Bivariate Description

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					Bivariate Description


           Heibatollah Baghi, and
               Mastee Badii
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 (Pearson’s r).

                                      2
Definitions
 Univariate: examination of variable’s
  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?

                                                           3
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



                                                   4
Cross Tabulation
 Describes relationship between two
  nominal variables
 Two dimensional frequency distribution
                             Gender
    Readmission status Male     Female Total
    Readmitted         15 (30%) 10 (20%) 25
    Not readmitted     35 (70%) 40 (80%) 75
    Total                    50       50 100

                                                        are
           Also appropriate if either or both variables 5
           ordinal-level with a small number of categories
Elements of Cross Tabulation




                         Column
 Row
                                  6
Elements of Cross Tabulation



                          Cell count
                           Row %
                          Column %
                           Total %



                                   7
Elements of Cross Tabulation




                              Marginal
                  Joint
               distribution


    Marginal
                                         8
Group Mean Comparison
   Describes a nominal variable and an
    interval/ratio variable




                                          9
           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)


                                            10
             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.

                                        11
Scatter Plot of Positively Correlated Data




                                        12
Scatter Plot of Negatively Correlated Data




                                       13
Scatter Plot of Non Linear Data




                                  14
Scatter Plot of Uncorrelated Data




                                15
Covariance Formula

      Cov. & Correlation


               (X - X)(Y - Y)
      SXY    
                     n -1

                SXY
      rXY   
              (SX )(SY )         16
               Cov. & Correlation
  Correlation Formula
                        (X - X)(Y - Y)
               SXY             of
                      Covariancen - 1
                         X&Y


                         SXY
 Standard      rXY                    Standard
Deviation of           (SX )(SY )     Deviation of
    X                                     Y




                                                     17
 Example Data           ID
                             GPA
                              Y
                                      SAT
                                       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
Sum    30.80   6550.0
                        I     2.4     650
Mean   2.57    545.80
                        J     3.4     650
S.D.   0.54    128.73
                        K     2.8     700
                        L         3   750

                                       18
STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
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
                                                           19
S.D.       0.54     128.73
STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
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
                                                           20
S.D.       0.54     128.73
STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
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
                                                           21
S.D.       0.54     128.73
STUDENTS   Y(GPA)   X(SAT)   (Y - Y ) (X - X )   (Y - Y )(X - X )
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
                                                            22
S.D.       0.54     128.73
Calculation of Covariance &
Correlation
  Covariance & Correlation


           (X - X)(Y - Y) 378.33
  SXY                            34.39
                 n -1        11

            SXY          34.39
  rXY                              0.50
          (SX )(SY ) (0.54)(128.73)

                                             23
Correlations in SPSS




                       24
Limitation of the Covariance


            It is metric-dependent
Properties of Pearson r
   r is metric-independent
   r reflects the direction of the
    relationship
   r reflects the magnitude of the
    relationship



                                      26
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



                                                 27
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.




                                             28
   Strength of association

r2 = Coefficient of determination
1 – r2 = Coefficient of non-determination




                                            29
Analysis of Relationships




                            30
    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


                                               31

				
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posted:2/21/2012
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