Assignment 3 – CSC/DSC 265/465 – Spring 2020 – Due April 29
Q1: Suppose a logistic regression model is fit with response Y = Minor injury during the past year and one predictor X = age. The sample size is n = 100 and the range of X in years is [3.2, 18.1]. The data was fit using the R glm() function, and produced the following output:
> fit = glm(y ~ x, family=’binomial’)
> summary(fit)
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glm(formula = y ~ x, family = “binomial”)
Deviance Residuals:
Min 1Q Median 3Q Max
-1.0094 -0.6021 -0.4758 -0.3770 2.3251
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -3.89379 1.09305 -3.562 0.000368 ***
x 0.19859 0.09147 2.171 0.029927 *
Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1
(Dispersion parameter for binomial family taken to be 1)
Null deviance: 84.542 on 99 degrees of freedom
Residual deviance: 79.500 on 98 degrees of freedom
Number of Fisher Scoring iterations: 5
What is the estimated odds ratio for probability of Minor injury during the past year between 15 year
old and 5 year old subjects? Give an approximate 95% confidence interval.
Q2: For this question, use the NCI60 data set from the ISLR package. From the help page: NCI microarray data. The data contains expression levels on 6830 genes
from 64 cancer cell lines. Cancer type is also recorded.
The object NCI60$data is a 64×6830 matrix. Each row represents a cancer cell line, and contains 6830 gene expression measurements. Assume that the measurements are already standardized. The object NCI60$labs is a vector of length 64 containing text identifying the type of cancer (OVARIAN, MELANOMIA, and so on).
(a) Use the hclust() function (from library stats) to create a hierarchical clustering of the cancer cell lines. Use the option method = ‘average’ to specify the average distance agglomeration method. Plot the dendogram using labels from the NCI60$labs object.
(b) From the dendogram, it can be seen that some cancer types cluster more definititively than others. The tendency of a cancer type to cluster can be quantified by comparing the maximum cophenetic distance between samples of this type, and the minimum cophenetic distance between a sample of this type and a sample not of this type. Determine these quantities for the MELANOMA, RENAL and COLON cancer types, and comment briefly on what you find.
Q3: For this question, use the mammals dataset from the MASS library. This data frame contains average body and brain weights for 62 species of land mammals (in kilograms and grams respectively). The names of the mammals can be accessed using row.names(mammals).
(a) First, log-transform the data. Then for each K = 1, . . . , 10 calculate a K -means cluster solution based on the two log-transformed features. Use option nstart=100. For each solution calculate R2 = 1 − SSwithin/SStotal, and plot these values against K. Identify the smallest value of K, say K∗, for which R2 ≥ 0.8.
(b) Draws a scatterplot of the features, and superimpose the centers output with the clustering solution for K∗. Distinguish the observations by using separate colors for the clusters identified by the solution. These clusters can be identified as follows:
fit = kmeans(x,centers=nc,nstart=100)
fit$cluster
Create separate lists of the species names for each cluster. Does the clustering make sense? Comment briefly.
Q4: For this question, use the Khan data set from the ISLR package. From the help page:
The data consists of a number of tissue samples corresponding to four distinct types
of small round blue cell tumors. For each tissue sample, 2308 gene expression
measurements are available.
The format is a list containing four components: xtrain, xtest, ytrain, and ytest.
xtrain contains the 2308 gene expression values for 63 subjects and ytrain records
the corresponding tumor type. ytrain and ytest contain the corresponding testing
sample information for a further 20 subjects.
Consolidate the training and test data into a single dataset:
> library(ISLR)
> library(class)
> x = rbind(Khan$xtrain,Khan$xtest)
> y = c(Khan$ytrain,Khan$ytest)
[1] 83 2308
> length(y)
There is now a single data set. The object x is an 83 × 2308 table, with 2308 gene expression measurements for each of 83 tissue samples. Then y is a vector of length 83 containing the tumor type, labeled 1 to 4.
(a) Use the prcomp() function (from library stats) to create a matrix of principal components, using the gene expressions as a feature set. Use centering, but not scaling.
(b) Build a KNN classifier for tumor type based on (a) the entire set of gene expressions; and (b) the first 10 principal components. Use the method of Q4 of Assignment 2, using k.list = seq(1,50,1). For each analysis plot CE against K (the neighborhood size). Report only CE. Which classifier is preferable (give several reasons for your answer)?
Q5: [For Graduate Students] Explain why scaling makes a difference for K-means clustering but not linear discriminant analysis.
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