Name:
Andrew ID:
Collaborated with:

On this homework, you can collaborate with your classmates, but you must identify their names above, and you must submit your own homework as an knitted HTML file on Canvas, by Monday 10pm, next week (July 9).

## For reproducibility --- don't change this!
set.seed(07012019)

# Some R basics

• 1a. Let’s start easy by working through some R basics, just to brush up on them. Define a variable x_vec to contain the integers 1 through 100. Check that it has length 100. Report the data type being stored in x_vec. Add up the numbers in x_vec, by calling a built-in R function.

• 1b. Convert x_vec into a matrix with 20 rows and 5 columns, and store this as x_mat. Here x_mat should be filled out in the default order (column major order). Check the dimensions of x_mat, and the data type as well. Compute the sums of each of the 5 columns of x_mat, by calling a built-in R function. Check (using a comparison operator) that the sum of column sums of x_mat equals the sum of x_vec.

• 1c. Extract and display rows 1, 5, and 17 of x_mat, with a single line of code. Answer the following questions, each with a single line of code: how many elements in row 2 of x_mat are larger than 40? How many elements in column 3 are in between 45 and 50? How many elements in column 5 are odd? Hint: take advantage of the sum() function applied to Boolean vectors.

• 1d. Using Boolean indexing, modify x_vec so that every even number in this vector is incremented by 10, and every odd number is left alone. This should require just a single line of code. Print out the result to the console. Challenge: show that ifelse() can be used to do the same thing, again using just a single line of code.

• 1e. Consider the list x_list created below. Complete the following tasks, each with a single line of code: extract all but the second element of x_list—seeking here a list as the final answer. Extract the first and third elements of x_list, then extract the second element of the resulting list—seeking here a vector as the final answer. Extract the second element of x_list as a vector, and then extract the first 10 elements of this vector—seeking here a vector as the final answer. Note: pay close attention to what is asked and use either single brackets [ ] or double brackets [[ ]] as appropriate.

x_list <- list(rnorm(6), letters, sample(c(TRUE,FALSE), size=4, replace=TRUE)) Prostate cancer data set ===

OK, moving along to more interesting things! We’re going to look again, as in lab, at the prostate cancer data set: 9 variables measured on 97 men who have prostate cancer (from the book The Elements of Statistical Learning):

1. lpsa: log PSA score
2. lcavol: log cancer volume
3. lweight: log prostate weight
4. age: age of patient
5. lbph: log of the amount of benign prostatic hyperplasia
6. svi: seminal vesicle invasion
7. lcp: log of capsular penetration
8. gleason: Gleason score
9. pgg45: percent of Gleason scores 4 or 5

To load this prostate cancer data set into your R session, and store it as a matrix pros_data:

pros_data <-
as.matrix(read.table("https://raw.githubusercontent.com/benjaminleroy/36-350-summer-data/master/Week1/pros.dat"))

# Computing standard deviations using iteration

• 2a. Using on-the-fly Boolean indexing, extract the rows of pros_data that correspond to patients with SVI, and the rows that correspond to patients without it. Call the resulting matrices pros_data_svi and pros_data_no_svi, respectively. Display the dimensions of these matrices. Compute the column means of pros_data_svi and pros_data_no_svi, stored into vectors called pros_data_svi_avg and pros_data_no_svi_avg, respectively. For each matrix, this should require just a single call to a built-in R function. Display these column means.

• 2b. Take a look at the starter code below. The first line defines an empty vector pros_data_svi_sd of length ncol(pros_data) (of length 9). The second line defines an index variable i and sets it equal to 1. Write a third line of code to compute the standard deviation of the ith column of pros_data_svi, using a built-in R function, and store this value in the ith element of pros_data_svi_sd.

pros_data_svi_sd <- vector(length = ncol(pros_data))
i <- 1
• 2c. Repeat the calculation as in the previous question, but for patients without SVI. That is, produce three lines of code: the first should define an empty vector pros_data_no_svi_sd of length ncol(pros_data) (of length 9), the second should define an index variable i and set it equal to 1, and the third should fill the ith element of pros_data_no_svi_sd with the standard deviation of the ith column of pros_data_no_svi.

• 2d. Write a for() loop to compute the standard deviations of the columns of pros_data_svi and pros_data_no_svi, and store the results in the vectors pros_data_svi_sd and pros_data_no_svi_sd, respectively, that were created above. Note: you should have a single for() loop here, not two for loops. And if it helps, consider breaking this task down into two steps: as the first step, write a for() loop that iterates an index variable i over the integers between 1 and the number of columns of pros_data (don’t just manually write 9 here, pull out the number of columns programmatically), with an empty body. As the second step, paste relevant pieces of your solution code from Q2b and Q2c into the body of the for() loop. Print out the resulting vectors pros_data_svi_sd and pros_data_no_svi_sd to the console. Comment, just briefly (informally), by visually inspecting these standard deviations and the means you computed in Q2a: which variables exhibit large differences in means between the SVI and non-SVI patients, relative to their standard deviations?

# Computing t-tests using vectorization

• 3a. Recall that the two-sample (unpaired) t-statistic between data sets $$X=(X_1,\ldots,X_n)$$ and $$Y=(Y_1,\ldots,Y_m)$$ is: $T = \frac{\bar{X} - \bar{Y}}{\sqrt{\frac{s_X^2}{n} + \frac{s_Y^2}{m}}},$ where $$\bar{X}=\sum_{i=1}^n X_i/n$$ is the sample mean of $$X$$, $$s_X^2 = \sum_{i=1}^n (X_i-\bar{X})^2/(n-1)$$ is the sample variance of $$X$$, and similarly for $$\bar{Y}$$ and $$s_Y^2$$. We will compute these t-statistics for all 9 variables in our data set, where $$X$$ will play the role of one of the variables for SVI patients, and $$Y$$ will play the role of this variable for non-SVI patients. Start by computing a vector of the denominators of the t-statistics, called pros_data_denom, according to the formula above. Take advantage of vectorization; this calculation should require just a single line of code. Make sure not to include any hard constants (e.g., don’t just manually write 21 here for $$n$$); as always, programmatically define all the relevant quantities. Check using all.equal() that your computed denominators match the “magic denominators” given below in magic_denom. Then compute a vector of t-statistics for the 9 variables in our data set, called pros_data_t_stat, according to the formula above, and using pros_data_denom. Again, take advantage of vectorization; this calculation should require just a single line of code. Print out the t-statistics to the console.
magic_denom <- c(0.19092077, 0.08803179, 1.91148819, 0.34076326, 0.00000000,
0.25730390, 0.15441770, 6.30903678, 0.23021447)
• 3b. Given data $$X$$ and $$Y$$ and the t-statistic $$T$$ as defined the last question, the degrees of freedom associated with $$T$$ is: $\nu = \frac{(\frac{s_X^2}{n}+\frac{s_Y^2}{m})^2}{\frac{(\frac{s_X^2}{n})^2}{n-1} + \frac{(\frac{s_Y^2}{m})^2}{m-1}}.$ Compute the degrees of freedom associated with each of our 9 t-statistics (from our 9 variables), storing the result in a vector called pros_data_df. This might look like a complicated calculation, but really, it’s not too bad: it only involves arithmetic operators, and taking advantage of vectorization, the calculation should only require a single line of code. Hint: to simplify this line of code, it will help to first set short variable names for variables/quantities you will be using, as in sx <- pros_data_svi_sd, n <- nrow(pros_data_svi), and so on. Print out these degrees of freedom values to the console.

• 3c. The function pt() evaluates the distribution function of the t-distribution. E.g.,

pt(x, df=v, lower.tail=FALSE)

returns the probability that a t-distributed random variable, with v degrees of freedom, exceeds the value x. Importantly, pt() is vectorized: if x is a vector, and so is v, then the above returns, in vector format: the probability that a t-distributed variate with v[1] degrees of freedom exceeds x[1], the probability that a t-distributed variate with v[2] degrees of freedom exceeds x[2], and so on.

Call pt() as in the above line, but replace x by the absolute values of the t-statistics you computed for the 9 variables in our data set, and v by the degrees of freedom values associated with these t-statistics. Multiply the output by 2, and store it as a vector pros_data_p_val. These are called p-values for the t-tests of mean difference between SVI and non-SVI patients, over the 9 variables in our data set. Print out the p-values to the console. Identify the variables for which the p-value is smaller than $$0.05/9$$ (hence deemed to have a significant difference between SVI and non-SVI patients)1. Identify the variable with the smallest p-value (the most significant difference between SVI and non-SVI patients).

# Computing t-tests using iteration

• 4. The function t.test() computes a two-sample (unpaired) t-test between two data sets. E.g.,

t_test_obj <- t.test(x = rnorm(10), y = rnorm(10))
names(t_test_obj)
## [1] "statistic"   "parameter"   "p.value"     "conf.int"    "estimate"
## [6] "null.value"  "alternative" "method"      "data.name"

computes a t-test between data sets x=rnorm(10) and y=rnorm(10) (here, just for the sake of example, these are just two sets of 10 randomly generated standard normals), and stores the output in a list called t_test_obj. The names of the list are then displayed. Note: the element named p.value contains the p-value.

Define an empty vector of length ncol(pros_data) (of length 9) called pros_data_p_val_master. Then write a for() loop to populate its entries with the p-values from calling t.test() to test the mean difference between SVI and non-SVI patients, over each of the 9 variables in our data set. Important: the t.test() function will throw an error when it tries to consider the mean difference in the SVI variable itself, across the two groups of SVI patients and non-SVI patients; this will occur at some value of i (i.e., the value for which pros_data[,i] is the SVI column). To avoid this error, use an if() statement to check if the current variable being considered is SVI, and in this case, just set the p-value equal to NaN (rather than calling t.test()). Check using all.equal() that the p-values stored in pros_data_p_val_master match the ones you computed in Q3c. Note: use check.names = FALSE as a third argument to all.equal(), which instructs it to ignore the names of its first two arguments.

# Shakespeare’s complete works

On to the more fun stuff! As in lab, we’re going to look at William Shakespeare’s complete works, taken from Project Gutenberg. The Shakespeare data file is up on our course website, and to load it into your R session, as a string vector called shakespeare_lines:

shakespeare_lines <-
readLines("https://raw.githubusercontent.com/benjaminleroy/36-350-summer-data/master/Week1/shakespeare.txt")

# Where are Shakespeare’s plays, in this massive text?

• 5a. Some lines in shakespeare_lines are empty, i.e., they are just equal to “”. How many such lines are there? Remove all empty lines from shakespeare_lines. Also, trim all “extra” white space characters in the lines of shakespear_lines using the trimws() function. Note: if you are unsure about what trimws() does, try it out on some simple strings/some simple vectors of strings.

• 5b. Visit https://raw.githubusercontent.com/benjaminleroy/36-350-summer-data/master/Week1/shakespeare.txt in your web browser and just skim through this text file. Near the top you’ll see a table of contents. Note that “THE SONNETS” is the first play, and “VENUS AND ADONIS” is the last. Using which(), find the indices of the lines in shakespeare_lines that equal “THE SONNETS”, report the index of the first such occurence, and store it as toc_start. Similarly, find the indices of the lines in shakespeare_lines that equal “VENUS AND ADONIS”, report the index of the first such occurence, and store it as toc_end.

• 5c. Define n = toc_end - toc_start + 1, and create an empty string vector of length n called titles. Using a for() loop, populate titles with the titles of Shakespeare’s plays as ordered in the table of contents list, with the first being “THE SONNETS”, and the last being “VENUS AND ADONIS”. Print out the resulting titles vector to the console. Hint: if you define the counter variable i in your for() loop to run between 1 and n, then you will have to index shakespeare_lines carefully to extract the correct titles. Think about the following. When i = 1, you want to extract the title of the first play in shakespeare.lines, which is located at index toc.start. When i = 2, you want to extract the title of the second play, which is located at index toc_start + 1. And so on.

• 5d. Use a for() loop to find out, for each play, the index of the line in shakespeare.lines at which this play begins. It turns out that the second occurence of “THE SONNETS” in shakespeare_lines is where this play actually begins (this first ocurrence is in the table of contents), and so on, for each play title. Use your for() loop to fill out an integer vector called titles_start, containing the indices at which each of Shakespeare’s plays begins in shakespeare_lines. Print the resulting vector titles_start to the console.

• 5e. Define titles_end to be an integer vector of the same length as titles_start, whose first element is the second element in titles_start minus 1, whose second element is the third element in titles_start minus 1, and so on. What this means: we are considering the line before the second play begins to be the last line of the first play, and so on. Define the last element in titles_end to be the length of shakespeare_lines. You can solve this question either with a for() loop, or with proper indexing and vectorization. Challenge: it’s not really correct to set the last element in titles_end to be length of shakespeare_lines, because there is a footer at the end of the Shakespeare data file. By looking at the data file visually in your web browser, come up with a way to programmatically determine the index of the last line of the last play, and implement it.

• 5f. In Q5d, you should have seen that the starting index of Shakespeare’s 38th play “THE TWO NOBLE KINSMEN” was computed to be NA, in the vector titles_start. Why? If you run which(shakespeare_lines == "THE TWO NOBLE KINSMEN") in your console, you will see that there is only one occurence of “THE TWO NOBLE KINSMEN” in shakespeare_lines, and this occurs in the table of contents. So there was no second occurence, hence the resulting NA value.

But now take a look at line 118,463 in shakespeare_lines: you will see that it is “THE TWO NOBLE KINSMEN:”, so this is really where the second play starts, but because of colon “:” at the end of the string, this doesn’t exactly match the title “THE TWO NOBLE KINSMEN”, as we were looking for. The advantage of using the grep() function, versus checking for exact equality of strings, is that grep() allows us to match substrings. Specifically, grep() returns the indices of the strings in a vector for which a substring match occurs, e.g.,

grep(pattern="cat",
x=c("cat", "canned goods", "batman", "catastrophe", "tomcat"))
## [1] 1 4 5

so we can see that in this example, grep() was able to find substring matches to “cat” in the first, fourth, and fifth strings in the argument x. Redefine titles_start by repeating the logic in your solution to Q5d, but replacing the which() command in the body of your for() loop with an appropriate call to grep(). Also, redefine titles_end by repeating the logic in your solution to Q5e. Print out the new vectors titles_start and titles_end to the console—they should be free of NA values.

# Extracting and analysing a couple of plays

• 6a. Let’s look at two of Shakespeare’s most famous tragedies. Programmatically find the index at which “THE TRAGEDY OF HAMLET, PRINCE OF DENMARK” occurs in the titles vector. Use this to find the indices at which this play starts and ends, in the titles_start and titles_end vectors, respectively. Call the lines of text corresponding to this play shakespeare_lines_hamlet. How many such lines are there? Do the same, but now for the play “THE TRAGEDY OF ROMEO AND JULIET”, and call the lines of text corresponding to this play shakespeare_lines_romeo. How many such lines are there?

• 6b. Repeat the analysis, outlined in Q10a of Lab 1, on shakespeare_lines_hamlet. That is:
• collapse shakespeare_lines_hamlet into one big string, separated by spaces;
• convert this string into all lower case characters;
• divide this string into words, by splitting on spaces or on punctuation marks, using split = "[[:space:]]|[[:punct:]]" in the call to strsplit();
• remove all empty words (equal to the empty string “”), and report how many words remain;
• compute the unique words, report the number of unique words, and plot a histogram of their numbers of characters;
• report the 5 longest words;
• compute a word table, and report the 25 most common words and their counts;
• finally, produce a plot of the word counts verus rank.
• 6c. Repeat the same task as in Q6b, but on shakespeare.lines.romeo. Comment on any similarities/differences you see in the answers.

1. Due to the fact that we are testing multiple hypotheses we are using a Bonferoni correction on the false positive rate of .05 (This will be cover in other classes - but feel free to ask me about it).