Bivariate Description

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Bivariate Description

• Heibatollah Baghi, and
• Mastee Badii

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OBJECTI VES

• Define bivariate and univariate statistical
  tests.
• Explain when to use correlational techniques to
  answer research questions.
• Understand measure of Pearson Product Moment
  Correlation Coefficient (Pearsons r).

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Definitions

• Univariate examination of variables frequency
  distribution, central tendency, and variability.
• Bivariate examination of two variables
  simultaneously.
• Is SES related to intelligence?
• Do SAT scores have anything to do with how well
  one does in college?
• The question is do these variables correlate or
  covary?

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Typical Situations

• Two nominal variables
• Gender and readmission status
• A nominal and interval/ratio variables
• Delivery type and weight of child
• Bed rest and weight gain during pregnancy
• Two interval ratio variables
• Respiratory function and extent of anxiety

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Cross Tabulation

• Describes relationship between two nominal
  variables
• Two dimensional frequency distribution

Also appropriate if either or both variables are
ordinal-level with a small number of categories
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Elements of Cross Tabulation
Column
Row
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Elements of Cross Tabulation
Cell count Row Column Total
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Elements of Cross Tabulation
Marginal
Joint distribution
Marginal
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Group Mean Comparison

• Describes a nominal variable and an
  interval/ratio variable

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Linear Association

• The correlation coefficient is a bivariate
  statistic that measures the degree of linear
  association between two interval/ratio level
  variables. (Pearson Product Moment Correlation
  Coefficient)

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Scatter plot

• Reveals the presence of association between two
  variables. The stronger the relationship, the
  more the data points cluster along an imaginary
  line.
• Indicates the direction of the relationship.
• Reveals the presence of outliers.

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Scatter Plot of Positively Correlated Data
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Scatter Plot of Negatively Correlated Data
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Scatter Plot of Non Linear Data
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Scatter Plot of Uncorrelated Data
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Covariance Formula
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Correlation Formula
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Example Data
GPA SAT
ID Y X
A 1.6 400
B 2 350
C 2.2 500
D 2.8 400
E 2.8 450
F 2.6 550
G 3.2 550
H 2 600
I 2.4 650
J 3.4 650
K 2.8 700
L 3 750
Sum 30.80 6550.0
Mean 2.57 545.80
S.D. 0.54 128.73
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STUDENTS Y(GPA) X(SAT)
A 1.6 400
B 2.0 350
C 2.2 500
D 2.8 400
E 2.8 450
F 2.6 550
G 3.2 550
H 2.0 600
I 2.4 650
J 3.4 650
K 2.8 700
L 3.0 750
Sum 30.80 6550.0
Mean 2.57 545.80
S.D. 0.54 128.73
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STUDENTS Y(GPA) X(SAT)
A 1.6 400 -0.97
B 2.0 350 -0.57
C 2.2 500 -0.37
D 2.8 400 0.23
E 2.8 450 0.23
F 2.6 550 0.03
G 3.2 550 0.63
H 2.0 600 -0.57
I 2.4 650 -0.17
J 3.4 650 0.83
K 2.8 700 0.23
L 3.0 750 0.43
Sum 30.80 6550.0
Mean 2.57 545.80
S.D. 0.54 128.73
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STUDENTS Y(GPA) X(SAT)
A 1.6 400 -0.97 -145.80
B 2.0 350 -0.57 -195.80
C 2.2 500 -0.37 -45.80
D 2.8 400 0.23 -145.80
E 2.8 450 0.23 -95.80
F 2.6 550 0.03 4.20
G 3.2 550 0.63 4.20
H 2.0 600 -0.57 54.20
I 2.4 650 -0.17 104.20
J 3.4 650 0.83 104.20
K 2.8 700 0.23 154.20
L 3.0 750 0.43 204.20
Sum 30.80 6550.0
Mean 2.57 545.80
S.D. 0.54 128.73
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STUDENTS Y(GPA) X(SAT)
A 1.6 400 -0.97 -145.80 141.43
B 2.0 350 -0.57 -195.80 111.61
C 2.2 500 -0.37 -45.80 16.95
D 2.8 400 0.23 -145.80 -33.53
E 2.8 450 0.23 -95.80 -22.03
F 2.6 550 0.03 4.20 0.13
G 3.2 550 0.63 4.20 2.65
H 2.0 600 -0.57 54.20 -30.89
I 2.4 650 -0.17 104.20 -17.71
J 3.4 650 0.83 104.20 86.49
K 2.8 700 0.23 154.20 35.47
L 3.0 750 0.43 204.20 87.81
Sum 30.80 6550.0 378.33
Mean 2.57 545.80
S.D. 0.54 128.73
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Calculation of Covariance Correlation
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Correlations in SPSS
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Limitation of the Covariance

• It is metric-dependent

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Properties of Pearson r

• r is metric-independent
• r reflects the direction of the relationship
• r reflects the magnitude of the relationship

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What does positive correlation mean?

• Scores above the mean on X tend to be associated
  with scores above the mean on Y
• Scores below the mean on X tend to be accompanied
  by scores below the mean of Y
• Note for this reason deviation score is an
  important part of Covariance

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What does negative correlation mean?

• Scores above the mean on X tend to be associated
  with scores below the mean on Y
• Scores below the mean on X tend to be accompanied
  by scores above the mean of Y.

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Strength of association

• r2 Coefficient of determination
• 1 r2 Coefficient of non-determination

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Analysis of Relationships
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Take Home Lessons

• Always make a scatter plot
• See the data first
• Examining the scatter plot is not enough
• A single number can represent the degree and
  direction of the linear relation between two
  variables