1Stat 511 MS Exam – Fall 2003 Page 1 of 3 This question concerns several analyses of a set of data from a field trial involving 5 Varieties of a crop and 5 Fertilizer regimens. The response variable is yield (in bushels per acre)y = and the 25 different combinations of Variety and Yield were each run once on a part (1/25th ) of a single rectangular field that had been divided evenly into 5 Rows and 5 Columns. The table below gives the yield data, where “Rows” are labeled 1-5 North to South and “Columns” are labeled 1-5 West to East. Case Row Column Variety Fertilizery , Yield1 1 1 1 1 163 2 1 2 4 4 216 3 1 3 2 2 173 4 1 4 5 5 222 5 1 5 3 3 198 6 2 1 5 2 183 7 2 2 3 5 224 8 2 3 1 3 203 9 2 4 4 1 135 10 2 5 2 4 203 11 3 1 4 3 208 12 3 2 2 1 141 13 3 3 5 4 210 14 3 4 3 2 167 15 3 5 1 5 216 16 4 1 2 5 220 17 4 2 5 3 198 18 4 3 3 1 148 19 4 4 1 4 205 20 4 5 4 2 165 21 5 1 3 4 210 22 5 2 1 2 177 23 5 3 4 5 217 24 5 4 2 3 199 25 5 5 5 1 144 Define 25 1× column vectors V2 V3 V4 V5 F2 F3 F4 F5,,,,,,,X XX XXXXX whose entries are all 0’s and 1’s by V1 if case is of Variety 0otherwisejiijX= and F1 if case uses Fertilizer regimen 0otherwisejiijX= Let Y be the 25 1× column vector of responses in last column of the data table above, and 1 be a 25 1× column of 1's .2 Stat 511 MS Exam -- Fall 2003 Page 2 of 3 To begin, we will ignore the possibility of any spatial effects on yield (fertility variations in the field) and suppose that only “Variety” and “Fertilizer” are important in determining response. a) Use the notation above and write out particular X matrices and parameter vectors β for the linear model =+YXβε describing the possibilities that i) only “Variety” effects on yield are important, ii) only “Variety” effects are important and varieties may be separated into the groups {}{}{}1, 2 , 3, 4 , and 5 with equivalent yields within a given group, iii) there are “Variety” and “Fertilizer” main effects on yield, but no interaction effects. b) In your model from a) iii) above, what matrix C could you use to write a hypothesis 0H:=Cβ 0 expressing the possibility that Varieties 1 and 2 have the same main effects on yield and (simultaneously) Varieties 3 and 4 have the same main effects on yield? For the yield for Variety and Fertilizer regimen ijyij= the first part of the R printout for this problem concerns analysis of the yield data under the model ij i j ijyvfµε=++ + (*) for constants 1234512345,,,,,,,,,,vvvvv f f f f fµ and iid()2N0,σ random variables ijε. c) Under model (*), what are 90% confidence limits for the difference in Variety 1 and Variety 2 main effects? (Plug correct numbers into a correct formula, but you need not evaluate the limits.) d) In model (*), what are the value of an F statistic and the degrees of freedom for that F statistic for testing 01 2 3 4 5H:fffff==== ? Now consider including spatial effects in an analysis of yield. Define (), the row number in which Variety appears with Ferilizer regimen rij i j= and (), the column number in which Variety appears with Ferilizer regimen cij i j= A model that allows for linear North-South and West-East fertility gradients in the test field is ()()row col,,ij i j ijyrijcijvfµγγ ε=+++++ (**) for constants row col and γγ. And for constants row row row row row col col col col col123451234 5, , , , ,,,,, and ββ β β β ββββ β, a model that allows arbitrary row and column fertility effects in the test field is () ()row col,,ij i j ijrij cijyvfµββ ε=+++++ (***) Notice that model equation (**) is the special case of model equation (***) where row row col col and llllβγβγ==3Stat 511 MS Exam -- Fall 2003 Page 3 of 3 The 2nd and 3rd parts of the R printout concern analyses of the yield data corresponding to model equations (**) and (***). Use them to answer the following questions. e) Considering fixed effects versions of models (*), (**), and (***), give values of F statistics and degrees of freedom for judging whether i) there are detectable “Row” or “Column” spatial effects on yield, ii) any “Row” or “Column” spatial effects on yield are adequately described as linear fertility gradients. f) Continuing to use fixed effects versions of models (**) and (***), say why model (**) will support prediction of Variety 1 with Fertilizer 1 yield in a plot located in Row 6 and Column 6 (adjacent to the test field) while model (***) will not. Then make 90% prediction limits for Variety 1 and Fertilizer regimen 1 in this location the season of the study. (Again, plug correct numbers into a correct formula, but you need not evaluate the limits.) Henceforth consider a version of model (***) where the rowlβ are iid ()2rowN0,σ, the collβ are iid ()2colN0,σ, and the row col, , and ij l lεββ are all independent. (Only , the , and the ijvfµ are fixed effects.) It is possible to verify (you need not do so) that the field trial was laid out so that each Variety appears once in each row and in each column, as does each Fertilizer regimen. g) Write the thl row average yield as a function of the terms appearing in (***). Do the same for the thl column average yield. What is the expected value of sample variance of the row average yields? Of the column average yields? Explain. h) Notice that the row mean square corresponding to equation (***) is in fact 5 times the sample variance of row average yields and the column mean square is in fact 5 times the sample variance of column average yields. How does it seem that 2rowσ compares to 2colσ? Explain. i) Let 55111111 ,11 1, , and 55 25ji ijji ijyyyyy y==== =∑∑ ∑ii ii Under this model it is possible to show that () ()22 211 11 11 row col119E and Var5525yyy vf yyyµσσ σ+− =++ +− = + +ii ii ii ii (You may take this as true without proof.) Suppose that *y is a 26th yield not in the original data set, corresponding to Variety 1 and Fertilizer 1 in Row 6 and Column 6 the season of the study. Give a standard error for ()11yyy+−ii ii as a predictor of *y , say predSE . Then briefly indicate how you would find a multiplier τ so that ()11 predSEyyyτ+− ±ii ii will serve as prediction limits for *y . (You need …