Assignment 7

Due Date: Friday, March 9, 2006

The Questions

As promised in Assignment 4 we'll now learn how to analyze these data more efficiently.

Question 1 Fit a single regression model (or at most two models) and test these data for a treatment effect. Use whatever methodology you think appropriate.

Hint 1 : You may use what you learned in Assignment 4 as a basis for choosing an appropriate model. I don't expect you to screen a large number of models here.

Hint 2 : By a treatment effect I'm referring to what might be called an omnibus test of treatment, i.e., does a model that includes treatment as part of the model do a better job of summarizing the data than one that ignores treatment altogether?

Question 2  I would now like you to fit a single regression model that not only tests for an overall treatment effect, but also tests the following three hypotheses simultaneously.

1. Treatment 4 involves a double application of fungal spores. Test whether a double application yields a different result than a single application (ignoring the timing of the single application). This should be a contrast of treatment 4 with treatments 2 and 3 combined.
2. Treatments 2 and 3 are both single applications of fungal spores, but their timing is different. Test whether the timing of the application makes a difference. This should be a contrast of treatment 2 with treatment 3.
3. Test whether any application of fungal spores at all yields a result different from no application. This should be a contrast of treatment 1 with treatments 2, 3, and 4 combined.

What conclusions do you draw for each of these tests?

Question 3  Are the contrasts specified in Question 2 orthogonal contrasts? Why or why not?

Question 4  Create an error bar plot in which you display 95% Wald confidence intervals for the mean much like you did in Question 3 of Assignment 4. This time though use only the results from the regression model you fit in Question 1 of this assignment.

Hint 3 : Start by writing out the equations for treatments 1 through 4 based on the regression model fit in Question 1 and the coding scheme that was used for the treatment regressors that appear in the model. This will give you the equations you need for calculating the treatment means that are predicted by the model.

Hint 4 : Now that you have formulas for each of the treatment means based on parameters in the regression model, use these formulas to obtain estimates of the standard errors of the individual means. Note: To calculate the standard errors you will need to extract various elements of the estimated covariance matrix of the parameter estimates.

Hint 5 : To obtain the covariance matrix I recommend applying the summary function to your model object and saving the results to a variable. If you then apply the names function to this variable you'll see both cov.unscaled and cov.scaled as list elements. These matrices should be identical here and either one can be used. If you use the ls.diag function on the other hand the cov.unscaled and cov.scaled elements are different, and they should not be. The values in the cov.unscaled element of the ls.diag object are correct. I don't know what the values in cov.scaled represent in this case. The vcov function also returns these same mysterious values.

Hint 6 : Don't worry about adjusting the α-level when constructing the confidence interval. Just use α = .05. If you don't know what I'm talking about here, then just ignore this hint.

Hint 7 : If there's a link function involved, then remember that all of your parameter values are on the scale of the link. In calculating confidence intervals calculate the endpoints of the intervals on the scale of the link (it's on this scale that the distributional assumptions are supposed to hold) and only at the very end back-transform using the inverse link to the original scale of the variables.

Question 5  How does the picture you drew in Question 4 compare to the comparable picture you constructed in Assignment 4?

Cited References

Beall, Geoffrey. 1940. The fit and significance of contagious distributions when applied to observations on larval insects. Ecology 21(4): 460–474.
Evans, D. A. 1953. Experimental evidence concerning contagious distributions in ecology. Biometrika 40: 186–211.

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