Chapters 5-7 Correlation/Linear Regression

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Chapters 5-7 Correlation/Linear Regression

• Linear Relationships: If the explanatory and response variables show a straight-line pattern, then we say they follow a linear relationship.

• Curved relationships and clusters are other forms to watch for.

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• Linear Relationships: If the explanatory and response variables show a straight-line pattern, then we say they follow a linear relationship.

• Curved relationships and clusters are other forms to watch for.

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• Direction: If the relationship has a clear direction, we speak of either positive association or negative association.

• Positive association: high values of the two variables tend to occur together

• Negative association: high values of one variable tend to occur with low values of the other variable.

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• Correlation is a number that determines the strength of a linear relationship between two quantitative variables.

• Correlation is always between -1 and 1 inclusive

• The sign of a correlation coefficient determines positive/negative association between the variables

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• Strong correlation: If r is between 0.8 and 1 and -0.8 and -1

• Moderate correlation: If r is between 0.5 and 0.8 and -0.8 and -0.5

• Weak correlation: If r is between 0 and 0.5 and -0.5 and 0

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• Correlation does not distinguish between X and Y

• Correlation is unitless

• Correlation measures the strength of linear relationship between two quantitative variables

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Choose the best description of the scatter plot

A. Moderate, negative, linear association

B. Strong, curved, association

C. Moderate, positive, linear association

D. Strong, negative, non-linear association

E. Weak, positive, linear association

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Which of the following values is most likely to represent the correlation coefficient for the data shown in this scatterplot?

A. r = -0.67

B. r = -0.10

C. r = 0.71

D. r = 0.96

E. r = 1.00

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A. r = -0.67

B. r = -0.10

C. r = 0.71

D. r = 0.96

E. r = 1.00

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A. r = -0.67

B. r = -0.10

C. r = 0.71

D. r = 0.96

E. r = 1.00

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Cautions about Correlation

• It should only be used– To describe the relationship between 2

QUANTITATIVE variables– When the association is “linear enough”– When there are no outliers

• Correlation does NOT imply causation

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A teacher at an elementary school measures theheights of children on the playground and then makes ascatter plot of the children’s heights and reading testscores. The data meet the conditions for correlation soshe calculates r = .79. Which conclusion is mostaccurate?

A. Being taller causes students to read betterB. Being shorter causes students to read betterC. Taller students tend to have better reading

scoresD. Shorter students tend to have better reading

scores

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• Easiest to understand and analyze

• Relationships are often linear

• Variables with non-linear relationship can often be transformed into linear relationship through an appropriate transformation

• Even when a relationship is non-linear, a linear model may provide an accurate approximation for a limited range of values.

• Strength: The strength of a linear relationship is determined by how close the points in the scatterplot lie to a straight line

Least Square Regression Line - Calculations

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• Not all data fall on a straight line!

• Residual = Data – Model or

• Residual = Observed Y – Predicted y

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Example

X= Fat Y= Calories

19 410

31 580

34 590

35 570

39 640

39 680

43 660

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454035302520

700

650

600

550

500

450

400

C1

C2

Scatterplot of C2 vs C1

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454035302520

700

650

600

550

500

450

400

C1

C2

S 27.3340R-Sq 92.3%R-Sq(adj) 90.7%

Fitted Line PlotC2 = 211.0 + 11.06 C1

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• S = 27.3340 R-Sq = 92.3% R-Sq(adj) = 90.7%

Residual Plot

7654321

3

2

1

0

-1

-2

-3

Observation Order

Residual

Versus Order(response is C1)

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• Extrapolation: Reaching beyond the data

• Outliers: Regression models are sensitive to outliers

• Leverage: An unusual data point whose x value is far from the mean of the x values

• A point with high leverage has the potential to change the regression line.

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• Influential: A point is influential if omitting it from the analysis gives a very different model.

• Influence depends on leverage and residual

• Lurking variables: A variable that is not included in the construction of the linear model/study.

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• Lurking variables may influence correlation and regression models.

• Association is not causations!!

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Summary

• r is a number between -1 and 1• r = 1 or r = -1 indicates a perfect

correlation case where all data points lie on a straight line

• r > 0 indicates positive association• r < 0 indicates negative association• r value does not change when units of

measurement are changed (correlation has no units!)

• Correlation treats X and Y symmetrically. The correlation of X with Y is the same as the correlation of Y with X

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Summary

• Quantitative variable condition: Do not apply correlation to categorical variables

• Correlation can be misleading if the relationship is not linear

• Outliers distort correlation dramatically. Report correlation with/without outliers.

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More Examples for Checking Linear Enough ConditionAll four data sets have r = .82

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In which case is a linear model appropriate?

6

8

10

12

6 8 10 12 14

X

Ya

8.0

10.0

6 8 10 12 14

X

Yb

6

8

10

12

6 8 10 12

X

Yc 7

9

11

13

10.0 14.0 18.0

Xd

Yd

A. B.

C. D.

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6

8

10

12

6 8 10 12 14

X

Ya -1

0

1

6 8 10 12 14

X

residuals(Y/X)

A. Linear model appropriate; residual plot shows no pattern

8.0

10.0

6 8 10 12 14

X

Yb

-1.50

-0.75

0.00

0.75

4 6 8 10 12

X

residuals(Y/X)

B. Linear model not appropriate; clear pattern of residuals

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C. Graph has an outlier; outlier is clear on the residual plot

D. Linear model not appropriate; clear pattern of residuals

6

8

10

12

6 8 10 12

X

Yc -0.00

1.25

2.50

4 6 8 10 12

X

residuals(Y/X)

7

9

11

13

10.0 14.0 18.0

Xd

Yd -1

0

1

10.0 14.0 18.0

Xd

residuals(Y/X)

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Calculating r with the TI-83/84

• The first time you do this:– Press 2nd, CATALOG (above 0)– Scroll down to DiagnosticOn– Press ENTER, ENTER– Read “Done”– Your calculator will remember this setting

even when turned off

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• Press STAT, ENTER• If there are old values in L1:

– Highlight L1, press CLEAR, then ENTER

• If there are old values in L2:– Highlight L2, press CLEAR, then ENTER

• Enter predictor (x) values in L1• Enter response (y) values in L2

– Pairs must line up– There must be the same number of predictor and

response values

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• Press STAT, > (to CALC)

• Scroll down to LinReg(ax+b), press ENTER, ENTER

• Read r at bottom of screen

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Re-Expression with the TI-83/84

• Most common re-expressions are built in.• To see what’s available, try

– STAT– CALC– Scroll down to see

• 5:QuadReg• 6:CubicReg• 9:LnReg• 0:ExpReg• A:PwrReg

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Example

• X: Age in months

• Y: Height in inches

• X: 18 19 20 21 22 23 24

• Y: 29.9 30.3 30.7 31 31.38 31.45 31.9

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• Linear Model: Height = 24.212 +.321 * Age

Correlation: r = .992

Examples

• Age = 24 months, Observed Height = 31.9

• Predicted Height = 31.916

• Residual = 31.9 – 31.916 = .016

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• Age = 20 years (20*12 = 240)

• Predicted Height ~ 8.5 ft!!

• Residual = BIG!

• Be aware of Extrapolation!

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Example

4. Relationship between calories and sugar content: A researcher tracked the sugar content and calorie of 15 baked goods and found the following information:

Average sugar content: 7.0 grams

Standard deviation of sugar content: 4.4 grams

Average calories: 107.0 grams

Standard deviation of calories: 19.5 grams

Correlation between sugar content and calories:

0.564

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Solution to Example

a) Find a linear model that describes this example:

b_{1}=r S_{y}/S_{x} = 0.564*19.5/4.4 = 2.5 calories per gram of sugar

b_{0}= mean of (Y) –b{1}mean of (X) = 107 -2.50*7 = 89.5

Linear Model: y = b_{0}+b_{1}x y= 89.5 + 2.5x or better

calories = 89.5 +2.50* sugar

b) How many calories are there in a muffin with 6.5 grams of sugar?

calories = 89.5 +2.50* 6.5 = 105.75

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Chapters 5-7 Correlation/Linear Regression: Re-expressing Data

• Example: The data shows the number of academic journals published on the Internet and during the last decade.

YEAR # of Journals

1991 27

1992 36

1993 45

1994 181

1995 306

1996 1093

1997 2459

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0

500

1000

1500

2000

2500

3000

1990 1991 1992 1993 1994 1995 1996 1997 1998

Year

Nu

mb

er

of

Jo

urn

als

Series1

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• Re-express data to linearize:

Year Log(# journals)

1991 1.431364

1992 1.556303

1993 1.653213

1994 2.257679

1995 2.485721

1996 3.03862

1997 3.390759

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0

0.5

1

1.5

2

2.5

3

3.5

4

1990 1992 1994 1996 1998

Year

Lo

g(N

um

be

r o

f J

ou

rna

ls

Series1

43

0

0.5

1

1.5

2

2.5

3

3.5

4

0 2 4 6 8

Year

Lo

g(#

of

Jo

urn

als

)

Series1

Series2

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Chapter 10 Re-expressing Data

• Least Square Regression Line has the following equation:

Log(journals) = 1.22 + 0.346 * Year

Problem:

How many journals will be published online in year 2000?

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Answer

Log(journals) = 1.22+ 0.346*9 =4.334

Answer: 21577.44 (10^(4.334))

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Why Re-expressing data?

1. Make a distribution of a variable more symmetric

2. Make the spread of several groups more alike, even if their centers differ

3. Make the form of a scatterplot more nearly linear

4. Make the scatter in a scatterplot spreadout more evenly rather than thickening at one end.

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The Ladder of Powers:

Power 2: the square of the data values y^2– Try this for unimodal distributions that are skewed to

the left.

Power 1: No change at all

Power ½: the square root of the data values

Y^(1/2)– Try this for counted data

Power 0: the logarithm of the data values y– Try this for measurements that cannot be negative– Especially those that grow by percentage increases– Salries and populations are good examples.

Chapters 5-7 Correlation/Linear Regression

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