Chapter 2: Looking at Data Relationships Section 9'1: Data Analysis for TwoWay Tables

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Chapter 2 Looking at Data RelationshipsSectio
n 9.1 Data Analysis for Two-Way Tables
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Relationships Between 2 Variables

• More than one variable can be measured on each
  individual.
• Examples
• Gender and Height
• Eye color and Major
• Size and Cost
• We want to look at the relationship among these
  variables.

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Relationships Between 2 Variables

• A response variable measures an outcome of a
  study. An explanatory variable explains or causes
  changes in the response variables.
• The explanatory variable influences the response
  variable.
• Examples
• Attendance for class and the grade of STAT 303
• Gender and Height
• Smoking and lung cancer

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Relationships Between 2 Variables

• We may be interested in relationships of
  different types of variables.
• Categorical and Numeric
• E.g. Gender and Height
• Categorical and Categorical
• E.g. Eye color and Major
• Numeric and Numeric
• E.g. Size and Cost

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1. 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 prices for no-airbag, one-airbag and
  two-airbag cars
• Compare GPR for 20 different majors

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1. Relationships Between Categorical and Numeric
Variables

• We could look at summary statistics for each
  group.
• Example prices for no airbags, one airbag and
  two airbag cars.
• Explanatory airbag
• Response price

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1. Relationships Between Categorical and Numeric
Variables

• Graphical Comparison

Side by Side Boxplots
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1. Relationships Between Categorical and Numeric
Variables

• Associations A categorical and numeric variable
  are associated if the distribution of the numeric
  variable is not the same for all populations.
  (The populations are defined by the values the
  categorical explanatory variable takes on.)
• Example of no association

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2. Relationships Between Two Categorical Variables

• Depending on the situation, one of the variables
  is the explanatory variable and the other is the
  response variable.
• Examples
• Gender and Tomatoes Preference
• Country of Origin and Marital Status
• Gender and Highest Degree Obtained
• Compare percentages in each level of one
  categorical variable across the levels of the
  other categorical variable

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Two-Way Tables

• A two-way (contingency) table can summarize the
  data for relationships between two categorical
  variables.
• - Example Response Tomatoes, Explanatory
  Gender

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2. Relationships Between Two Categorical Variables

• Associations Two categorical variables are
  associated if the relative frequencies in the
  response variable are not the same for all
  populations. (The populations are defined by the
  values the categorical explanatory variable takes
  on.)
• Percentages for the joint, marginal, and
  conditional distributions
• Joint Distribution How likely are you to like
  tomatoes and be a male? Ans13/38
• Marginal Distribution What is the percentage of
  people who like tomatoes? Ans21/38
• Conditional Distribution If you are a female,
  how likely are you to like tomatoes? Ans8/19

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2. Relationships Between Two Categorical Variables

• Example 9.10

• Example 9.9

Eg 9.9 Hospital A loses 3, B loses 2. Choose
B. Eg 9.10 Good condition A loses 1, B
1.3. Poor condition A loses 3.8, B 4. Choose
A.
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2. Relationships Between Two Categorical 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.
• E.g. Good/ Poor Condition.
• 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.
• E.g. We chose Hospital B in 9.9, but chose
  Hospital A in 9.10.

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3. Relationships Between Two Numeric Variables

• Depending on the situation, one of the variables
  is the explanatory variable and the other is the
  response variable.
• Examples
• Height and Weight
• Income and Age
• Time and Growth
• Amount of time spent studying for STAT303 and
  exam scores

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3. Relationships Between Two Numeric Variables

• Example
• Response MPG
• Explanatory Weight

This is called a scatter plot. Each individual in
the data appears as one point in the plot.
Response Variable (y-axis)
Explanatory Variable (x-axis)
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3. Relationships Between Two Numeric Variables

• Example
• Response MPG
• Explanatory Weight

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3. Relationships Between Two Numeric Variables

• Example
• Response Horsepower
• Explanatory Weight

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3. Relationships Between Two Numeric Variables

• Correlation or r measures the direction and
  strength of the linear relationship between two
  numeric variables.
• If X represents the explanatory and Y represents
  the response, the correlation is calculated as

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3. Relationships Between Two Numeric Variables

• General Properties of Correlation
• It must be between -1 and 1, or (-1lt r lt 1).
• If r is negative, the relationship is negative.
• If r is -1, there is a perfect negative linear
  relationship.
• If r is positive, the relationship is positive.
• If r is 1, there is a perfect positive linear
  relationship.
• If r is 0, there is no 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
• r measures ONLY linear relationships.

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3. Relationships Between Two Numeric Variables
r 1
r 0
r -1
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3. Relationships Between Two Numeric Variables
r 0.0489
r 0.04
r 0.4306
r -0.8428
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3. Relationships Between Two 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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3. Relationships Between Two Numeric Variables

• Important notes
• Association/Correlation does not imply causation
• Slope is not correlation
• scale change does not affect correlation, but
  affects slope.
• For correlation, it doesnt matter which is x,
  which is y.
• But for slope, it does matter.
• Correlation doesnt measure the strength of a
  non-linear relationship
• r 0.46

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3. Relationships Between Two Numeric Variables

• Association does not imply causation
• For the worlds nations,
• variable X number of TV sets per person
• variable Y average life expectancy
• There is high positive correlation nations with
  more TV sets have higher life expectancies.
• Is there causation? Can we lengthen the lives of
  people in poor nations by shipping them TV sets?
  No!

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3. Relationships Between Two Numeric Variables

• Regression Line a straight line that describes
  how a response variable Y changes as an
  explanatory variable X changes.
• General form
• y a bx, where a is the intercept, b is the
  slope.
• Least Squares Regression best fit
• Associations Two numerical variables are
  associated if the distribution of the response is
  not the same for each value of the explanatory
  variable.
• 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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3. Relationships Between Two Numeric Variables
Horsepower -10.78 0.04weight (Equation of
the line.)
Intercept y-value or response (horsepower) when
line crosses the y-axis.
Slope 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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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, b, 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, b, is the amount by which y changes
  when x increases by one unit.
• The intercept, a, 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 variance
  explained

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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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3. Relationships Between Two Numeric Variables

• What is the effect of an outlier on correlation?
• Adding a point that is not near the line and is
  far from the other points (an outlier in the y
  direction)

r 0.53 Note This point does not greatly
affect the estimated regression equation.
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3. Relationships Between Two Numeric Variables

• What is the effect of an outlier on correlation?
• Adding a point that is near the line and is far
  from the other points (an outlier in the x
  direction)

r 0.94 Note This point greatly influences the
estimated regression equation (an influential
point.)
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• How does the correlation change when adding a
  point to data set?
• Adding a point at the mean doesnt change
  anything (not even intercept or slope )
• The further a point is from the mean, the more
  the correlation changes.

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SUMMARY Descriptive Statistics

• ONE POPULATION (Chapter 1)
• Describing the distribution of a single variable
• Categorical variable
• Frequency table
• Pie chart
• Bar chart
• Relative frequencies, Mode
• Numeric variable
• Measures of location (center(mean, median), Q1,
  Q3, min, max)
• Measures of spread (standard deviation, range,
  IQR)
• Frequency table
• Stemplot
• Histogram
• Boxplot
• Normal quantile plot

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SUMMARY Descriptive Statistics

• COMPARING POPULATIONS (Chapter 2 and 9.1)
• Looking for associations between two variables 
• Explanatory (independent) variable is
  categorical, Response (dependent) variable is
  numeric
• Measures of location and spread by category
• Side by side boxplot
• Explanatory variable is categorical, Response
  variable is also categorical
• Two-way table
• Simpsons Paradox
• Lurking variable
• Explanatory variable is numeric, Response
  variable is also numeric
• Scatter plot
• Correlation, r