Part 1: Loading and cleaning (20 points)

Read the data census.RData which can be found on Canvas. Check that it has 74020 rows and 31 columns. Each row represents a census tract, and each column a variable that has been measured. Load the plyr library. (Hint: it will be very useful!)

1. (2 points) How many states are represented among the 74020 census tracts? How many counties?

2. (1 point) Columns 8 through 31 of the census data frame represent numeric variables, but columns 8 and 9 are not stored as such. These two are measured in US dollars: median household income (Med_HHD_Inc_ACS_09_13) and median house value (Med_House_value_ACS_09_13). What are the classes of these columns?

3. (5 points) Convert columns 8 and 9 into numbers (in whole US dollars). For example, $63,030 should be converted into the integer 63030. (Hint: you may first convert them into strings, then remove any non-numeric characters using substr() or gsub(), then convert into numbers.) Check your answer by printing out the summary() of these two new columns. Make sure that empty entries (“”) are properly converted to NA.

4. (5 points) Several entries are missing in this data set, including the ones you discovered in the previous question. Compute the number of missing entries in each row, and save the vector as num.na.row. Then, obtain the indices of rows containing any missing values and save them in a vector named contains.na. What is the average number of missing values among the rows that contain at least one missing entry?

5. (2 points) Are there any states with no missing values? If so, print out the names of all such states.

6. (5 points) Redefine the census data frame by removing rows that have missing values, as per the contains.na vector computed in Part 1 Question 4. Check that the new census data frame has now 70877 rows. How many states and counties are represented in this new data frame? What states (if any) have been thrown out compared to the original data frame?

Part 2: Exploratory statistics (20 points)

For all questions below, we will use the cleaned census data frame derived from Part 1 Question 6.

1. (5 points) There are several variables which count the percentage of respondents in various age categories (they all start with pct_Pop_). (Hint: you may access these columns using grep().) Use these to determine, for each census tract, the percentages of the population that are less than 5 years old, and append these values in a new column called pct_Pop_0_4_ACS_09_13 to the census data frame. Then, use this new column, along with the existing Tot_Population_ACS_09_13 column, to determine the number of 0-4 year olds in each state. Which state has the highest number, and what is its value?

2. (5 points) Using your answer from the last question, determine the percentage of 0-4 year olds in each state. Which state has the highest percentage, and what is its value?

3. (5 points) Calculate the correlation between each of the numeric variables, columns 8 through 31 of the census data frame. Which two variables have the highest positive correlation? Which two variables have the lowest negative correlation? Do these relationships make sense?

4. (2 points) Plot a histogram of Med_House_value_ACS_09_13, and label the axes appropriately. See that small bump at the value 1000001? This is a due to a common practice in public data sets of “top-coding”, i.e., censoring unusually large values to protect the anonymity of survey respondents.

5. (3 points) It is possible that the tracts that have been top-coded differ significantly from the other tracts, in other ways than the median house value. The following code computes a t-test between two groups of census tracts, on any given variable. The two groups being compared are: all tracts with median house value equal to the max (1000001) and all tracts with median house value less than the max (<1000001). It then returns a p-value for the test. Note that a lower p-value means a more significant difference between top-coded and non-top-coded tracts, according to the variable in question.

{ group = census$Med_House_value_ACS_09_13 == p.val = t.test([group], [!group])$p.value (p.val)}

Apply the function my.test() to the variables in columns 10 through 31 of the census data frame. What are the two smallest p-values, and which variables do these correspond to? Does this make sense?

Part 3: Sampling and plotting (30 points)

Plotting data of this size is tricky just because the sheer number of datapoints. For instance, you can try on your own to plot(census) to see your computer wheeze, cough, crash and burn. In this part, we will writea suite of functions that allow you to more efficiently plot relationships between pairs of variables in your dataset.

- (10 points) Writea function plot.sample() that takes in five arguments: x, y, nsample, xlab, and ylab. The first two arguments are variables to be plotted. The third is the number of points to be randomly sampled, with a default of 500. (Hence, if x and y are vectors of length, say, 5000, then a random 500 of the total 5000 x-y pairs will be plotted, by default.) The last two are the x and y labels, both with defaults of “” (the empty string). A few notes:

- check that x,y have the same length, and if not, throw an error (using stop() or stopifnot());
- check that the number of requested samples does not exceed the total number of data points, otherwise throw an error;
- the plot should not have a title.

After writing this function, you can try it out using the following code (specify eval=TRUE).

, census, xlab=, ylab=)

- (10 points) Next, writea function add.lineartrend.info(), which is designed to be called after plot.sample(). This takes as input x, y for variables that already appear on the x-and y-axes, and does two things:

- adds the best fit line (produced by lm()) function in red, and with twice the default width (using the lwd=2 option);
- adds a title to the plot with the numeric correlation of the two variables, using 3 significant digits (see signif()).

To reiterate, this function assumes there is a plot already in place; it simply adds the line and the title. Again, after writing this function, you can try it out using the following code (specify eval=TRUE).

, census, xlab=, ylab=)add.lineartrend.info(census, census)

- (10 points) Lastly, writea function plot.all() which takes as input dataset and nsample, a data frame, and the number of points to be sampled. This function will mimick the behavior of plot() when applied to data frames. In other words, if dataset has p columns, then plot.all() should produce a p by p retangular grid of plots, where the plot in entry i, j of the grid uses column i for the x-axis data, and column j for the y-axis data. Some notes:

- in each grid entry, the plot should be produced by your previous function, plot.sample();
- in each grid entry, a best fit line should be added and the correlation should be displayed in the title, using your previous function, add.lineartrend.info();
- in the diagonal grid entries, instead of plotting the data (and adding linear trend info), simply use text() to write the variable names in the middle of otherwise empty plots.

A code template for plot.all() is found below. (Hint: using a for() loop for the grid entries is perfectly fine.) Fill it out, and then run the code that follows (specify eval=TRUE) to plot the pairwise relationships between the 4 census variables Med_HHD_Inc_ACS_09_13, Med_House_value_ACS_09_13, pct_College_ACS_09_13, and Mail_Return_Rate_CEN_2010. Comment on the results; do the relationships make sense to you?

) { p = ncol(dataset) orig.par = par() par(mar=c(,,,)) par(mfrow=c(p,p)) your plotting code goes here par(mar=orig.par$mar) par(mfrow=orig.par$mfrow)}mydat = census[,c(, , ,)]plot.all(mydat)

Part 4: Permute away! (25 points + 10 points Bonus)

Here, we will use what is called permutation test to answer the following question: are the linear regression coefficients between two different states substantially different? Note: a permutation test is a fairly advanced statistical technique, but not much statistical knowledge is required to answer this question. For a bit more (optional) discussion on the permutation test, see the end of this part.

1. (3 points) From the census data frame, extract all rows that correspond to tracts in Virginia or West Virginia, and call the resulting data frame census.vw. Verify that this has 2321 rows.

2. (6 points) Now perform two separate linear regressions using census.vw. The first is a linear regression of Mail_Return_Rate_CEN_2010 (as the response) on pct_NH_White_alone_ACS_09_13 and pct_Renter_Occp_HU_ACS_09_13 (as the predictors), using only the tracts in Virginia; the second is again a linear regression of Mail_Return_Rate_CEN_2010 (as the response) on pct_NH_White_alone_ACS_09_13 and pct_Renter_Occp_HU_ACS_09_13 (as the predictors), but now using only the tracts in West Virginia. Ignoring the intercept terms, each linear regression model here gives you two coefficients (one for pct_NH_White_alone_ACS_09_13 and one for pct_Renter_Occp_HU_ACS_09_13). Compare the two sets of coefficients. Are they the same?

3. (6 points) Even though the two coefficients for Virginia and West Virginia may be different, it is hard to tell just how different they are. To help answer this question, we will look at the linear regression coefficients if we were to randomly scramble census tracts between Virginia and West Virginia, then refit the linear regression models. This is the idea behind a permutation test. Vaguely speaking, if the two sets of coefficients were truly the same, then shuffling the labels should have no real effect on our estimated coefficients. First, make a copy of census.vw called census.vw.perm. Then, randomly permute the entries in census.vw.perm$State_name. (Hint: recall the function sample().) Once this is done, rerun the two regression models from Part 4 Question 2 using census.vw.perm, and report the coefficients for Virginia, and for West Virginia. What are their values now? Are the differences between coefficients for Virginia and West Virginia much smaller than they were in Part 4 Question 2?

4. (10 points) You will writea function to encapsulate the tasks you performed in the previous questions. The function will be called reg.coef(), and it will take in five arguments:

- census.subset, a data frame, whose rows are a subset of the full census data frame;
- x1, x2, two strings that match column names in census.subset, representing predictor variables for the regression;
- y, another string that matches a column name in census.subset, representing a response variable for the regression;
- shuffle, a Boolean variable, whose default value is FALSE.

Your function should perform the following. If shuffle is TRUE, then the entries in census.subset$State_name are randomly permuted. If shuffle is FALSE, then this step is not performed (and census.subset$State_name is left alone). Next, for each state in the census.subset data frame, run a regression of the response, represented by y, on predictor variables, represented by x1 and x2. (Hint: you may create the formula in lm() using paste().) A matrix with 2 columns is returned, and one row for each state in census.subset. Each row gives the coefficients for x1 and x2 in the linear regression model computed for that particular state (though they are included in the regressions, the intercept terms here are ignored). (Hint: for running regressions at a state-by-state level, use daply().) Recreate the results of Part 4 Questions 2&3 with your function, to check that it gives the same answers. Be sure to use set.seed() function to enforce reproducibility.

**Bonus (10 points)**Lastly, we will finally implement our permutation test. At a high-level, the idea is to judge differences in coefficients, say, from Part 4 Question 2 between Virginia and West Virginia, to those in Part 4 Question 3 computed using scrambled data. To make our comparisons to more meaningful, we will repeat the scrambling of state labels (the permutations) multiple times. Writea function permutation.test() that takes in the following seven arguments:

- census, the census data frame;
- state1 and state2, two strings that represent state names;
- x1, x2, y, as in reg.coef();
- num.perm, an integer with a default value of 100.

This function will carry out the following tasks, in order:

- ensure that the columns represented by x1, x2 and y are distinct numeric columns of census; also make sure that state1 and state2 are valid states;
- extract the rows from census where each row contains only observations with State_name being equal to state1 or state2, and call this smaller data frame census.subset;
- run reg.coef() from above with arguments census.subset, x1, x2 and y, and set shuffle to FALSE; store your results in obs.coef;
- run reg.coef() as in the last step, but now with shuffle set to TRUE, and repeat for a total of num.perm times; stack the results into one big matrix has 2 columns, and 2*num.perm rows, called perm.coef;
- plot the first column of perm.coef (x-axis) versus the second column (y-axis); there should be 2*num.perm points on this plot; in addition, draw on top the first column of obs.coef versus the second column; this should give you 2 points, and color them in red; also, label the axes according to the names of the variables x1 and x2;
- lastly, return a list containing obs.coef and perm.coef.

Once written, run the code below (specify eval=TRUE) and comment on the results. In each case, do the red dots lie roughly in the point cloud of black dots? If so, it implies that the differences between states are insignificant; and if not, it implies that the differences between states are significant.

.seed()out1 = permutation.test(census, , , , , )out2 = permutation.test(census, , , , , )

**Statistical background.** For the statistically curious. Note that the permutation test we described is one of many ways to determine how substantially different regression coefficients are. Students who have taken other statistical courses might know that using standard errors is another way. The permutation test is in some sense more robust because it assumes less about the data-generating distribution.