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E.g., using the worksheet in Figure 1 of Regression Analysis, we note that SS Reg= DEVSQ(K5:K19) and SS Res= DEVSQ(L5:L19). Observation: There are many ways of calculating SS Reg, SS Resand SS T. 05 = α, and so once again we reject the null hypothesis. Alternatively, we note that p-value = FDIST( F, df Reg, df Res) = FDIST(13.5, 1, 13) = 0.0028 <. Since F crit= FINV( α, df Reg, df Res) = FINV(.05, 1, 13) = 4.7 < 13.5 = F, we reject the null hypothesis, and so accept that the regression line is a good fit for the data (with 95% confidence). Fitting a line to your log-log plot by least squares is a bad idea. We now calculate the test statistic F = MS Reg/ MS Res= 857.0/63.6 = 13.5. The purpose of regression analysis is to evaluate the effects of one or more independent variables on a single dependent variable. Abusing linear regression makes the baby Gauss cry. From these values, it is easy to calculate MS Regand MS Res. By Property 1 of Regression Analysis, SS Res= SS T – SS Reg= 1683.7 – 857.0 = 826.7. We note that SS T= DEVSQ(B4:B18) = 1683.7 and r = CORREL(A4:A18, B4:B18) = -0.713, and so by Property 3 of Regression Analysis, SS Reg= r 2 Observation: Linear regression can be effective with a sample size as small as 20.Įxample 1: Test whether the regression line in Example 1 of Method of Least Squares is a good fit for the data.įigure 1 – Goodness of fit of regression line for data in Example 1 The homogeneity of the variance assumption is equivalent to the condition that for any values x 1 and x 2 of x, the variance of y for those x are equal, i.e. See Multivariate Normal Distribution for more information about this distribution.
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In fact the normality assumption is equivalent to the condition that the sample comes from a population with a bivariate normal distribution. Normality of the error term distribution.Observation: The use of the linear regression model is based on the following assumptions: To test the above null hypothesis we set F = MS Reg/ MS Res and use df Reg, df Resdegrees of freedom. We now express the null hypothesis in a way that is more easily testable:Īs described in Two Sample Hypothesis Testing to Compare Variances, we can use the F test to compare the variances in two samples. If we reject the null hypothesis it means that the line is a good fit for the data. H 0: the regression line doesn’t capture the relationship between the variables In this section we test the following null hypothesis: