We finished the basic analysis of variance now for our vascular graft experiment. We've conducted a residual analysis that seems to indicate that there are no problems with our underlying assumptions. Once we've completed that, now we're interested in doing some additional testing to try to discover the specific differences between the tips that led us to believe that we should reject that null hypothesis. The way we typically do that is with the Fisher LSD method. There are other methods that could be used, but the Fisher method is one that's quite easy to use, and here's the output of the Fisher method for one particular package design expert. Some of this information was also on an earlier slide, when we saw some of the computer output from this package in looking at the ANOVA results. The upper part of this display shows you the individual treatment mean, and then the standard error of each treatment mean. Then the bottom part of the display shows you the actual Fisher LSD, in which we compare all pairs of means. So we start off by comparing one against two, one against three, one against four, then two against three, two against four, and then finally, three against four. So this column are the differences in those means. This column is the differences in those means. Then those differences in means are divided by the standard error, and that produces the t-statistic that you see in this column. So these are the pairs of means that are different. The ones that have the small p-values. One and three appear to be different, and one and four appear to be different, two and four appear to be different, three and four possibly different. The p-value is 0.06. I would consider that to be different. But one and two are the same because the p-value is quite large. Then this pair, two and three, are probably the same because they have a p-value that's just a little bit less than 0.1. So we can isolate specific differences between individual tips by using the Fisher LSD method. We could also use Tukey's method, we could also use the graphical method that we talked about before. Now, some other aspects of the randomized complete block design. The RCBD uses an additive model. It assumes that there is no interaction between the treatments and the blocks. This is a pretty strong assumption, and if that assumption is violated, then there's the likelihood that it's going to affect your experiment adversely. In fact, the error term will be inflated because of that. There is a test for additivity. It's the Tukey one-degree of freedom test for additivity, which is mentioned in the book. We've considered the treatments and the blocks to be fixed effects. Well, it's entirely possible that treatment effects and/or the block effects might be random. Well we'll talk about that. Missing values. Sometimes in a randomized block design, a value is missing. It's either lost or something happened and the experiment was not able to be completed at that test combination. With modern computer software, missing values are really not a big problem. The non orthogonality that that introduces is handled by the least squares methods that are used inside the software to do the analysis. But there are ways to estimate missing values, and there's discussion of that in the book. Finally, what are the consequences of not blocking if we should have? Well, if you didn't block when you should have, all of the variability associated with the blocks ends up in the error term, and that can inflate the error term. It can make the error mean square bigger than it should be, and of course then you're F-ratio, mean square treatments over mean square error is going to be smaller than it would be if you had block. That could lead you to an erroneous conclusion. It could lead you to conclude that there's no difference in the treatments, when in fact there really is. What about if we have random blocks or treatments? Let's talk about that. Just for a simple case, let's assume the blocks are random and the treatments are fixed. Well, that now introduces a covariant structure into the data, that's a little bit different. Now the variance of an individual observation, that is the variance of the observations on the ith treatment in the jth block, is Sigma square Beta, which is now a variance component due to blocks, plus the error variance Sigma squared. Observations that are on different treatments in different blocks are not correlated, but observations on different treatments that are in the same block are correlated, and the variance is Sigma square Beta, the block variance component. It turns out that under that situation, the expected mean squares are as in equation 415. The mean square for error estimate Sigma square, the mean square for blocks estimate Sigma square plus a component containing your variance component, a times Sigma square Beta. Then the mean square for treatments, it's expected value is Sigma square plus a fixed type effect that involves those treatment effects, the tall sub i's. The appropriate test statistic for the hypothesis of no treatment effects is still mean square treatments over mean square. It's exactly the same as it was before. All the computing is done exactly the same way. So the sums of squares for treatments and blocks and error all exactly the same, degrees of freedom are exactly the same. If you wanted to estimate the variance component due to blocks, then we could produce an analysis of variance type estimator very easily. The way we do that is we simply equate the mean square for blocks to its expected value. Then we solve that for the variance component estimate Sigma square hat sub Beta, and is simply mean square blocks minus mean square error over little a. So for example, in our vascular graft experiment, mean square blocks was 38.45, mean square error was 7.33. We take the difference in those two mean squares and divide by little a, which is the number of treatments, and for the block variance component, we get 7.78. We could also do the analysis in JMP, and designate blocks as random, and then we would get the residual error maximum likelihood estimates of these variance components. Here's the JMP output. There are the variance component estimates, and here are the lower and upper confidence limits on our variance component estimates. You notice that we have a negative lower bound on blocks. Its minus four roughly, and the upper bound is almost 20, a very wide confidence interval that includes zero. The reason that interval is so wide is because the number of blocks was relatively small. But the block variance component is about 51 percent of the total variability. So it is a very significant factor. If we had not included that as a blocking factor in the design, it would've inflated the experimental error quite a bit.