BIVARIATE DATA

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Chapter 5

• BIVARIATE DATA

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BIVARIATE DATA

• The study designs considered so far investigated
  only one characteristic of a population. They are
  single variable studies.
• Many study designs aim to look for an association
  between two quantitative variables measured on
  the same subject. These are bivariate study
  designs. The prefix bi means two.

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Scatterplots

• Scatterplots are the most useful graphical device
  to examine the possible association between two
  quantitative variables.
• Scatterplots help us identify
• Trends
• Outliers

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Linear Relationship

• A scatterplot of systolic blood pressure and age
  for 29 subjects.

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Constructing a Scatterplot

• A scatterplot is a two dimension display
• The data for one variable is plotted on the
  horizontal axis and the data for the other
  variable is plotted on the vertical axis.
• The convention is to place the studied variable
  (response variable) on the vertical (y-axis). The
  variable that is used to do the predicting (the
  explanatory variable) is placed on the horizontal
  (x-axis).

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Response Explanatory Variables

• Some studies are conducted to predict the value
  of one variable using the value of anther
  associated variables. In these studies we can
  identify response and explanatory variables.
• Others are conducted to simply look for potential
  associations between two bivariate variables. In
  these studies the choice of response and
  explanatory variables is arbitrary.

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Examples

• Decide which variable is the response and which
  is the explanatory variable
• Serving size of an ice cream cone and the
  calories of the ice cream cone
• Explanatorysize Response calories
• The gas mileage of an automobile and the weight
  of the automobile
• Explanatory weight Response mpg
• The price of a theatre ticket and the number of
  ticket sales
• Explanatory price Response sold
• The age at marriage of the women and the age at
  marriage of the man
• No clear choice just ? association

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Linear Relationship

• A scatterplot of systolic blood pressure and age
  for 29 subjects.
• Points rise to the right
• Positive association

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Example Car Fuel Efficiency
Fuel_efficiency.xls
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Scatterplot for Fuel Efficiency

• Points fall to the right Negative association

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Positive and Negative Trends

• Two variables are said to have a positive (or
  direct) association if larger values of one
  variable occur with larger values of the other
  variable
• They are said to have a negative (or inverse)
  association is smaller values of one variable
  occur with larger values of the other variable

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Examples

• Classify each association in the following slides
  as a strong, weak or no association
• If there is an association go onto classify
  the association as either a positive or negative
  association

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Classify Graph 1

• Moderate, positive association

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Classify Graph 2

• Moderate, negative association

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Classify Graph 3

• Strong, positive association

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Classify Graph 4

• Strong, negative association

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Classify Graph 5

• No association

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Non-linear Relationship

• Monthly temperatures in Raleigh, N.C.

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Measuring the Strength of a Linear Relationship

• We have seen that associations can be
• Strong, weak or non-existent
• Positive or negative
• Linear or non-linear
• Now we want to numerically measure the
  strength of linear relationships

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Correlation Coefficient

• The correlation coefficient, r, measures the
  strength of a linear association
• It is always the case that
• If r 1 then there is perfect positive
  association
• If r -1 then there is perfect negative
  association
• We only quote a correlation coefficient if
  there is a linear relationship!

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Correlation Guessing Game

• http//www.stat.uiuc.edu/courses/stat100/java/GCAp
  plet/GCAppletFrame.html

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Computational Formula

• The computational formula for the sample
  correlation coefficient is

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Good News

• Graphical and statistical calculators as well as
  programs like Excel and StatCrunch provide the
  correlation coefficient as one of their options
  when doing bivariate analysis

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Example Car Fuel Efficiency
Fuel_efficiency.xls
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Scatterplot for Fuel Efficiency

• Describe the association and estimate r

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Results
r -0.816
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Association Does Not Mean Causation!
Life_expectancy.xls
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Results
r -0.789
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Question

• The Number of TVs per Person has a strong
  negative correlation with Life Expectancy
• Does this mean buying more TVs will increase
  life expectancy?
• Association does not mean Causation!

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Linear Models

• The correlation coefficient measures the strength
  of the linear relationship between two
  quantitative variables x and y.
• A linear equation describing how an dependant
  variable, y, is associated with an explanatory
  variable, x, looks like
• y b mx

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Example

• A college charges a basic fee of 100 a semester
  for a meal plan plus 2 a meal. The linear
  equation describing the association between the
  cost of the meal plan, y, and the number of meals
  purchased, x, is
• y 100 2x

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Linear Equations

• A linear equation takes the form
• y b mx
• m slope
• b y-intercept
• The slope measures the rate of change of y
  with respect to x
• The y-intercept measures the initial value of y
  (value of y when x 0)

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Linear Modeling

• Rarely does an exact linear relationship exist
  between two studied variables.
• The correlation coefficient and the scatter plot
  help us decide if there is a reasonably strong
  linear relationship between two studied variables.

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Airfare from Baltimore, MD (1995 data)
Airfare.xls
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What is the Line of Best Fit?

• What properties do we want the line we fit to the
  data to have?
• It should be as close as possible to the data
• To decide if one line is better than another we
  could measure how far data values are from the
  line
• We will use the symbol, , to designate a
  predicted y-value

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Scatter Plot
Y 278
.
Residual for Dallas
The airfare for Dallas is 52 higher than
predicted
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Residuals

• To find a residual we take
• observed y-value and subtract the predicted
• y-value
• Positive residuals imply the observed value is
  higher than predicted
• Negative residuals imply the observed value is
  less than predicted

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Minimize the Residuals

• It would seem like a good criterion would be to
  have the line of best fit be one that the
  residuals.
• Since residuals can be both positive and negative
  we dont want to just add them up . Cancellation
  of positive and negative residuals would occur.

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Remember the Standard Deviation?

• Remember how when we defined the standard
  deviation we squared the difference between a
  data value and the mean to create a non-negative
  quantity?

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Do it Again

• We apply the same process to create positive
  residuals . We square them
• Then we look for the line than minimizes the sum
  of these squared residuals

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How it Works

• Minimize
• Its a good plan but you need calculus to carry
  it out.
• The end results are actually quite easy. All
  thats important is that you realize the formulas
  that will be presented minimize the residuals

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The Formulas

• The methods of calculus can be used to find
  equations for the slope and y-intercept of the
  least squares line. Here are the results.

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Least Squares Line for Airfare Data

• Distance (x)
• Airfare (y)
• r .795

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Prediction Equation

• Airfare 83.53 (.117)distance
• Each additional 100 miles costs an additional
  11.70

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A Few Residuals
Pittsburgh Distance from Baltimore 210
miles Airfare 138 Predicted airfare .117(210)
83.53 107.92 Residual 138 107.92
30.08 The airfare from Pittsburgh to Baltimore
is 30.08 more than you would expect based on
the distance between these cities
St. Lois Distance from Baltimore 737
miles Airfare 98 Predicted airfare .117(737)
83.53 169.77 Residual 98 169.77 -
71.77 The airfare from St. Lois to Baltimore is
71.77 less than you would expect based on the
distance between these cities
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Coefficient of Determination

• The coefficient of determination is the
  correlation coefficient squared
• It can help us determine how good the least
  squares line is as a prediction equation
• It is the fractional amount of total variation in
  y that can be explained by the linear
  relationship with x

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Airfare Data

• r .795
• r2 .63
• 63 of the variability in airfare cost can be
  explained by a linear relationship with distance
• 37 of the variability in airfare cost is due to
  factors other than distance

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Age Systolic Blood Pressure for 30 Adults
SBP.xls
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Approximate Positive Linear Relationship
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Equation of Fitted Line
SBP 98.7 0.97(AGE)y 98.7 0.97 x
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Interpretation of the Slope

• The slope of the SBP vs Age fitted equation is
  0.97
• 0.97 rate of change of SBP with respect to age
• Every year a subjects blood pressure rises
  approximately 0.97 units.

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Analysis of Variability

• r .657 and r2 .43
• 43 of the variability in SBP can be explained by
  a linear relationship with age
• 57 of the variability in SBP is due to factors
  other than age

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Interpreting Residuals
Subject 2 Age 47 SBP 220 Predicted SBP
98.71 .97(47) 144.35 Residual 220 144.35
75.65 This subjects SBP is 75.65 units higher
than would be expected for his age
Subject 23 Age 39 SBP 120 Predicted SBP
98.71 .97(39) 136.58 Residual 120 136.58
- 16.58 This subjects SBP is 16.58 units lower
than would be expected for his age
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More Good News

• Many computer programs including Excel and
  StatCrunch as well as graphing calculators
  provide the slope and
• y-intercept of the least squares line

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Only selected items are relevant
Regression equation
Correlation coefficient
Coefficient of determination
Not relevant
Not relevant
Look for these in the data table
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Notice, the dialog box that presents the summary
statistics has a Next button. Press next to get
the scatterplot
Look back at the data table and notice
the predicted (fitted) and residual values
are now included