Simple Linear Regression explanation.pptx

These slides describe simple linear regression. Very detail explanation on it. I hope students get more benefits about it.

1. Program: Pharm-D 4 th Semester: II ( 2019-20) Course : Bio-Statistics Course code: PHAR-03428 Class Teacher: DR . LIAQUAT AHMAD email: [email protected] Lecture # 25-27

2.   Simple Regression Definition A regression model is a mathematical equation that describes the relationship between two or more variables. A simple regression model includes only two variables: one independent and one dependent. The dependent variable is the one being explained, and the independent variable is the one used to explain the variation in the dependent variable.

3. Linear Regression Definition   Simple regression model that gives a straight-line relationship between two variables is called a linear regression model . For example the relationship between food expenditure ( d ependent variable) and income (independent variable) There is a relationship between CGPA and the time to study. Linear  means straight line

4. Multiple Regression Multiple regression analysis is a straight forward extension of simple regression analysis which allows more than one independent variable. OR When we are considering the relationship between one dependent variable and more than one independent variable, we use Multiple Regression.

5. Calculate regression line for the following data using reticulocytes (x) and lymphocytes (Y). Also calculate standard error of estimate and r2.

6. Linear Regression Model The regression model is y = a + bx + Є , a is called the y-intercept or constant term, b is the slope, and Є is the random error term. The dependent and independent variables are y and x, respectively.

7. SIMPLE LINEAR REGRESSION ANALYSIS y = a + bx Y - is the dependent variable a is the constant term known as intercept B is slope of the line or the regression coefficient x – is the independent variable

8. SIMPLE LINEAR REGRESSION ANALYSIS In the model ŷ = a + bx , a and b, which are calculated using sample data, are called the estimates of a and b =

9. Purpose of Simple Linear Regression The purpose of simple linear regression analysis is to answer three questions : Is there a relationship between the two variables ? How strong is the relationship (e.g. trivial, weak, or strong)? What is the direction of the relationship (high scores are predictive of high or low scores)?

10. Interpreting the Regression Equation: the Intercept The intercept is the point on the vertical axis where the regression line crosses the axis. It is the predicted value for the dependent variable when the independent variable has a value of zero . This may or may not be useful information depending on the context of the problem.

11. Interpreting the Regression Equation: the Slope The slope is interpreted as the amount of change in the predicted value of the dependent variable associated with a one unit change in the value of the independent variable . If the slope has a negative sign, the direction of the relationship is negative or inverse, meaning that the scores on the two variables move in opposite directions . If the slope has a positive sign, the direction of the relationship is positive or direct, meaning that the scores on the two variables move in the same direction.

12. Example Using X a s the independent and Y the dependent variable

13.   Interpretation of a and b cont. Interpretation of b The value of b in the regression model gives the change in y due to change of one unit in x We can state that, on average, a 1 increase in age of a person will increase the weight by Weight = 4.675 + 0.92 (1) = 5.595

14. Interpreting the Regression Equation: the Slope equals 0 If there is no relationship between two variables, the slope of the regression line is zero and the regression line is parallel to the horizontal axis . A slope of zero means that the predicted value of the dependent variable will not change, no matter what value of the independent variable is used.

15. Error Sum of Squares (SSE) The error sum of squares, denoted SSE, is the values of a and b that give the minimum SSE are called the least square estimates of A and B, and the regression line obtained with these estimates is called the least square line . the least squares regression line ŷ = a + bx

16. Standard Deviation of Regression Line or Standard error of estimate

17. Standard Error of estimate

18. COEFFICIENT OF DETERMINATION The coefficient of determination, denoted by r2, represents the proportion of SST that is explained by the use of the regression model. The computational formula for r2 is and 0 ≤ r2 ≤ 1

19. Example # 2

20. Statistics