STA6166 F07-22 Means Comparisons

STA6166 F07-22 Means Comparisons - Topic 22 - ANOVA (II)...

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Topic 22 - ANOVA (II) 22-1 Topic 22 – Inference of Means in a One-Way ANOVA When Variance is Constant for All Treatments Once we have rejected the null hypothesis that all means are equal, and we have checked the assumptions of the testing procedure, we usually wish to do some specific tests that can elucidate the relationships among the means. Note how this differs from regression. There, we assumed the relationship among the means was linear and so the only test of interest was of the slope. Here we made no such assumption mainly because X is categorical and so regression would make no sense. So, we want to do specific comparisons between the levels of X. These are variously called multiple comparisons tests , contrasts , or tests of linear combinations of means. A priori Hypotheses : hypotheses about population means that are decided during the planning of the experiment and prior to any data analysis. They are the reason for performing the experiment! A posteriori Hypotheses: hypotheses generated as a result of looking at the data after the experiment has been performed. Also called data snooping or data dredging . This is almost ALWAYS inappropriate and to be avoided. The only valid
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Topic 22 - ANOVA (II) 22-2 reason for doing so is as an exploratory analysis that will guide future experimentation. Example (a posteriori testing) : suppose a 1-way ANOVA is performed and the results are obtained. The analyst looks over the results and decides to test 2 means because they appear to be very different (e.g. the smallest and largest ones). Now, the effect could be due to a real difference in population means or to random occurrence due to sampling that makes them appear different. Investigating only comparisons for which the effect appears large leads to a true confidence level for a conclusion that is lower than the stated confidence level. In other words you are more likely to reject H 0 : not different. It can be shown that the actual confidence is 60% (!!!!) when 6 levels are used in an experiment and the statistical analysis always includes testing the difference between the largest and smallest means using a stated 95% confidence. Note also that that treatments compared each time need not be the same ones since the largest and smallest means could be for different treatments. There are times when it is possible to do a posteriori testing – BUT the statistical method needs to be modified appropriately to account for the data snooping (see later).
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22-3 1) Estimation of a Treatment Mean The population mean for the i th treatment i is estimated using the sample mean i i y ˆ with a standard error of i i n MSE y SE ) ( Under our assumptions of normality and random sampling, the (1– )100% Confidence Interval of the i th population mean is ) ( , 2 i t N i y SE t y where t N t , 2 is the critical value for the upper tail of a t- distribution on N ) is done using a t-test as is usual for a single population mean.
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STA6166 F07-22 Means Comparisons - Topic 22 - ANOVA (II)...

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