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Quantitative Methods

Quantitative Methods. Varsha Varde. Quantitative Methods. Models for Data Analysis & Interpretation: Regression Analysis. Cause and Effect. The Present Contains Nothing More Than The Past, and What Is Found In The Effect Was Already In The Cause . - Henri Bergson

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Quantitative Methods

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  1. Quantitative Methods Varsha Varde

  2. Quantitative Methods Models for Data Analysis & Interpretation: Regression Analysis

  3. Cause and Effect The Present Contains Nothing More Than The Past, and What Is Found In The Effect Was Already In The Cause. - Henri Bergson (19th Century French Philosopher) Varsha Varde

  4. Regression Model • A Statistical Model which Depicts the Influence of One Cardinal Variable (The Cause) on Another Cardinal Variable (The Effect). • These Models Have a Wide Variety of Forms and Degrees of Complexity. Varsha Varde

  5. Regression • The Step Logically Next To Correlation. • Situation: Usually, Correlation Between Two Variables Is Not Mere Benign Association. But, It Is In Fact Causation. • It Is a Cause and Effect Relationship, Where X Influences Y. • X is the Cause Variable. • Y is the Effect Variable. Varsha Varde

  6. Some Examples Varsha Varde

  7. Regression • Dictionary Says: The Act of Returning or Stepping Back to a Previous Stage. • Query: Do Quantitative Methods Force Us to Regress instead of Progress? • Or, Is It Back to the Future? • Answer: Statistics, Like Any Other Field, Adopts Crazy Names Arising from Some Important Historical Events. • Soap Opera. Varsha Varde

  8. Story of Regression • Sir Francis Galton Studied the Heights of the Sons in Relation to the Heights of Their Fathers. • His Conclusion: Sons of Tall Fathers Were Not So Tall and Sons of Short Fathers Were Not So Short as their Fathers. • Path Breaking Finding: Human Heights Tend To Regress Back To Normalcy. Varsha Varde

  9. Evolution of the Term ‘Regression’ • Since Then (1880), Similar Studies on Nature and Extent of Influence of One or More Variables on Some Other Variable Acquired the Name ‘Regression Analysis’. • In Quantitative Methods, Regression Means a ‘Cause and Effect Relationship’. • Cause Variable = Independent Variable • Effect Variable = Dependent Variable Varsha Varde

  10. Scatter PlotHorizontal Axis: Reasoning Scores Vertical Axis: Creativity Scores Varsha Varde

  11. Scatter PlotHorizontal Axis: Cause Variable: Reasoning Scores Vertical Axis: Effect Variable: Creativity Scores Varsha Varde

  12. Regression CurveHorizontal Axis: Cause Variable: Reasoning Scores Vertical Axis: Effect Variable: Creativity Scores Varsha Varde

  13. Regression Analysis • A Quantitative Method which Tries to Estimate the Value of a Cardinal Variable (the Effect) by Studying Its Relationship with Other Cardinal Variables (the Cause). • This Relationship is Expressed by a Custom-Designed Statistical Formula Called Regression Equation. Varsha Varde

  14. Purpose of Regression Analysis • To Establish Exact Nature of Influence of Cause Variable on Effect Variable. • To Determine the Quantum of Influence. • To Estimate an Unknown Value of Effect Variable from Value of Cause Variable. • To Forecast Future Values of Effect Variable from Info about Cause Variable Varsha Varde

  15. Patterns of Regression Curves • Pattern: Upward Sloping Straight Line • Mathematical Model: Y = a + bX (b > 0) Varsha Varde

  16. Estimating Regression Parameters a & b • Formula for Regression Coefficient b : Mean of Products of Values – Product of the Two Means = -------------------------------------------------------------------------- Variance of Cause Variable • Formula for Regression Constant a : a = Mean of Effect Variable Minus b times Mean of Cause Variable • Don’t Worry. This is the Job of SPSS. Varsha Varde

  17. Estimating Correlation Coefficient • Recall the Formula for Correlation Coeff. • Pearson’s Correlation Coefficient • Formula: Mean of Products of Values – Product of the Two Means = -------------------------------------------------------------------------- Product of the Two Standard Deviations • Spot the Similarity and the Difference. Varsha Varde

  18. A Simple Example Varsha Varde

  19. Regression Model • Formula for Regression Coefficient b : Mean of Products of Values – Product of the Two Means = -------------------------------------------------------------------------- Variance of Cause Variable (911 / 5) – (5 x 34) 182.2 – 170 12.2 = ----------------------- = ------------- = ----------- = 2.30 2.3 x 2.3 5.3 5.3 • Formula for Regression Constant a : a = Mean of Effect Variable Minus b times Mean of Cause Variable = 34 – 2.3 x 5 = 22.5 • Regression Model: Y = 22.5 + 2.3 X Varsha Varde

  20. Check Goodness of the Model Varsha Varde

  21. Check Goodness of the Model Varsha Varde

  22. Concept: Error of Estimation • Note the Difference Between the Actual Values of Effect Variable (Salary) and the Values Estimated by the Regression Model • This is the Error of Estimation • Less the Error, Better the Model. Ideally 0. • Statistical Model: Y = a + b X + e • If Correlation is Perfect (+1 or -1), e = 0. Varsha Varde

  23. Q5. Given below are five observations collected in a regression study on two variables, x (independent variable) and y (dependent variable). x y 2 4 3 4 4 3 5 2 6 1 a. Develop the least squares estimated regression equation. b. Estimate value of y for x=7. Varsha Varde

  24. Exercise: Fit a Regression Model to Reasoning & Creativity Scores Varsha Varde

  25. Exercise • Does Your Model Look Like What I Got?: Creativity Scores = 5.23 + 0.65 x Reasoning Scores + e • Test the Goodness of Your Regression Model • How Bad are the Errors? Varsha Varde

  26. Other Patterns of Regression Curves • Pattern: Downward Sloping Straight Line • Statistical Model: Y = a - bX + e (b > 0) Varsha Varde

  27. Other Patterns of Regression Curves • Pattern: Simple Exponential Model: Log Y = a + bX + e (b > 0) • Pattern: Negative Exponential Model: Log (1/Y) = a + bX + e (b > 0) • Pattern: Upward Curvilinear Model: Y = a + b Log X + e (b > 0) • Pattern: Downward Curvilinear • Pattern: Logistic or S Curve Varsha Varde

  28. Your Role as a Manager • Grasp the Situation Thoroughly. (Qualitative) • Identify Related Cardinal Variables. (DIY) • Obtain Quantitative Data on Them. • Draw Scatter Plot. Your Asstt Will Do It For You • If It Shows a Pattern, Compute Correlation Coefficient. (Use SPSS or YAWDIFY) • If It Is High (+ or -), Draw a Free Hand Curve and Identify the Pattern of Regression Curve. • Compute Regression Parameters for the Pattern and Fit Regression Model. (SPSS or YAWDIFY) Varsha Varde

  29. A Word of Caution • Undertake Regression Analysis Only For Cardinal Variables. • Select the Variables Only If You Logically Suspect Influence of One Over the Other. • Carry Out Regression Analysis Only After Completing Correlation Analysis AND Only If The Selected Cause and Effect Variables Are Highly Correlated. Varsha Varde

  30. Simple and Multiple Regression • Simple Regression: One Cause Variable Influences the Effect Variable. • This is What We Focused On So Far. • Regression Models Have a Wide Variety of Forms and Degrees of Complexity. • Multiple Regression: Several Cause Variables Jointly Influence Effect Variable. Varsha Varde

  31. Multiple Regression • Multiple Regression Analysis is a Method to Analyze the Effect of Joint Influence of Many Cause Variables on Effect Variable. • Multiple Regression Model: Y = a + b1X1 + b2X2 + - - - - +bnXn + e • Caution: Cause Variables X1, X2, - - - -, Xn Should Not Be Inter-Correlated. • Otherwise You Face Multicollinearity. Varsha Varde

  32. Exercise: Select Cause Variables Varsha Varde

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