between X and Y. Stablize the variance of the residuals Normalize the residuals. Transforming Variables 62 Common transformation include, but not limited to, the square, square root, log, and reciprocal of the response. Not all transformation will correct a given situation. The analysis is performed on the transformed variable, and the result must be interpreted t accordingly. 31

32 Transforming Variables Questions? 32

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