ASSOCIATION: CONTINGENCY, CORRELATION, AND REGRESSION

Chapter 3

3.1 The Association between Two Categorical Variables

Response and Explanatory Variables

Response variable (dependent, y) outcome variable

Explanatory variable (independent, x) defines groups

Response/Explanatory1. Grade on

test/Amount of study time

2. Yield of corn/Amount of rainfall

Association

Association – When a value for one variable is more likely with certain values of the other variable

Data analysis with two variables 1. Tell whether there is an association and 2. Describe that association

Contingency Table

Displays two categorical variables

The rows list the categories of one variable; the columns list the other

Entries in the table are frequencies

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Contingency Table

What is the response (outcome) variable? Explanatory?

What proportion of organic foods contain pesticides?Conventionally grown?

What proportion of all sampled foods contain pesticides?

Proportions & Conditional Proportions

Side by side bar charts show conditional proportions and allow for easy comparison

Proportions & Conditional Proportions

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If no association, then proportions would be the same

Proportions & Conditional Proportions

Since there is association, then proportions are different

3.2 The Association between Two Quantitative Variables

Internet Usage & GDP Data Set

INTERNET GDP INTERNET GDPAlgeria 0.65 6.09 J apan 38.42 25.13Argentina 10.08 11.32 Malaysia 27.31 8.75Australia 37.14 25.37 Mexico 3.62 8.43Austria 38.7 26.73 Netherlands 49.05 27.19Belgium 31.04 25.52 New Zealand 46.12 19.16Brazil 4.66 7.36 Nigeria 0.1 0.85Canada 46.66 27.13 Norway 46.38 29.62Chile 20.14 9.19 Pakistan 0.34 1.89China 2.57 4.02 Philippines 2.56 3.84Denmark 42.95 29 Russia 2.93 7.1Egypt 0.93 3.52 Saudi Arabia 1.34 13.33Finland 43.03 24.43 South Africa 6.49 11.29France 26.38 23.99 Spain 18.27 20.15Germany 37.36 25.35 Sweden 51.63 24.18Greece 13.21 17.44 Switzerland 30.7 28.1India 0.68 2.84 Turkey 6.04 5.89Iran 1.56 6 United Kingdom 32.96 24.16Ireland 23.31 32.41 United States 50.15 34.32Israel 27.66 19.79 Vietnam 1.24 2.07

Yemen 0.09 0.79

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INTERNET GDP INTERNET GDPAlgeria 0.65 6.09 J apan 38.42 25.13Argentina 10.08 11.32 Malaysia 27.31 8.75Australia 37.14 25.37 Mexico 3.62 8.43Austria 38.7 26.73 Netherlands 49.05 27.19Belgium 31.04 25.52 New Zealand 46.12 19.16Brazil 4.66 7.36 Nigeria 0.1 0.85Canada 46.66 27.13 Norway 46.38 29.62Chile 20.14 9.19 Pakistan 0.34 1.89China 2.57 4.02 Philippines 2.56 3.84Denmark 42.95 29 Russia 2.93 7.1Egypt 0.93 3.52 Saudi Arabia 1.34 13.33Finland 43.03 24.43 South Africa 6.49 11.29France 26.38 23.99 Spain 18.27 20.15Germany 37.36 25.35 Sweden 51.63 24.18Greece 13.21 17.44 Switzerland 30.7 28.1India 0.68 2.84 Turkey 6.04 5.89Iran 1.56 6 United Kingdom 32.96 24.16Ireland 23.31 32.41 United States 50.15 34.32Israel 27.66 19.79 Vietnam 1.24 2.07

Yemen 0.09 0.79

Scatterplot

Graph of two quantitative variables: Horizontal Axis:

Explanatory, x Vertical Axis:

Response, y

Interpreting Scatterplots

The overall pattern includes trend, direction, and strength of the relationship Trend: linear, curved,

clusters, no pattern Direction: positive,

negative, no direction Strength: how closely

the points fit the trend Also look for outliers

from the overall trend

Used-car Dealership

What association would we expect between the age of the car and mileage?

a) Positiveb) Negativec) No association

Linear Correlation, r

Measures the strength and direction of the linear association between x and y

Correlation coefficient: Measuring Strength & Direction of a Linear Relationship

Positive r => positive associationNegative r => negative associationr close to +1 or -1 indicates strong linear associationr close to 0 indicates weak association

3.3 Can We Predict the Outcome of a Variable?

Regression Line

Predicts y, given x:

The y-intercept and slope are a and b

Only an estimate – actual data vary

Describes relationship between x and estimated means of y

bxay ˆ

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Residuals

Prediction errors: vertical distance between data point and regression line

Large residual indicates unusual observation

Each residual is:

Sum of residuals is always zero

ˆy y

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Goal: Minimize distance from data to regression line

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Least Squares Method

Residual sum of squares:

Least squares regression line minimizes vertical distance between points and their predictions

2 2ˆ( ) ( )residuals y y

Regression Analysis

Identify response and explanatory variables Response

variable is y Explanatory

variable is x

Anthropologists Predict Height Using Remains?

Regression Equation:

is predicted height and x is the length of a femur, thighbone (cm)

Predict height for femur length of 50 cm

xy 4.24.61ˆ y

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Interpreting the y-Intercept and slope

y-intercept: y-value when x = 0

Helps plot line Slope: change in y

for 1 unit increase in x

1 cm increase in femur length means 2.4 cm increase in predicted height

xy 4.24.61ˆ

Slope Values: Positive, Negative, Zero

Slope and Correlation

Correlation, r:

Describes strength

No units Same if x and y

are swapped

Slope, b:

Doesn’t tell strength

Has units Inverts if x and y

are swapped

Proportional reduction in error, r2

Variation in y-values explained by relationship of y to x

A correlation, r, of .9 means

81% of variation in y is explained by x

%8181.9. 22 r

Squared Correlation, r2

3.4 What Are Some Cautions in Analyzing Associations?

Extrapolation

Extrapolation: Predicting y for x-values outside range of data Riskier the

farther from the range of x

No guarantee trend holds

Neil Weiss, Elementary Statistics, 7th Edition

Outliers and Influential Points

Regression outlier lies far away from rest of data

Influential if both:

1. Low or high, compared to rest of data

2. Regression outlier

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Correlation Does Not Imply Causation

Strong correlation between x and y means

Strong linear association between the variables

Does not mean x causes y

Ex. 95.6% of cancer patients have eaten pickles, so do pickles cause cancer?

Lurking Variables & Confounding

1. Ice cream sales & drowning => temperature2. Reading level & shoe size => age

Confounding – two explanatory variables both associated with response variable and each other

Lurking variables – not measured in study but may confound

Simpson’s Paradox Example

Probability of Death of Smoker = 139/582 = 24%

Probability of Death of Nonsmoker = 230/732 = 31%

Simpson’s Paradox:

Association between two variables reverses after third is included

Break out Data by Age

Simpson’s Paradox Example

Associations look quite different after adjusting for third variable

Simpson’s Paradox Example