Correlation AnalysisCorrelation AnalysisCorrelation meas.docx
- 1. Correlation Analysis Correlation Analysis Correlation measures the relationship between two quantitative variables Linear correlation measures if the ordered paired data follow a straight-line relationship between quantitative variables. The correlation coefficient (r) computed from the sample data measures the strength and the direction of a linear relationship between two variables. The range of correlation coefficient is -1 to +1. When there is no linear relationship between the two variables or only a weak relationship, the value of correlation coefficient will be close to 0. Things to Remember Correlation coefficient cutoff points +0.30 to + 0.49 weak positive association. + 0.5 to +0.69 medium positive association.
- 2. +0.7to + 1.0 strong positive association. - 0.5 to - 0.69 medium negative association. - 0.7 to - 1.0 strong negative association. - 0.30 to - 0.49 weak negative association. 0 to - 0.29 little or no association. 0 to + 0.29 little or no association. Relationships of Linear Correlation As x increases, no definite shift in y: no correlation. As x increase, a definite shift in y: correlation. Positive correlation: x increases, y increases. Negative correlation: x increases, y decreases. If the points exhibit some other nonlinear pattern: no linear relationship.
- 3. Example: No correlation. As x increases, there is no definite shift in y. Example: Positive/direct correlation. As x increases, y also increases. Example: Negative/indirect/inverse correlation. As x increases, y decreases. Coefficient of linear correlation: r, measures the strength of the linear relationship between two variables. Pearson Correlation formula: Note: r = +1: perfect positive correlation r = -1 : perfect negative correlation Use the calculated value of the coefficient of linear correlation, r, to make an inference about the population correlation coefficient r.
- 4. Example 1: Is there a relationship between age of the children and their score on the Child Medical Fear Scale (CMFS), using the data shown in Table 1? H0: There is no significant relationship between the age of the children and their score on the CMFS Or H0: r = 0 IDAge (x)CMFS (y)18312925394041027511356929782589349844101119117281 2647136421483715935161216171512181323191026201036 Table 1 Scattergram (Scatterplot) Age (x) = Independent variable, CMFS (y)= Dependent variable
- 5. Correlation Coefficient The Results: a. Decision: Reject H0. b. Conclusion: There is evidence to suggest that there is a significant linear relationship between the age of the child and the score on the CMFS. Answers the question of whether there is a significant linear relationship or not Simple Linear Regression Analysis Linear Regression Analysis Linear Regression analysis finds the equation of the line that predicts the dependent variable based on the independent variable. 210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150 Drug A (dose in mg) Symptom Index 210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150 Drug A (dose in mg)
- 6. Symptom Index y = dependent (predicted )variable a = y intercept (constant) b = slope (regression coefficient) of line x = independent (predictor) variable y = a + bx Y X Simple Linear Regression Assumptions: Normality Equal variances Independence Linear relationship Regression analysis establishes a regression equation for predictions For a given value of x, we can predict a value of y How good is the predictor? Very good predictor Moderate predictor 210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150
- 7. Drug A (dose in mg) Symptom Index 210 190 165 150 130 115 100 90 70 60 40 50 30 60 80 60 110 90 140 110 140 130 Drug B (dose in mg) Symptom Index How good is the predictor? R2 For simple regression, Coefficient of determination (R2) is the square of the correlation coefficient Reflects variance accounted for in data by the best-fit line Takes values between 0 (0%) and 1 (100%) Frequently expressed as percentage, rather than decimal Coefficient of Determination (R2) High variance explained Less variance explained 210 190 165 150 130 115 100 90 70 60 40 25 35 60 75 85 100 110 120 130 140 150 Drug A (dose in mg) Symptom Index 210 190 165 150 130 115 100 90 70 60 40 50 30 60 80 60 110 90 140 110 140 130
- 8. Drug B (dose in mg) Symptom Index Previous example: Scatter gram (Scatterplot) Age (x) = Independent variable, CMFS (y)= Dependent variable 95% Confidence Interval Regression Line Correlation Coefficient Coefficient of Determination (R2) (Gives the % of variation) Example 2: A recent article measured the job satisfaction of subjects. The data below represents the job satisfaction scores, y, and the salaries (in thousands), x, for a sample of similar individuals.
- 9. 1. Draw a scatter diagram for this data. 2. Find the equation of the line of best fit. IDSalaries (x)Scores (y)1311723320322134241553518629177231283721 Scatter gram: The Regression Equation: Thus a salary of $30,000 will result in a score of 17. Example 3: Is high school GPA a useful predictor of college GPA, using the data shown in Table 2?IDHS GPACollege GPA14.003.8023.702.7032.202.3043.803.2053.803.5062.802.40 73.002.6083.403.0093.302.70103.002.80 Table 2: Scattergram:
- 10. Results: Correlation Analysis There is a significant linear relationship between high school GPA and College GPA. Regression Analysis 72.1% of the variation in college GPA is explained by high school GPA Regression equation: College GPA = 0.50 + 0.73 HS GPA Conclusion: High school GPA is a useful predictor of college GPA 3 0 2 0 1 0 5 5 4 5 3 5 I n p u t O u t p u t 5
- 14. .560 .535 6.3442 Model 1 R R Square Adjusted R Square Std. Error of the Estimate Predictors: (Constant), Age a. 0 20 40 60 80 100 120 140 160 180 200 050100150200250 1.49.517() 1.49.517(30) Scoresalaries Score =+ =+ Coefficients a 1.490 2.327 .640
- 15. .546 .517 .078 .938 6.613 .001 (Constant) SALARIES Model 1 B Std. Error Unstandardized Coefficients Beta Standardi zed Coefficien ts t Sig. Dependent Variable: Job Satisfaction a. High school GPA 4.54.03.53.02.52.0 College GPA 4.0 3.8 3.6 3.4 3.2 3.0 2.8 2.6 2.4 2.2
- 16. Model Summary .849 a .721 .687 .2678 Model 1 R R Square Adjusted R Square Std. Error of the Estimate Predictors: (Constant), High school GPA a. ANOVA b 1.486 1 1.486 20.725 .002 a .574 8 7.171E-02 2.060 9 Regression Residual Total Model 1 Sum of Squares
- 17. df Mean Square F Sig. Predictors: (Constant), High school GPA a. Dependent Variable: College GPA b. Coefficients a .496 .535 .927 .381 .729 .160 .849 4.552 .002 (Constant) High school GPA Model 1 B Std. Error Unstandardized Coefficients Beta Standardi zed Coefficien ts t Sig. Dependent Variable: College GPA a.