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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 |

language: | English |

pages: | 31 |

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