Association: Quantitative Variables

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Unit 7

• Association Quantitative Variables

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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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Scatterpolts

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

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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 and 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
• The gas mileage of an automobile and the weight
  of the automobile
• The price of a theatre ticket and the number of
  ticket sales
• The age at marriage of the women and the age at
  marriage of the man

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What to Look for in a Scatterplot

• Linear or straight line trends
• Non-linear or curved trends
• No obvious relationship
• Clusters
• Outliers

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Linear Relationships

• A scatterplot of systolic blood pressure and age
  for 29 subjects. Notice the points are not
  exactly a straight line fit, but approximately a
  straight line fit.

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Non-Linear Relationships

• Monthly temperatures in Raleigh NC.

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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 as a strong, weak or no association
• then if either strong or weak
• Classify as either a positive or negative
  association

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Classify Graph 1
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Classify Graph 2
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Classify Graph 3
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Classify Graph 4
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Classify Graph 5
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Clusters

• The next interactive examines the concept of data
  clusters
• Unit A-7, Uses, Uses 1

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Quantifying the Strength of a Quantitative
Association

• We have seen that associations can be
• Strong, weak or non-existent
• Positive or negative
• Linear or non-linear
• Now we want to 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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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 MINITAB provide the
  correlation coefficient as one of their options
  when doing bivariate analysis

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An Example

• Car Fuel Efficiency

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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!
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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!