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PUAD 6060 - Fall 2016Homework 41. What is linear regression? Explain the difference between a bivariate, orsimple, linear regression analysis and a multiple regression analysis.Linear regression is the most commonly used technique for predictive analysis. Theregression is used to explain the relationship between a dependent variable and one or moreindependent variables when the relationship between the variables can be described using alinear model. The value of the dependent variable is predicted from that of the independentvariable (CAMO, 2016).In a simple or bivariate linear regression analysis, a single independent variable is used todetermine the value of the dependent variable. It is called bivariate linear regression as itinvolves only two variables, one dependent and one independent variable.In a multiple linear regression analysis, two or more independent variables are used todetermine the value of the dependent variable. It is multiple regression analysis as it involvestwo or more independent variables and one dependent variable.2. What do we mean by the term “model” or “regression model?”A regression model is used to relate the dependent variable (usually represented as Y) withindependent variable (usually represented as X) and unknown parameters (usuallyrepresented as β). A regression model relates Y to a function of X and β,Y=f(X,β)The model includes certain statistical assumptions and the model provides tools for predictionof the independent variables as well as for finding a solution for the unknown parameters.3. What are regression coefficients? What does it mean when acoefficient is statistically significant?Regression coefficient is the constant which represents the rate of change of the dependentvariable (usually y) as a function of the change in the independent variable (say x). It is theslope of the regression line.(y=ax+b)
If a coefficient is statistically significant it means that there exists a significant relationshipbetween the dependent and the independent variables that is, the change in independentvariable will result in a significant change in the dependent variable.4 What is the R-square statistic? Explain why it can be interpreted as apercentage.R-square statistic is a measure of how close the data is to the fitted regression line. It isgenerally known as the coefficient of determination.R-square can be determined as a percentage as it is defined by the formula, R-Square = Explained Variation / Total VariationIt is the percentage of response variable variation as explained by the linear model.A 0% value of R-square indicates that none of the variability of the response data is aroundits mean. A value of 100% indicates that all the variability of the response data is around themean (Frost, 2013). 5. Using the model and variable definitions presented below, answerthe following questions:5a. How much variation of the dependent variable does your modelexplain? How do you know?12.3% of the variation of the dependent variable (Index of Stability) is explained by themodel. We can get this value by looking at the R square value. R square is a statistical measure todetermine how close the data is to the fitted regression line. It can also be defined aspercentage of variation in dependent variables which are explained by independentvariables in a model.5b. Are any of the regression coefficients statistically significant? Whichcoefficients? What is the probability that these relationships are notrandom? How do you know?Yes, 7 out of 8 variables are statistically significant. Taking significance level at 0.05, we can find the p value from t score. T score can becalculated by dividing the Variable coefficients by respective SE coefficient.P values of statistically significant variables are mentioned along with their respective t scorebelow:
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