Ch 2 and 9.1 Relationships Between 2 Variables

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Ch 2 and 9.1Relationships Between 2 Variables

• More than one variable can be measured on each
  individual.
• Examples
• Gender and Height
• Size and Cost
• Eye color and Major
• We want to look at the relationship among these
  variables.
• Is there an association between these two
  variables?
• Two variables measured on the same individuals
  are associated if some values tend to occur more
  often with some values of the second variable
  than with other values of that variable.

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Relationships Between 2 Variables

• If we expect one variable to influence another,
  we call it the ___________ variable.
• Explains or influences changes in the response
  variable
• The variable that is influenced is called the
  ____________ variable.
• Measures an outcome of a study
• In each of the following examples, identify the
  explanatory and response variables
• Gender and blood pressure
• Class attendance and course grade
• Number of beers and BAC

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Relationships Between 2 Variables

• We may be interested in relationships of
  different types of variables.
• Categorical and Numeric
• Categorical and Categorical
• Numeric and Numeric

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Relationships between Categorical and Numeric
Variables

• We are interested in comparing the numerical
  variable across each of the levels of the
  categorical variable.
• Examples
• Compare high speeds for 4 different car brands
• Compare sucrose levels for 5 different types of
  fruit
• Compare GPR for 20 different majors

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Relationships between Categorical and Numeric
Variables

• Graphical Comparison
• Example Sucrose levels of fruits (fictitious
  data)

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Relationships between Categorical and Numeric
Variables

• Numerical Comparison
• We could also look at summary statistics for each
  group.

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Ch 9.1Relationships Between Two Categorical
Variables

• Depending on the situation, one of the variables
  is the explanatory variable and the other is the
  response variable.
• In this case, we look at the percentages of one
  variable for each level of the other variable.
• Examples
• Gender and Soda Preference
• Country of Origin and Marital Status
• Smoking Habits and Socioeconomic Status

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Two-Way Tables

• Two-way tables come about when we are interested
  in the relationship between two categorical
  variables.
• One of the variables is the _____________.
• The other is the _______________.
• The combination of a row variable and a column
  variable is a ______________.

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Two-Way Tables

• Example

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Relationships between two categorical variables

• Example Gender and Highest Degree Obtained
• Joint Distribution How likely are you to have a
  bachelors degree and be a male? _____________
• Marginal Distribution What is the least likely
  highest degree obtained? _____________
• Conditional Distribution If you are a female,
  how likely are you to have obtained a graduate
  degree? ______________

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Relationships between two categorical variables

• Shows the percentages
• for the joint, marginal,
• and conditional distributions.

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Ch 2 Relationships Between 2 Numeric Variables

• Depending on the situation, one of the variables
  is the explanatory variable and the other is the
  response variable.
• There is not always an explanatory-response
  relationship.
• Examples
• Height and Weight
• Income and Age
• SAT scores on math exam and on verbal exam
• Amount of time spent studying for an exam and
  exam score

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Relationships between 2 numeric variables

• Scatterplots
• Look for overall pattern and any striking
  deviations from that pattern.
• Look for outliers, values falling outside the
  overall pattern of the relationship
• You can describe the overall pattern of a
  scatterplot by the form, direction, and strength
  of the relationship.
• Form Linear or clusters
• Direction
• Two variables are _____________________ when
  above-average values of one tend to accompany
  above-average values of the other and likewise
  below-average values also tend to occur together.
• Two variables are _____________________ when
  above-average values of one variable accompany
  below-average values of the other variable, and
  vice-versa.
• Strength-how close the points lie to a line

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Relationships between 2 numeric variables

• Example
• Response MPG
• Explanatory Weight

Response Variable (y-axis)
Explanatory Variable (x-axis)
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Relationships between 2 numeric variables

• Relationships between two numeric variables
• Example
• Vehicle Weight
• Horsepower

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Relationships between 2 numeric variables

• ___________ or r measures the direction and
  strength of the linear relationship between two
  numeric variables
• General Properties
• It must be between -1 and 1, or (-1 r 1).
• If r is negative, the relationship is negative.
• If r 1, there is a perfect negative linear
  relationship (extreme case).
• If r is positive, the relationship is positive.
• If r 1, there is a perfect positive linear
  relationship (extreme case).
• If r is 0, there is no linear relationship.
• r measures the strength of the linear
  relationship.
• If explanatory and response are switched, r
  remains the same.
• r has no units of measurement associated with it
• Scale changes do not affect r

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Relationships between 2 numeric variables

• Examples of extreme cases

r 1
r 0
r -1
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Relationships between 2 numeric variables

• Match the correlation with to the scatterplot

r 0.04
r 0.43
r -0.84
r 0.76
r 0.21
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Relationships between 2 numeric variables

• It is possible for there to be a strong
  relationship between two variables and still have
  r 0.
• EX.

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Relationships between 2 numeric variables

• Important notes
• Association does not imply causation
• Correlation does not imply causation
• Slope is not correlation
• A scale change does not change the correlation.
• Correlation doesnt measure the strength of a
  non-linear relationship

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Regression Line

• A regression line is a straight line that
  describes how a response variable y changes as an
  explanatory variable x changes.
• A regression line summarizes the relationship
  between two variables, but only in a specific
  setting when one of the variables helps explain
  or predict the other.
• We often use a regression line to predict the
  value of y for a given value of x.
• Regression, unlike correlation, requires that we
  have an explanatory variable and a response
  variable

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Regression Line

• Fitting a line to data means drawing a line that
  comes as close as possible to the points.
• Extrapolation-the use of a regression line for
  prediction far outside the range of values of the
  explanatory variable x that you used to obtain
  the line.
• Such predictions are often not accurate.

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Least-Squares Regression Line

• The least-squares regression line of y on x is
  the line that makes the sum of squares of the
  vertical distances of the data points from the
  line as small as possible.
• These vertical distances are called the
  residuals, or the error in prediction, because
  they measure how far the point is from the line
  
• where y is the point and
  is the predicted point.

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Least-Squares Regression Line

• The equation of the least-squares regression line
  of y on x is

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Least-Squares Regression Line

• The expression for slope, b1, says that along the
  regression line, a change of one standard
  deviation in x corresponds to a change of r
  standard deviations in y.
• The slope, b1, is the amount by which y changes
  when x increases by one unit.
• The intercept, b0, is the value of y when
• The least-squares regression line ALWAYS passes
  through the point

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r2 in Regression

• The square of the correlation, r2, is the
  fraction of the variation in the values of y that
  is explained by the least-squares regression of y
  on x.
• Use r2 as a measure of how successfully the
  regression explains the response.
• Interpret r2 as the percent of variation
  explained
• For Simple Linear Regression, r2 is simply the
  square of the correlation coefficient.

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Relationships between 2 numeric variables

• Example
• How much of the variation is explained
• by the least squares line of y on x? ______
• What is the correlation coefficient? ______

Horsepower -10.78 0.04weight (Equation of
the line.)
__________ y-value or response (horsepower) when
line crosses the y-axis.
_______ increase in response for a unit increase
in explanatory variable.
So if weight increases by one pound, horsepower
increases by 0.04 units (on average).
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Relationships between 2 variables

• Lurking Variable A variable that is not among
  the explanatory or response variables in a study
  and yet may influence the interpretation of
  relationships among those variables.
• Simpsons Paradox An association or comparison
  that holds for all of several groups can reverse
  direction when the data are combined to form a
  single group. This reversal is called Simpsons
  Paradox. This can happen when a lurking variable
  is present. Please see Examples 9.9 and 9.10 in
  the text.

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Outliers and Influential Observations in
Regression

• An outlier is an observation that lies outside
  the overall pattern of the other observations.
• An observation is influential for a statistical
  calculation if removing it would markedly change
  the result of the calculation.
• Points that are outliers in the x direction of a
  scatterplot are often influential for the
  least-squares regression line.

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Outliers and Influential Observations in
Regression
Child 18 is an outlier in the x direction.
Because of its extreme position on the age scale,
this point has a strong influence on the position
of the regression line.
r2 is also affected by the influential
observation. With Child 18, r2 41, but without
Child 18, r2 11. The apparent strength of the
association was largely due to a single
influential observation.
The dashed line was calculated leaving out Child
18. The solid line is with Child 18.