ANOVA is comparison of means Each possible value of a factor or combination of factor is a treatment The ANOVA is a powerful and common statistical procedure in the social sciences It can handle a variety of situations ID: 1001298
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1. ANOVA
2. ANOVA is an abbreviation for the name of the method: Analysis of Variance introduced by R A Fisher in 1920.ANOVA is comparison of means. Each possible value of a factor or combination of factor is a treatment.The ANOVA is a powerful and common statistical procedure in the social sciences. It can handle a variety of situations.
3. Analysis of Variance (ANOVA)ANOVA is used to test hypotheses about differences between two or more means.The t-test can only be used to test differences between two means.When there are more than two means, it is possible to compare each mean with each other mean using t-test.However, conducting multiple t-tests can lead to severe inflation of the type-I error.ANOVA can be used to test differences among several means for significance without increasing the Type I error rate.
4. Why ANOVA?Widely used in testing of significance of results of test conducting in Agri & Bio Sci.In real life things do not typically result in two groups being compared.Two-sample t-tests are problematic Increasing the risk of a Type I errorAt .05 level of significance, with 100 comparisons, 5 will show a difference when none exists (experiment wise error)So the more t-tests you run, the greater the risk of a type I error (rejecting the null when there is no difference)ANOVA allows us to see if there are differences between means with an OMNIBUS test
5. When ANOVA?Data must be experimentalIf you do not have access to statistical software, an ANOVA can be computed by hand With many experimental designs, the sample sizes must be equal for the various factor level combinationsA regression analysis will accomplish the same goal as an ANOVA. ANOVA formulas change from one experimental design to another
6. MethodologyHo : µ1=µ2=µ3=µ4……..µkH1 : Two of the means are not equalDetermination of Variance Ratio of Variance (F)= Greater Variance/Lesser VarianceF = Variance between the samples/ Variance within the samplesF-test calculated value< Table valueMeaning thereby the evidence against the null hypothesis is not significant.
7. Eg. The figures of yield of crop on 12 plots of land for 3 varieties of Rice. Find out whether there is significant difference in the yield of crop of different varieties. PlotVarietiesAVarietiesBVarietiesC1121810208121631614184121220
8. Table of squaresPlotVarietiesAA2BB2CC2total11214418324101002864121441625631625614196183244121441214420400Total485664T-168Sum of Squares6088081080SOS=2496Squares of sums230431364096TSOS=9536Correction Factor = T2/n168x168/12 =2352Total SOS = Total SOS- Correction factor2496-2352 = 144SOS (between samples) = TSOS/ N- C.F.9536/4-2352 = 32SOS ( within samples) = TSOS- SOS(between Samples)144-32 = 112
9. Cont’dSOSD.f.VarianceBetween samples32216With in samples112912.44Total14411Ratio of Variance (F) = Variance (between samples)/ Variance( within samples) = V1/V2 = 16/12.44 F = 1.37 (Calculated Value)Since table value for 2 and 9 degrees of freedom at 5 % level of significance is 4.26.Thus F value 1.37( calculated value) < 4.26 (table value) so the evidence against the null hypothesis is not significant.
10. Variance to compare MeansWe are applying the variance concept to meansHow do means of different groups compare to the overall meanDo the means vary so greatly from each other that they exceed individual differences within the groups?
11. Between/Within GroupsVariance can be separated into two major componentsWithin groups – variability or differences in particular groups (individual differences)Between groups - differences depending what group one is in or what treatment is receivedFormulas: page 550
12. Fundamental ConceptsYou are able to compare MULTIPLE meansBetween-group variance reflects differences in the way the groups were treatedWithin-group variance reflects individual differencesNull hypothesis: no difference in meansAlternative hypothesis: difference in means
13. Other applicationsANOVA for three or more independent variables.One way ANOVATwo way ANOVARepeated ANOVA
14. Thank you