Stat 112: Lecture 16 Notes

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Stat 112 Lecture 16 Notes

• Finish Chapter 6
• Influential Points for Multiple Regression
  (Section 6.7)
• Assessing the Independence Assumptions and
  Remedies for Its Violation (Section 6.8)
• Homework 5 due next Thursday. I will e-mail it
  tonight.
• Please let me know of any ideas you want to
  discuss for the final project.

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Multiple regression, modeling and outliers,
leverage and influential pointsPollution Example

• Data set pollution.JMP provides information about
  the relationship between pollution and mortality
  for 60 cities between 1959-1961.
• The variables are
• y (MORT)total age adjusted mortality in deaths
  per 100,000 population
• PRECIPmean annual precipitation (in inches)
• EDUCmedian number of school years completed for
  persons 25 and older
• NONWHITEpercentage of 1960 population that is
  nonwhite NOXrelative pollution potential of Nox
  (related to amount of tons of Nox emitted per
  day per square kilometer)
• SO2relative pollution potential of SO2

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Multiple Regression Steps in Analysis

• Preliminaries Define the question of interest.
  Review the design of the study. Correct errors
  in the data.
• Explore the data. Use graphical tools, e.g.,
  scatterplot matrix consider transformations of
  explanatory variables fit a tentative model
  check for outliers and influential points.
• Formulate an inferential model. Word the
  questions of interest in terms of model
  parameters.

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Multiple Regression Steps in Analysis Continued

1. Check the Model. (a) Check the model assumptions
   of linearity, constant variance, normality. (b)
   If needed, return to step 2 and make changes to
   the model (such as transformations or adding
   terms for interaction and curvature) (c) Drop
   variables from the model that are not of central
   interest and are not significant.
2. Infer the answers to the questions of interest
   using appropriate inferential tools (e.g.,
   confidence intervals, hypothesis tests,
   prediction intervals).
3. Presentation Communicate the results to the
   intended audience.

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Scatterplot Matrix

• Before fitting a multiple linear regression
  model, it is good idea to make scatterplots of
  the response variable versus the explanatory
  variable. This can suggest transformations of
  the explanatory variables that need to be done as
  well as potential outliers and influential
  points.
• Scatterplot matrix in JMP Click Analyze,
  Multivariate Methods and Multivariate, and then
  put the response variable first in the Y, columns
  box and then the explanatory variables in the Y,
  columns box.

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Scatterplot Matrix
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Crunched Variables

• When an X variable is crunched meaning that
  most of its values are crunched together and a
  few are far apart there will be influential
  points. To reduce the effects of crunching, it
  is a good idea to transform the variable to log
  of the variable.

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• 2. a) From the scatter plot of MORT vs. NOX we
  see that NOX values are crunched very tight. A
  Log transformation of NOX is needed.
• b) The curvature in MORT vs. SO2 indicates a Log
  transformation for SO2 may be suitable.
• After the two transformations we have the
  following correlations

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Influential Points, High Leverage Points,
Outliers in Multiple Regression

• As in simple linear regression, we identify high
  leverage and high influence points by checking
  the leverages and Cooks distances (Use save
  columns to save Cooks D Influence and Hats).
• High influence points Cooks distance gt 1
• High leverage points Hat greater than (3( of
  explanatory variables 1))/n is a point with
  high leverage. These are points for which the
  explanatory variables are an outlier in a
  multidimensional sense.
• Use same guidelines for dealing with influential
  observations as in simple linear regression.
• Point that has unusual Y given its explanatory
  variables point with a residual that is more
  than 3 RMSEs away from zero.

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New Orleans has Cooks Distance greater than 1
New Orleans may be influential.
3 RMSEs 108 No points are outliers in residuals
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Labeling Observations

• To have points identified by a certain column, go
  the column, click Columns and click Label (click
  Unlabel to Unlabel).
• To label a row, go to the row, click rows and
  click label.

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Dealing with New Orleans

• New Orleans is influential.
• New Orleans also has high leverage,
  hat0.45gt(36/60)0.2.
• Thus, it is reasonable to exclude New Orleans
  from the analysis, report that we excluded New
  Orleans, and note that our model does not apply
  to cities with explanatory variables in the range
  of New Orleans.

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Leverage Plots

• A simple regression view of a multiple
  regression coefficient. For xj
• Residual y (w/o xj) vs. Residual xj (vs the
  rest of xs)
• (both axes are recentered)
• Slope in leverage plot
• coefficient for that variable in the multiple
  regression
• Distances from the points to the LS line are
  multiple regression residuals. Distance from
  point to horizontal line is the residual if the
  explanatory variable is not included in the
  model.
• Useful to identify (for xj)
• outliers
• leverage
• influential points
• (Use them the same way as in a simple regression
  to identify the effect of points for the
  regression coefficient
• of a particular variable)
• Leverage plots are particularly useful for points
  which are influential for a particular
  coefficient in the regression.

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The enlarged observation New Orleans is an
outlier for estimating each coefficient and is
highly leveraged for estimating the coefficients
of interest on log Nox and log SO2. Since New
Orleans is both highly leveraged and an outlier,
we expect it to be influential.
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Analysis without New Orleans
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Checking the Model
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Linearity, constant variance and normality
assumptions all appear reasonable.
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Inference About Questions of Interest

• Strong evidence that mortality is positively
  associated with S02 for fixed levels of
  precipitation, education, nonwhite, NOX.
• No strong evidence that mortality is associated
  with NOX for fixed levels of precipitation,
  education, nonwhite, S02.

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Multiple Regression and Causal Inference

• Goal Figure out what the causal effect on
  mortality would be of decreasing air pollution
  (and keeping everything else in the world fixed)
• Lurking variable A variable that is associated
  with both air pollution in a city and mortality
  in a city.
• In order to figure out whether air pollution
  causes mortality, we want to compare mean
  mortality among cities with different air
  pollution levels but the same values of the
  confounding variables.
• If we include all of the lurking variables in the
  multiple regression model, the coefficient on air
  pollution represents the change in the mean of
  mortality that is caused by a one unit increase
  in air pollution.
• If we omit some of the lurking variables, then
  there is omitted variables bias, i.e., the
  multiple regression coefficient on air pollution
  does not measure the causal effect of air
  pollution.

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Time Series Data and Autocorrelation

• When Y is a variable collected for the same
  entity (person, state, country) over time, we
  call the data time series data.
• For time series data, we need to consider the
  independence assumption for the simple and
  multiple regression model.
• Independence Assumption The residuals are
  independent of one another. This means that if
  the residual is positive this year, it needs to
  be equally likely for the residuals to be
  positive or negative next year, i.e., there is no
  autocorrelation.
• Positive autocorrelation Positive residuals are
  more likely to be followed by positive residuals
  than by negative residuals.
• Negative autocorrelation Positive residuals are
  more likely to be followed by negative residuals
  than by positive residuals.

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Example Melanoma Incidence

• Is the incidence of melanoma (skin cancer)
  increasing over time? Is melanoma related to
  solar radiation?
• We address thse questions by looking at melanoma
  incidence among males from the Connecticut Tumor
  Registry from 1936 to 1972. Data is in
  melanoma.JMP

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Residuals suggest positive autocorrelation.
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Test of Independence

• The Durbin-Watson test is a test of whether the
  residuals are independent. The null hypothesis
  is that the residuals are independent and the
  alternative hypothesis is that the residuals are
  not independent (either positively or negatively)
  autocorrelated.
• To compute Durbin-Watson test in JMP, after Fit
  Model, click the red triangle next to Response,
  click Row Diagnostics and click Durbin-Watson
  Test. Then click red triangle next to
  Durbin-Watson to get p-value.
• For melanoma data,
• Remedy for autocorrelation Add lagged value of Y
  to model.