4'1: Cautions on Regression

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4.1 Cautions on Regression
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• Extrapolation use of a regression line to
  predict far outside the domain of values
• - Often very inaccurate
• - Ex. Using a linear model of age vs. height for
  the first ten years, to predict at age 30
• Lurking Variables a variable not among the
  explanatory or response variables that may
  influence the interpretation of the relationship

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• a. Falsely suggest a strong relationship
• Ex. More men complaining of chest pains are
  likely to get detailed tests and treatment than
  women
• Lurking Variable Age of patient. Women tend
  to have heart problems later in life where
  aggressive tests and treatments might not be an
  option

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• Hide a relationship that exists
• Ex. A British study compared the relationship
  between amount of overcrowding vs. the lack of
  indoor toilets and found no relationship.
• Lurking Variable Public Housing. A high amount
  of public housing (where there is indoor toilets)
  would hide the relationship.

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• Finding lurking variables is often a matter of
  common sense and content knowledge.
• Many lurking variables will present themselves if
  you sort the data over time vs. sorting by x
• If there is a possible lurking variable, it does
  not mean the data are bad but that you should
  reconsider how you report and interpret your
  results.

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• For each of the following, determine a possible
  lurking variable as well as the effect it is
  having on the relationship.
• People who use artificial sweeteners in place of
  sugar tend to be heavier than people who use
  sugar. You conclude that increased use of
  sweeteners will increase your weight.
• A study showed that women who worked in a
  manufacturing plant have an increased number of
  miscarriages. The union concludes that there are
  unsafe work conditions.
• A study finds that students who take algebra and
  geometry in high school are more successful in
  college. The researcher concludes that taking
  more math classes will increase your college
  grades.

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• Using averaged data
• If a study uses the average of many individuals
  as one data pt. you must be cautious of making
  predictions for individuals.
• Ex. If we plot age vs. the average height of
  children, then we would expect to see a very
  strong positive correlation.
• It doesnt make sense to use this relationship to
  predict the height of one individual child.
• If we looked at each individual data point, you
  would see much more scatter and variability.
• What effect would this have on r?

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Causation

• Think about this A researcher randomly selects
  20 children and finds the height of their
  parents. He performs a linear regression of
  mothers height vs. fathers height and finds a
  strong positive correlation. Does he conclude
  that one variable caused or explains the other?
• What are some possible factors that might help
  explain these results?

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I. Causation

• When two variables have a direct cause and effect
  link.
• Ex. Mothers Body Mass Index vs. Daughters Body
  Mass Index.
• Heredity is the causal link
• Note Even when direct causation is present, it
  is rarely a complete explanation of the
  association between two variables.

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• Ex. An experiment on the amount of saccharin in a
  rats diet vs. the number of tumors in a rats
  bladder has shown a causal relationship
• Note We cannot generalize a causal relationship
  to other settings than the one observed!
• Best way to show evidence of causation is
  from experiments that actually change one
  variable while keeping all others fixed

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II. Common Response

• Observed association between two variables x and
  y is actually explained by a lurking variable z.
• Could be any number of lurking variables, often
  use common sense to come up with another
  explanation.

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• Common Response can create an association where
  one may not exist.
• Ex. x SAT score
• y Fresh. Yr. GPA
• There would appear to be a positive association,
  but what is a lurking variable that could be
  having an effect on this relationship?

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• Ex. x of children
• y income
• Possible lurking variables?
• Ex. x of years of education
• y of books in a personal library
• Possible lurking variables??

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III. Confounding

• Two variables are confounded when their effects
  on a response variable cannot be distinguished
  from each other. The confounded variables may be
  either explanatory or lurking variables.
• What are some possible confounding variables in
  the mother vs. daughter BMI case?

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• Ex. What might a confounding variable be in each
  case?
• x whether a person regularly attends
  religious services
• y lifespan
• Confounding variable
• 2. x number of pieces of fruit eaten regularly
• y number of cavities
• Confounding variable

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IV. Establishing Causation

• Conduct a carefully designed experiment in which
  the effects of possible lurking variables are
  controlled. (Chapter 5)
• Social/Political Issues often hard to establish
  causal relationship since an experiment is not
  possible from an ethical standpoint.

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• Ex. Power lines and Leukemia there is no way
  to conduct any experiment. But, through research
  and case studies there has only been a chance
  connection shown.
• Ex. Smoking vs. Lung cancer certain benchmarks
  have been met to infer causation.
• Association is strong
• Association is consistent in multiple countries
  and at different times
• Higher doses stronger response
• Cause precedes the effect in time
• Cause is plausible