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Chapter 2: Looking at Data - Relationships

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General Procedure

1. Plot the data.2. Look for the overall pattern.3. Calculate a numeric summary.4. Answer the question (which will be defined

shortly)

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2.1: Relationships - Goals• Be able to define what is meant by an association

between variables.• Be able to categorize whether a variable is a response

variable or a explanatory variable.• Be able to identify the key characteristics of a data set.

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Questions

• What objects do the data describe?• What variables are present and how are they

measured?• Are all of the variables quantitative?• Are the variables associated with each other?

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Association (cont.)

Two variables are associated if knowing the values of one of the variables tells you something about the values of the other variable.1. Do you want to explore the association?2. Do you want to show causality?

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Variable Types

• Response variable (Y): outcome of the study• Explanatory variable (X): explains or causes

changes in the response variable

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Key Characteristics of Data• Cases: Identify what they are and how many• Label: Identify what the label variable is (if

present)• Categorical or quantitative: Classify each

variable as categorical or quantitative. • Values. Identify the possible values for each

variable.• Explanatory or Response: Classify each

variable as explanatory or response.

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2.2: Scatterplots - Goals• Be able to create a scatterplot (lab)• Be able to interpret a scatterplot–Pattern–Outliers– Form, direction and strength of a relationship

• Be able to interpret scatterplots which have categorical variables.

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Scatterplot - Procedure

1. Decide which variable is the explanatory variable and put on X axis. The response variable goes on the Y axis.

2. Label and scale your axes.3. Plot the (x,y) pairs.

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Example: Scatterplot

The following data is to determine the relationship between age and change in systolic blood pressure (BP, mm Hg) after 24 hours in response to a particular treatment.

a) Draw a scatterplot of this data.

Obs 1 2 3 4 5 6 7 8 9 10 11Age 70 51 65 70 48 70 45 48 35 48 30BP -28 -10 -8 -15 -8 -10 -12 3 1 -5 8

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Example: Scatterplot (cont)

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Pattern

• Form• Direction• Strength• Outliers

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PatternLinear

Nonlinear

No relationship

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Outliers

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Example: Scatterplot (cont)

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Scatterplot with Categorical Variables

http://statland.org/Software_Help/Minitab/MTBpul2.htm

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I am a Turkey, not Tukey!Thank you for not eating me!

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2.3: Correlation - Goals• Be able to use (and calculate) the correlation to

describe the direction and strength of a linear relationship.

• Be able to recognize the properties of the correlation.

• Be able to determine when (and when not) you can use correlation to measure the association.

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Sample correlation, r(Pearson’s Sample Correlation Coefficient)

𝑟=∑ [ (𝑥𝑖−𝑥 ) (𝑦 𝑖− 𝑦 ) ]

√∑ (𝑥𝑖−𝑥 )2√∑ ( 𝑦 𝑖− 𝑦 )2=

𝑆𝑆𝑥𝑦

√𝑆𝑆𝑥𝑥√𝑆𝑆𝑦𝑦

=∑ [ (𝑥𝑖−𝑥 ) ( 𝑦 𝑖− 𝑦 ) ]

(𝑛−1)𝑠𝑥 𝑠 𝑦

= 1𝑛−1∑ [(𝑥 𝑖−𝑥

𝑠𝑥 )( 𝑦 𝑖− 𝑦𝑠 𝑦

)]

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Sum of Squares

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Properties of Correlation

• r > 0 ==> positive associationr < 0 ==> negative association

• r is always a number between -1 and 1.• The strength of the linear relationship

increases as |r| moves to 1.– |r| = 1 only occurs if there is a perfect linear

relationship– r = 0 ==> x and y are uncorrelated.

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Positive/Negative Correlation

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Example: Positive/Negative Correlation

1) Would the correlation between the age of a used car and its price be positive or negative? Why?

2) Would the correlation between the weight of a vehicle and miles per gallon be positive or negative? Why?

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Properties of Correlation

• r > 0 ==> positive associationr < 0 ==> negative association

• r is always a number between -1 and 1.• The strength of the linear relationship

increases as |r| moves to 1.– |r| = 1 only occurs if there is a perfect linear

relationship– r = 0 ==> x and y are uncorrelated.

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Variety of Correlation Values

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Value of r

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Properties of Correlation

• r > 0 ==> positive associationr < 0 ==> negative association

• r is always a number between -1 and 1.• The strength of the linear relationship

increases as |r| moves to 1.– |r| = 1 only occurs if there is a perfect linear

relationship– r = 0 ==> x and y are uncorrelated.

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Comments about Correlation• Correlation makes no distinction between

explanatory and response variables.

• r has no units and does not change when the units of x and y change.

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Cautions about Correlation• Correlation requires that both variables be

quantitative.• Correlation measures the strength of LINEAR

relationships only.

• The correlation is not resistant to outliers.• Correlation is not a complete summary of

bivariate data.

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Datasets with r = 0.816

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Questions about Correlation

• Does a small r indicate that x and y are NOT associated?

• Does a large r indicate that x and y are linearly associated?

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2.4: Least-Squares Regression - Goals• Be able to generally describe the method of

‘Least Squares Regression’• Be able to calculate and interpret the

regression line.• Using the least square regression line, be able

to predict the value of y for any appropriate value of x.

• Be able to calculate r2.• Be able to explain the meaning of r2.–Be able to discern what r2 does NOT explain.

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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.

We can use a regression line to predict the value of y for a given value of x.

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Idea of Linear Regression

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Linear Regression

i i xy y

1 2xx xi

x x y y SS sb r

SS sx x

b0 = y - b1x�

y = b0 + b1x

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Example: Regression Line

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BP

Age

y = 20.11 - 0.526x

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Example: Regression LineThe following data is to determine the relationship

between age and change in systolic blood pressure (BP, mm Hg) after 24 hours in response to a particular treatment.

= 52.727, y = -7.636, sx� x = 14.164, sy = 9.688, r = -0.76951

b) What is the regression line for this data?c) What would the predicted value be for someone who is

51 years old?

Obs 1 2 3 4 5 6 7 8 9 10 11Age 70 51 65 70 48 70 45 48 35 48 30BP -28 -10 -8 -15 -8 -10 -12 3 1 -5 8

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Facts about Least Square Regression

1. Slope: A change of one standard deviation in x corresponds to a change of r standard deviations in y.

2. Intercept: the value of y when x = 0.3. The line passes through the point ( ,y).x�4. There is an inherent difference between x

and y.

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r2

• Coefficient of determination.• Fraction of the variation of the values of y that

is explained by the least-squares regression of y on x.

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Example: Regression LineThe following data is to determine the

relationship between age and change in systolic blood pressure (BP, mm Hg) after 24 hours in response to a particular treatment.

d) What percent of variation of Y is due to the regression line?

Obs 1 2 3 4 5 6 7 8 9 10 11Age 70 51 65 70 48 70 45 48 35 48 30BP -28 -10 -8 -15 -8 -10 -12 3 1 -5 8

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Beware of interpretation of r2

• Linearity• Outliers• Good prediction

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2.5: Cautions about Correlation and Regression - Goals

• Be able to calculate the residuals.• Be able to use a residual plot to assess the fit of a

regression line.• Be able to identify outliers and influential observations

by looking at scatterplots and residual plots.• Be able to determine when you can predict a new

value.• Be able to identify lurking variables that can influence

the relationship between two variables.• Be able to explain the different between association

and causation.

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Residuals

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Example: Regression LineThe following data is to determine the

relationship between age and change in systolic blood pressure (BP, mm Hg) after 24 hours in response to a particular treatment.

e) What is the residual for someone who is 51 years old?

Obs 1 2 3 4 5 6 7 8 9 10 11Age 70 51 65 70 48 70 45 48 35 48 30BP -28 -10 -8 -15 -8 -10 -12 3 1 -5 8

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Residual PlotsGood

Linearity Violation

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Residual PlotsGood

Constant variance violation

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Residual Plots – Bp

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Original Y outlier

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Residual Plots – Bp

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Original X outlier

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Influential Point

• 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.

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Cautions about Correlation and Regression: Extrapolation

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Cautions about Correlation and Regression:

• Both describe linear relationship.• Both are affected by outliers.• Always PLOT the data.• Beware of extrapolation.• Beware of lurking variables– Lurking variables are important in the study,

but are not included.–Confounding variables confuse the issue.

• Correlation (association) does NOT imply causation!

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Lurking VariablesIn each of these cases, identify the lurking variable.

1. For children, there is an extremely strong correlation between shoe size and math scores.

2. There is a very strong correlation between ice cream sales and number of deaths by drowning.

3. There is very strong correlation between number of churches in a town and number of bars in a town.

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What is the lurking variable?

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2.6: Data Analysis for Two-Way Tables - Goals

Statements• The distribution of a two random variables

(bivariate) is called a joint distribution.• Two random variables are similar to two events in

that they can have conditional probabilities and be independent of each other.

Goal• Interpret examples of Simpson’s paradox

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Simpson’s 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 Simpson’s paradox.

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Simpson’s Paradox

Consider the acceptance rates for the following groups of men and women who applied to college.

Counts Accepted Notaccepted Total

Men 198 162 360

Women 88 112 200

Total 286 274 560

Percents Accepted Notaccepted

Men 55% 45%

Women 44% 56%

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Simpson’s Paradox

• Business School

• Art School

Counts Accepted Notaccepted Total

Men 18 102 120

Women 24 96 120

Total 42 198 240

Percents Accepted Notaccepted

Men 15% 85%

Women 20% 80%

Counts Accepted Notaccepted Total

Men 180 60 240Women 64 16 80

Total 244 76 320

Percents Accepted Notaccepted

Men 75% 25%

Women 80% 20%

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2.7: The Question of Causation - Goals• Be able to explain an association–Causation–Common response–Confounding variables

• Apply the criteria for establishing causation.

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Causation

Association does not mean causation!

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Establishing CausationPerform an experiment!What do we need for causation?1. The association is strong.2. The association is consistent.

The connection happens in repeated trialsThe connection happens under varying conditions

3. Higher doses are associated with strong responses.

4. Alleged cause precedes the effect.5. The alleged cause is plausible.