Relations between 2 quantitative variables

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Relations between 2 quantitative variables
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Example

• Students who apply to MBA programs must write
  the Graduate Management Admission Test (GMAT).
  University admission committees use the GMAT
  scores as one of the critical indicators of how
  well a student is likely to perform in the MBA
  program. However, the GMAT may not be a very
  strong indicator for all MBA programs. Suppose
  that an MBA program designed for middle managers
  who wish to upgrade their skills was launched 3
  years ago. To judge how well the GMAT score
  predicts MBA performance, a sample of 12
  graduates was taken. Their grade point average in
  the MBA program (values from 0 to 12) and the
  GMAT score (values from 200 to 800) are listed in
  the table below and stored in file Xm04-16.

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Scatterplots

• Show the relationship between two paired
• quantitative variables (X and Y).
• Examples
• Variable X age Variable Y height
• Variable X SAT score Variable Y academic
  success
• Variable X alcohol level Variable Y no. of
  heart diseases

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• Data for variable X x1,x2,,xn
• Data for variable Y y1, y2,,yn
• Plot (x1,y1), (x2,y2),, (xn,yn)

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Interpreting scatterplots

• Three important aspects of pattern
• Form
• Direction
• Strength

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• For the GMAT-GPA
• Form
• Direction
• Strength

Close to linear association
positive
Moderate to low
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Relationship does not necessarily mean that one
variable causes the other!

• Two levels of relationship
• 1. Simple association (and Response variable and
  explanatory variable)
• 2. Causality

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Two levels of relationship between variables

• 1. Simple association
• Two variables are associated if some values of
  one variable tend to occur more often with some
  values of the second variable than with other
  values of that variable.
• Examples
• Statistics course grade and
• American history course grade
• Cholesterol level and blood pressure
• No. of babies and number of storks
• Heights of fathers and their sons
• GMAT scores and GPA
• The association between the variables means
• that we can predict one variable using the other.
  

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• Response variable and explanatory variable
• If we think that a variable x can explain
  changes in variable y, we call x an explanatory
  variable and y a response variable.

For each of the following, if possible, determine
which variable is the explanatory and which
variable is the response Statistics course
grade and American history course
grade Cholesterol level and blood pressure No.
of babies and number of storks Heights of
fathers and their sons SAT scores and
performance at University
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• 2. Causality
• Changes in one (or more) explanatory variable
  cause changes in a response variable.
• Examples
• Time spent studying for an exam and the grade on
  the exam
• smoking and life expectancy

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How to quantify the relationship?
The most common way of quantifying relationship
is due primarily to
Karl Pearson
? Pearson correlation coefficient
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Correlation coefficient

• The correlation coefficient r measures the
  strength of the linear relationship between two
  quantitative variables.
• To compute r for two variables X and Y we need
• Mean and standard deviation of X ,
• Mean and standard deviation of Y ,
• r is given by

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• Note that r a dimensionless number.
• That is, its value depends only on the
  association between the two variables, not on
  their units or the magnitude their values

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Interpreting the Correlation Coefficient
The correlation coefficient is between -1 to 1 A
positive value for r indicates a positive
relationship A negative value for r a negative
relationship.
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
X
r -1 Perfect negative linear association
r 0 No linear association
r 1 Perfect positive linear association
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Correlation r-0.3
Correlation r0
Correlation r0
Correlation r0.5
Correlation r-0.7
Correlation r0.9
Correlation r-0.99
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Outliers affect correlation
r
r
0.5
0.8
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• The correlation coefficient r ONLY measures a
  linear relationship. If all of data points lie on
  a circle, you will get an r 0, even though the
  nonlinear relationship is perfect
• You MUST examine the scatterplot first in order
  to get a sense of what type of relationship, if
  any, is present.

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Assigning categorical variables to scatterplots
Example
Scatterplot of income versus age
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? To add a categorical variable to a
scatterplot, use a different color or
symbol for each category
Scatterplot of income versus age classified by sex
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Regression

• Correlation measures the direction and the
  strength of linear relationship between 2
  quantitative variables
• Regression allows us to predict y from the
  knowledge of x.

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Example
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We want a line that is as close as possible to
all points. This line will be used to predict y
from x.
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Fitting a straight line using least squares
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The method of least squares chooses the line that
makes the sum of squares of these errors as small
as possible. To find this line we must find the
a and b that minimize
Or
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• The equation of the straight line is

Where is the predicted value of y
What is a? Intercept -value of y when x0 What
is b? Slope amount of increase in y for
every unit change in x
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Fitting a least square line to the GMAT data
632.3
43.6
1.120
8.867
The correlation between GMAT and GPA is r0.536
Slope
Intercept
The equation of the least square line
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Using the least square line to calculate
predicted y values
8.408
9.649
8.201 8.849 8.339 9.015 9.194 8.339 9.939 8.574 8.
326 9.566
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How far is the fitted line from the observations?

Error observed value of y predicted value of y
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Computing the errors
Predicted GPA (
Error (y-
1.192
-0.849
-0.801 1.151 -0.539 0.185 0.406 0.061 1.261 -0.974
0.474 -1.566
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Example of correlation and regression

• The manager of a small restaurant studied the
  absentee rate of employees. Whenever employees
  called in sick, or simply didnt appear, the
  restaurant had to find replacements in a hurry,
  or else work short-handed. The manager had data
  on the total number of absences of each employee
  per month (y) and the number of months
  experience at the restaurant (x) for 10
  employees. The manager expected that longer-term
  employees would be more reliable and absent less
  often.

The data
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Scatterplot of the data
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• Correlation between x and y
• X Y
• Sx Sy

22.44
5.9
2.658
2.99
r -0.875
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• Fitting a straight line

Slope
Intercept
Minitab CORRELATION restaurants.MPJ
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Minitab output statgtregressiongtregression Regres
sion Analysis absences versus experience The
regression equation is absences 28.01 - 0.985
experience
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Minitab output statgtregressiongtfitted-line plot

• The regression line

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• What number of absences would the model predict
  for a worker that has 22 months experience?
• The value of experience is x22
• The predicted absences according to the model
  are

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• The actual number of absences for an employer
  who has 22 months experience was 7 days. What is
  the residual (error) for this observation?
• 7 6.334 0.666 days

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• For what experience level was the residual the
  largest?
• Was the Y larger or smaller than what the
  model predicted?
• hint To find the largest residual, look for the
  point on the scatterplot that is furthest from
  the regression line in the vertical direction.
• For 20 months experience the absences were
  larger than predicted

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• Describe how the model relates the experience
  to
• the absences
• The slope is -0.985, or about one
• For each extra month of experience we would
  predict the number of absences to decrease by
  about one day relative to the month before.

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• Would you recommend using this model to
  predict the absence days for employees that are
  less than one year in the restaurant? Why or why
  not?
• The answer is no.
• Explanation
• The predictions are known to be valid only in
  the range of the explanatory variable used to
  create the prediction equation. In this case,
  that is from 18 to 27.3 months.

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More regression questions

• Suppose we developed the following least
  squares regression equation Y 14 1.5X. Which
  of the following statements is correct?
• A. The equation crosses the Y axis at 14 B. The
  equation crosses the Y axis at 1.5
• C. the equation crosses the Y axis at 0
• D. None of the above

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• Couples who share more similar attitudes
  indicate that they are more satisfied with their
  relationship.
• This reflects a ___________ correlation.
• positive / negative

positive
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A study was conducted to examine whether the age
at which a child begins to talk can predict later
Gessel score on a test of mental ability.
Age score 15 95 26 71 10 83 9 91 15 102 20
87 18 93 11 100 8 104 20 94 7 113 9 96 10
83 11 84 11 102 10 100 12 105 42 57 17
121 11 86 10 100
The regression equasion is score 109.874 -
1.12699 age. Draw the straight line. Hint
in order to draw a straight line you need 2
points. Use ages 10 and 40 as your points Age
predicted score 10 ____________ 40
____________
98.604
64.7944
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Regression linescore 109.874 - 1.12699 age
Draw the straight line
(we need just 2 points to draw the line)
Age predicted score 10
109.874-1.12699(10) 98.604 40
109.8741.12699(40) 64.7944
r -0.640
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Right or wrong?

• When correlation between variables x and y is
  very high, it implies that variable x causes y.
• It is o.k. to predict the y value of a point that
  is outside the range of the x observations.
• A correlation of -0.93 means a very weak linear
  relationship.
• An influential observation should always be
  removed from the data.
• The following linear equation describes the
  relationship between sales of a product (in
  thousands of dollars) and days of advertising
  sales32x. This means that for every extra day
  of advertising we obtain extra 3 thousand
  dollars.