Regression Analysis: Methods and Examples
Learn about regression analysis, including linear regression, simple and multiple regression, estimating regression functions, and more. Explore examples of independent and dependent variables to gain insights into relationships among variables.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author. If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
statistics For first stage For first stage prepared by prepared by Aseel Aseel Ali Ali
Regression Regression analysis is a statistical method that helps us to analyses and understand the relationship between two or more variables of interest. The process that is adapted to perform regression analysis helps to understand which factors are important, which factors can be ignored and how they are influencing each other. Linear Regression The simplest of all regression types is Linear Regression where it tries to establish relationships between Independent and Dependent variables. The Dependent variable considered here is always a continuous variable.
Examples of Independent & Dependent Variables: x is Rainfall and y is Crop Yield x is Advertising Expense and y is Sales x is sales of goods and y is GDP
If the relationship with the dependent variable is in the form of single variables, then it is known as Simple Linear Regression Simple Linear Regression X > Y If the relationship between Independent and dependent variables are multiple in number, then it is called Multiple Linear Regression Multiple Linear Regression
Simple Linear Regression Model As the model is used to predict the dependent variable, the relationship between the variables can be written in the below format. ??= ??+ ????+ ?? Where, ?? Dependent variable ?? Intercept ?? Slope Coefficient ?? Independent Variable ?? Random Error Term
Estimating linear regression function Estimating linear regression function That the most used method for estimating parameters is the OLS Ordinary least square method. This method is based on the principle of ((minimizing the sum of squares of errors)) as it seeks to find the parameters that make the sum of squares of error as minimal as possible. ? = ??+ ??? ??= ???? ? ?? ?? ??? ? ??? ??= ? ?? ?
??? ???, , ???= ?? ? ? ? ??= = (?? ?)(?? ?) , , ???= ?? ( ?)? = (?? ?)?. . ?? ? ? ? ?? ( ?)? ? ? and and ? are the mean values of are the mean values of ? and ? respectively . respectively . and the estimate of the error variance and the estimate of the error variance is is ??? ? ? ? (?? ?)(?? ?) (?? ?)? ??= = and ??=
Where ? ? ? ??2 ?0 ?? ?1 ??? = ???? ?=1 ?=1 ?=1 Example Example : : Car Plant Electricity Usage The manager of a car plant wishes to investigate how the plant s electricity usage depends upon the plant s production.
For a specific value of the explanatory variable ?, this equation provides a fitted value ? |? = ?0+ ?1? for the dependent variable ? , as illustrated in Figure:
Production Production ($ million) ($ million) 4.51 3.58 4.31 5.06 5.64 4.99 5.29 5.83 4.70 5.61 4.90 4.20 Electricity Usage Electricity Usage (million k (million kWh 2.48 2.26 2.47 2.77 2.99 3.05 3.18 3.46 3.03 3.26 2.67 2.53 Wh) ) January February March April May June July August September October November December
For this example ?=? ????= ?.?? + + ?.?? = ??.?? ?? ??= ?.?? + + ?.?? = ??.?? ?=? ?? ???= (?.??)?+ + ?.???= ???.???? ?=? ?? ???= (?.??)?+ + ?.???= ??.???? ?=? ?? ????= ?.?? ?.?? + + ?.?? ?.?? = ???.???? ?=?
?? ?? ?? ??)/ ??? ? ??? ??= ? ???? ( ??)( ?=? ?=? ?=? ?? ???.???? (??.?? ??.??) ?? ???.???? (??.??)? = = ?.????? ??.?? ?? ?.????? ??.?? ??= ? ?? ? = = ?.???? ??
The fitted regression line : The fitted regression line : ? = ??+ ??? = ?.??? + ?.???? ? |? = ??+ ??? ? |?.?= ?.??? + ?.??? ?.? = ?.???? ? ? ? ??? ?? ?=? ?? ?? ?=? ? ? ?=? ???? ??= ??.???? ?.???? ??.?? (?.????? ???.????) ?? ?.???? = ?.???? = ? = X= 10000 . 1000 : x=10
