In a randomized block experiment we have to find a way to remove the variability due to the blocks from the analysis so that we can test the equality of the treatment means, the analysis of variance can be extended to handle that problem. Let's suppose that in general we have little a treatments, that is little a levels of our single factor, and little b blocks. Then a statistical model that we could use for the RCBD is the equation you see in the middle of the slide, yij. That is the observation, I'm sorry, that's the observation on the treatment in the block. Mu is an overall mean, is the treatment effect due to the ithe factor level or treatment, beta sub j is the block effect and epsilon ij is a random error term, which we're going to make our usual assumption that those errors are normally and independently distributed with mean 0 and constant variant sigma Square. Then the appropriate hypotheses for us to test would be that the means of the treatment effects are all 0, that is mu 1 is equal to mu 2 is equal to all the way on out to mu sub a. And the equality of those means, as I say, is equivalent to testing the hypothesis that the treatment effects are all equal to 0. And of course the mean is equal to mu plus sub i. The model we've used here is the effects model, okay? And we've assumed that the factor levels and the blocks are fixed. So how does the analysis of variance accomplish this? Well, basically what the analysis of variance does is it takes the total sum of squares of the observations and it partitions it into three components, one component is a sum of squares for treatments due, is a sum of squares due to treatment, that's this one. The second one is a sum of squares due to blocks, that's this one. And then the third one Is a sum of squares due to error, which is the last one that you see here. If you look at the sum of squares for treatments, the first one that I circled. Notice that we're doing is we're taking the factor level average or treatment average and subtracting the grand average from it. And then for the block affect, the block sum of squares, we're taking the block average, subtracting the grand average from it, squaring the differences and added them up. And then the the error sum of squares turns out to be essentially the variability that's left over, okay? After we explained those two. And the equation there looks a little odd, yij is the observation on treatment i block j. We subtract out the grand average for each treatment, we subtract out the average for each block and then we add the grand average back. And what that effectively does is that uses the residual unexplained variability as an estimate of error. So, SS total is decomposed into a sum of squares for treatments to sum of squares for blocks and an error sum of squares. We can also partition up the degrees of freedom. There are little a treatments and little b blocks. So there are a,b minus 1 total degrees of freedom. a minus one of those are associated with the treatments, b minus 1 of those are associated with the blocks and a minus 1 times b minus 1 of those degrees of freedom are associated with error. Ratios of the sums of squares to the degrees of freedom result in mean squares, and these mean squares have essentially chi-square distributions. And so the ratio of mean square for treatments to the error mean square is an F statistic that is distributed as an F statistic under the null hypothesis of no difference in treatment means and that F statistic can be used to test the hypothesis that the treatment means are all equal. In fact, the ratio of sum of squares treatment or mean square treatments over mean square error should be approximately 1 if there's no difference in treatment means, and if it's greater than 1 that's an indication that there are differences in treatment means. So an upper tailed, 1 tailed F test is the appropriate reference distribution for the test. Usually, the ANOVA is displayed in a table. And the table that you see here is very typical of what you will see in most computer output. There will be, in the source of variation column, there will be sources for treatments, blocks, error and total. The sums of squares will typically display, be displayed next as well the degrees of freedom, and then the mean squares are simply the sum of squares divided by the degrees of freedom. And then the F statistic is mean square treatments over mean square error. And large values of that ratio are an indication that the treatment means are different. The degrees of freedom there, the F statistic would have a minus 1 numerator degrees of freedom and a minus 1 times b minus 1 denominator degrees of freedom. You can do the computing manually. Notice I said ugh, [LAUGH]. Because the manual computing is, begins to get a little tedious now in this experiment, but it can be done without terrible difficulty by hand. I'll show you the equations there in the book. But you typically use software, either Design Expert or Jump or Minitab, to do the analysis. Here are the manual computing equations from the textbook equations 49 through 412. The total sum of squares is computed in the usual way, that is we take each individual observation, square it, add them all up and subtract off the usual correction factor, which is the grand total squared divided by the total number of runs. And then SS treatments is found by taking each treatment total, squaring it, adding them up, divided by the number of observations in each treatment, which by the way will be the number of blocks, subtracting the correction factor. SS blocks is found by taking the total for each block, squaring it, adding it up, dividing by the number of observations in each block and then subtracting the correction factor. And then finally the error sum of squares we get by subtraction. So the arithmetic is not complicated but it is notoriously error-prone and it's really easy to make a mistake. So, typically we should rely on using the computer to do this.