Correlation and Regression Wisdom

1
Lesson 3 - 3

• Correlation and Regression Wisdom

2
Knowledge Objectives

• Recall the three limitations on the use of
  correlation and regression.
• Explain what is meant by an outlier in bivariate
  data.
• Explain what is meant by an influential
  observation and how it relates to regression.
• Define a lurking variable.
• Give an example of what it means to say
  association does not imply causation.

3
Construction Objectives

• Given a scatterplot in a regression setting,
  identify outliers and influential observations
• Explain how correlations based on averages differ
  from correlations based on individuals

4
Vocabulary

• Influential Observation an observation that if
  removed would markedly change the result of the
  regression calculation

5
Limitations

• Correlation and regression describe only linear
  relationships
• Extrapolation (using model outside range of the
  data) often produces unreliable predications
• Correlation is not resistant (to outliers!)

6
Outliers vs Influential Observation

• Outlier is an observation that lies outside the
  overall pattern of the other observations
• Outliers in the Y direction will have large
  residuals. but may not influence the slope of the
  regression line
• Outliers in the X direction are often influential
  observations
• Influential observation is one that if by
  removing it, it would markedly change the result
  of the regression calculation

7
Example 1

• Does the age at which a child begins to talk
  predict later score on a test of metal ability?
  A study of the development of 21 children
  recorded the age in months at which they spoke
  their first word and their later Gesell Adaptive
  Score (GAS).

Child Age GAS Child Age GAS Child Age GAS
1 15 95 8 11 100 15 11 102
2 26 71 9 8 104 16 10 100
3 10 83 10 20 94 17 12 105
4 9 91 11 7 113 18 42 57
5 15 102 12 9 96 19 17 121
6 20 87 13 10 83 20 11 86
7 18 93 14 11 84 21 10 100
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Example 1 cont

1. What is the equation of the LS regression line
   used to model this data?
2. What is the interpretation of this data?

y-hat 109.8738 1.127x r -0.64
The scatter plot and the slope of the regression
line indicates a negative association. Children
who begin to speak later tend to have lower test
scores than early talkers. The slope suggests
that for every month older a child is when they
begin to speak, their score on the Gesell test
will decrease by about 1.13 points. The
y-intercept has no real meaning in this case.
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Example 1 cont

1. Are there any outliers?
2. Are there any influential observations?

Child 19 is an outlier in the Y-direction and
child 18 is an outlier in the X-direction.
Child 18 is an outlier in the X-direction and
also an influential observation because it has a
strong influence on the positioning of the
regression line.
10
Example 1 cont
Scatterplot w/ Regression Line
Residual Plot
11
Lurking or Extraneous Variable

• The relationship between two variables can often
  be misunderstood unless you take other variables
  into account
• Association does not imply causation!
• Instances of Rocky Mt spotted fever and drownings
  reported per month are highly correlated, but
  completely without causation

12
Remember Sampling Distributions

• When we looked at individual values, they had
  much broader spreads (variances) than when we
  looked at the distributions of x-bar
• Same is true with correlations based on averaged
  data strong correlations may exist between
  averages, but individuals will have much greater
  variances
• Correlations based on averages are usually too
  high when applied to individuals.

13
Summary and Homework

• Summary
• Correlation and regression must be interpreted
  with caution
• Plot data to be sure that the relationship is
  roughly linear and to detect outliers
• Check for influential observations that
  substantially change the regression line
• Lurking variables may explain the relationship
  between the explanatory and response variables
• Homework
• pg 242-3 3.63-67