POLI 443 Applied Political Research Session 12: Linear Regression Analysis Lecturer: Prof. A. Essuman-Johnson , Dept. of Political Science Contact Information: aessuman-johnson@ug.edu.gh College of Education School of Continuing and Distance Education 2014/2015 – 2016/2017
Linear Regression Analysis • Regression analysis deals with how changes in one variable affect changes in another variable. In other words regression is about predicting the value of one variable on the basis of the known values of the other variable. Of the two variables, one is called the dependent and the other the independent. The dependent variable is the variable to be predicted and the independent variable provides the basis for the prediction. The dependent variable is denoted as the Y variable and the independent variable is denoted as the X variable. Slide 2
• The prediction is based on the least square lines or the lines of best fit. This is the line for which the sum of the squared deviation is minimum. The equation for the line of best fit for the prediction of Y is given by the following: • Y = a + bX • Where: • a = the point at which on the graph at which the regression line crosses the Y axis or the Y intercept. Slide 3
a • b = the slope of the regression line i.e. the amount by which Y increases or decreases when X increases by one unit. Y b a X Slide 4
• The regression line of Y on X is given by the equation • Y = a + bX • Where: • b = N∑XY - ∑X∑Y/N∑X 2 – (∑X) 2 a = ∑Y - b∑X /N • Y scores are here predicted from X scores i.e. Y is the dependent variable. • We can also predict X based on Y values in which case the predicting equation will be • X = a + bY • b = N∑XY - ∑X∑Y/N∑Y 2 – (∑Y) 2 a = ∑X - b∑Y /N • The normal prediction equation is predicting Y on X Slide 5
Illustration • Calculate the regression line of Y on X for the following data on the relationship between Income (Y) per annum (‘000) and number of years in school (X) for 8 persons. Predict the income of someone who had 12 years of school? Person A B C D E F G H Income (Y) 1 2 4 4 5 7 8 9 Years in School(X) 1 3 4 6 8 9 11 14 Slide 6
• Working Sheet X Y X 2 Y 2 XY 1 1 1 1 1 3 2 9 4 6 4 4 16 16 16 6 4 36 16 24 8 5 64 25 40 9 7 81 49 63 11 8 121 64 88 14 9 196 81 126 ∑X =56 ∑Y= 40 ∑X 2 =524 ∑Y 2 =256 ∑XY= 364 Slide 7
• N = 8 • b = N∑XY - ∑X∑Y/N∑X 2 – (∑X) 2 • 8(364) – (56)(40)/8(524) – (56) 2 • 2912 – 2240/4192- 3136 • 672/1056 • 0.636 • 0.64 • a = ∑Y - b∑X /N • 40-(0.64)(56)/8 • 40-35.84/8 • 4.16/8 • 0.52 Slide 8
• The predicting equation will be as follows: • Y = a+bX • 0.52 + 0.64X • For someone who had 12 years of school then X = 12, therefore the predicted income will be 0.52 + 0.64 (12) = 0.52 + 7.68 = 8.2 • The predicted income for someone who has had 12 years schooling will be GH¢8,200 per annum. Slide 9
Summary • In this section, you have learned how to compute regression analysis and its application. It should now be possible to make political predictions as it relates to election polls. Slide 10
• Introduction • The ability to predict the outcome of political events has engaged the attention of Political Scientists especially when it relates to elections. Regression analysis is about the use of statistics to help Political Scientists to do prediction. Slide 11
Assignment The following table gives the marks obtained by 10 students in URC : 200 (X) together with the marks obtained in the exam in POLI 308 (Y). • URC 200 (X) 8 8 9 10 10 11 12 13 13 11 14 • POLI 308 (Y) 7 11 8 7 12 11 10 12 14 17 15 • State the equation of the regression line of Y on X. • What does the Y-intercept of the regression line indicate? • What does the slope of the regression line indicate? • If a student was absent from POLI 308 but scored 18 in URC 200, state the regression line, which would be suitable for estimating his/her possible mark in POLI 308 and work out a fair estimate for his /her possible mark in POLI 308. Slide 12
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