DOC PREVIEW
WCU ECO 252 - Analysis of Variance

This preview shows page 1-2-3-4 out of 12 pages.

Save
View full document
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
View full document
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience
Premium Document
Do you want full access? Go Premium and unlock all 12 pages.
Access to all documents
Download any document
Ad free experience

Unformatted text preview:

1 252anova 1 26 07 Open this document in Outline view Roger Even Bove F ANALYSIS OF VARIANCE 1 1 Way Analysis of Variance a The ANOVA model relation to regression The one way ANOVA model is used to compare the means of more than two samples taken from populations that are all assumed to have the same variance Each sample called a treatment is usually represented as a column but there is no requirement that each column have the same number of items in it xij a j eij where i 1 through n j and n n j We thus have m We will assume the model j 1 m treatments n j items in each column and a total of n observations Thus x ij should be the number in column i and row j b An ANOVA problem The following data describes monthly expenses for energy in three random samples of essentially identical homes Each column represents expenses on one fuel 05 Fuel 1 2 3 Sum Our hypotheses are 89 101 87 87 364 104 120 98 110 432 86 98 100 96 380 H 0 1 2 3 In the notation used here H 1 Not all equal that a mean has been taken that is x j is the mean of column x 1 i is replaced by a dot to indicate j in particular the mean of column 1 is 364 432 380 91 x 2 108 and x 3 95 The overall or grand mean is the mean of 4 4 4 all the numbers in the problem and is often indicated by x but x seems to be a more appealing 364 432 380 98 12 We compute three sums of squares i The total sum of squares is the same thing as the numerator of the sample variance of the numbers in notation x the problem SST x ij x j i 2 89 98 2 101 98 2 2 87 98 87 98 2 104 98 2 120 98 2 98 98 2 110 98 2 86 98 2 98 98 2 1148 100 98 2 96 98 2 2 ii The sum of squares within treatments has the same number of terms but highlights the contribution to the total sum of squares generated by the difference between the individual numbers and the column treatment means xij x j 2 SSW j i 89 91 2 101 91 2 2 87 91 87 91 2 104 108 2 120 108 2 98 108 2 110 108 2 86 95 2 98 95 2 516 iii The sum 100 95 2 96 95 2 of squares between treatments also has the same number of terms but it highlights the contribution to the total sum of squares generated by the difference between the column treatment means and the overall mean x SSB j i j x 2 91 98 2 91 98 2 2 91 98 91 98 2 108 98 2 108 98 2 108 98 2 108 98 2 95 98 2 95 98 2 632 95 98 2 95 98 2 2 But because of the repetition of the column mean this can be simplified to SSB n j x j x j 4 91 98 4 108 98 4 95 98 632 But note that SSB SSW SST so that the computation of one of the three sums of squares is unnecessary The material is summarized in a table like the one below Source SS DF MS F SSB MSB Between SSB m 1 MSB m 1 F MSW SSW Within SSW n m MSW n m Total SST n 1 We fill in the table with the numbers we have computed and compare the F that we have computed with an F with the appropriate significance level and degrees of freedom shown in the DF column If the F that we have computed is larger than the table F reject the null hypothesis F Source SS DF MS F 05 H0 2 9 Between 632 2 316 5 51 Column means equal F 4 26 2 2 2 s Within 516 9 57 333 Total 1148 11 The s for significant difference indicates that the null hypothesis of equality of means has been rejected ns for no significant difference would indicate that the null hypothesis has not been rejected 3 c A format for ANOVA If we use the same simplifications that we use in calculating a sample variance we can get the tableau below Sum nj x j SS x 2j x SSB x SST 1 89 101 87 87 364 Fuel 2 104 120 98 110 432 3 86 98 100 96 380 Sum 1176 4 4 4 12 n x 91 00 108 00 95 00 1176 98 x 12 33260 46920 36216 116396 8281 11664 9025 ij x 28970 x 2 ij 2 j 2 x ij2 nx 2 116396 12 98 2 1148 2 x n j x 2j nx 2 4 91 2 4 108 2 4 95 2 12 98 2 ij x j 4 28970 12 98 2 632 Source SS Between 632 DF MS 2 316 F 5 51 F 05 H0 F 2 9 4 26 Column means equal s Within 516 9 57 333 Total 1148 11 Explanation Since the Sum of Squares SS column must add up 516 is found by subtracting 632 from 1148 Since n 12 the total degrees of freedom are n 1 11 Since there are 3 random samples or columns the degrees of freedom for Between is 3 1 2 Since the Degrees of Freedom DF column must add up 9 11 2 The Mean Square MS column is found by dividing the SS column by MSB the DF column 316 is MSB and 57 333 is MSW F and is compared with F 05 from MSW the F table df 1 2 df 2 9 To see this as Minitab output go to 252anovaex1 d Confidence Intervals i A single Confidence Interval If we desire a single interval we use the formula for the difference between two means when the variance is known For example if we want the difference between means of column 1 and column 2 1 1 1 2 x 1 x 2 t n m s where s MSW 2 n1 n 2 4 ii Scheff Confidence Interval If we desire intervals that will simultaneously be valid for a given confidence level for all possible intervals between column means use 1 2 x 1 x 2 m 1 F m 1 n m s 1 1 n1 n 2 iii Bonferroni Confidence Interval n m If we only need k different intervals use 1 2 x 1 x 2 t 2 k s 1 1 n1 n 2 iv Tukey Confidence Interval This also applies to all possible differences s 1 1 1 2 x 1 x 2 q m n m This gives rise to Tukey s HSD Honestly Significant 2 n1 n 2 Difference procedure Two sample means x 1 and x 2 are significantly different if x 1 x 2 …


View Full Document

WCU ECO 252 - Analysis of Variance

Documents in this Course
Load more
Download Analysis of Variance
Our administrator received your request to download this document. We will send you the file to your email shortly.
Loading Unlocking...
Login

Join to view Analysis of Variance and access 3M+ class-specific study document.

or
We will never post anything without your permission.
Don't have an account?
Sign Up

Join to view Analysis of Variance 2 2 and access 3M+ class-specific study document.

or

By creating an account you agree to our Privacy Policy and Terms Of Use

Already a member?