Class 17: Tuesday, Nov. 9

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Class 17 Tuesday, Nov. 9

• Another example of interpreting multiple
  regression coefficients
• Steps in multiple regression analysis and example
  analysis
• Omitted Variables Bias
• Discuss final project

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Interpreting Multiple Regression Coefficients
Another Example

• A marketing firm studied the demand for a new
  type of personal digital assistant (PDA). The
  firm surveyed a sample of 75 consumers. Each
  respondent was initially shown the new device and
  then asked to rate the likelihood of purchase on
  a scale of 1 to 10, with 1 implying little chance
  of purchase and 10 indicating almost certain
  purchase. The age (in years) and income (in
  thousands of dollars) were recorded for each
  respondent. The data are in pda.JMP.

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Simple Regressions to Predict Rating (Likelihood
of Purchase)

• As income rises, the likelihood of purchase also
  increases specifically a 10,000 increase in
  income is associated with a 0.7 increase in
  rating.
• As age increases, the likelihood of purchase also
  increases specifically a 10-year increase in age
  is associated with a 0.9 increase in rating.

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Multiple Regression

• For any fixed level of income, the average rating
  decreases by 0.7 if Age increases by 10 years.
• For all fixed income levels, old consumers have
  higher ratings on average than young consumers
  and at all fixed age levels, average ratings
  increase as income rises.
• Positive association between age and rating is a
  result of positive association between age and
  income.

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Air Pollution and Mortality

• 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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Air Pollution and Mortality

• Question of interest What is the association
  between the air pollution variables (NOX and S02)
  once environmental variables (precipitation) and
  demographic variables have been taken into
  account?

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Curvature in relationship between Mortality and
S02. Tukeys Bulging Rule suggests transforming
S02 to log S02 as a possible remedy. The
scatterplot of Mortality vs. NOX is crunched.
When a scatterplot between a response and
explanatory variable crunched, transforming the
explanatory variable to log(explanatory variable)
is a good idea.
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Initial Model
Checking for influential points New Orleans
has Cooks distance of 1.75 and
leverage 0.45gt(36/60). We should remove New
Orleans, noting that it has unusual explanatory
variables and that our conclusions do not apply
to explanatory variables in the range of New
Orleans.
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Because New Orleans is an influential point and
has leverage 0.45gt(36/60)0.30, we remove it and
note that our model does apply to observations in
the range of explanatory variables of New
Orleans.
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Checking the Model
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Model Building

• Model Parsimony If a variable is not of central
  interest and is not significant, we remove it
  from the model.
• We can remove Education. We dont remove log NOX
  since it is of central interest.

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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.

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Omitted Variables

• What happens if we omit a lurking variable from
  the regression, e.g., percentage of smokers?
• Suppose we are interested in the causal effect of
  on y and believe that there are lurking
  variables
• and that
  
• is the causal effect of on y. If we
  omit the confounding variable, , then the
  multiple regression will be estimating the
  coefficient as the coefficient on .
  How different are and .

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Omitted Variables Bias Formula

• Suppose that
• Then
• Formula tells us about direction and magnitude of
  bias from omitting a variable in estimating a
  causal effect.
• Omitted variable bias
• Formula also applies to least squares estimates,
  i.e.,