4.2 - RCBD and RCBD's with Missing Data

The residual analysis for the Vascular Graft example is shown:

The pattern does not strike me as indicating an unequal variance.

Another way to look at these residuals is to plot the residuals against the two factors. Notice that pressure is the treatment factor and batch is the block factor. Here we'll check for homogeneous variance. Against treatment these look quite homogeneous.

Plotted against block the sixth does raise ones eyebrow a bit. It seems to be very close to zero.

Basic residual plots indicate that normality, constant variance assumptions are satisfied. Therefore, there seems to be no obvious problems with randomization. These plots provide more information about the constant variance assumption, and can reveal possible outliers. The plot of residuals versus order sometimes indicates a problem with the independence assumption.

In the example dataset above, what if the data point 94.7 (second treatment, fourth block) was missing? What data point can I substitute for the missing point?

If this point is missing we can substitute x, calculate the sum of squares residuals, and solve for x which minimizes the error and gives us a point based on all the other data and the two-way model. We sometimes call this an imputed point, where you use the least squares approach to estimate this missing data point.

After calculating x, you could substitute the estimated data point and repeat your analysis. Now you have an artificial point with known residual zero. So you can analyze the resulting data, but now should reduce your error degrees of freedom by one. In any event, these are all approximate methods, i.e., using the best fitting or imputed point.

Before high-speed computing, data imputation was often done because the ANOVA computations are more readily done using a balanced design. There are times where imputation is still helpful but in the case of a two-way or multiway ANOVA we generally will use the General Linear Model (GLM) and use the full and reduced model approach to do the appropriate test. This is often called the General Linear Test (GLT).

Let's take a look at this in Minitab now (no sound)...

The sum of squares you want to use to test your hypothesis will be based on the adjusted treatment sum of squares, \(R( \tau_i | \mu, \beta_j) \) using the notation for testing:

\(H_0 \colon \tau_i = 0\)

The numerator of the F-test, for the hypothesis you want to test, should be based on the adjusted SS's that is last in the sequence or is obtained from the adjusted sums of squares. That will be very close to what you would get using the approximate method we mentioned earlier. The general linear test is the most powerful test for this type of situation with unbalanced data.

The General Linear Test can be used to test for significance of multiple parameters of the model at the same time. Generally, the significance of all those parameters which are in the Full model but are not included in the Reduced model are tested, simultaneously. The F test statistic is defined as

\(F^\ast=\dfrac{SSE(R)-SSE(F)}{df_R-df_F}\div \dfrac{SSE(F)}{df_F}\)

Where F stands for “Full” and R stands for “Reduced.” The numerator and denominator degrees of freedom for the F statistic is \(df_R - df_F\) and \(df_F\) , respectively.

Here are the results for the GLM with all the data intact. There are 23 degrees of freedom total here so this is based on the full set of 24 observations.

When the data are complete this analysis from GLM is correct and equivalent to the results from the two-way command in Minitab. When you have missing data, the raw marginal means are wrong. What if the missing data point were from a very high measuring block? It would reduce the overall effect of that treatment, and the estimated treatment mean would be biased.

Above you have the least squares means that correspond exactly to the simple means from the earlier analysis.

We now illustrate the GLM analysis based on the missing data situation - one observation missing (Batch 4, pressure 2 data point removed). The least squares means as you can see (below) are slightly different, for pressure 8700. What you also want to notice is the standard error of these means, i.e., the S.E., for the second treatment is slightly larger. The fact that you are missing a point is reflected in the estimate of error. You do not have as many data points on that particular treatment.

The overall results are similar. We have only lost one point and our hypothesis test is still significant, with a p-value of 0.003 rather than 0.002.

Here is a plot of the least squares means for Yield with all of the observations included.

Here is a plot of the least squares means for Yield with the missing data, not very different.

Again, for any unbalanced data situation, we will use the GLM. For most of our examples, GLM will be a useful tool for analyzing and getting the analysis of variance summary table. Even if you are unsure whether your data are orthogonal, one way to check if you simply made a mistake in entering your data is by checking whether the sequential sums of squares agree with the adjusted sums of squares.