Okay, we're now going to turn to a different type of experimental setting involving our single factor experiment. This is the case that we call the random effects model. In the random effects model, instead of having factor level specifically chosen by the experimenter because he or she is interested in those levels, the specific levels, the levels are actually chosen at random from a large population of levels. In fact, theoretically, there's an infinite number of levels that these individual factor levels come from. The experimental chooses a of these levels at random. Now we want the conclusions, the inference to be valid about the entire population of levels, not just the ones that were specifically chosen. The statistical model for the random effects experiment is exactly the same as it is for the fixed effects case. That is, there's a overall mean Mu, there's a treatment effect toss a bar, and a random error term Epsilon ij. But now there's a difference here because the treatment effects toss a bar are random, they're chosen at random. So they are random variables. So there are two random variables in this experiment, toss a bar and Epsilon ij. So once again, we're going to assume that our errors are NID 0 Sigma square, and that the treatment effects are also in NID with mean 0 and variance Sigma square Tau. Okay? So if the treatment levels differ, it's because Sigma square Tau is not 0. The variance of any single observation is Sigma square Tau plus Sigma square. These are called variance components and we typically want to estimate those. Many people refer to the random effects model as the variance components model. There is a covariant structure here and this explains the covariant structure for you. The covariance between any two observations that are in different treatments is Sigma square Tau. But the covariance between any different observations into different treatments is 0 not equal to i prime. Oh, I'm sorry. Observations within a specific factor level, all have the same covariance. Why is this true? Well, because the experiments, before it's conducted, we expect the observations at that factor level to be similar because they all have the same random component. Once the experiment is conducted, we can assume on all other observations are independent because the parameter tall has been determined and the observations in that treatment differ only because of random error. So that leads to an interesting situation. Let's take a simple example, three treatments of two replicates. So there are six observations. The six by six covariance matrix of those observations is not diagonal. In a fixed effects model, it would be diagonal and the variances would all be equal to Sigma square. But now, the main diagonal elements are Sigma square Tau plus Sigma square and then the off diagonal, one position adjacent to that is Sigma square Tau. The main diagonals are the same at every off-diagonal element is the covariance between a pair of observations. Now, in the random model, it turns out that the ANOVA Identity is still valid. The calculations for sums of squares are exactly as they are in the fixed effects case. But testing hypotheses about the individual means doesn't really tell me Much. Why? Well, because the treatment levels that you obtained were chosen random, you're more interested in the population of treatments. So in the random model, typically we do not test hypotheses about means. We test hypotheses about the variance components Sigma square Tau. So the null hypothesis is H naught Sigma square Tau equal to 0. Against the alternative that says Sigma-squared tall is greater than 0. If you reject that null hypothesis, Sigma squared tall is non-zero, and that is capturing the variability in the population of treatments. It turns out that SS era over Sigma square is Chi-square with N minus eight degrees of freedom and SS treatments over Sigma square is also chi-square but with a minus one degrees of freedom. Both of these random variables are independent. So under the null hypothesis that Sigma squared Tau is equal to 0, we have the same F statistic that we had before. That is Mean Square treatments over mean-square era is the F statistic for testing the hypothesis of Sigma square Tau equal to 0 in equation 3.49. It turns out that if you look at the expected value of the mean squares, they are quite interesting. The expected value of mean-square treatments is Sigma square plus N times Sigma square Tau. The expected value of mean square error is exactly equal to Sigma square. So the logic behind the F-test for the random model is identical to the logic for the fixed effects case. That is, if mean square treatments is larger than mean square era, it's an indication that Sigma squared tall is not 0 and we can use the F statistic just as we did in the fixed effects case for this hypothesis. The expected value of the Mean Squares give us a basis for estimating the variance components. For example, we know that mean square treatments estimate Sigma square plus N times Sigma square Tau and we know that mean square error estimate Sigma square. So it's logical to use mean square error as an estimate of Sigma square. We can solve this equation for Sigma square Tau to get an estimate of the variance component for treatments and there it is. Mean Square treatments over mean square error divided by N is a estimate of the variance components for treatments. Now, these are what we call moment estimators because they used the moments, the expected values of the mean squares. These estimators are unbiased. They don't require the normality assumption, but they're a little clumsy. One of the things that's clumsy about them is trying to find confidence intervals on the variance components. There are ways to do it but, they're not as pretty as we would like. The other thing that's sometimes troubling is we often or sometimes can get negative estimates for these variance components. For example, you could get a negative estimate for Sigma squared tall simply because mean square error is larger than mean square treatments, that would cause a negative estimate and negative estimates are kind of embarrassing really, because variances have to be non-negative. So conducting an experiment and finding a negative estimate of a variance component is definitely troubling.
