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1、Regression Analysis,Chapter 11,Regression and Correlation,Techniques that are used to establish whether there is a mathematical relationship between two or more variables, so that the behavior of one variable can be used to predict the behavior of others. Applicable to “Variables” data only. “Regres
2、sion” provides a functional relationship (Y=f(x) between the variables; the function represents the “average” relationship. “Correlation” tells us the direction and the strength of the relationship. The analysis starts with a Scatter Plot of Y vs X.,The analysis starts with a Scatter Plot of Y vs X,
3、Regression and Correlation,Excel will do Regression analysis and Correlation analysis: Tools Data analysis Regression (Correlation),Simple Linear Regression,Is there a Relationship Between the Variables? What Direction is the Relationship? How Strong is the Relationship?,Low,High,Low,High,. . . . .
4、. . . . . . . . . . .,Y,X,Simple Linear Regression,Is there a Relationship Between the Variables? What Direction is the Relationship? How Strong is the Relationship?,Low,High,Low,High,. . . . . . . . . . . . . . . .,Y,X,?,?,?,Simple Linear Regression,b = Y intercept = the Y value at point that the l
5、ine intersects Y axis.,m = slope =,rise run,Y,X,0,b,rise,run,A simple linear relationship can be described mathematically by Y = mX + b,Simple Linear Regression,Y,X,0,10,5,5,0,Y = 0.5X + 1,Simple regression example,Step 1: Scatter plot,Scatter plot suggests that there is a linear relationship betwee
6、n Rent and Size,Step 2: Analysis via EXCEL,Interpreting EXCEL output,Regression Equation Rent = 177.12082+1.0651439*Size,Meaning of the regression coefficient,What does the coefficient of Size mean? For every additional square feet, Rent goes up by $1.0651493,Using regression for prediction,Predict
7、monthly rent when apartment size is 1000 square feet:,Regression Equation Rent = 177.12082+1.0651439*Size Thus Rent = 177.12082+1.0651439*1000 Rent = $1242.26472,Using regression for prediction Caution!,Regression equation is valid only over the range over which it was estimated! Do not use the equa
8、tion in predicting Y when X values are not within the range of data used to develop the equation. Thus, we should not use the equation to predict rent for an apartment whose size is 500 square feet, since this value is not in the range of size values used to create the regression equation.,Correlati
9、on (r),“Correlation coefficient”, r, is a measure of the strength and the direction of the relationship between two variables. Values of r range from +1 (very strong direct relationship), through “0” (no relationship), to 1 (very strong inverse relationship). It measures the degree of scatter of the
10、 points around the “Least Squares” regression line.,Correlation Levels,Correlation tells us how much linear association there is between two variables.,Coefficient of correlation from EXCEL,The sign of r is the same as that of the coefficient of X (Size) in the regression equation (in our case the s
11、ign is positive). Also, if you look at the scatter plot, you will note that the sign should be positive. R=0.85 suggests a fairly strong correlation between size and rent.,Coefficient of determination (r2),“Coefficient of Determination”, r-squared, (sometimes R- squared), defines the amount of the v
12、ariation in Y that is attributable to variation in X.,Getting r2 from EXCEL,It is important to remember that r-squared is always positive. It is the square of the coefficient of correlation r. In our case, r2=0.72 suggests that 72% of variation in Rent is explained by the variation in Size. The high
13、er the value of r2, the better is the simple regression model.,Standard error (SE),Standard error measures the variability or scatter of the observed values around the regression line.,Getting the standard error (SE) from EXCEL,In our example, the standard error associated with estimating rent is $1
14、94.60.,Using the standard error,For an apartment that is 1000 square feet, we predicted rent to be $1242.26 We can now develop an approximate confidence interval for rent as follows:,Hint: Use the t-distribution when developing confidence intervals for specific levels (such as 90% or 95%). In simple
15、 regression, t has n-2 degrees of freedom.,Is the simple regression model statistically valid?,It is important to test whether the regression model developed from sample data is statistically valid. For simple regression, we can use 2 approaches to test whether the coefficient of X is equal to zero:
16、 using t-test, or, by using ANOVA.,Is the coefficient of X equal to zero?,The hypothesis we want to test is:,Using EXCEL for testing if slope=0,t-stat=7.740398 and P-value=7.52E-08. P-value is very small. If it is smaller than our a level, then, we reject null; not otherwise. If a=0.05, we would reject null and conclude that slope is not zero. Same result hol
