STAT 511 Assignment 4 Name ________________ Spring 2002 Reading Assignment: Rencher: Read Chapters 11, 4 and 5. Chapter 11 was part of an earlier reading assignment. It covers the basics on estimable functions and testable hypotheses. Chapter 4 reviews some properties of the multivariate normal distribution, and Chapter 5 considers distributions of quadratic forms and related F and t distributions. The introduction to regression analysis in Chapters 6 and 7 and the regression diagnostics in Chapter 9 were covered in Stat 500, and we will not repeat the coverage of this material in Stat 511. You can review as much of this material as you find useful. Chapter 8 covers tests of hypotheses and confidence intervals for parameters in regression models. Reading Chapter 8 will help you apply the general linear model theory we are developing in Stat 511 to the analysis of multiple regression models. Written Assignment: On-campus students: Due Friday, February 15, in class. Distance students: Put it in the mail or e-mail or FAX by February 22. 1. A food scientist performed the following experiment to study the effects of combining three different fats and three different surfactants on the specific volume of bread loaves. Four batches of dough were made for each of the six combinations of fat and surfactant. Ten loaves of bread were made from each batch of dough and the average volume of the ten loaves was recorded for each batch. Unfortunately, some of the yeast used to make some batches of dough was ineffective and data from the loaves made from those batches had to be removed from the analysis. Fortunately, all six combinations of the levels of fat and surfactant were observed at least once. The data (average volume of 10 loaves) are shown below. Surfactant A B C 6.7 4.3 5.7 7.1 5.9 5.6 5.5 6.4 5.8 Fat 1 Fat 2 5.9 7.4 7.1 5.6 6.8 6.4 5.1 6.2 6.32Consider the model Yijk = µ + αi + βj + γij + εijk where εijk ~ NID(0, σ2) and Yijk denotes the average of the volumes of ten loaves of bread made from the k-th batch of dough using the i-th fat and the j-th surfactant. For each of the following linear functions of the parameters, determine if it is estimable. If it is estimable, give a vector a such that )(ETYa is equal to that linear combination of parameters. If it is estimable, describe what that linear combination of parameters represents with respect to the effects of the fats and surfactants on mean bread volume. (a) µ (b) 2α (c) 32β−β (d) 23γ (e) 2332γ+β+α+µ (f) 1211γ−γ (g) 23211311γ+γ−γ−γ (h) )(21)(2313221232γ−γ−γ+γ+β−β (i) 23132212γ−γ−γ+γ 2. Data were collected to study the effect of temperature on the yield of a chemical process. Two different catalysts A and B, were used in the study. Yields were measured under 5 different temperatures for each catalyst. The data are as follows: Run Yield (grams) Y Temperature (°°C) T Catalyst 3 20 90 A 10 24 95 A 4 27 100 A 8 33 105 A 5 38 110 A 9 25 90 B 2 29 95 B 6 32 100 B 1 37 105 B 7 41 110 B3 Each run can be considered as an independent observation. The order in which the runs were made was randomized. Consider the linear model ijijiij)100T(Y ε+−β+α+µ= , for i = 1, 2 and j = 1, 2, ..., 5 where ijY = the observed yield for the run using the i-th catalyst and the j-th temperature level. catalystthithetoscorrespond i−α ijT = the temperature under which the process was run. (a) For this linear model the vector of mean responses can be written as XYE=)( ββ. Write out the model matrix X corresponding to the parameter vector T21) ( βααµ=â . (b) Determine which, if any, of the following quantities are estimable. For each estimable quantity, report the value of a vector a such that aTY satisfies the definition of an estimable function of ββ. (i) µ (ii) µ +α2 (iii) β (iv) α1 - α2 (v) µ + βT where T is any specified temperature (vi) µ + α1 + β(T-100) where T is any specified temperature (c) The data are posted in the file hw402p2.txt on the course web page. This file has five columns. The first two columns match the first two columns in the table shown above. The third and fourth column use dummy variables to indicate which catalyst was used. The third column is coded “1” when catalyst A was used and coded “0” when the other catalyst was used. The fourth column is coded “1” when catalyst B was used and coded “0” when the other catalyst is used. The fifth column contains the temperature values minus 100. Use the command4 W <- read.table("c:/stat511/hw402p2.txt",header= T) to enter these data into a data frame in S-PLUS. Of course, you should replace "c:/stat511/hw402p2.txt" with the name of the file in which you stored these data. Use the command Y <- as.matrix(W[ ,2 ]) to create a vector of observed responses. Use the command X <- as.matrix(cbind(rep(1,length(Y)),W[ ,3:5])) to construct the model matrix for the model in part (a). If you are using S-PLUS version 6 for Windows or the UNIX version 5.1 of S-PLUS (available on the system of VINCENT workstations at ISU), use the ginverse( ) function to compute a generalized inverse of XTX and report the result. If you are using an older version of S-PLUS for windows, the ginverse( ) function will not be available. Those users can use the ginv( ) function in the MASS library of functions. The MASS library comes with any Windows version of S-PLUS and you downloaded it onto you computer when you installed S-PLUS. To use a function in the MASS library, you must first establish a path to the library by issuing the command library(MASS) from the command window. Then when you issue the command M <- ginv(t(X)%*%X) S-PLUS will looking the MASS library for the ginv( ) function. If you fail to issue the library(MASS) command, S-PLUS will only look for the ginv( ) in the default library of functions. (d) Use S-PLUS to check if the generalized inverse computed in part (c) satisfies the four properties of the Moore-Penrose inverse. State your conclusion. (e) Use the generalized