STAT3006代写、R程序设计代写

日期：2022-10-28 09:57

STAT3006/7305 Assignment 4, 2022

High-Dimensional Analysis

Weighting: 20%

Due: Monday 14/11/2022

This assignment involves the analysis of a high-dimensional dataset. Here we focus on an

early microarray dataset analysed by Alon et al (1999), which involves predicting whether a

given tissue sample is cancerous or not, as well as trying to determine which genes are

expressed differently between the two classes.

You should select one classifier for the task of classification, which you have not used in

previous assignments. Probability-based classifiers discussed in this course include linear,

quadratic, mixture and kernel density discriminant analysis. Non-probability-based classifiers

discussed include k nearest neighbours, neural networks, support vector machines and

classification trees. All of these are implemented via various packages available in R. If you

wish to use a different method, please check with the coordinator. In addition, you will make

use of lasso-penalised logistic regression.

It will quickly become apparent that the number of observations is less than the number of

variables, and so some form of dimensionality reduction is needed for most forms of

probability-based classifier and can be used if desired with the non-probability-based

classifiers.

The Alon dataset contains measurements of the expression levels of 2000 genes from 62

colon tissue samples, 40 of which are labelled as being from tumours (cancers) and 22 normal

(non-cancerous). Here we consider analysis of this data to (i) develop a model which is

capable of accurately predicting the class (cancer or normal) of new observations, without the

need for examination by clinicians (ii) determine which genes are expressed differently

between the two groups. Discriminant analysis/supervised classification can be applied to

solve (i), and in combination with feature (predictor) selection, can be used to provide a

limited solution to (ii) also. Other methods such as single-variable analysis can also be

applied to attempt to answer (ii). You should use R for the assignment.

Tasks:

1. (5 marks) Perform principal component analysis of the Alon dataset and report and

comment on the results. Detailed results should be submitted via a separate file, including

what each principal component direction is composed of in terms of the original

explanatory variables, with some explanation in the main report about what is in the file.

Give a plot or plots which shows the individual and cumulative proportions of variance

explained by each component. Also produce and include another plot about the principal

components which you think would be of interest to biologists such as Alon et. al, along

with some explanation and discussion. The R package FactoMineR is a good option for

PCA.

2. (4 marks) Perform single variable analysis of the Alon dataset, looking for a relationship

with the response variable (the class). Use the Benjamini-Hochberg (1995) or BenjaminiYekutieli (2001) approach to control the false discovery rate to be at most 0.01. Explain

the assumptions of this approach and whether or not these are likely to be met by this

dataset, along with possible consequences of any violations. Also explain how the method

works mathematically, but leave out why (i.e. give something equivalent to pseudocode).

Report which genes are then declared significant along with the resulting threshold in the

original p-values. Also give a plot of gene order by p-value versus unadjusted p-value (or

the log of these), along with a line indicating the FDR control.

Within the stats package is the function p.adjust, which offers this method. More advanced

implementations include the fdrame package in Bioconductor.

3. (3 marks) Define binary logistic regression with a lasso penalty mathematically, including

the function to be optimised and briefly introduce a method than can be used to optimise

it. Note that this might require a little research.

4. (3 marks) Explain the potential benefits and drawbacks of using PCA to reduce the

dimensionality of the data before attempting to fit a classifier. Explain why you have

chosen to reduce the dimensionality or not to do so for this purpose.

5. Apply each classification method (your choice and lasso logistic regression) using R to the

Alon dataset, report the results and interpret them.

For lasso logistic regression, I suggest you use the glmnet package, available in CRAN, and

make use of the function cv.glmnet and the family=“binomial” option. If you are interested,

there is a recording of Trevor Hastie giving a tutorial on the lasso and glmnet at

http://www.youtube.com/watch?v=BU2gjoLPfDc .

Results should include the following:

a) (1 mark) characterisation of each class: parameter estimates.

b) (2 marks) cross-validation (CV)-based estimates of the overall and class-specific error

rates: obtained by training the classifier on a large fraction of the whole dataset and then

applying it to the remaining data and checking error rates. You may use 5-fold, 10-fold or

leave-one-out cross-validation to estimate performance.

c) (2 marks) For lasso logistic regression, you will need to use cross-validation to estimate of

the optimal value of λ. Explain how you plan to search over possible values. Then produce

and explain a graph of your cost function versus λ. You should also produce a list of the

genes included as predictor variables in the optimal classifier, along with their estimated

coefficients.

6. (5 marks) Compare the results from all approaches to analysis of the Alon dataset (PCA,

single-variable analysis and the two classifiers). Explain what each approach seems to offer,

including consideration of these results as an example. In particular, if you had to suggest 10

genes for the biologists to study further for possible links to colon cancer, which ones would

you prioritise, and what makes you think they are worth studying further?

Notes:

(i) R commands you might find useful:

objects() – gives the current list of objects in memory.

attributes(x) – gives the set of attributes of an object x.

(ii) How to open the Alon data file in R:

File → Load Workspace → Files of type: All files (*.*), select alon.rda > Open. This

should load the Alon data into memory as x, xm and y. x is the predictor variable data for 62

subjects, y lists the labels (classes). You can ignore xm, but there are some clues on what

these values are in the Alon et al. paper.

(iii) Please put all the R commands in a separate text file or files and submit these separately

via a single text file or a zip file. You should not give any R commands in your main report

and should not include any raw output – i.e. just include figures from R (each with a title,

axis labels and caption below) and put any relevant numerical output in a table or within the

text.

(iv) As per http://www.uq.edu.au/myadvisor/academic-integrity-and-plagiarism, what you

submit should be your own work. Even where working from sources, you should endeavour

to write in your own words. Equations are either correct or not, but you should use consistent

notation throughout your assignment and define all of it.

(v) Please name your files something like student_number_STAT3006_A4.pdf to assist with

marking. You should submit your assignment via two links on Blackboard: one for your pdf

report and the other for your .txt or .zip file containing R code in text file(s).

(vi) Some references:

R

Maindonald, J. and Braun, J. Data Analysis and Graphics Using R - An Example-Based

Approach, 3rd edition, Cambridge University Press, 2010.

Venables, W.N. and Ripley, B.D., Modern Applied Statistics with S, Fourth Edition, Springer,

2002.

Wickham, H. and Grolemund, G. R for Data Science, O'Reilly, 2017.

High-dimensional Analysis

Bishop, C. Pattern Recognition & Machine Learning, Springer, 2006.

Buhlmann, P. and van de Geer, S. Statistics for High-Dimensional Data, Springer, 2011.

Efron, B. and Hastie, T. Computer Age Statistical Inference, Cambridge University Press,

2016.

Hastie, T., Tibshirani, R. and Friedman, J. The Elements of Statistical Learning: Data

Mining, Inference, and Prediction, 2nd edition, Springer, 2009.

Hastie, T., Tibshirani, R. and Wainwright, M. Statistical Learning with Sparsity, CRC Press,

2015.

McLachlan, G.J., Do, K.-A. and Ambroise, C. Analyzing Microarray Gene Expression Data,

Wiley, 2004.

Other references

Alon, U. et al. Broad Patterns of Gene Expression Revealed by Clustering Analysis of Tumor

and Normal Colon Tissues Probed by Oligonucleotide Arrays. Proceedings of the National

Academy of Sciences of the United States of America, 96(12), 6745–6750, 1999.

Lazar, C. et al. A Survey on Filter Techniques for Feature Selection in Gene Expression

Microarray Analysis, IEEE/ACM Transactions on Computational Biology and

Bioinformatics, 9, 1106-1119, 2012.

Note: Lazar et al. is just an example overview of the range of techniques used in this field. It

is also worth noting that microarray experiments have largely been superseded by more

recent technology such as RNA-Seq. However, the methods of analysis are similar.