Bivariate Data analysis

Bivariate Data

In this PowerPoint we look at sets of data which contain two variables.

Scatter plots Correlation

Outliers Causation

Variables

DiscreteContinuous 

Quantitative(Numerical)

(measurements and counts)

Qualitative(categorical)

(define groups)

Ordinal(fall in natural order)

Categorical(no idea of order)

We are only going to consider quantitative variables in this AS

Quantitative

Discrete• Many repeated values• Age groups• Marks

Continuous• Few repeated values• Height• Length• Weight

Qualitative

Categorical• Gender• Religious

denomination• Blood types• Sport’s numbers (e.g.

He wears the number ‘8’ jersey)

Ordinal• Grades• Places in a race (e.g.

1st, 2nd, 3rd)

We often want to know if there is a relationship between two

numerical variables.

A scatter plot, which gives a visual display of the relationship between two variables, provides a good starting point.

In a relationship involving two variables, if the values of one variable ‘depend’ on the values of another variable, then the former variable is referred to as the dependent (or response) variable and the latter variable is

referred to as the independent (or explanatory) variable.

y - axis dependent (response) variable

x - axis independent (explanatory) variable

Consider data on ‘hours of study’ vs ‘ test score’

Hours Score Hours Score Hours Score

18 59 14 54 17 59

16 67 17 72 16 76

22 74 14 63 14 59

27 90 19 72 29 89

15 62 20 58 30 93

28 89 10 47 30 96

18 71 28 85 23 82

19 60 25 75 26 35

22 84 18 63 22 78

30 98 19 61

We may want to see if we could predict

the test score (response variable) based on the

hours of study (explanatory variable).

y - axis: Test score

x - axis: Hours of study

We look for a pattern in the way

the points lie

Certain patterns tell us about

the relationship

This is called

correlation

This pointis an outlier

We could describe the rest of the data as having a linear form.

Scatter plots• Use hollow circles for points• Label axes correctly with units• What you want to predict goes on the y-axis

(response variable)• Title of graph• No background; No gridlines• Unless you need to show categories- no legend• Show different categories on a single graph in

different colours rather than on separate graphs.• Adjust scale and size of font (14pt for pasting)

What to look for in your plot?

• Direction of the relationship - positive or negative• Form of the graph - linear or curved• The strength - whether it is strong, moderate or

weak• Scatter - constant scatter, a fan effect…• Outliers• Groupings

Page 22

What do you see in this scatter plot?

• There appears to be a linear trend.

• There appears to be moderate constant scatter.

• Negative Association.

• No outliers or groupings visible.

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Latitude (°S)   

Mean January Air Temperatures for 30 New Zealand Locations

Tem

pera

ture

(°C

)

What do you see in this scatter plot?

• There appears to be a non-linear trend.

• There appears to be non-constant scatter about the trend line.

• Positive Association.

• One possible outlier (Large GDP, low % Internet Users).

0 10 20 30 40

GDP per capita (thousands of dollars)

0

10

20

30

40

50

60

70

80

Inte

rnet

Use

rs (

%)

% of population who are Internet Users vs GDP per capita for 202 Countries

What do you see in this scatter plot?

• Two non-linear trends (Male and Female).

• Very little scatter about the trend lines

• Negative association until about 1970, then a positive association.

• Gap in the data collection (Second World War).

Year

1990198019701960195019401930

30

28

26

24

22

20

Ag

e

Average Age New Zealanders are First Married

Rank these relationships from weakest (1) to strongest (4):

1

2

3

4

Describe these relationshipsPerfect, negative, linear

relationship

Perfect, positive, linear

relationship

Norelationship

Moderate,negative

linearrelationship

Weak,positivelinear

relationship

Describe this relationship.

As the hours of study increase, the test score . . . .? . . .

Pearson’s product-moment correlation coefficient, r

Correlation measures the strength of the linear association between two quantitative variables.

r = -1 r = -0.7 r = -0.4 r = 0 r = 0.3 r = 0.8 r = 1

Points fall exactly on a straight line

No linear relationship

(uncorrelated)

Points fall exactly on a straight line

The correlation coefficient may take any value between -1.0 and +1.0

How close the points in the scatter plot come to lying on the line.

r - what does it tell you?

r = 0.99

x

y

**** ** ** ** * **** **** *

r = 0.57

x

y

*

**

*

** *

*

**

*

****

*

*

*

* *r = 0.99 r = 0.57

Interpreting r

• 0.75-1 Strong positive linear association• 0.5-0.75 Moderate positive linear association• 0.25-0.5 Weak positive linear association• -0.25-0.25 No association or weak linear

association• -0.5--0.25 Weak negative linear association• -0.75--0.5 Moderate negative linear association• -1 - -0.75 Strong negative linear association

Useful websites

• http://www.ruf.rice.edu/~lane/stat_sim/reg_by_eye/index.html Regression by eye

• http://istics.net/stat/Correlations/ Guessing

• http://illuminations.nctm.org/LessonDetail.aspx?ID=L455#whatif effect of outliers

Assumptions

• linear relationship between x and y

• continuous random variables• The residuals must be normally distributed• x and y must be independent of each other• all individuals must be selected at random

from the population

• all individuals must have equal chance of being selected

What is correlation?

A measure of the strength of a LINEAR association between two

quantitative variables.

Sure you can calculate a correlation coefficient for any

pair of variables but correlation measures the strength only of the linear association and will be

misleading if the relationship is not linear.

Do you know that:

• Correlation applies only to quantitative variables. Check you know the units and what they measure.

• Outliers can distort the correlation dramatically.

Some facts about the correlation coefficient

• The sign gives the direction of the association.• Correlation is always between -1 and 1.• Correlation treats x and y symmetrically. The correlation

of x and y is the same as the correlation of y with x.• Correlation has no units and is generally given as a

decimal.• r is a multiple of the slope• Note: variables can have a strong association but still have

a small correlation if the association isn’t linear.• Correlation is sensitive to outliers. A single outlying value

can make a small correlation large or make a large one small.

The sign gives the direction of the association.

Positive Negative

Correlation treats x and y symmetrically. The correlation of x and y is the same as

the correlation of y with x.

r is a multiple of the slope

Variables can have a strong association but still have a small correlation if the association isn’t

linear.

Always plot the data before looking at the correlation!

Would it be OK to use a correlation coefficient to

describe the strength of the relationship?

9876543210

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3000

2000

1000

0

Position Number

Dis

tan

ce (

mill

ion

mile

s)

Distances of Planets from the Sun

√

Reaction Times (seconds) for 30 Year 10 Students

0

0.2

0.4

0.6

0.8

0 0.2 0.4 0.6 0.8 1

Non-dominant Hand

Dom

inan

t H

an

d

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Latitude (°S)   

Mean January Air Temperatures for 30 New Zealand Locations

Tem

pera

ture

(°C

)

√Female ($)

Average Weekly Income for Employed New Zealanders in 2001

Male

($

)

0

200

400

600

800

1000

1200

0 200 400 600 800

XX

Correlation is sensitive to outliers. A single outlying value can make a small correlation large or make a

large one small.

You should be cautious in interpreting the correlation - these

graphs all have the same correlation coefficient (0.817)

Data set 1

Data set 2

Data set 3

Data set 4

Outliers can distort the correlation dramatically. An

outlier can make an otherwise small correlation look big or hide

a large correlation. It can even give an otherwise positive

association a negative correlation coefficient (and vice versa).

What do you see in this scatterplot?

22 23 24 25 26 27 28 29

150

160

170

180

190

200

Foot size (cm)

Heig

ht

(cm

)

Height and Foot Size for 30 Year 10 Students •Appears to be

a linear trend, with a possible outlier (tall person with a small foot size.)

•Appears to be constant scatter.

•Positive association.

What will happen to the correlation coefficient if the tallest Year 10

student is removed?

22 23 24 25 26 27 28 29

150

160

170

180

190

200

Foot size (cm)

Heig

ht

(cm

)

Height and Foot Size for 30 Year 10 Students

•It will get smaller

•It will get bigger

•It will stay the same

What do you see in this scatter plot?

•Appears to be a strong linear trend.

•Outlier in X (the elephant).

•Appears to be constant scatter.

•Positive association.

6005004003002001000

40

30

20

10

Gestation (Days)

Life

Exp

ect

an

cy (

Years

)

Life Expectancies and Gestation Period for a sample of non-human Mammals

Elephant

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40

30

20

10

Gestation (Days)

Life

Exp

ect

an

cy (

Years

)

Life Expectancies and Gestation Period for a sample of non-human Mammals

Elephant

What will happen to the correlation coefficient if the

elephant is removed?

•It will get smaller

•It will get bigger

•It will stay the same

How does the outlier affect the r - value?

How does the outlier affect the r - value?

How does the outlier affect the r - value?

How does the outlier affect the r - value?

How does the outlier affect the r - value?

How does the outlier affect the r - value?

When you see an outlier, it’s often a good idea to report the

correlations with and without the point.

Don’t confuse Correlation with causation. Scatterplots and

correlation never prove causation.

Using the information in the plot, can you

suggest what needs to be done in a country to increase the life expectancy?

Explain.

400003000020000100000

80

70

60

50

People per Doctor

Life

Exp

ect

an

cy

Life Expectancy and Availability of Doctors for a Sample of 40 Countries

Perhaps if you have less people per Doctor (i.e. more Doctors per person), then the life expectancy will increase.

Using the information in this plot, can you make another suggestion as

to what needs to be done in a country to increase life expectancy?

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80

70

60

50

People per Television

Life

Exp

ect

an

cy

Life Expectancy and Availability of Televisions for a Sample of 40 Countries It looks like if

you decrease the number of people per television (i.e. have more TVs per person), then the life expectancy will increase!

Can you suggest another variable that is linked to life expectancy and

the availability of doctors (and televisions) which explains the

association between the life expectancy and the availability of

doctors

(and televisions)?

Some measure of wealth of a country.

Eg Average income per person or GDP.

Damaged for life by too much TV

• Watching too much television as a child causes serious health problems years later, and raises the risk of heart disease, a New Zealand study of 1000 children has found….

• It links the amount of time spent in front of the box as a child with obesity, high cholesterol, poor fitness and smoking….

Damaged for life by too much TV

Damaged for life by too much TV

Hea

lth

Sco

re

TV watching

r = - 0.93

Causal relationships

• Two general types of studies: experiments and observational studies

• In an experiment, the experimenter determines which experimental units receive which treatments.

• In an observational study, we simply compare units that happen to have received each of the treatments.

• Only properly designed and carefully executed experiments can reliably demonstrate causation.

• An observational study is often useful for identifying possible causes of effects, but it cannot reliably establish causation

Causal relationships

• In observational studies, strong relationships are not necessarily causal relationships.

• Correlation does not imply causation.

• Be aware of the possibility of lurking variables.

Causal relationships

Watch out for lurking variables. Damage ($) vs number of firemen would show a strong correlation, but damage doesn’t cause firemen

and firemen do seem to cause damage (spraying water and

chopping holes). The underlying variable is the size of the blaze.

Although there was plenty of evidence that increased smoking was associated with increased levels of lung cancer, it took

years to provide evidence that smoking actually causes lung

cancer.

It would be a good idea to read the two pages of notes you have

that discusses correlation and causation!

So now you want to know how to calculate the correlation

coefficient, r.Here is one version of the

formula!

Luckily the computer will calculate R2 and you can square

root this to get r. Remember only when the

association is linear.

r measures the strength of the relationship NOT R2!!!!

r measures the strength of the relationship NOT R2!!!!

r measures the strength of the relationship NOT R2!!!!

The words you use

• There is a strong, positive, linear relationship between ‘x’ and ‘y’ and when the x- values increase, the y-values increase also. This is indicated by the value of the correlation coefficient i.e. r = 0.85 which is close to 1.

• (Note: Do not use ‘x’ and ‘y’ use what they represent.)