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Treatment Comparisons ANOVA can determine if there are differences among the treatments, but what is the nature of those differences? Are the treatments measured on a continuous scale? Look at response surfaces (linear regression, polynomials) Is there an underlying structure to the treatments? Compare groups of treatments using orthogonal contrasts or a limited number of preplanned mean comparison tests Use simultaneous confidence intervals on preplanned comparisons Are the treatments unstructured? Use appropriate multiple comparison tests (todays topic)

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Variety Trials In a breeding program, you need to examine large numbers of selections and then narrow to the best In the early stages, based on single plants or single rows of related plants. Seed and space are limited, so difficult to have replication When numbers have been reduced and there is sufficient seed, you can conduct replicated yield trials and you want to be able to pick the winner

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Comparison of Means Pairwise Comparisons –Least Significant Difference (LSD) Simultaneous Confidence Intervals –Tukeys Honestly Significant Difference (HSD) –Dunnett Test (making all comparisons to a control) May be a one-sided or two-sided test –Bonferroni Inequality –Scheffés Test – can be used for unplanned comparisons Other Multiple Comparison Tests - Data Snooping –Fishers Protected LSD (FPLSD) –Student-Newman-Keuls test (SNK) –Waller and Duncans Bayes LSD (BLSD) –False Discovery Rate Procedure Often misused - intended to be used only for data from experiments with unstructured treatments

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Multiple Comparison Tests Fixed Range Tests – a constant value is used for all comparisons –Application Hypothesis Tests Confidence Intervals Multiple Range Tests – values used for comparison vary across a range of means –Application Hypothesis Tests

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Type I vs Type II Errors Type I error - saying something is different when it is really the same (false positive) (Paranoia) –the rate at which this type of error is made is the significance level Type II error - saying something is the same when it is really different (false negative) (Sloth) –the probability of committing this type of error is designated –the probability that a comparison procedure will pick up a real difference is called the power of the test and is equal to 1- Type I and Type II error rates are inversely related to each other For a given Type I error rate, the rate of Type II error depends on –sample size –variance –true differences among means

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Nobody likes to be wrong... Protection against Type I is choosing a significance level Protection against Type II is a little harder because –it depends on the true magnitude of the difference which is unknown –choose a test with sufficiently high power Reasons for not using LSD to make all possible comparisons –the chance for a Type I error increases dramatically as the number of treatments increases

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Pairwise Comparisons Making all possible pairwise comparisons among t treatments –# of comparisons: If you have 10 varieties and want to look at all possible pairwise comparisons –that would be t(t-1)/2 or 10(9)/2 = 45 –thats quite a few more than t-1 df = 9

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Comparisonwise vs Experimentwise Error Comparisonwise error rate ( = C ) –measures the proportion of all differences that are expected to be declared real when they are not Experimentwise error rate ( E ) –the risk of making at least one Type I error among the set (family) of comparisons in the experiment –measures the proportion of experiments in which one or more differences are falsely declared to be significant –the probability of being wrong increases as the number of means being compared increases –Also called familywise error rate (FWE)

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Experimentwise error rate ( E ) Probability of no Type I errors = (1- C ) x where x = number of pairwise comparisons Max x = t(t-1)/2, where t=number of treatments Probability of at least one Type I error E = 1- (1- C ) x Comparisonwise error rate C = 1- (1- E ) 1/x if t = 10, Max x = 45 E = 1-(1-0.05) 45 = 90% Comparisonwise vs Experimentwise Error

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Least Significant Difference Calculating a t for testing the difference between two means –Any difference for which the t calc > t would be declared significant Further, is the smallest difference for which significance would be declared –Therefore –For equal replication, where r is the number of observations forming each mean

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Dos and Donts of using LSD LSD is a valid test when –Making comparisons planned in advance of seeing the data (this includes the comparison of each treatment with the control) –Comparing adjacent ranked means The LSD should not (unless F test for treatments is significant**) be used for –Making all possible pairwise comparisons –Making more comparisons than df for treatments **Some would say that LSD should never be used unless the F test from ANOVA is significant

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Pick the Winner A plant breeder wanted to measure resistance to stem rust for six wheat varieties –planted 5 seeds of each variety in each of four pots –placed the 24 pots randomly on a greenhouse bench –inoculated with stem rust –measured seed yield per pot at maturity

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Ranked Mean Yields (g/pot) Mean YieldDifference VarietyRank F195.3 D E B A C

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ANOVA SourcedfMSF Variety52, Error Compute LSD at 5% and 1%

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Back to the data... Mean YieldDifference VarietyRank F195.3 D E * B A * C ** LSD =0.05 = LSD =0.01 = 22.29

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Fishers protected LSD (FPLSD) Uses comparisonwise error rate Computed just like LSD but you dont use it unless the F for treatments tests significant So in our example data, any difference between means that is greater than is declared to be significant

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Tukeys Honestly Significant Difference (HSD) From a table of Studentized range values (see handout), select a value of Q which depends on p (the number of means) and v (error df) Compute: For any pair of means, if the difference is greater than HSD, it is significant Uses an experimentwise error rate Use the Tukey-Kramer test with unequal sample size

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Student-Newman-Keuls Test (SNK) Rank the means from high to low Compute t-1 significant differences, SNK j, using the studentized values for the HSD Compare the highest and lowest –if less than SNK, no differences are significant –if greater than SNK, compare next highest mean with next lowest using next SNK Uses experimentwise for the extremes Uses comparisonwise for adjacent means where j=1,2,..., t-1; k=2,3,...,t k = number of means in the range

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Using SNK with example data: Mean Yield VarietyRank F195.3 D294.0 E375.0 B469.0 A550.3 C624.0 k23456 Q SNK = 15 comparisons 18 df for error SNK=Q*se

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Duncans New Multiple-Range Test Critical value varies depending on the number of means involved in the test Alpha 0.05 Error Degrees of Freedom 6 Error Mean Square Number of Means Critical Range Means with the same letter are not significantly different. Duncan Grouping Mean N variety A A A A B A B A B A B B C

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Waller-Duncan Bayes LSD (BLSD) Do ANOVA and compute F (MST/MSE) with q and f df (corresponds to table nomenclature) Choose error weight ratio, k –k=100 corresponds to 5% significance level –k=500 for a 1% test Obtain t from table (A7 in Petersen) –depends on k, F, q (treatment df) and f (error df) Compute Any difference greater than BLSD is significant Does not provide complete control of experimentwise Type I error Reduces Type II error BLSD = t 2MSE/r

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Bonferroni Inequality Theory E X * C where X = number of pairwise comparisons To get critical probability value for significance C = E / X where E = maximum desired experimentwise error rate Alternatively, multiply observed probability value by X and compare to E (values >1 are set to 1) Advantages –simple –strict control of Type I error Disadvantage –very conservative, low power to detect differences

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False Discovery Rate Reject H 0 Bars show P values for simple t tests among means – Largest differences have the smallest P values Line represents critical P values = (i/X)* E

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Most Popular FPLSD test is widely used, and widely abused BLSD is preferred by some because –It is a single value and therefore easy to use –Larger when F indicates that the means are homogeneous and small when means appear to be heterogeneous The False Discovery Rate (FDR) has nice features –Good experimentwise Type I error control –Good power (Type II error control) –May not be as well-known as some other tests Tukeys HSD test –Widely accepted and often recommended by statisticians –May be too conservative if Type II error has more serious consequences than Type I error

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