Comparison Wise Type Error Rate
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the simple question posed by an analysis of variance - do at least two treatment means differ? It may be that embedded in a group of treatments there is only one "control" treatment to which every experiment wise error rate other treatment should be compared, and comparisons among the non-control treatments may be uninteresting. One
Type I Error Rate Calculation
may also, after performing an analysis of variance and rejecting the null hypothesis of equality of treatment means want to know exactly what is the definition of type i error which treatments or groups of treatments differ. To answer these kinds of questions requires careful consideration of the hypotheses of interest both before and after an experiment is conducted, the Type I error rate selected for
Comparisonwise Error Rate
each hypothesis, the power of each hypothesis test, and the Type I error rate acceptable for the group of hypotheses as a whole. Comparisons or Contrasts If we let represent a treatment mean and ci a weight associated with the ith treatment mean then a comparison or contrast can be represented as: , where It can be seen that this contrast is a linear combination of treatment means (other contrasts experiment wise error anova such as quadratic and cubic are also possible). All of the following are possible comparisons: because they are weighted linear combinations of treatment means and the weights sum to zero. For example, previously we have performed comparisons between two treatment means using the t - statistic: with (n1 + n2) - 2 degrees of freedom. This statistic is a "contrast." The numerator of this expression follows the general form of the contrast outlined above with the weights c1 and c2 equal to 1 and -1, respectively: However, we also see that this contrast is divided by the pooled within cell or within group variation. So, a contrast is actually the ratio of a linear combination of weighted means to an estimate of the pooled within cell or error variation in the experiment: with degrees of freedom. For a non - directional null hypothesis t could be replaced by F: with 1, and degrees of freedom. In general, a contrast is the ratio of a linear combination of weighted means to the mean square within cells times the sum of the squares of the weights assigned to each mean divided by the sample size within cells: where the cI' s are
the experimentwise error rate is: where αew http://online.sfsu.edu/efc/classes/biol458/multcomp/multcomp.htm is experimentwise error rate αpc is the per-comparison error rate, and c is the number of comparisons. For example, if 5 independent comparisons http://davidmlane.com/hyperstat/A43646.html were each to be done at the .05 level, then the probability that at least one of them would result in a Type I error is: 1 - (1 - .05)5 = 0.226. If the comparisons are not independent then the experimentwise error rate is less than . Finally, regardless of whether the comparisons are independent, αew ≤ (c)(αpc) For this example, .226 < (5)(.05) = 0.25.
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