Contrasts and multiple comparisons.

We continued our discussion of multiple comparison procedures including Tukey’s, Dunnett’s, and Scheffe’s methods. We also looked at the implementation of these methods as well as non-pairwise contrasts in SAS. The examples we looked at were anorexia_multcomp.sas and rats_multcomp.sas.

Reading: Contrasts and multiple comparisons are covered in Chapter 9, but we will not be discussing nonparametric methods so you can skip Section 9.9.

Inferences for contrasts and multiple comparisons procedures.

We continued covering (linear) contrasts and outlined how to make inferences (estimation, confidence intervals, and tests) concerning contrasts. We also made the distinction between pairwise and non-pairwise contrasts/comparisons. I then discussed the problem of an inflated experimentwise Type I error rate and introduced multiple comparison procedures that are designed to address this problem. So far we covered the Bonferroni adjustment method and Fisher’s LSD (Least Significant Difference) procedure.

Here is the silver_solutions.sas file which contains the solutions to the silver_practice.sas practice problems.

ANOVA assumptions and alternatives, and introduction to linear contrasts.

We discussed three alternatives to a standard ANOVA analysis when assumptions are violated — namely the assumptions of population normality and homogeneity of variance. We focused largely on transformations (also known as variance stabilizing transformations) as perhaps the simplest way to deal with situations where we don’t have near homogeneity of variance.

This concludes the material for the first examination. We then started talking about the concept of a linear contrast in preparation for a discussion of multiple comparison procedures.

Read: Chapter 9 (skip 9.9), but this material will be on the second examination.

Here is a small set of practice problems concerning ANOVA: silver_practice.sas.

Introduction to ANOVA — continued.

Today we focused on the use of the ratio of the between-groups and within-groups mean squares as a test statistic to determine if a significant amount of variability in the data is due to differences between groups. This is a test of the null hypothesis that the t population means are equal. We talked about this test, its relationship with the two-sample t-test for independent samples, and its assumptions.

Here is the anorexia_anova.pdf handout and the corresponding SAS data file anorexa_anova.sas.

Also here are the solutions to the darwin and dopamine practice problem sets.

Introduction to ANOVA.

We finished our discussion of the factors that affect Type II error probabilities and power. We also saw that some of those factors also influence the margin of error of a confidence interval.

We’ve begun looking at ANOVA which can be thought of a an extension of the two-group case to the case where there is an arbitrary number of groups. We discussed the decomposition of the total sum of squares into within-groups and between-groups sums of squares, and the construction of within-groups and between-groups mean squares. These will be the building blocks of inferences from the ANOVA.

Reading: Chapter 8 except 8.4 and 8.6. We will be skipping Chapter 7 entirely.

Assumptions and the decision theoretic framework of statistical tests.

We completed our discussion of the technical assumptions underlying the one- and two-group cases for inferences concerning a mean or difference between two means, respectively. We then started discussing Type I and Type II errors and what factors determine the probabilities of these two types of errors.

The two-group case, continued, and assumptions.

Today we finished our discussion of using PROC TTEST to conduct inferences concerning the difference between two means. Before we move forward we have a few remaining details to review. One is the relationship between confidence intervals and statistical tests — that a test can be conducted using the appropriate confidence interval. We also started discussing the underlying technical assumptions for inferential tools concerning means and differences between means.

Here are your first set of practice problems. The files darwin_practice.sas and dopamine_practice.sas are just text files, but can be read by SAS. In a few days I will post solutions.

The two-population/sample case.

Today we reviewed inferences concerning the difference between two population means. Specifically we discussed estimation, confidence intervals, and tests — both when we assume that the populations have equal variances and when we don’t make that assumption.

I also started to introduce SAS and how to use it for the one-group and two-group cases. Here is the handout, and the SAS commands including the data.

One-sample/population inference for a population mean.

Today we reviewed inference for a population mean using the sample mean. We discussed the properties of the sampling distribution of the sample mean and how these can be used for inferences such as a confidence interval or  a statistical test. This allowed us to review some basic concepts regarding confidence intervals and statistical tests.

Next time we will discuss inferences concerning the difference between two sample means.

Introduction & review.

Today I introduced the course and started to review some fundamental concepts — things that you should have seen in an introductory course but may be a little fuzzy. I discussed the idea of a population versus a sample of observations, parameters versus statistics, and description versus inference. To review fundamental concepts in statistical inference (i.e., estimation, confidence intervals, and statistical tests) I started reviewing the case where we wish to make inferences concerning the value of a population mean from a sample mean with one population/sample (afterwards we will consider the two-population/sample case). We also discussed the important theoretical concepts of the sampling distribution and the standard error of a statistic.

You can review from Chapter 1, 2, 3, 5 (skip 5.8, 5.9) and 6 (skip 6.3, 6.5). These readings are perhaps not as critical as others later in the book since they are review and there is more detail there than we need for our discussion, but you might skim them and/or use them for review.