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Statistical Techniques I EXST7005 Miscellaneous ANOVA Topics & Summary

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LSMeans calculation n The calculations of LSMeans is different. For a balanced design, the results will be the same. However, for unbalanced designs the results will often differ. n The MEANS statement in SAS calculates a simple mean of all available observations in the treatment cells. n The LSMeans statement will calculate the mean of the treatment cell means.

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LSMeans calculation (continued) n Example: n The MEAN of 4 treatments, where the observations are 3,4,8 for a1, 3,5,6,7,9 for a2, 7,8,6,7 for a3 and 3,5,7 for a4 is 5.8667. n The individual cells means are 5, 6, 7 and 5 for a1, a2, a3 and a4 respectively. The mean of these 4 values is 5.75. This would be the LSMean.

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Treatments a1a2a3Means b1 569 78 6.5 4 b2 755 9 76.6 Means 75.757 Raw means

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Treatments a1a2a3Means b1 6697 b2 8566.33 Means 75.57.5 LSMeans means Treatments a1a2a3Means b1 6.5 b2 6.6 Means 75.757

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Confidence Intervals on Treatments Like all confidence intervals on normally distributed estimates, this will employ a t value and will be of the form Mean ta/2(S Y) n The treatment mean can be obtained from a means (or LSMeans) statement, but the standard deviation provided is not the correct standard error for the interval.

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Confidence Intervals on Treatments (continued) n The standard error is the square root of MSE/n, where n is the number of observations used in calculating the mean. n The degrees of freedom for the tabular t value is the d.f. from the MSE used to calculate the standard error.

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Confidence Intervals on Treatments (continued) n If there are several error terms (e.g. experimental error and sampling error) use the one that is appropriate for testing the treatments.

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Exam Coverage n ANOVA (one-way and two-way) will be covered. n Be aware of similarities and differences with the t-test. n HOV tests and tests of normality will be included n Factorial treatment arrangement (two-way) with interpretation interactions will be covered

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Exam Coverage (continued) n RBD will be covered only as the concepts. How is the linear model different, why do we block, what is a block, etc. No SAS output on RBD. n Be able to interpret and discuss Post-ANOVA tests è contrasts è range tests

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Exam Coverage (continued) n Be able to place a confidence interval on a treatment mean. n Recognize designs and treatment arrangements from a described problem. è be able to determine the experimental unit, sampling unit, and get the d.f. error. n Answers to questions will be on the net. n Good Luck.

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