Math 150 - Methods in Biostatistics - Homework 1

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Your Name Here

Published

January 25, 2023

Assignment Summary (Goals)

Practice using R to run t-tests and linear models
Provide details about what the models mean

Note on R:

At the bottom of this assignment, I’ve given an R Tutorial of sorts with some of the main ideas that we’ll be using this semester. You might want to read over them and try out some of the commands yourself.

Please ask questions as you go along! Asking questions about the R code (and about the course in general) is your key to success.

Look carefully at the line of code in the R chunk immediately below. Notice that a dataset from the textbook is being loaded in so that it can be analyzed. The datasets for the textbook are all provided to you on Canvas. You’ll need to get the data from Canvas (the file for this HW is called C2 Games1.csv), and copy it either to your own computer or to the file system on the Pomona R Server (if you are using the server). Then you need to point to the location of that dataset.

If it is difficult to find the location, look at the upper right side of the R Studio window for the icon that says Import Dataset. By clicking through, import the dataset into R. But wait, you aren’t done! After you import the data, you’ll see the correct path in the Console. That path must be written into the R chunk which imports the data.

You will know that you have successfully imported the data if you can knit this file to a pdf without any errors. After you’ve imported the data (and before you move on to the rest of the assignment), try to knit to pdf. Does it work? Great! Does it not work? Read the paragraph above again (then ask questions on Slack).

games1 <- readr::read_csv("~/Dropbox/teaching/MA150/PracStatCD/Data Sets/Chapter 02/CSV Files/C2 Games1.csv")

Q0. Collaborative Learning

Describe one thing you learned from someone (a fellow student or mentor) in our class this week (it could be: content, logistical help, background material, R information, etc.) 1-3 sentences.

Q1. Chapter 2, A10.

Use R to calculate a two-sample test statistic (assuming equal variances) and find the p-value corresponding to this statistic. In addition, calculate a 95% confidence interval for the difference between the two means (\(\mu_1 - \mu_2\)). The end of chapter exercises will provide details on conducting this calculation by hand. If \(H_0: \mu_1 = \mu_2\) is true, the p-value states how likely that just random sampling variability would create a difference between two sample means (\(\overline{y}_1 - \overline{y}_2\)) at least as large as we observed. Based on the p-value, what can you conclude about these two types of games?

games1 %>%
    t.test(Time ~ Type, data=., var.equal = TRUE)

games1 %>%
    t.test(Time ~ Type, data=., var.equal = TRUE) %>%
    tidy()

Q2. Chapter 2, A11

To fit a linear model, the Type variable will need to be binary. Fit a linear model in R using lm() and notice which level of Type gets set to 0 and which gets set to 1. How can you tell?

Develop a regression model using Time as the response and the indicator on Type as the explanatory variable.

Create a linear model (lm()) and then tidy() the model. The following example code might help.

games1 %>%
  lm(Time ~ Type, data = .) %>%
  summary()

games1 %>%
  lm(Time ~ Type, data = .) %>%
  tidy()

Q3. Chapter 2, A12

Use R to calculate the t-statistic and p-value for the hypothesis test \(H_0: \beta_1 = 0\) vs \(H_a: \beta_1 \ne 0\). In addition, construct at 95% confidence interval for \(\beta_1\). Based on these statistics, can you conclude that the coefficient \(\beta_1\) is significantly different from zero? How is the test of \(\beta_1\) equivalent to the t.test() in the earlier question?

The argument conf.int = TRUE inside tidy() on the linear model will find confidence intervals for the coefficients.

Q4. Chapter 2, E1

Assume you are conducting a t-test to determine if there is a difference between two means. You have the following summary statistics: \(\overline{x}_1 = 10, \overline{x}_2 = 20\) and \(s_1=s_2=10\). Without completing the hypothesis test, explain why \(n_1=n_2=100\) would result in a smaller p-value than \(n_1=n_2=16\).

Q5. Chapter 2, E2

If the hypothesis test \(H_0: \beta_1 = 0\) vs \(H_a: \beta_1 \ne 0\) results in a small p-value, can we be confident that the regression model provides a good estimate of the response value for a given value of \(x_i\)? Provide an explanation for your answer.

Q6. Chapter 2, E3

What model technical conditions (if any) need to be satisfied in order to calculate \(b_0\) and \(b_1\) in a simple linear regression model?

Q7. Chapter 2, E4

Explain why the model notation \(y_i = \beta_0 + \beta_1 x_i\) is not appropriate, but \(\hat{y}_i = b_0 + b_1 x_i\) is appropriate.

HW assignments will be graded out of 5 points, which are based on a combination of accuracy and effort. Below are rough guidelines for grading.

Score & Description

5 points: All problems completed with detailed solutions provided and 75% or more of the problems are fully correct.

4 points: All problems completed with detailed solutions and 50-75% correct; OR close to all problems completed and 75%-100% correct. An assignment will earn a 4 if there is superfluous information printed out on the assignment.

3 points: Close to all problems completed with less than 75% correct

2 points: More than half but fewer than all problems completed and > 75% correct

1 point: More than half but fewer than all problems completed and < 75% correct; OR less than half of problems completed

0 points: No work submitted, OR half or less than half of the problems submitted and without any detail/work shown to explain the solutions.

General notes on homework assignments (also see syllabus for policies and suggestions):

please be neat and organized, this will help me, the grader, and you (in the future) to follow your work.
be sure to include your name on the assignment
please include at least the number of the problem, or a summary of this question (this will also be helpful to you in the future to prepare for exams).
for R problems, it is required to use R Markdown. You can write out other problems with pencil and combine pdf as appropriate.
please do not print errors, messages, warnings, or anything else that makes your homework unwieldy. You will be graded down for superfluous printouts.
in case of questions, or if you get stuck please don’t hesitate to email me or DM on Slack! The sooner (and more often) questions get asked, the better for everyone.

In the examples below, we focus on a few R ideas which should become familiar to you as we go forward this semester.

the pipe command %>%
data wrangling verbs: filter(), summarize(), group_by(), mutate(), arrange()
ifelse()

Getting started

Load packages

Here we will explore the data using functions that can be found in the tidyverse package. The data can be found in the package nycflights13.

Let’s load the packages.

library(tidyverse)
library(nycflights13)

The data

The Bureau of Transportation Statistics (BTS) is a statistical agency that is a part of the Research and Innovative Technology Administration (RITA). As its name implies, BTS collects and makes available transportation data, such as the flights data we will be working with here.

We begin by loading the flights data frame. Run the following command by clicking on the green triangle:

data(flights)

The data set flights that shows up in your workspace is a data matrix, with each row representing an observation and each column representing a variable. R calls this data format a data frame, which is a term that will be used throughout the course. For this data set, each observation is a single flight.

To view the names of the variables, run the command

names(flights)

 [1] "year"           "month"          "day"            "dep_time"      
 [5] "sched_dep_time" "dep_delay"      "arr_time"       "sched_arr_time"
 [9] "arr_delay"      "carrier"        "flight"         "tailnum"       
[13] "origin"         "dest"           "air_time"       "distance"      
[17] "hour"           "minute"         "time_hour"

This returns the names of the variables in this data frame. The codebook (description of the variables) can be accessed by pulling up the help file (run this line with the green triangle and then look at the box on the bottom right of the RStudio screen):

?flights

One of the variables refers to the carrier (i.e. airline) of the flight, which is coded according to the following system.

carrier: Two letter carrier abbreviation.
- 9E: Endeavor Air Inc.
- AA: American Airlines Inc.
- AS: Alaska Airlines Inc.
- B6: JetBlue Airways
- DL: Delta Air Lines Inc.
- EV: ExpressJet Airlines Inc.
- F9: Frontier Airlines Inc.
- FL: AirTran Airways Corporation
- HA: Hawaiian Airlines Inc.
- MQ: Envoy Air
- OO: SkyWest Airlines Inc.
- UA: United Air Lines Inc.
- US: US Airways Inc.
- VX: Virgin America
- WN: Southwest Airlines Co.
- YV: Mesa Airlines Inc.

A very useful function for taking a quick peek at your data frame and viewing its dimensions and data types is glimpse().

glimpse(flights)

Rows: 336,776
Columns: 19
$ year           <int> 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2013, 2…
$ month          <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ day            <int> 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1…
$ dep_time       <int> 517, 533, 542, 544, 554, 554, 555, 557, 557, 558, 558, …
$ sched_dep_time <int> 515, 529, 540, 545, 600, 558, 600, 600, 600, 600, 600, …
$ dep_delay      <dbl> 2, 4, 2, -1, -6, -4, -5, -3, -3, -2, -2, -2, -2, -2, -1…
$ arr_time       <int> 830, 850, 923, 1004, 812, 740, 913, 709, 838, 753, 849,…
$ sched_arr_time <int> 819, 830, 850, 1022, 837, 728, 854, 723, 846, 745, 851,…
$ arr_delay      <dbl> 11, 20, 33, -18, -25, 12, 19, -14, -8, 8, -2, -3, 7, -1…
$ carrier        <chr> "UA", "UA", "AA", "B6", "DL", "UA", "B6", "EV", "B6", "…
$ flight         <int> 1545, 1714, 1141, 725, 461, 1696, 507, 5708, 79, 301, 4…
$ tailnum        <chr> "N14228", "N24211", "N619AA", "N804JB", "N668DN", "N394…
$ origin         <chr> "EWR", "LGA", "JFK", "JFK", "LGA", "EWR", "EWR", "LGA",…
$ dest           <chr> "IAH", "IAH", "MIA", "BQN", "ATL", "ORD", "FLL", "IAD",…
$ air_time       <dbl> 227, 227, 160, 183, 116, 150, 158, 53, 140, 138, 149, 1…
$ distance       <dbl> 1400, 1416, 1089, 1576, 762, 719, 1065, 229, 944, 733, …
$ hour           <dbl> 5, 5, 5, 5, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 5, 6, 6, 6…
$ minute         <dbl> 15, 29, 40, 45, 0, 58, 0, 0, 0, 0, 0, 0, 0, 0, 0, 59, 0…
$ time_hour      <dttm> 2013-01-01 05:00:00, 2013-01-01 05:00:00, 2013-01-01 0…

The flights data frame is a massive trove of information (336,776 observations!!!). Notice also that the glimpse() function let’s you know how the variables are stored: integer, double (a fancy way to say decimal number), character string, date/time, etc. Let’s think about some questions we might want to answer with these data:

How delayed were flights that were headed to Los Angeles?
How do departure delays vary over months?
Which of the three major NYC airports has a better on time percentage for departing flights?

Tidy Structure of Data

For plotting, analyses, model building, etc., the data should be structured according to certain principles.

Tidy Data: rows (cases/observational units) and columns (variables).
The key is that every row is a case and every column is a variable.
No exceptions.
Creating tidy data is often not trivial.

Within R (really within any type of computing language, Python, SQL, Java, etc.), it is important to understand how to build data using the patterns of the language.
Some things to consider:

object_name <- anything is a way of assigning anything to the new object_name.
object_name <- function_name(data_frame, arguments) is a way of using a function to create a new object.
object_name <- data_frame %>% function_name(arguments) uses chaining syntax as an extension of the ideas of functions.
In chaining, the value on the left side of %>% becomes the first argument to the function on the right side.

object_name <- data_frame %>%
                    function_name(arguments) %>% 
                    another_function_name(other_arguments)

is extended chaining. %>% is never at the front of the line, it is always connecting one idea with the continuation of that idea on the next line. * In R, all functions take arguments in round parentheses (as opposed to subsetting observations or variables from data objects which happen with square parentheses). Additionally, the spot to the left of %>% is always a data table. * The pipe syntax should be read as then, %>%.

Using the pipe to chain

The pipe syntax (%>%) takes a data frame (or data table) and sends it to the argument of a function. The mapping goes to the first available argument in the function.
For example:

x %>% f(y) is the same as f(x, y)

y %>% f(x, ., z) is the same as f(x,y,z)

Pipes are used commonly with functions in the dplyr package and they allow us to sequentially build data wrangling operations. We’ll start with short pipes and throughout the course build up to longer pipes that perform multiple operations.

Analysis

Departure delays

Let’s start by examining the distribution of departure delays of all flights using the summary function. The first item (on the left) is the data set. Subsequently, two functions are applied, (1) select() to get only the dep_delay variables, and (2) summary() to produce the numerical summary.

flights %>% 
  dplyr::select(dep_delay) %>% 
  summary()

   dep_delay      
 Min.   : -43.00  
 1st Qu.:  -5.00  
 Median :  -2.00  
 Mean   :  12.64  
 3rd Qu.:  11.00  
 Max.   :1301.00  
 NA's   :8255

`filter` (create a smaller dataset)

If we want to focus only on departure delays of flights headed to Los Angeles, we need to first filter() the data for flights with that destination (dest == "LAX"). The departure delay for the LAX flights can then be summarized.

lax_flights <- flights %>%
  dplyr::filter(dest == "LAX")

lax_flights %>% 
  dplyr::select(dep_delay) %>% 
  summary()

   dep_delay      
 Min.   :-16.000  
 1st Qu.: -4.000  
 Median : -1.000  
 Mean   :  9.401  
 3rd Qu.:  7.000  
 Max.   :800.000  
 NA's   :98

Let’s decipher these two commands (It’s common to add a break to a new line after %>% to help readability).

Command 1: Take the flights data frame, filter() for flights headed to LAX, and save the result as a new data frame called lax_flights.
- == means “if it is equal to”. (notice that there are TWO equals signs)
- LAX is in quotation marks since it is a character string.
Command 2: Basically the same call for summarizing the departure delay.

Notice that if we only want the summary of dep_delay for the LAX flights (and we don’t need to keep a copy of the dataset), we can perform the above tasks by combining them into fewer steps:

flights %>%
  filter(dest == "LAX") %>%
  dplyr::select(dep_delay) %>% 
  summary()

   dep_delay      
 Min.   :-16.000  
 1st Qu.: -4.000  
 Median : -1.000  
 Mean   :  9.401  
 3rd Qu.:  7.000  
 Max.   :800.000  
 NA's   :98

Logical operators: Filtering for certain observations (e.g. flights from a particular airport) is often of interest in data frames where we might want to examine observations with certain characteristics separately from the rest of the data. To do so we use the filter() function and a series of logical operators. The most commonly used logical operators for data analysis are as follows:

== means “equal to”
!= means “not equal to”
> or < means “greater than” or “less than”
>= or <= means “greater than or equal to” or “less than or equal to”

`summarize` (calculate statistics)

We can also obtain numerical summaries for these flights:

lax_flights %>%
  summarize(mean_dd = mean(dep_delay, na.rm=TRUE), 
            median_dd = median(dep_delay, na.rm=TRUE), n_dd = n())

# A tibble: 1 × 3
  mean_dd median_dd  n_dd
    <dbl>     <dbl> <int>
1    9.40        -1 16174

Note that in the summarize() function we created a list of three different numerical summaries that we were interested in. The names of these elements are user defined, like mean_dd, median_dd, n_dd, and you could customize these names as you like (just don’t use spaces in your names). Calculating these summary statistics also require that you know the function calls. Note that n() reports the sample size.

Summary statistics: Some useful function calls for summary statistics for a single numerical variable are as follows:

mean
median
sd
IQR
min
max

Note that each of these functions take a single vector as an argument, and returns a single value.

Functions you may not be familiar with (and that we will see in more detail in coming weeks) include: \[\mbox{sd} = \sqrt{\frac{1}{n-1} \sum_{i=1}^n (X_i - \overline{X})^2}\] \[\mbox{IQR} = 75\% - 25\%\]

We can also filter based on multiple criteria. Suppose we are interested in flights headed to San Francisco (SFO) in February:

sfo_feb_flights <- flights %>%
  filter(dest == "SFO", month == 2)

Note that we can separate the conditions using commas if we want flights that are both headed to SFO and in February. If we are interested in either flights headed to SFO or in February we can use the | instead of the comma.

Practice

Create a new data frame that includes flights headed to SFO in February, and save this data frame as sfo_feb_flights. How many flights meet these criteria?

sfo_feb_flights %>%
  summarize(n())

# A tibble: 1 × 1
  `n()`
  <int>
1   791

Describe the distribution of the arrival delays of these flights using summary and/or appropriate summary statistics.

sfo_feb_flights %>%
  summarize(arrdelmean = mean(arr_delay, na.rm=TRUE), 
            arrdelsd = sd(arr_delay, na.rm=TRUE),
            arrdelmed = median(arr_delay, na.rm=TRUE), 
            arrdeliqr = IQR(arr_delay, na.rm=TRUE))

# A tibble: 1 × 4
  arrdelmean arrdelsd arrdelmed arrdeliqr
       <dbl>    <dbl>     <dbl>     <dbl>
1      -9.14     31.4       -13        29

`group_by` (group before summarizing)

Another useful technique is quickly calculating summary statistics for various groups in your data frame. For example, we can modify the above command using the group_by() function to get the same summary stats for each origin airport:

sfo_feb_flights %>%
  group_by(origin) %>%
  summarize(median_dd = median(dep_delay, na.rm=TRUE), 
            iqr_dd = IQR(dep_delay, na.rm=TRUE), n_flights = n())

# A tibble: 2 × 4
  origin median_dd iqr_dd n_flights
  <chr>      <dbl>  <dbl>     <int>
1 EWR            0      9       194
2 JFK           -2      8       597

Here, we first grouped the data by origin, and then calculated the summary statistics.

Practice

Calculate the median and interquartile range for arr_delay of flights in in the sfo_feb_flights data frame, grouped by carrier. Which carrier has the most variable arrival delays (as measured by IQR)?

sfo_feb_flights %>% 
  group_by(carrier) %>%
  summarize(arrdelmed = median(arr_delay, na.rm=TRUE), 
            arrdeliqr = IQR(arr_delay, na.rm=TRUE)) %>%
  arrange(desc(arrdeliqr))

# A tibble: 5 × 3
  carrier arrdelmed arrdeliqr
  <chr>       <dbl>     <dbl>
1 AA             -7      35  
2 DL            -24      27.5
3 UA             -9      27  
4 B6            -11      25.5
5 VX            -20      23

`arrange()` departure delays over months

Practice

Which month would you expect to have the highest average delay departing from an NYC airport?

Let’s think about how we would answer this question:

First, calculate monthly averages for departure delays. With the new language we are learning, we need to
- group_by() months, then
- summarize() mean departure delays.
Then, we need to arrange() these average delays in desc()ending order

flights %>%
  group_by(month) %>%
  summarize(mean_dd = mean(dep_delay, na.rm=TRUE)) %>%
  arrange(desc(mean_dd))

# A tibble: 12 × 2
   month mean_dd
   <int>   <dbl>
 1     7   21.7 
 2     6   20.8 
 3    12   16.6 
 4     4   13.9 
 5     3   13.2 
 6     5   13.0 
 7     8   12.6 
 8     2   10.8 
 9     1   10.0 
10     9    6.72
11    10    6.24
12    11    5.44

On time departure rate for NYC airports

Suppose you will be flying out of NYC and want to know which of the three major NYC airports has the best on time departure rate of departing flights. Suppose also that for you a flight that is delayed for less than 5 minutes is basically “on time”. You consider any flight delayed for 5 minutes of more to be “delayed”.

In order to determine which airport has the best on time departure rate, we need to

first classify each flight as “on time” or “delayed”,
then group flights by origin airport,
then calculate on time departure rates for each origin airport,
and finally arrange the airports in descending order for on time departure percentage.

`mutate()` (create a new variable)

Practice

Let’s start with classifying each flight as “on time” or “delayed” by creating a new variable with the mutate() function.

flights <- flights %>%
  mutate(dep_type = ifelse(dep_delay < 5, "on time", "delayed"))

The first argument in the mutate() function is the name of the new variable we want to create, in this case dep_type. Then if dep_delay < 5 we classify the flight as "on time" and "delayed" if not, i.e. if the flight is delayed for 5 or more minutes.

Note that we are also overwriting the flights data frame with the new version of this data frame that includes the new dep_type variable.

We can handle all the remaining steps in one code chunk:

flights %>%
  group_by(origin) %>%
  summarize(ot_dep_rate = mean(dep_type == "on time", na.rm=TRUE)) %>%
  arrange(desc(ot_dep_rate))

# A tibble: 3 × 2
  origin ot_dep_rate
  <chr>        <dbl>
1 LGA          0.728
2 JFK          0.691
3 EWR          0.639

Practice

If you were selecting an airport (of the three NYC airports in the dataset) simply based on on time departure percentage, which NYC airport would you choose to fly out of? (How did you define “on time”? 0 min? 5 min? Something else?)

LGA seems to have the highest on time departure percentage, so that’s the airport I would choose.

Reuse

https://creativecommons.org/licenses/by/4.0/

Assignment Summary (Goals)

Note on R:

Q0. Collaborative Learning

Q1. Chapter 2, A10.

Q2. Chapter 2, A11

Q3. Chapter 2, A12

Q4. Chapter 2, E1

Q5. Chapter 2, E2

Q6. Chapter 2, E3

Q7. Chapter 2, E4

Score & Description

General notes on homework assignments (also see syllabus for policies and suggestions):

Getting started

Load packages

The data

Tidy Structure of Data

Using the pipe to chain

Analysis

Departure delays

filter (create a smaller dataset)

summarize (calculate statistics)

Practice

group_by (group before summarizing)

Practice

arrange() departure delays over months

Practice

On time departure rate for NYC airports

mutate() (create a new variable)

Practice

Practice

Reuse

`filter` (create a smaller dataset)

`summarize` (calculate statistics)

`group_by` (group before summarizing)

`arrange()` departure delays over months

`mutate()` (create a new variable)