ScatterPlots, Correlation, Regression

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ScatterPlots, Correlation, Regression
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• Annual wine consumption (liters of alcohol from
  drinking wine/person) and deaths/100,000

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More Examples

• Whenever any reasonable attempt can be made of
  using one variable (characteristic) to explain
  another variable (characteristic) of the
  subjects, the first variable is called
  explanatory and the latter is called response
  variable.
• However this does NOT imply the explanatory
  variable is the actual cause of the response
  (variable) even though they may have a strong
  relationship.
• IQ scores and school grades (strong relationship,
  not cause).
• Family income level and education level (either
  could be explanatory, the other is response
  variable).
• Someones height and grades (not reasonable?).
• Husbands age and wifes age (strong
  relationship, not cause).

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More Examples

• In the study of the effect of alcohol on the
  change of body temperature.
• Intake of carbohydrate and weight.
• Overweight and mortality rate.
• study on Antibiotics and breast cancer.
• Heights of parents and heights of children.
• Explanatory or response variables dont have to
  be quantitative. But often they are quantitative.

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More Examples

• Observed meditation practice and age-related
  enzyme level (general concern for ones well
  being may also be affecting the response (and the
  decision to try meditation)Noetic Sciences
  Review, Summer 1993, page 28
• Lack of social relationship and frequency of
  illness (other factor that predisposes people
  both to have lower social activity and become
  ill?) Science, Vol. 241 (1988), pp 540-545

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ScatterPlots

• If we have two quantitative variables
  (characteristics) of subjects, we plot one
  variable against the other. The result is called
  ScatterPlot (graph representation of the two
  quantitative variables).
• If one of the two quantitative variables is
  explanatory, the other is response variable, the
  explanatory variable is on horizontal axis, the
  response variable on the vertical axis.

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How to draw a scatterplot an example
Miles/gal
A cars speed mile/hr and Miles/gal at different
times
Speed
Horizontal Explanatory Vertical Response
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One of the reasons we care about overall pattern
is that we want to make predictions! (Example
Previous slide)
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If the overall pattern of the scatterplot
approaches a straight line, the two variables are
linearly associated otherwise they are not
linearly associated. The strength of the
association, whether linear or non-linear, can be
roughly described and compared. Linear is the
simplest.
A relationship is strong if the points lie close
to the overall pattern curve, and weak if they
are widely scattered about the overall pattern
curve
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More Examples of Relationships
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Introducing categorical variable in Scatterplots
South Rising Page 82
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Linear AssociationCorrelation

• Correlation is a number that measures the
  direction and strength of linear association.
  Notation r.
• The number r is always between 1 and 1.
• If r is negative, the two quantitative variables
  are negatively associated otherwise, positively
  associated.
• The association is strong if r is close to 1 or
  1 (or r is close to 1) Weak if r is close to
  0.
• Only for linear association. In nonlinear
  situation, even if there is a strong association,
  the correlation r can be 0. Studying linear
  association simply because its simple. r doesnt
  depend on the units used.

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Correlation Calculation

• Suppose we have data on variables X and Y for n
  individuals
• x1, x2, , xn and y1, y2, , yn
• Each variable has a mean and std dev

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Correlation 0 Nonlinear Relationship
Miles/gallon Versus Speed

• Linear relationship?
• Correlation is close to zero.
• Strong association!

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Correlation 0 Nonlinear Relationship
Miles/gallon Versus Speed

• Curved (nonlinear) relationship.

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A Simple Example Some students midterm grades
and their final grades
Its a linear relationship!!!
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A Simple Example Some students midterm grades
and their final grades
Sum 85
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Which Relation Is Highly Correlated?

• Husbands versus wifes ages.
• Husbands versus wifes heights.
• Professional golfers putting success distance
  of putt in feet versus percent success.

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Problems With Correlations

• Outliers can inflate or deflate correlations (see
  next slide)
• Groups combined inappropriately may mask
  relationships (a third variable)
• groups may have different relationships when
  separated

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Outliers and Correlation
A
B
For each scatterplot above, how does the outlier
affect the correlation?
A outlier decreases the correlation (weaken
the trend) B outlier increases the
correlation (strengthen the trend) Example 2004
Election Kerry-Edwards
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Linear Regression and Prediction

• If two quantitative variables are linearly
  associated (the overall pattern is close to a
  straight line), what line is the best line to
  describe the data? (want to do predictions!).
• The best line is called the least-squares
  regression line. This line helps the prediction
  of the value of response variable (y) for a given
  value of explanatory variable (x).
• The slope of the regression line and correlation
  r always have the same sign.

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Least-squares Regression Line

• Regression equation y bx a

• x is the value of the explanatory variable
• y-hat is the average value of the response
  variable (predicted response for a value of x)
• a is the intercept
• b is the slope of the straight line
• note that r and b are not the same thing, but
  their signs will agree

sx and sy are the standard deviations of the two
variables, and r is their correlation.
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Prediction Via Regression Line Number of New
Birds and Percent Returning

• The regression equation is y-hat
  31.9343 ? 0.3040x
• y-hat is the average number of new birds for all
  colonies with percent x returning
• For all colonies with 60 returning, we predict
  the average number of new birds to be 13.69
• 31.9343 ? (0.3040)(60) 13.69 birds
• Suppose we know that an individual colony has 60
  returning. What would we predict the number of
  new birds to be for just that colony?

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Midterm Grades and Final Grades
The slope of the regression line is
The y-intercept is
The regression line is
The predicted final grade for Peter (if his
midterm is 70)?
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Residuals

• A residual is the difference between an observed
  value of the response variable and the value
  predicted by the regression line
• residual y ? y

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Case Study Residuals
Gesell Adaptive Score and Age at First Word
Draper, N. R. and John, J. A. Influential
observations and outliers in regression,
Technometrics, Vol. 23 (1981), pp. 21-26.
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CautionsAbout Correlation and Regression

• only describe linear relationships
• are both affected by outliers
• always plot the data before interpreting
• beware of extrapolation
• predicting outside of the range of x
• beware of lurking variables
• have important effect on the relationship among
  the variables in a study, but are not included in
  the study
• association does not imply causation

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Excel Instructions

• Download/open excel file GDPnLifeE.xls
• ScatterPlot only Select Data Block (B3C12) ?
  Insert ? Chart ? XY (Scatter) ? Choose Scatter
  subtype ? Next ? Series in Column ? Next ? Change
  Title etc.? Next and Finish ? Enhancements
• Correlation only Enter correl(Xarray,Yarray)
  (correl(B3B12,C3C12))
• Scatter and Regression Line Tools ? Data
  Analysis ? Regression ? Next ? Enter Y data
  (C3C12) ? Enter X data (B3B12) ? Enter output
  position ? Check Line Fit Plot (or residual
  plots) ? Ok ? Cursor on predicted value ? right
  click ? add Trendline (no trendline for residual
  plots) ? Enhancements