1. Section 1 Exercises (all) Appendix A.1 of Vardeman and Jobe (pages ).

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1 Stat 40B Homework/Fall 07 Please see the HW polcy on the course syllabus. Every student must wrte up hs or her own solutons usng hs or her own words, symbols, calculatons, etc. Copyng of the work of others s a volaton of academc ethcs and wll be handled as such. Homework # (Due September 8). Secton Eercses (all) Append A. of Vardeman and Jobe (pages ).. Secton Eercses,, 4, 5, 6, 7, 8, 9 Secton 5. of Vardeman and Jobe (pages 43-44). Homework # (Due September 5). Secton Eercses (all) Secton 5. of Vardeman and Jobe (page 63).. Secton 4 Eercses, 3, 4, 5 Secton 5.4 of Vardeman and Jobe (pages ). 3. Secton 5 Eercses, 3, and 4 Secton 5.5 of Vardeman and Jobe (pages (3 and 3). Homework #3 (Not to be Collected but Covered on Eam September ). Secton Eercses (all) Secton 6. of Vardeman and Jobe (page 344).. Secton eercses (all) Secton 6. of Vardeman and Jobe (page 36). Homework #4 (Due October 6). Secton 3 Eercses (all) Secton 6.3 of Vardeman and Jobe (page 385). For parts a) and d) of Eercse, "normal probablty plot" of Sectons 3. and 5.3 are ntended. (Read those sectons.) There s code on the course Web page for usng R to make normal plots on the same set of aes (and add approprate lnes to the plots). The data vectors n the code are named and and you wll have to replace the eample data vectors wth ones of nterest. (You can make also the "obvous" edts to the code to make a sngle plot or more than plots when such are needed.) For Eercse 4 part b) do the testng frst assumng that the two types of tres have the same varablty of skd length (as n Eample 8 page 38 of V&J). Then redo t droppng that assumpton and operatng as ndcated on the course summary of one and two sample nference formulas and usng the smplfed degrees of freedom formula. For Eercse 4 part d) frst use the unpleasant "Satterthwate appromate degrees of freedom formula" and then redo the problem usng the smplfed degrees of freedom.. Secton 4 Eercses (all) Secton 6.4 of Vardeman and Jobe (page 399).

2 3. Secton 5, Eercses (all) Secton 6.5 of Vardeman and Jobe (page 43). For the all confdence lmts problems, use the modfcaton of the book's formulas provded on the course pˆ pˆ n the "plus and mnus" formula summary sheets. Ths means that n place of products ( ) parts of the formulas (NOT n place of center values ˆp ) use products p ( p). (Ths wll have the effect of makng the confdence ntervals a bt larger/more conservatve so that they wll have actual confdence at least the nomnal ones. Wthout ths "f" the formulas n the tet wll tend to be a bt too short and have actual confdence levels smaller than the nomnal ones for etreme values of the unknown p 's.) 4. Secton 6 Eercses and 3 Secton 6.6 of Vardeman and Jobe (pages 46 and 47). Homework #5 (Due October 3). Secton Eercses (all) Secton 7. of Vardeman and Jobe (page 460).. Secton Eercses and, Secton 7. of Vardeman and Jobe (pages 47 and 47). 3. Secton 4 Eercses b) and b), Secton 7.4 of Vardeman and Jobe (page 495). 4. Secton Eercses,, and 3, Secton 4. of Vardeman and Jobe (pages 39-40). 5. Chapter 4 Eercse of Vardeman and Jobe (page 03). 6. Secton Eercses a), b), c), d), f), h), and a) through f), Secton 9. of Vardeman and Jobe (page 674). Homework #6 (Due November 0). On the UCI machne Learnng Repostory there s a "Glass Identfcaton data Set" at Download that data set and use the frst 46 cases (those correspondng to Glass # and Glass #) n the followng. Suppose that samples of glass are presented to an analyst n about the same relatve frequences as n the UCI data set (at fractons 70/46 of type and 76/46 of type ). Based on,,, 9 we want to say wth what probablty a specmen n hand s of glass type. The varables n the data set are

3 = refractve nde = Na (sodum) ode weght percent = Mg (magnesum) ode weght percet = Al (alumnum) ode weght percent = S (slcon) ode weght percent = K (potassum) ode weght percent = Ca (calcum) ode weght percent = Ba (barum) ode weght percent = Fe (ron) ode weght percent y = Type of glass (class attrbute) (I'll put a verson of the data set on the 40B web page.) a) Import the data to R. Help s here or here or here or you can copy to the clpboard and use the psych package as descrbed here b) Ft a logstc regresson model based on all 9 predctors to the data. Identfy any of the predctors that (n the presence of all others) mght be dropped from the model. What s your ratonale for choosng/dentfyng these? c) Make sde-by-sde bo plots for the ftted probabltes of beng a Glass # specmen for the groups of (actual Glass # and actual Glass #) specmens based on the full model of b). (Remember that graphcs lke ths were produced on Lab #5.) Does t seem lke logstc regresson provdes a sensble way to tell whether a specmen s of type #? Eplan. d) Now ft a logstc regresson model that ncludes only the predctors Na, Mg, S, and Ca to the data. Make a plot of the type requested n part c). Does the plot here look much worse than the plot n c)? Eplan. The "resdual devance" for a ftted logstc regresson model s more or less an analogue of the "error sum of squares" for ordnary MLR. Locate and report values for ths for both the full model of b) and for ths reduced logstc regresson model. Is the ncrease n "resdual devance" observed here consstent wth the plots here and n c)? Eplan. Do you consder the ncrease to be "severe"? (The "null devance" s more or less the analogue of the "total sum of squares" for ordnary MLR.) e) Interpret the sgns of the coeffcents n the ft of part d). (Relatve to Glass#, do Glass# specmens tend to have hgher or to have lower values of the predctors?) f) Purely for purposes of eercse, not because t's a good model here, ft a logstc regresson model that ncludes only the predctors Na and Ca to the data. Then make a plot on the Na-Ca 3

4 plane locatng the lnes (the sets of ( Na, Ca ) pars) for whch the model says that the probabltes of a specmen beng a Glass # specmen are.,.3,.5,.7, and.9.. The book Nonlnear Regresson Analyss and ts Applcatons by Bates and Watts contans a small data set taken from an MS thess of M.A. Treloar "Effects of Puromycn on Galactosyltransferase of Golg Membranes." It s reproduced below. y s reacton velocty (n counts/mn ) for an enzymatc reacton and s substrate concentraton (n ppm) for untreated enzyme and enzyme treated wth Puromycn y Untreated 67, 5 84, 86 98, 5 3, 4 44, Treated 76, 47 97, 07 3, 39 59, 5 9, 0 07, 00 Apparently a standard model here (for ether the untreated enzyme or for the treated enzyme) s the "Mchaels-Menten model" y θ = + θ + Note that n ths model, ) the mean of y s 0 when = 0, ) the lmtng (large ) mean of y s θ, and 3) the mean of y reaches half of ts lmtng value when = θ. Begn by consderng only the "Treated" part of the data set (and an ( ) ε (*) d N 0, σ ε 's verson of the model). Of course, use R to help you do all that follows. Begn by readng n vectors y and. a) Plot y vs and make "eye-estmates" of the parameters based on your plot and the nterpretatons of the parameters offered above. (Your eye-estmate of θ s what looks lke a plausble lmtng value for y, and your eye-estmate of θ s a value of at whch y has acheved half ts mamum value.) b) Add the stats package to your R envronment. Then ssue the commands > <-c(.0,.0,.06,.06,.,.,.,.,.56,.56,.,.) > y<-c(76,47,97,07,3,39,59,5,9,0,07,00) > REACT.fm<-nls(y~theta*/(theta+),start=c(theta=#,theta=##),trace=T) 4

5 where n place of # and ## you enter your eye-estmates from a). Ths wll ft the nonlnear model (*) va least squares. What are the least squares estmate of the parameter vector and the "devance" (error sum of squares) ( ( θ ( θ ) )) ˆ ˆ θ θ ˆ ˆ OLS = and SSE = y / + ˆ θ = c) Re-plot the orgnal data wth a supermposed plot of the ftted equaton. d) Get more complete nformaton on the ft by typng > summary(react.fm) e) The concentraton, say 00, at whch mean reacton velocty s 00 counts/mn s a functon of θ and θ. Fnd a sensble pont estmate of 00. f) As a means of vsualzng what functon the R routne nls mnmzed n order to fnd the least squares coeffcents, use the followng code to make a contour plot for the error sum of θ, θ. squares as a functon of ( ) > theta<-coef(react.fm) > ss<-functon(t,t){sum((y-t*/(t+))^)} > ss(theta[],theta[]) > se<-sqrt(dag(vcov(react.fm))) > dv<-devance(react.fm) > dv > gsze<-0 > th<-theta[]+seq(-4*se[],4*se[],length=gsze) > th<-theta[]+seq(-4*se[],4*se[],length=gsze) > z<-outer(th,th,vectorze(functon(,y) ss(,y))) > contour(th,th,z,levels=seq(000,4000,00)) What contour on ths plot corresponds to an appromately 90% appromate confdence regon for the parameter vector θ? (Use the Beale regon. The F degrees of freedom wll be and 0, and the upper 0% pont of the dstrbuton s employed.) 5

6 g) Now redo the contour plottng, placng only two contours on the plot usng the followng code. > plot(th,th,type="n",man="error Sum of Squares Contours") > contour(th,th,sumofsquares,levels=dv*c((+.*qf(.95,,0)),(+.*qf(.95,,0)))) Identfy on ths plot an appromately 95% (jont) confdence regon for θ and ndvdual 95% confdence ntervals for θ and θ. (If you draw a rectangle wth sdes parallel to the coordnate aes around the regon defned by the frst (lower) contour, the etent of the bo n each drecton gves the 95% ndvdual confdence lmts. The second (hgher) contour defnes the jont 95% confdence regon.) (By the way, t would have been possble to smply add these contours to the frst plot, by makng the second call to contour() as above, ecept for settng "add=t" as a parameter of the call.) h) Use the standard errors for the estmates of the coeffcents produced by the routne nls() and make 95% t ntervals forθ and θ. How much dfferent are these from your ntervals n g)? (Notce that the sample sze n ths problem s small and relance on any verson of large sample theory to support nferences s tenuous. I would take any of these nferences above as very appromate. ) Make an appromate 95% confdence ntervals forσ by carryng over the MLR type result that SSE / σ ~ χ n k. j) Use the R functon confnt() to get 95% ntervals for θ and θ. That s, add the MASS package n order to get access to the functon. Then type > confnt(react.fm, level=.95) How do these ntervals compare to the ones you found n part g)? k) Apparently, scentfc theory suggests that treated enzyme wll have the same value of θ as does untreated enzyme, but that θ may change wth treatment. That s, f a possble model s z 0 f treated (Puromycn s used) = otherwse y ( θ + θ z ) 3 = + θ + ε 6

7 and the parameter θ 3 then measures the effect of the treatment. Go back to the data table and now do a ft of the (3 parameter) nonlnear model ncludng a possble Puromycn effect usng all 3 data ponts. Make dfferent appromately 95% confdence ntervals for θ 3. Interpret these. (Do they ndcate a statstcally detectable effect? If so, what does the sgn say about how treatment affects the relatonshp between and y?) Plot on the same set of aes the curves ( ˆ θ+ ˆ θ3) ˆ θ = and = for 0 < < y y ˆ θ ˆ + θ + 3. Ths queston concerns the analyss of a set of home sale prce data obtaned from the Ames Cty Assessor s Offce. Data on sales May 00 through June 003 of and story homes bult 945 and before, wth (above grade) sze of 500 sq ft or less and lot sze 0,000 sq ft or less, located n Low- and Medum-Densty Resdental zonng areas. (The data are n an Ecel spreadsheet on the Stat 40B Web page. These need to be loaded nto R for analyss.) n = 88 dfferent homes fttng ths descrpton were sold n Ames durng ths perod. ( were actually sold twce, but only the second sales prces of these were ncluded n our data set.) For each home, the value of the response varable Prce = recorded sales prce of the home and the values of 4 potental eplanatory varables were obtaned. These varables are Sze Land Bedrooms Central Ar Freplace Full Bath Half Bath Basement Fnshed Bsmnt Bsmnt Bath Garage Multple Car the floor area of the home above grade n sq ft, the area of the lot the home occupes n sq ft, a count of the number n the home a dummy varable that s f the home has central ar condtonng and s 0 f t does not, a count of the number n the home, a count of the number of full bathrooms above grade, a count of the number of half bathrooms above grade, the floor area of the home's basement (ncludng both fnshed and unfnshed parts) n sq ft, the area of any fnshed part of the home's basement n sq ft, a dummy varable that s f there s a bathroom of any sort (full or half) n the home's basement and s 0 otherwse, a dummy varable that s f the home has a garage of any sort and s 0 otherwse, a dummy varable that s f the home has a garage that holds more than one vehcle and s 0 otherwse, 7

8 Style ( Story ) a dummy varable that s f the home s a story (or a home and s 0 otherwse, and Zone ( Town Center ) a dummy varable that s f the home s n an area zoned as "Urban Core Medum Densty" and 0 otherwse. story) (The effect of usng a dummy varable n a MLR contet s to add the correspondng beta to the mean functon for those cases where the dummy varable s. So, for eample, you are allowed to just add a fed amount for havng a garage.) Use leaps to fnd the sngle models wth hghest R values for all numbers of predctors through 4. Then compare these 4 models on the bass of -fold cross valdaton. Ultmately, what model do you lke best? Ft t to the entre data set. Interpret ts ftted coeffcents n the contet of the problem and estmate the standard devaton of sellng prce for fed values of the predctors (usng 95% confdence lmts). Homework #7 (Due November 7). Secton 3 Eercses and, Secton 4.3 of Vardeman and Jobe (page 90-9).. Secton Eercse parts a) through f) Secton 8. of Vardeman and Jobe (pages ). 3. Secton Eercses a, (don't answer the second part of c)), and 3, Secton 8. of Vardeman and Jobe (pages ). Homework #8 (Due December 8). Return to the contet of problem of HW 6 and the glass type data. a) Begn by fttng an ordnary logstc regresson (as n HW 6). Ths produces an estmate of p( ) = p(,,, 9) = P[ y =,,, 9] Then a best classfer classfes to glass type f p ( ) > Then wth ˆp ( ) the ftted logstc regresson an appromately optmal classfer s to classfy as glass f p ˆ ( ) > What s the tranng classfcaton error rate for ths classfer? (Note that glm() produces ftted values of p ( ) for all the tranng cases, so you can fgure out how each case would be classfed.) b) Use 0-fold cross-valdaton to choose a best number of neghbors for k-nearest neghbor classfcaton between glasses and based on these tranng data. What s the best cross- 8

9 valdaton error rate avalable? Apply ths number of neghbors to do knn classfcaton for the tranng set. What s the correspondng tranng classfcaton error rate? c) Use 0-fold cross-valdaton to choose a best penalty factor "α " (or cp n the rpart package) for prunng a large classfcaton tree. What s the cross-valdaton classfcaton error rate for ths penalty factor? Apply ths penalty factor to the entre data set to dentfy a classfcaton tree to use n dstngushng between glass and glass. Make a graph of the bnary tree and dentfy ts correspondng tranng classfcaton error rate. d) Use the out-of-bag classfcaton error rate to choose a best "mtry" to use wth a random forest classfer based on ths tranng set. How does the best out-of-bag classfcaton error rate compare wth the cross-valdaton classfcaton error rates you found n parts b) and c) of ths queston? e) How do the tranng classfcaton error rates for parts a), b), and c) compare? (Because n classfcaton problems the default choce of the mnmum number of cases n a fnal bootstrap tree s, tranng error for a classfcaton random forest s 0 and completely unrelable as an ndcator of lkely predcton performance. So there s no pont n ncludng the random forest of d) n ths comparson.) 9