Power analysis and the future.

We finished up the power analysis examples featuring PROC GLMPOWER. I then discussed other areas of statistics and how they related to the material covered in Statistics 401. I discussed other courses in applied statistics that might interest students who have taken Statistics 401.

No class on Friday.

Power analysis.

Today we reviewed the concepts of Type I and Type II errors and power, and we talked about what factors influence the power of a test. Power analyses involve calculating under hypothetical conditions the likelihood of rejecting a false null hypothesis. Power analyses can also be used to solve for the necessary sample size to achieve a desired power. I demonstrated how to use PROC GLMPOWER to compute the power or sample size requirements for a designed experiment. The example power.sas was borrowed from the documentation for PROC GLMPOWER.

Here is the handout regarding the final examination.

Repeated measures and crossover designs.

Today we focused on repeated measures and crossover designs. The single-factor repeated measures design and the two-factor repeated measures designs are similar to randomized complete block and split-plot designs, respectively except that there is no randomization within units. Repeated measurements can sometime induce correlations that are in violation of the compound symmetry assumption. One approach to deal with this situation is to generalize the model, but this is beyond this course. Another approach is to adjust inferences to compensate. This is the approach we examined.

Crossover designs are like a repeated measures and latin square hybrid. Treatments vary within subjects, but there are different sequences of treatments so that the same treatment is not always observed at the same period. Crossover designs are very efficient, but carryover effects can complicate the analysis. Fortunately we can test statistically for the presence of carryover effects.

Here is the crossover.sas file illustrating the analysis of the crossover study featured in the textbook.

Split-plot and repeated measures designs.

We finished our discussion of split-plot designs and turned our attention to repeated measures designs. Repeated measures designs are similar in structure and analysis to randomized block and split-plot designs. The key feature of a repeated measures design is that repeated observations are made of each “unit” over time and/or space.

Here are the solutions to the homework problems concerning random and mixed models. And here are the heartrate.sas and diet.sas files illustrating the analysis of data from repeated measures designs.

Split-plot designs.

Today we focused on split-plot designs, distinguishing between whole-plots and sub-plots, and the factors associated with them. We saw how to implement split-plot designs in the ANOVA linear-model framework.

Here are the anorexia.sas and phosphorous.sas examples.

Nested factors and an introduction to split-plot designs.

Today we discussed the concept of nested factors in comparison to crossed factors in designs with two (or more) factors. This led toan introduction to split-plot designs. We will continue to discuss split-plot designs on Friday.

Here is the tablets.sas example illustrating the analysis of data with a nested factor.

The file randommixed.sas contains some practice problems on random-effects and mixed-effects designs/models.

Read: Chapter 18.

Implementation of random and mixed-effects models.

Today we focused on how to implement model with random effects including models with only random effects (i.e., random effects models) and models with both fixed and random effects (i.e., mixed effects models). As we saw the expected mean squares depend on the design and whether factors are fixed or random, and inferences (e.g., test statistics) sometimes differ depending on how we specify our effects. Fortunately good statistical packages allow us to specify which effects are fixed and which are random and will assist us in construction the appropriate inferences.

We also saw how we can estimate the variance components of a random effects model and use these to estimate the proportion of variance accounted for by each random effect.

Here are the examples I used in class: paint.sas, labs.sas, and sunscreen.sas.

Next time we’ll move on to nested factors and then split-plot designs.

Designs with fixed versus random effects.

Today we looked at several designs and compared the situations where all of the effects were fixed (fixed-effects models) versus those where all the effects were random (random-effects models). We saw that although this changes the expected mean squares, in many cases the hypotheses are conceptually similar and the test statistics are identical. However, we also saw that in some designs, like a factorial design with two random factors and replications, differences in the structure of the expected mean squares implies different test statistics for the fixed- versus random-effects models. Next time we will consider mixed effects models in which some factors are fixed and some are random.

Read: 17, but you can skip or just skim section 17.5

 

Relative efficiency, and an introduction to random effects.

We discussed the concept of relative efficiency as a method to compare the statistical precision of inferences of two difference designs. A design that is more efficient will result in analyses with higher power and smaller confidence intervals.

This concludes the material for Wednesday’s quiz.

I introduced the concept of a fixed versus random effects in a designed experiment. Some designs include random effects, or a mixture of fixed and random effects. The combination of fixed and random effects is relevant to how we analyze the data as we will see later.

 

Analysis of covariance.

Today we covered the analysis of covariance design and analysis. This isn’t as new a topic as it might first seem since statistically we already discussed how to accommodate a categorical and a quantitative variable in a regression model. ANCOVA is simply an application of that model to situations where we wish to control for a covariate statistically when making inferences about a treatment.

Here are the links to the peanuts_ancova.sas example, and the esteem.sas example.

Here are the solutions to blocking.sas: blocking_solutions.sas.