1. Statistical Techniques I EXST7005 Miscellaneous ANOVA Topics & Summary

2. 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.

3. 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.

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

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

6. 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.

7. 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.

8. 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.

9. 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

10. 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

11. 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.