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1 Valid Analysis Depends on Valid Assumptions Section 4.1 p 123 & Cochran 1947 Biometrics 3(1):22-36 The purposes of the ANOVA are. .. 1) To estimate certain treatment differences that are of interest to us. We want such estimates to be efficient which is to say that we want the estimate of the difference to have as small a variance as we can get from the data and design we are working with. 2) To have an idea of the accuracy of our estimates and that they be unbiased. We do this with estimates of the standard errors or confidence intervals. 3) To perform tests of significance. This is usually an F-test or t- test. We want the probability associated with those tests to valid and we would the tests be powerful (1- \$ ). In setting up an ANOVA we recognize three types of effects. 1) treatment effects - we introduce these. Treatment effects can be either random or fixed. 2) environmental effects - examples are years, locations, blocks, rows, columns and numerous others. 3) experimental errors - this is all of the contributors to variability that are not accounted for in the treatment and environment effects.

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Chapter 4 Diagnosing Agreement Between the Data and the Model 2 To achieve those purposes we must meet several assumption that are implicit in the ANOVA. The key assumption is that experimental errors are. .. 1) the treatment and environmental effects must be additive and errors have a mean of zero. This is shown in the following linear model. ij where : is the overall mean, J is the effect of the i treatment, D th is the effect of the j replication and e of the j experimental unit th receiving the i treatment. 2) the experimental errors must be independent. This means that the probability that the error of any observation has a particular value does not depend on the values of the errors of the other observations. 3) the experimental errors must have a common variance. This means all of the error variance are the same size which we designate as F . 2 4) the experimental errors should be normally distributed. These assumptions are commonly shown as. .
Chapter 4 Diagnosing Agreement Between the Data and the Model 3 The Effects of Departures from the Assumptions of ANOVA Section 4.2 pp 123-124 If normality is not achieved. .. if the errors are not distributed normally then the probabilities given for the F and t-tests are not right, but fortunately there is usually not a large difference. Cochran suggests that as a rule of thumb non-normal errors may have a tabular value of 5% while the true probability is between 4 and 7%. Similarly a tabular value of 1% may have a true probability of between ½% and 2%. Ito showed similar results in 1980. As a general rule, the tabular probability is too small and we tend to announce too many significant results. If the errors are heterogeneous (i.e. they don’t have common variance). In general there is a loss of efficiency or precision (i.e. the standard errors of the treatment means will be greater than if the variances are constant over all observations). The stated significance of the

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