STAT 420: Homework 12Spring 2019, R. ZhuDue: Tuesday, April 23 by 11:30 PM CTContentsDirections 1Assignment 2Exercise 1 (Simulating Wald and Likelihood Ratio Tests) . . . . . . . . . . . . . . . . . . . . . . . 2Exercise 2 (Surviving the Titanic) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3Exercise 3 (Breast Cancer Detection) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4DirectionsStudents are encouraged to work together on homework. However, sharing, copying or providing any part ofa homework solution or code is an infraction of the University’s rules on Academic Integrity. Any violationwill be punished as severely as possible.Final submissions must be uploaded to our Compass 2g site on the Homework page. No email, hardcopy, orlate submissions will be accepted. Your assignment must be submitted through the submission link on Compass 2g. You have twooptions to submit your file:– [PDF format] Attach two files: Your RMarkdown file, saved as hw11_yourNetID.Rmd. For example hw11_rqzhu.Rmd. The result of knitting your RMarkdown file as hw11_yourNetID.pdf. For examplehw10_rqzhu.pdf.– [HTML format] Attach one .zip file, named hw11_yourNetID.zip, which contains: Your RMarkdown file which should be saved as hw11_yourNetID.Rmd. The result of knitting your RMarkdown file as hw11_yourNetID.html. Your resulting .html/.pdf file will be considered a “report” which is the material that will determinethe majority of your grade. Be sure to visibly include all R code and output that is relevant to answeringthe exercises. (You do not need to include irrelevant code you tried that resulted in error or did notanswer the question correctly.) You are granted an unlimited number of submissions, but only the last submission before the deadlinewill be viewed and graded. If you use this .Rmd file as a template, be sure to remove the directions section. Consider removingeval = FALSE from any code chunks provided in the template, if you would like to run that code aspart of your assignment. Your .Rmd file should be written such that, if it is placed in a folder with any data your are asked toimport, it will knit properly without modification. Unless otherwise stated, you may use R for each of the exercises. Be sure to read each exercise carefully! Include your Name and NetID in the final document, not only in your filenames.1AssignmentExercise 1 (Simulating Wald and Likelihood Ratio Tests)In this exercise we will investigate the distributions of hypothesis tests for logistic regression. For this exercise,we will use the following predictors.sample_size = 150set.seed(114)x1 = rnorm(n = sample_size)x2 = rnorm(n = sample_size)x3 = rnorm(n = sample_size)Recall thatp(x) = P[Y = 1 | X = x]Consider the true model�(a) To investigate the distributions, simulate from this model 2500 times. To do so, calculateP[Y = 1 | X = x]for an observation, and then make a random draw from a Bernoulli distribution with that success probability.(Note that a Bernoulli distribution is a Binomial distribution with parameter n = 1. There is no directionfunction in R for a Bernoulli distribution.)Each time, fit the model:�Store the test statistics for two tests: The Wald test for H0 : β2 = 0, which we say follows a standard normal distribution for “large” samples The likelihood ratio test for H0 : β2 = β3 = 0, which we say follows a χ2 distribution (with some degreesof freedom) for “large” samples(b) Plot a histogram of the empirical values for the Wald test statistic. Overlay the density of the truedistribution assuming a large sample.(c) Use the empirical results for the Wald test statistic to estimate the probability of observing a test statisticlarger than 1. Also report this probability using the true distribution of the test statistic assuming a largesample.(d) Plot a histogram of the empirical values for the likelihood ratio test statistic. Overlay the density of thetrue distribution assuming a large sample.2(e) Use the empirical results for the likelihood ratio test statistic to estimate the probability of observinga test statistic larger than 5. Also report this probability using the true distribution of the test statisticassuming a large sample.(f) Repeat (a)-(e) but with simulation using a smaller sample size of 10. Based on these results, is thissample size large enough to use the stanSTAT 420作业代做、R程序语言作业调试、R实验作业代写、Likelihood Ratio作业代做 调试Matlabdard normal and χ2 distributions in this situation? Explain.sample_size = 10set.seed(114)x1 = rnorm(n = sample_size)x2 = rnorm(n = sample_size)x3 = rnorm(n = sample_size)Exercise 2 (Surviving the Titanic)For this exercise use the ptitanic data from the rpart.plot package. (The rpart.plot package depends onthe rpart package.) Use ?rpart.plot::ptitanic to learn about this dataset. We will use logistic regressionto help predict which passengers aboard the Titanic will survive based on various attributes.# install.packages(rpart)# install.packages(rpart.plot)library(rpart)library(rpart.plot)data(ptitanic)For simplicity, we will remove any observations with missing data. Additionally, we will create a test andtrain dataset.ptitanic = na.omit(ptitanic)set.seed(114)trn_idx = sample(nrow(ptitanic), 300)ptitanic_trn = ptitanic[trn_idx, ]ptitanic_tst = ptitanic[-trn_idx, ](a) Consider the model β0 + β1x1 + β2x2 + β3x3 + β4x4 + β5x3x4wherep(x) = P[Y = 1 | X = x]is the probability that a certain passenger survives given their attributes and x1 is a dummy variable that takes the value 1 if a passenger was 2nd class. x2 is a dummy variable that takes the value 1 if a passenger was 3rd class. x3 is a dummy variable that takes the value 1 if a passenger was male. x4 is the age in years of a passenger.Fit this model to the training data and report its deviance.(b) Use the model fit in (a) and an appropriate statistical test to determine if class played a significant rolein surviving on the Titanic. Use α = 0.01. Report: The null hypothesis of the test The test statistic of the test3 The p-value of the test A statistical decision A practical conclusion(c) Use the model fit in (a) and an appropriate statistical test to determine if an interaction between ageand sex played a significant role in surviving on the Titanic. Use α = 0.01. Report: The null hypothesis of the test The test statistic of the test The p-value of the test A statistical decision A practical conclusion(d) Use the model fit in (a) as a classifier that seeks to minimize the misclassification rate. Classify each ofthe passengers in the test dataset. Report the misclassification rate, the sensitivity, and the specificity of thisclassifier. (Use survived as the positive class.)Exercise 3 (Breast Cancer Detection)For this exercise we will use data found in wisc-train.csv and wisc-test.csv, which contain train and testdata, respectively. wisc.csv is provided but not used. This is a modification of the Breast Cancer Wisconsin(Diagnostic) dataset from the UCI Machine Learning Repository. Only the first 10 feature variables havebeen provided. (And these are all you should use.) UCI Page Data DetailYou should consider coercing the response to be a factor variable if it is not stored as one after importing thedata.(a) The response variable class has two levels: M if a tumor is malignant, and B if a tumor is benign. Fitthree models to the training data. An additive model that uses radius, smoothness, and texture as predictors An additive model that uses all available predictors A model chosen via backwards selection using AIC. Use a model that considers all available predictorsas well as their two-way interactions for the start of the search.For each, obtain a 5-fold cross-validated misclassification rate using the model as a classifier that seeks tominimize the misclassification rate. Based on this, which model is best? Relative to the best, are the othertwo underfitting or over fitting? Report the test misclassification rate for the model you picked as the best.(b) In this situation, simply minimizing misclassifications might be a bad goal since false positives and falsenegatives carry very different consequences. Consider the M class as the “positive” label. Consider each of theprobabilities stored in cutoffs in the creation of a classifier using the additive model fit in (a).cutoffs = seq(0.01, 0.99, by = 0.01)That is, consider each of the values stored in cutoffs as c. Obtain the sensitivity and specificity in the testset for each of these classifiers. Using a single graphic, plot both sensitivity and specificity as a function ofthe cutoff used to create the classifier. Based on this plot, which cutoff would you use? (0 and 1 have notbeen considered for coding simplicity. If you like, you can instead consider these two values.)�转自：http://www.7daixie.com/2019042612619813.html
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