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Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA). 2014/04/28. Purpose of ANOVA. The purpose of ANOVA (Analysis of Variance) is to test the equality of three or more population means. The general form of the hypothesis test: Example. P492. Chemitech , Inc. H 0 :  1  =   2  =   3  =  . . . =  k.

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Analysis of Variance (ANOVA)

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  1. Analysis of Variance (ANOVA) 2014/04/28

  2. Purpose of ANOVA • The purpose of ANOVA (Analysis of Variance) is to test the equality of three or more population means. • The general form of the hypothesis test: • Example. P492. Chemitech, Inc. H0: 1=2=3=. . . = k Ha: Not all population means are equal

  3. Assumptions of ANOVA • For each population, the dependent variable is normally distributed. • The variance of the dependent variable is the same for all of the populations. • The observations must be independent. • Note • Response variable is referred as dependent variable. • Factor or treatment is referred as independent variable.

  4. Concepts of ANOVA • If null hypothesis is true, all treatment populations have the dame distribution. =>(1)Between treatment estimate of population variance • Each of the sample variances provides an unbiased estimate of population variance. =>(2)Within treatment estimate of population variance • If H0 is true, (1) is close to (2), • F=(1)/(2), with k-1, n-k degree of freedom • If F is too large, then reject H0

  5. Equations of ANOVA • Mean square due to treatments (MSTR): • Mean square due to error (MSE):

  6. ANOVA Table p- Value Source of Variation Sum of Squares Degrees of Freedom Mean Square F Treatments k - 1 SSTR Error SSE nT - k Total SST nT - 1

  7. Multiple Comparison Procedure • Suppose that ANOVA provides statistical evidence to reject the null hypothesis of equal population means, In this case, Fisher’s LSD procedure can be used to determine where the differences occur. • Fisher’s LSD procedure is based on the t test statistic presented for the two population case.

  8. Fisher’s LSD Procedure • Hypotheses • Test Statistic

  9. Randomized Block Design • An experimental design, known as a randomized block design, is to control some of the extraneous sources of variation. • Ex. P 515 Table 13.5 • The randomized block design calls for a single sample of controllers. Each controller is tested with each of the alternatives.

  10. Randomized Block Design • To compute the F statistic for a difference among treatment means with a randomized block design: SST = SSTR + SSBL + SSE

  11. Randomized Block Design p- Value Source of Variation Sum of Squares Degrees of Freedom Mean Square F Treatments k - 1 SSTR Blocks SSBL b - 1 Error SSE (k – 1)(b – 1) Total SST nT - 1

  12. Factorial experiment • A factorial experiment is an experimental design that allows simultaneous conclusions about two or more factors. • Ex. P521 Table 13.9 • Interaction effects are also studied in factorial experiments.

  13. Factorial experiment • To compute the F statistic for a difference among treatment means with a randomized block design: SST = SSA + SSB + SSAB + SSE

  14. Factorial experiment p- Value Source of Variation Sum of Squares Degrees of Freedom Mean Square F SSA Factor A a - 1 Factor B SSB b - 1 Interaction SSAB (a – 1)(b – 1) Error SSE ab (r – 1) Total SST nT - 1

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