Chapter 8 CorrelationLinear Regression

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Chapter 8 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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Chapter 8 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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Chapter 8 Correlation/Linear Regression

• 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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Chapter 8 Correlation/Linear Regression

• 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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Chapter 8 Correlation/Linear Regression

• 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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Chapter 8 Correlation/Linear Regression

• 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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Chapter 8 Correlation/Linear Regression
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Choose the best description of the scatter plot

• Moderate, negative, linear association
• Strong, curved, association
• Moderate, positive, linear association
• Strong, negative, non-linear association
• 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?

• r -0.67
• r -0.10
• r 0.71
• r 0.96
• r 1.00

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

• r -0.67
• r -0.10
• r 0.71
• r 0.96
• r 1.00

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

• r -0.67
• r -0.10
• r 0.71
• r 0.96
• 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 the
• heights of children on the playground and then
  makes a
• scatter plot of the childrens heights and
  reading test
• scores. The data meet the conditions for
  correlation so
• she calculates r .79. Which conclusion is most
• accurate?
• Being taller causes students to read better
• Being shorter causes students to read better
• Taller students tend to have better reading
  scores
• Shorter students tend to have better reading
  scores

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Chapter 8 Linear Models

• 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

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Least Square Regression Line - Calculations
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Chapter 8 Linear Models

• Not all data fall on a straight line!
• Residual Data Model or
• Residual Observed Y Predicted y

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Chapter 8 Linear Models

• Example
• X Fat Y Calories
• 19 410
• 31 580
• 34 590
• 35 570
• 39 640
• 39 680
• 43 660

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Chapter 8 Linear Models
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Chapter 8 Linear Models
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Chapter 8 Linear Models

• S 27.3340 R-Sq 92.3 R-Sq(adj) 90.7
• Residual Plot

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Chapter 9 Regression Wisdom

• 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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Chapter 9 Regression Wisdom

• 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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Chapter 9 Regression Wisdom

• 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 gt 0 indicates positive association
• r lt 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?
B.
A.
C.
D.
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A. Linear model appropriate residual plot shows
no pattern
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
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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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Calculating r with the TI-83/84

• 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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Calculating r with the TI-83/84

• Press STAT, gt (to CALC)
• Scroll down to LinReg(axb), 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 whats available, try
• STAT
• CALC
• Scroll down to see
• 5QuadReg
• 6CubicReg
• 9LnReg
• 0ExpReg
• APwrReg

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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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Chapter 9 Prediction, Residuals, Influence

• 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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Chapter 9 Prediction, Residuals, Influence

• Age 20 years (2012 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_1r S_y/S_x 0.56419.5/4.4 2.5
  calories per gram of sugar
• b_0 mean of (Y) b1mean of (X) 107 -2.507
  89.5
• Linear Model y b_0b_1x
• 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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Chapter 10 Re-expressing Data

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

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Chapter 10 Re-expressing Data
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Chapter 10 Re-expressing Data

• Re-express data to linearize

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Chapter 10 Re-expressing Data
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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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Chapter 10 Re-expressing Data

• Answer
• Log(journals) 1.22 0.3469 4.334
• Answer 21577.44 (10(4.334))

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Chapter 10 Re-expressing Data

• Why Re-expressing data?
• Make a distribution of a variable more symmetric
• Make the spread of several groups more alike,
  even if their centers differ
• Make the form of a scatterplot more nearly linear
• Make the scatter in a scatterplot spreadout more
  evenly rather than thickening at one end.

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Chapter 10 Re-expressing Data

• The Ladder of Powers
• Power 2 the square of the data values y2
• 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.