Introduction. Chapter 8

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Chapter 8 Introduction In general, a researcher wants to compare one treatment against another.

When the null hypothesis of equality of treatment means are ed at some level of significance, we conclude that there exist significant differences

Typically, investigators want more detailed information about which treatments differ from which other treatments The analysis of variance does

classified generally as: pairwise or grouped Pairwise mean comparison can be classified as: planned or unplanned Planned (priori) pairwise

, control vs other treatments) done without doing F-test Unplanned (posteriori) pairwise comparison no specific comparison is chosen in advance

1 Chapter 8 Introduction In general, a researcher wants to compare one treatment against another. The analysis of variance (ANOVA) is a general test for comparing treatment means. When the null hypothesis of equality of treatment means are ed at some level of significance, we conclude that there exist significant differences among the treatment means. Typically, investigators want more detailed information about which treatments differ from which other treatments The analysis of variance does not tell us where these differences lie or which particular treatments have different means Introduction Multiple comparison procedures can be classified generally as: pairwise or grouped Pairwise mean comparison can be classified as: planned or unplanned Planned (priori) pairwise comparison specific pair of treatments to be compared was identified before the start of the experiment (e.g., control vs other treatments) done without doing F-test Unplanned (posteriori) pairwise comparison no specific comparison is chosen in advance every possible pair of treatment means is compared to identify pairs of treatment means that are significantly different done if F-test results in ing the null hypothesis of no difference Introduction The commonly used procedures for pairwise mean comparison are: Least significant difference (LSD) test Honest significant difference (HSD) test Scheffe s (S) test Student-Newman-Keul (SNK) test Duncan s multiple range test (DMRT) 4 1

Used only if the F-test in the ANOVA is significant May not be used when there are more than four treatments to test, but maybe used for preplanned comparisons regardless of the number of treatments

2 Used only if the F-test in the ANOVA is significant May not be used when there are more than four treatments to test, but maybe used for preplanned comparisons regardless of the number of treatments Used for experiments with equal or unequal replications Uses the T table relatively a liberal one since it makes a lot of ions of the null hypothesis Procedure: 1. Arrange the treatment means in descending order.. Compute the difference (d ij ) between the means of the i th and j th treatments: d xi x j. Compute the LSD value at α LSD t ij, v 1 1 MSE ri rj v=error df 5 Ho: μ i = μ j Ha: μ i μ j Decision Rule: Reject Ho if d ij LSD α ; else, do not Ho. This means that the difference between a specific pair of treatment means is significant at the α level of significance if it exceeds LSD α. Example: Replication r i Total

3 ANOVA Table Source of Variation SS df MS F F 0.05 s Error Total.0 16 r i T 5.4 T4 4.5 T1 5 T 1.7 s d ij LSD 0.05 Decision on Ho T vs T Conclusion T vs T Reject Significantly different T vs T Reject Significantly different T4 vs T T4 vs T Reject Significantly different T1 vs T r i T 5.40 a T4 4.5 ab T T 1. s with at least a common letter are not significantly different bc c Example: Consider the RCBD experiment conducted to determine the effect of fertilizer level on the yield of wheat involving three varieties. Four fertilizer levels were tested on wheat planted on pots placed in a greenhouse. Fertilizer Level T Block Total Block Variety of Wheat V 1 V V Total y.. = 5

4 ANOVA Table Source of variation df Sum of Squares Squares Fertilizer Levels Varieties 8 1. Error (Residual) Total F c F LSD t ,6 n i Sample T 5 s dij LSD0.05 Decision on Ho Conclusion T1 vs T Reject Significantly different T1 VS T Reject Significantly different T1 VS T Reject Significantly different T VS T T VS T T4 VS T n i Sample T a b b 5 b s with at least a common letter are not significantly different 4

5 Tukey s Honest Significant Difference Used only if the F-test in the ANOVA is significant Used for experiments with equal replications Uses the studentized range table More conservative than LSD Test statistic Let q α,t,v be the value of the studentized range statistic at a specified α, v error df and t treatments. Then Tukey-Kramer test Tukey (195) and Kramer (19) provided a modification of Tukey s HSD test when the number of replications are unequal Test statistic TK q, t, v MSE 1 1 ri rj HSD q, t, v 1 MSE r Tukey-Kramer test Example1: Refer to the first example used in LSD For this data, MSE=0.078, v=1 and t=4. r i T 5.4 T4 4.5 T1 5 T 1.7 Tukey-Kramer test s d ij TK Decision on Ho Conclusion T vs T T vs T T vs T Reject Significantly different T4 vs T T4 vs T Reject Significantly different T1 vs T

6 Tukey-Kramer test r i T 5.40 a T4 4.5 a T ab T 1.7 b s with at least a common letter are not significantly different Tukey s HSD test Example: Refer to the nd example used in LSD For this data, MSE=18, v=6 and p=4. r i T 5 HSD q 1 18, t, v MSE r Tukey s HSD test s dij HSD Decision on Ho Conclusion T1 vs T T1 VS T Reject Significantly different T1 VS T Reject Significantly different T VS T T VS T T4 VS T Tukey s HSD test r i T 5 a ab b b s with at least a common letter are not significantly different 6

Lec 1: An Introduction to ANOVA

Lec 1: An Introduction to ANOVA Ying Li Stockholm University October 31, 2011 Three end-aisle displays Which is the best? Design of the Experiment Identify the stores of the similar size and type. The displays are randomly assigned to