write down the assumed 4 parameter model function that the researchers fit to the data see the right hand side of the model statement

INSTRUCTIONS TO CANDIDATES

ANSWER ALL QUESTIONS

Spring ’22 STAT-498 Applied Statistics

Homework 3

Directions: Answer each of the four exercises below showing all relevant work. Conclusions and justifications are to be given in clear detailed English. Please type up your solutions or write very neatly.

1. Huet, Bouvier, et al (Statistical Tools for Nonlinear Regression, p.2) use the Pasture Regrowth dataset from Ratkowsky (Nonlinear Regression, p.88) to fit a certain four-parameter sigmoidal growth model. In the dataset, 𝑌 = pasture regrowth since last grazing, and 𝑋 = time, and for our present purposes, let’s assume that the data are independent measurements. The nonlinear model function that these authors used to fit the data is somewhat complicated – and coming up with starting values for the model parameters is not easy and comes only come after we understand the roles these parameters play. These data are given, plotted, and analyzed in the Appendix.

(a) List all the needed assumptions for the given proc nlin analysis. Give an example of conditions where – in the context of this situation – the above required independentmeasurements assumption would not be met.

(b) After examining SAS Program B (proc nlin), write down the assumed 4-parameter model function that the researchers fit to the data; see the right-hand side of the model statement.

(c) Assuming that 𝜃4 is positive and using algebra and one ‘limit’, clearly give the roles of the 𝜃1 and 𝜃2 parameters. (Hint: Which parameters – or functions of parameters – are the upper and lower asymptotes for this model?) Upon examining the graph of the data below, what are your “eyeball estimates” of these two parameter values?

(d) To obtain NLIN starting values for 𝜃3 and 𝜃4, we use the following approach: write down the expression with ‘y’ on the left-hand side and the above assumed nonlinear model function on the right-hand side (with no error term for now), substitute in our eyeball estimates for the upper and lower asymptotes and solve so that the new right-hand expression is a linear model in 𝑙𝑜𝑔(𝑥). Next, relate what you have found above to the simple linear regression (proc reg) performed in SAS Program A, and use SAS Output A to report the starting values for 𝜃3 and 𝜃4. Verify that these starting values (or approximations to these) are used in SAS Program B.

(e) Using SAS Output B (proc nlin), report the estimate of 𝜎 2 here.

(f) Using SAS Output B (proc nlin), do a two-sided Wald test that 𝜃4 = 3 using 𝛼 = 1%. Redo this two-sided Wald test using 𝛼 = 5%. Clearly report your test statistics, p-values, and conclusions in both cases.

(g) Repeat both tests done in part (f) but using Likelihood Ratio tests instead.

(h) In examining the listing of the residuals in Output C and the Residual Plot, it is apparent that one of the residuals (at 𝑥 = 21) may be ‘large’. If the proc nlin were to be rerun with this potential outlier removed, would the estimate of the lower asymptote increase or decrease?

Attachments:

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