---
title: "Math 150 - Methods in Biostatistics - Homework 1"
author: "Your Name Here"
date: "Due: Wednesday, January 25, 2023"
output: pdf_document
---

```{r global_options, include=FALSE, message=FALSE, warning=FALSE}
knitr::opts_chunk$set(message=FALSE, warning=FALSE, fig.height=3, fig.width=5, 
                      fig.align = "center")
library(tidyverse)
library(broom)
```


### 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).

```{r}
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. 

* * *

\newpage

**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.

***

\newpage

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.

```{r}
library(tidyverse)
library(nycflights13)
```

### The data

The [Bureau of Transportation Statistics](http://www.rita.dot.gov/bts/about/) 
(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:

```{r}
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

```{r}
names(flights)
```

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):

```{r}
?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()`.

```{r}
glimpse(flights)
```

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.

```{r}
flights %>% 
  dplyr::select(dep_delay) %>% 
  summary()
```


#### `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.

```{r}
lax_flights <- flights %>%
  dplyr::filter(dest == "LAX")

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

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:

```{r}
flights %>%
  filter(dest == "LAX") %>%
  dplyr::select(dep_delay) %>% 
  summary()
```

<div id="boxedtext">
**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"
</div>

#### `summarize` (calculate statistics)

We can also obtain numerical summaries for these flights:

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

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.

<div id="boxedtext">
**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. 
</div>

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:

```{r}
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? 
    
```{r}
sfo_feb_flights %>%
  summarize(n())
```    
    
Describe the distribution of the **arrival** delays of these flights using summary and/or appropriate summary statistics. 

```{r}
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))
``` 

#### `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:

```{r}
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())
```

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

```    

### `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

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


##### 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.

```{r}
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:

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

##### 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.

