Statistics-Correlation and Regression Analysis
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The document discusses correlation and regression analysis, including definitions, properties, and formulas relevant to these statistical concepts. It covers topics such as partial correlation, multiple regression, and the coefficient of determination, detailing methods for calculating relationships between variables. Additionally, it addresses error determination, auto correlation, and provides a summary of related calculations and their implications.
Statistics-Correlation and Regression Analysis
- 1. CORRELATION AND REGRESSION ANALYSIS 1 Presented by: Group-E (The Anonymous) Rabin BK (Editor) Bikash Dhakal Bimal Pradhan Gagan Puri Bikram Bhurtel
- 2. Contents Meaning of correlation What is partial correlation? Properties of partial correlation coefficient Co-efficient of Partial determination Regression and Multiple Regression equation Co-efficient of determination What is an error? Determining F-ratio Calculating Co-efficient of Multiple Auto Correlation Preferences 2
- 3. Meaning of correlation Relation or connection between two or more things (in general sense) Interdependence of variable quantities (statistically) 3
- 4. Order of correlation Zero order correlation First order correlation Second order correlation 4 𝑟12, 𝑟13, 𝑟23, ……… 𝑟12.3, 𝑟13.2, 𝑟23.1, ……… 𝑟12.34, 𝑟13.24, ………
- 5. For Zero order correlation, And 5 𝑟12 = 𝑛 𝑋1 𝑋2− 𝑋1 𝑋2 𝑛 𝑋1 2 − 𝑋1 2 𝑛 𝑋2 2 − 𝑋2 2 𝑟13 = 𝑛 𝑋1 𝑋3− 𝑋1 𝑋3 𝑛 𝑋1 2 − 𝑋1 2 𝑛 𝑋3 2 − 𝑋3 2 𝑟23 = 𝑛 𝑋2 𝑋3− 𝑋2 𝑋3 𝑛 𝑋2 2 − 𝑋2 2 𝑛 𝑋3 2 − 𝑋3 2
- 6. What is partial correlation co-efficient? Relationship between two variables keeping the other variable constant/fixed • Relation between X1 and X2 keeping X3 constant is denoted by: • Similarly, means relation between X1 and X3 keeping X2 and so on 6 𝑟12.3 𝑟13.2
- 7. Calculating partial correlation Based on zero order correlation Formulas: 7 𝑟12.3 = 𝑟12 − 𝑟13.𝑟23 1− 𝑟13 2 1− 𝑟23 2 𝑟13.2 = 𝑟13 − 𝑟12.𝑟23 1− 𝑟12 2 1− 𝑟23 2 𝑟23.1 = 𝑟23 − 𝑟12.𝑟13 1− 𝑟12 2 1− 𝑟13 2
- 8. Properties of Correlation co-efficient Its value lies between -1 to +1. 𝑟12 = 𝑟21, 𝑟13 = 𝑟31 and 𝑟23 = 𝑟32 𝑟12.3 = 𝑟21.3, 𝑟13.2 = 𝑟31.2 and 𝑟23.1 = 𝑟32.1 8
- 9. Co-efficient of partial determination Square of partial correlation coefficient Also known as the percent of variation Used to measure variation in one variable explained by other variable keeping next variable constant Example: If 𝑟12.3 = 0.5, then partial determination is: 𝑟12.3 2 = 0.25 = 25% Variation in 𝑋1, Explained by 𝑋2 , Constant = 𝑋3 9
- 10. Multiple correlation Relation between three/more variables at the same time Denoted by R If R<1 (r<1), more consistent If R>1 (r>1), less consistent 10
- 11. Calculating Multiple Correlation If 𝑋1, 𝑋2 and 𝑋3 are three variable then, 11 𝑅1.23 = 𝑟12 2 + 𝑟13 2 − 2𝑟13.𝑟23 𝑟12 1− 𝑟23 2 𝑅2.13 = 𝑟12 2 + 𝑟23 2 − 2𝑟13.𝑟23 𝑟12 1− 𝑟13 2 𝑅2.13 = 𝑟12 2 + 𝑟23 2 − 2𝑟13.𝑟23 𝑟12 1− 𝑟13 2
- 12. Regression A statistical tool used to find the nature of relationship Estimates the value of a dependent variable with the help of an independent variable Types: Regression of y on x is, y = a + bx (a and b are constants) Regression of x on y is, x = a + by (a and b are constants) 12 𝒀 = 𝒏𝒂 + 𝒃 𝑿 𝑿𝒀 = 𝒂 𝑿 + 𝒃 𝑿 𝟐 𝑿 = 𝒏𝒂 + 𝒃 𝒀 𝑿𝒀 = 𝒂 𝒀 + 𝒃 𝒀 𝟐
- 13. Multiple Regression Analysis An extension of simple linear regression Estimates the value of a dependent variable with the help of two independent variables If 𝑋1 is dependent and 𝑋2 and 𝑋3 are independent variables then, 13 𝑿𝟏 = 𝒂 + 𝒃𝟏 𝑿𝟐 + 𝒃𝟐 𝑿𝟑 𝑿𝟏 𝑿𝟐 = 𝒂 𝑿𝟐 + 𝒃𝟏 𝑿𝟐 𝟐 + 𝒃𝟐 𝑿𝟐 𝑿𝟑 𝑿𝟏 𝑿𝟑 = 𝒂 𝑿𝟑 + 𝒃𝟏 𝑿𝟑 𝟐 + 𝒃𝟏 𝑿𝟐 𝑿𝟑
- 14. Multiple Regression equation using simple correlation and standard deviations Multiple regression equation of 𝑋1 on 𝑋2 and 𝑋3 is: Where, Linear equation of 𝑥1 on 𝑥2 and 𝑥3 is: 14 𝑋1 − 𝑋1 = 𝑏12.3 𝑋2 − 𝑋2 + 𝑏13.2 𝑋3 − 𝑋3 𝑏12.3 = 𝜎1 𝜎2 𝑟12 − 𝑟23 𝑟13 1 − 𝑟23 2 𝑏13.2 = 𝜎1 𝜎3 𝑟13 − 𝑟23 𝑟12 1 − 𝑟23 2 𝑥1 = 𝑏12.3 𝑥2 + 𝑏13.2 𝑥3
- 15. Co-efficient of determination It is the degree of explained variation Denoted by 𝑅2 If 𝑅2 > 0.5, it is good fit If 𝑅2 < 0.5, it is less fit 15
- 16. Total variation = Explained variation + Unexplained variation Co − efficient of determination 16 Explained variation = SSR= TSS – SSE SSR= sum of square due to regression TSS= total sum of square SSE= Sum of square due to error R = 𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑡𝑜𝑡𝑎𝑙 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛
- 17. What is an error? Difference between true value and estimated value Degree of freedom (df) = n-1 For standard error we have, OR 17 Standard error = 𝐸𝑟𝑟𝑜𝑟 𝑜𝑓 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 𝐸𝑟𝑟𝑜𝑟 𝑜𝑓 𝑑𝑒𝑔𝑟𝑒𝑒 𝑜𝑓 𝑓𝑟𝑒𝑒𝑑𝑜𝑚 Standard error = 𝑈𝑛𝑒𝑥𝑝𝑙𝑎𝑖𝑛𝑒𝑑 𝑣𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 𝑛 − 3
- 18. Determining F-ratio 1. 2. 18 F − ratio = 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑚𝑒𝑎𝑛 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 1 𝑠𝑡 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑚𝑒𝑎𝑛 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 𝑜𝑓 2 𝑛𝑑 1 𝑠𝑡 𝑀𝑒𝑎𝑛 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 = 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑑𝑓 2 𝑛𝑑 𝑀𝑒𝑎𝑛 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 = 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑞𝑢𝑎𝑟𝑒 𝑅𝑒𝑔𝑟𝑒𝑠𝑠𝑖𝑜𝑛 𝑑𝑓
- 19. Calculating Co-efficient of Multiple determination in terms of Multiple regression 19 𝑅1.23 2 = 𝑎 𝑋1 + 𝑏1 𝑋1 𝑋2 + 𝑏2 𝑋1 𝑋3 − 𝑛 𝑋1 2 𝑋1 2 − 𝑛 𝑋1 2 𝑅2.13 2 = 𝑎 𝑋2 + 𝑏1 𝑋1 𝑋2 + 𝑏2 𝑋2 𝑋3 − 𝑛 𝑋2 2 𝑋2 2 − 𝑛 𝑋2 2 𝑅3.12 2 = 𝑎 𝑋3 + 𝑏1 𝑋1 𝑋3 + 𝑏2 𝑋2 𝑋3 − 𝑛 𝑋3 2 𝑋3 2 − 𝑛 𝑋3 2
- 20. Auto Correlation • Error terms are assumed independent in regression • Difference between the observed value and estimated value is known as error term • Error terms are correlated instead of being independent is known as auto correlation • Formula: 2 22 t tt e ee 2 1 t tt e ee
- 21. Reference: Statistics-I (Vikash Raj Satyal) Teachers note 21
- 22. Queries 22