6.4.6.1 SAS Analysis 6.4 ANOVA approach 6.4.5 Example - Comparing

6.4.6 Example - Cuckoo eggs - A model based approach

Reference: L.H.C. Tippett, The Methods of Statistics, 4th Edition, Williams and Norgate Ltd (London), 1952, p. 176.

L.H.C. Tippett (1902-1985) was one of the pioneers in the field of statistical quality control, This data on the lengths of cuckoo eggs found in the nests of other birds (drawn from the work of Latter, O.M. 1902. The egg of Cuculus canorus Biometrika 1, 164) is used by Tippett in his fundamental text.

Cuckoos are knows to lay their eggs in the nests of other (host) birds. The eggs are then adopted and hatched by the host birds.

That cuckoo eggs were peculiar to the locality where found was already known in 1892. A study by E.B. Chance in 1940 called The Truth About the Cuckoo demonstrated that cuckoos return year after year to the same territory and lay their eggs in the nests of a particular host species. Further, cuckoos appear to mate only within their territory. Therefore, geographical sub-species are developed, each with a dominant foster-parent species, and natural selection has ensured the survival of cuckoos most fitted to lay eggs that would be adopted by a particular foster-parent

Here is the raw cuckoo egg length data. When analyzed using Analyze->Fit Y-by-Xplatform, the following output was obtained.

There appears to be statistically significance difference between the mean lengths among the species that the cuckoo parasitizes.

As noted in previous sections, model building is an important part of the analysis of a complex design.

In this case the model for this experiment is:

Y_ij = $$\mu$$+ $$\tau_{i}^{}$$+ $$\epsilon_{ij}^{}$$

where

y_ij is the egg length for the j^th egg of species i;

$\mu$is the grand mean

$\tau_{i}^{}$is the treatment effect of species i; and

$\epsilon_{ij}^{}$is the unexplained, residual error.

A more compact notation is

Length = Species

where the grand mean and residual variation are assumed to be present and not listed.

This model is specified in JMP using the Analyze->Fit Modelplatform. Enter the length as the Y variable, and the species into the Model Effects area by selecting the variable in the left column and then clicking on the relevant button. The panel will then look like:

After pressing the Run Model button, a large amount of output is generated.

On the left side, under the Whole Model section, is the same anova table as above, residual plots, and leverage plots.

Ignore the section of the output on parameter estimates - these are useful in more advanced classes.

There should be a section entitled Effect Test which is a test for each factor in the model. As there is only one factor, this does not present any new information. [This section of the output will be more useful in multifactor designs.]

Of more interest is the section of the output on the Species factor. Here you can obtain a table of estimated means - these are identical to those from the Analyze->Fit Y-by-Xplatform. [The reason these are called LSMeans will be explained when we examine two-factor designs]. In this case the estimated LSMeans are identical to the raw sample means.

This platform can also construct a table of estimated differences and confidence intervals for the difference between each pair of means. Select the t-difference options to get the following output which estimates the difference between each pair of population means, its standard error, and a 95% confidence interval for the difference. [The two different methods of comparing the pairs of means is related to multiple comparisons which is dealt with later in this chapter.]

Finally, the platform allows the construction of any contrast. For example, suppose that the difference in length between the average egg length of Meadow and Tree Pipit vs the average egg length of Robins. This is specified using the contrast box by clicking on the `+' column for the Meadow and Tree species and the `-' column for the Robin species. This creates the contrast of

.5$$\mu_{Meadow}^{}$$ + 0.5$$\mu_{Tree}^{}$$ - $$\mu_{Robin}^{}$$

This gives the output below which includes the estimated value of the contrast, its estimated standard error, and a hypothesis test that the contrast has the value 0, i.e. of no difference.

 

• 6.4.6.1 SAS Analysis

6.4.6.1 SAS Analysis 6.4 ANOVA approach 6.4.5 Example - Comparing