Week6 - Goodness of Fit Test(All Slopes equal 0 ANOVA F...

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Unformatted text preview: Goodness of Fit Test (All Slopes equal 0) ANOVA F test is the overall test of the goodness of fit of the model. ANOVA Table Just like before we can obtain an ANOVA table to summarize the different sources of variation in our model. The ANOVA table is given by Source df SS MS F Regression Error p 1-- p n SSR SSE p SSR MSR / = 1 /-- = p n SSE MSE MSE MSR F / = Total 1- n SST Definition: Sums of Squares • Regression Sum of Squares: SSR= ∑ =- n i i y y 1 2 ) ˆ ( • Error Sum of Squares : SSE= ∑ =- n i i i y y 1 2 ) ˆ ( • Total Sum of Squares : SST= ∑ =- n i i y y 1 2 ) ( Recall ANOVA identity: SST=SSR+SSE 1 We can use the F statistic to test the null hypothesis ... : 2 1 = = = = p H β β β . H : ... 2 1 = = = = p β β β (All slope terms equal 0) Ha: ≠ i β for at least one i (At least one of the slope terms does not equal) Test Statistic: MSE MSR F = To compute the p-value of the test we use the 1 ,-- p n p F distribution. Sampling Distribution: F (p, n-p-1) where p= # of predictors Small p-values support Ha (at least one of the x’s is helpful in predicting y) 2 Hypothesis Test for One of the Slopes being equal to Zero Ho: i β = 0 Ha: i β ≠ Test Statistic: i s T i β β ˆ ˆ- = Sampling Distribution: t with df 1 n p υ =-- What is the difference between the ANOVA F test and T test? • ANOVA F test o all β’s together o Determines if the model is good or bad • T test –individual o Individual β’s o Does this prediction variable contribute any significant information after all other predictors are used in the model? 3 Confidence interval for the average y at a certain combination of X’s Formula: Sampling Distribution: t with 1 n p υ =-- Prediction interval formula for the next y at a certain combination of x’s Formula: Sampling Distribution: t with 1 n p υ =-- 4 What is R 2 ? R-square is the square of the sample correlation coefficient r between the observed y values and the predicted y-hat values. What is R 2 (adjusted)? R 2 is measuring how much of the variation of y is explained by all x terms. R 2 gets bigger every time you add a term. However, we want to have an optimum no. of terms as possible. So, R 2 (adjusted) takes out the automatic inflation. If you have multiple x’s, use the R 2 (adjusted). If you have a single x, use the R 2 . ( 29 2 2 2 1 1 k Adjusted R R R n k = - - - - k: no. of independent variables in the model. 5 Example 1: Screws that were threaded into metal blocks were getting stripped. It was thought that the depth that the stud was screwed into the block might affect the torque at which the stud stripped out. In the table, x is measured in 10-3 of and inch and torque is measured in lbs/in....
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This note was uploaded on 08/05/2011 for the course STA 3032 taught by Professor Kyung during the Summer '08 term at University of Florida.

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Week6 - Goodness of Fit Test(All Slopes equal 0 ANOVA F...

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