Regression

Statistics II

1. REGRESSION Desmond Ayim-Aboagye , Ph.D. LINEAR AND MULTIVARIATE

2. LINEAR REGRESSION Regression is a technique for predicting a score on variable Y based on what we already know to be true about the value of some variable X. Use one variable (i.e., mid term grade) to predict the value of another variable (i.e., final course grade) If correlation is equal to association, then, regression is equal to prediction

3. Bivariate or 2 V ariable Regression Regression analysis is based on correlational analysis, and it involves examining changes in the level of Y relative to changes in the level of X Variable Y is the dependent measure – criterion variable Variable X is independent measure – predictor variable

4. Z-Score Approach to Regression A variable Y can be predicted from X using the Z score regression equation, which is, Z ŷ = rxyZx Whre Z ŷ is a predicted score variable Y. Here ŷ (“Y caret“ or "Y hat") will be used to indicate a predicted or estimated value for Y. The correlation between variables X and Y is denoted rxy and Zx is an actual z score based on variable X.

5. Importance of Z score Equation 1. When rxy is positive in value, zx will be multiplied by a positive number, thus Z ŷ will be positive when Zx is positive and it will be negative when Zx is negative. The characteristic above is important. When rxy is positive, then Z ŷ will have the same sign as Zx, so that a high score will covary with high scores and low scores will do so with low scores. When rxy is negative, however, the sign of Z ŷ will be opposite of Zx ; low scores will be associated with high scores and high scores with low scores. 2. rxy = 1.00, Z ŷ will have the same score as Zx When < 1.00, Z ŷ will be closer to 0.0 than Zx  

6. The role of the Mean When two variables are uncorrelated with one another, the best predictor of any individual score on one of the variables is the mean. The mean is the predicted value of X or Y when the correlation between these variables is 0.

7. Computational Approaches to Regression Linear relationships between variables X and Y Y = a + b (X), Y = criterion variable (trying to predict) a and b constants fixed values X = predictor variable

8. Slope of the line B is also called the slope of the line B = change in Y change in X A = the intercept of the line or y intercept The intercept is the point in a regression of Y and X where the line crosses the Y axis

9. Regression Line A regression line is a straight line projecting through a given set of data, one designed to represent the best fitting linear relationship between variables X and Y

10. Regression toward the mean Regression toward the mean refers to situations where initially high or low observations are found to move closer to or "regress toward" their mean after subsequent measurement.

11. Research tool In practice, we forget that there is really little to be gained from thinking about regression as a way to predict Y from X when we have all the actual values of Y. Regression is really for predicting the behaviour of individuals in samples beyond the original sample. 1. Economists: income as a predictor variable and criterion variables like consumption and savings 2. Management professionals rely on regression to link skills, effort, responsibility, and job conditions to wages. 3. Verify personnel decision 4. Computer science instructor: how doing homework actually predicts their exams performance.

12. Multivariate Regression Multiple regression is a statistical technique for exploring the relationship between one dependent variable (Y) and more than one independent variable (X₁, X₂, … Xn ). Used in behavioral and natural sciences Y = a + b₁ (X₁) + b₂ (X₂).

13. Multiple Regression Analysis A. multiple correlation coefficient B. Symbolized by letter R C. Indicates the relationship between a given criterion variable (Y) and a set of predictor variables (X) . D. As R increases in magnitude , the multiple regression equation is said to perform a better job of predicting the dependent measure from the independent variables. R percentage of variance in Y that is accounted for by the set of predictors, that is, X variables.  

14. Regression Class TEST 1. What is the nature of the relationship between correlation and regression? ( 2 marks ) 2. Define each of the variables and constants in the formula Y= a + bX. ( 3 marks) 3. A student wonders if birth order (first born, second born, and so on) predicts shyness, such that first or only children tend to be shyer than later born children are. The student gives a standardized measure of shyness to 60 participants (30 males, 30 females), asking each one to indicate their number of siblings and their birth orders. Is a regression analysis appropriate? ( 3 marks ) 4. Describe the importance of SPSS to behavioural science research. ( 2 marks)