Association Between Two Variables

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Chapter 3

• Association Between Two Variables

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True Interest of Statistics

• Examine relationship between two or more
  variables.
• Does an association exist? Does the value of one
  variable relate to an other?

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Terms

• Dependent (or response) variable
• Independent (or explanatory) variable

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I) Two Categorical Variables

• Describe relationship using Contingency Table
• Each cell records the number of observations
  meeting criteria set by variables.

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Example Contingency Table

• Question Are pesticides less likely in organic
  food (p. 95)?
• 1) Identify explanatory and response variables.
• 2) Make a Table
• What do cell values represent?
• 3) Sum totals of each explanatory variable. WHY?

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Example Contingency Table

• 4) Conditional upon being a certain type of
  produce (Type X), what is the chance of finding
  pesticides?
• 5) Conclude

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Excel Example Contingency Table
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Other Examples of Two Categorical Variables

• Condition of Home vs. Home Ownership
• Gender vs. Smoking
• Others?

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Identifying Explanatory and Response Variables

• THINK of cause and effect Response variable is
  the effect
• Direction of arrow
• Income Political ID
• Greenhouse Gas Global Temperature

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But

• CORRELATION (ASSOCIATION) DOES NOT IMPLY
  CAUSATION.
• Reverse Causality (Feedback Loop)
• Omitted Variables
• This course Association/Correlation
• Advanced course Causation
• Be careful about language
• Side note Time component helps est. causation

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II) Comparing a Categorical and a Quantitative
Variable

• Easier to analyze if the categorical variable is
  the explanatory variable.
• Conditional upon being in group X, what are the
  characteristics of Y?
• Gender and height
• Race and income

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Options for Comparing

• 1) Compare summary statistics for two groups.

• 2) Recode categorical data as zeros and ones,
  then draw a scatterplot.

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Excel Example Ch3-1.xls

• 1) Sort data
• 2) Create table
• 3) Since womens heights are in Column B, Rows
  2-263, type average(B2B263) in the table.
  For men, type average(B264 B382).

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Excel Example Ch3-1.xls

• 4) Standard deviation (of a sample) is
  stdev(B2B263). To find this formula
• Excel 2007 Formulas ? More Functions ?
  Statistical ? stdev
• ALL Excel Go to Function Wizard ? More Functions
  ? Standard Deviation, then select one of the
  possibilities.
• 5) Coefficient of Variation?
• Advantage of CV over s?

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Excel Example Ch3-1.xls

• 6) Scatterplot
• i) Type if(A2Female,1,0) in Female
  Column. Why?
• ii) Double click crosshair on lower-right corner
  of the cell. Why?
• iii) Make Chart
• Excel 2007 Highlight Female and Height Data, Go
  to Insert ? Chart ? Scatter
• Old Excel Chart Wizard ? XY(Scatter) ? Finish
• What is true of all observations at X 0 ?
• What can we learn from this chart?

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III) Comparing Two Quantitative Variables

• Examples?

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Options for Comparing

• Correlation, r
• A single number (numerical summary) describing
  the strength of a linear relationship between X
  and Y.
• Range of values
• Strong Positive Relationship r
• Strong Negative Relationship r
• Insensitive to which of the two variables is X
  and which is Y.
• Unit-Free (Insensitive to units of measurement)

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Options for Comparing

• Correlation, r
• Where ZX deviations from the mean

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Options for Comparing

• Correlation, r
• Correlation does not imply causation
• Reverse Causality
• Omitted (Lurking) Variables
• Scatterplot
• Does a relationship exist?
• Is the relationship linear?
• Is the relationship positive or negative?

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Options for Comparing

• Scatterplot
• Example

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Options for Comparing

• Scatterplot
• Example r

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Options for Comparing

• Scatterplot
• Example r

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Options for Comparing

• Scatterplot
• Example r

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Options for Comparing

• Scatterplot
• Example Most of the time, when X is above its
  mean, Y is above its mean, so ZX ZY gt 0 most of
  the time.
• This implies that r gt 0
• See Correlation by Eye applet for more
  correlation examples.

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Options for Comparing

• REGRESSION (Line of Best Fit)
• Suppose we believe a linear relationship between
  X and Y exists such that
• Y a bX error
• We can develop statistics (a b) to estimate
  population parameters (a b) that predict Y
  values given particular X values.

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Options for Comparing

• REGRESSION (Line of Best Fit)

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Excel Example Ch3-3.xls

• 3rd Party Votes in Florida, 1996 vs. 2000.
• What is X What is Y?
• Why is this interesting?
• 1) Correlation
• i) correl(B2B68, C2C68)
• ii) Tools ? Data Analysis ? Correlation

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Excel Example Ch3-3.xls

• 2) Scatterplot
• i) Highlight columns
• ii) Choose Insert ? Chart ? Scatter (or Chart
  Wizard ? XY(Scatter))
• 3) Regression / Line of Best Fit (all XLS
  versions)
• i) Right-Click scatterplot data
• ii) Choose Add Trendline, Select Display
  equation on chart option.
• iii) Interpretation of coefficients?
• iv) Much more on this later

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Regression Intuition

• We observe Y, but develop statistics to predict Y
  for given X values.
• The difference (Y Y) is called a residual
• A regression line of best fit minimizes the sum
  of squared residuals. That is, it makes the
  errors associated with our predictions as small
  as possible.

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Regression Intuition

• What do residuals look like?
• Least squares regressions find a b values that
  minimize the sum of squared residuals.

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Regression Notes

• Size of b says nothing about the strength of the
  statistical relationship. If you change the units
  of X, you will change the size of b, but not r.
• Although b r (SY/SX), you can have r 1 but
  still find b to be meaningless.
• A regression of Annual Wages on Number of
  Children for men earning 20,000 to 40,000
  reveals
• Wage 29600 178 Children
• What does this mean?

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Regression Potential Problems

• Extrapolation
• Previous example What about men who earn more
  than 100,000 per year?
• Time Series ESPNs Sports Figures episode
  starring Marion Jones. Let the Y-variable
  represent Ms. Joness fastest time in the 100m
  Dash. Let X Years Since 1989, the year she
  started competing. What will happen 120 years
  after her first race?

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Regression Potential Problems

• Correlation does not imply causation.
• Reverse Causality
• Omitted Variables
• Book Example A study found that, over a three
  year period, the proportion of smokers who died
  was less than the proportion of non-smokers who
  died. Is smoking good for you?