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# Multiple Linear Regression

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Multiple Linear Regression

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### Multiple Linear Regression

1. 1. • Many applications of regression analysis involve situations in which there are more than one regressor variables. • A regression model that contains more than one regressor variables is called a multiple regression model. Introduction Multiple Linear Regression Model Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com Contact me on Email Address to make Project Reports or using SPSS to evaluate results
2. 2.  A study conducted to assess the satisfaction levels of staff from an educational institution with branches in a number of locations across the country.  Staff were asked to complete a short, anonymous questionnaire (shown later in this Appendix) containing questions about their opinion of various aspects of the organization and the treatment they have received as employees. Multiple Linear Regression Model Problem Statement Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
3. 3. • There are many variables exists in the research therefore factor analysis has been conducted to make decision which variables are best fit for the model by Extraction Method. • The factor analysis extracted Q1,Q2,Q3 but when Including the variables Q4,Q5and Q6 also gave the right results Factor Analysis of Variable to fit into the model Multiple Linear Regression Model Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
4. 4.  The error term has a normal distribution with a mean of 0.  The variance of the error term is constant across cases and independent of the variables in the model.  There is no multicolinearity Multiple Linear Regression Model Assumptions: Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
5. 5. Multiple Linear Regression Model • For example, suppose that the Satisfaction level of staff is depend on the Q1,Q2,Q3,Q4,Q5 and Q6. A possible multiple regression model could be Model Y=β0+ β1Q1+ β2Q2+ β3Q3 +β4Q4+β5Q5+β6Q6+E Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
6. 6. Multiple Linear Regression Model where Y - Total Staff Satisfaction Scale (Dependent) Q1 - Is it clear what is expected of you at work? (Independent) Q2 - At work have you been provided with all the equipment and materials required for you do your work efficiently? (Independent) Q3 - Does the organization keep you up to date with information concerning development and changes? (Independent) Q4 - Do you receive recognition from the organization for doing good Work ?(Independent) Q5 - Does your manager or supervisor encourage your development at work?(Independent) Q6 - Do you feel that your opinions seem to count to the organization? (Independent) E - Error Term (Influence of other Variables/factors) Introduction Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
7. 7. Multiple Linear Regression Model Example: Table 1.1 http://www.allenandunwin.com/spss/data_files.html Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
8. 8. Multiple Linear Regression Model Example: Table 1.2 Education Institution Staff Survey conducted in across Australia Example. 536 Observations has been taken to check the reliability of the Multiple Linear Regression Model. By taking Staff Total Satisfaction as dependent variable and Q1,Q2,Q3,Q4,Q5 and Q6 as Independent Variables Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
9. 9. Multiple Linear Regression Model For running the Multiple Linear Regression and testing the result in SPSS the variables has been set according their dependency and independency. By Using Enter Method Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
10. 10. Multiple Linear Regression Model Interpreting Regression Output •The Coefficient of determination or adjusted R2 can be interpreted because of more than one independent variables. •This is the percentage(%) of total variation in the dependent variable due to the independent variable. •We can see that Adjusted R square value 0.920 that our independent variables explain 92.% of the variability of our dependent variable. •Std. Error of the estimate shows that the other variables, that have not been taken in the model, have influence 1.995%. The std Error is low due the number of variables. The larger the size of variables the smaller the std.Error.Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
11. 11. Multiple Linear Regression Model Interpreting Regression Output •The F-ratio in the ANOVA table tests whether the overall regression model is a good fit for the data. The table shows that the independent variables statistically significantly predict the dependent variable, F = 944.111 Sig is < .05 (i.e., the regression model is a good fit of the data). Analysis of variance (ANOVA) Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
12. 12. Multiple Linear Regression Model Co-linearity diagnostics /Co-linearity issue Interpreting Regression Output • Colinearity (Multicolinearity) is the undesirable situation where the correlations among the independent variable are strong. •When two X variables are highly correlated, they both convey essentially the same information. When this happens, the X variables are colinear and the results show multicolinearity. •To check the issue Colinearity diagnostics has been selected. The Variation Inflation Factor and Significant values determine the multicolinearity issue. If VIF value is greater than >2 or >3 then the Multicolinearity exists and if Significant value >0.05 shows which independent variable highly insignificant. The higher the significant value the higher the insignificant independent variable. Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
13. 13. Multiple Linear Regression Model Interpreting Regression Output •The higher the beta value shows the more influential the independent variable and sign (Positive,Negative) shows the nature of the relationship between independent and dependent variable. The Q5a has higher influence. •By having a glance on the sig column we can interpret. The significance level should be below the <0.05. •We can see that the all independent variables are <0.05 means that they are statistically significance. •The VIF of the all independent variables are <3 which shows that the multicolinearity issue does not exist whereas the VIF should be near to 2 or <3. Interpretation of Coefficients Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
14. 14. Multiple Linear Regression Model Histogram showing the normality of the data Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
15. 15. Multiple Linear Regression Model The PP Plot showing the best fit of linearity of the data Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
16. 16.  1. Increasing the sample size is a common first step since when sample size is increased, standard error decreases (all other things equal). This partially offsets the problem that high multicollinearity leads to high standard errors .  2. Remove the most intercorrelated variable(s) from analysis. This method is misguided if the variables were there due to the theory of the model, which they should have been.  3. Take transformation of variables which is the best fir for model. Multiple Linear Regression Model Interpreting Regression Output Remedies of Multicolinearity Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
17. 17. Multiple Linear Regression Model Interpreting Regression Output Remedy of Colinearity 1. Enter Method: It embraces all the variables that exist in the model and determines whether the variables are significant or insignificant. 2. Stepwise Method: it embraces significant variables and excludes insignificant variable. 3. Forward Method: It embraces the highly significant variables and order/rank them but not to insignificant variables. 4. Backward Method: It will consider the highly insignificant variables and order/rank them but not to significant variables. Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com
18. 18.  Using Enter method in Multiple Linear Regression, I have selected a "best" model for predicting Satisfaction level of staff of Educational Institution across Australia.  With this model, I found two Q4 and Q5 in the model were high performing in all the across the institution, while no predictors are insignificant.  The histogram showing the normality of data and PP Plots also reflects the best fit of the model because all the points are on the linear line Multiple Linear Regression Model Summary: Rizwan Manzoor (MSc Finance) rizman2004@yahoo.com