This lab introduces R and R Studio as well as a few procedures we’ll use in future class sessions.

Directions (read before starting)

  1. Please work together with your assigned partner(s). Make sure you all fully understand each topic or example before moving on.
  2. Record your answers to lab questions separately from the lab’s examples. Everyone should submit a copy of the group’s responses, even if you work on a single PC and all turn in identical copies.
  3. Ask for help, clarification, or even just a check-in if anything seems unclear.

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Onboarding

Packages

While functions we’ll use are contained in the default installation of R, others are stored in external libraries known as “packages”.

The first package we’ll use is ggplot2, which is used to create professional quality data visualizations. To use a package it must be installed onto the computer you’re working on. You can do this using install.packages():

install.packages("ggplot2")

Installing a package is similar to downloading an app - you only ever need to download it once, but you’ll need to load it each time you want to use it. As a result, you’ll frequently encounter code that looks like this:

# install.packages("ggplot2")
library(ggplot2)

Here the installation command will not be run due to the comment, but the library() function, which loads the ggplot2 package into your current R session, will be run.

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The “Grammar of Graphics”

The ggplot2 package uses a layer-based framework to build data visualizations. We can understand layering via the following sequence of examples:

## Read the data from our last lab
college_majors <- read.csv("https://remiller1450.github.io/data/majors.csv")

## Example #1.1 - nothing (just a base layer)
ggplot(data = college_majors, mapping = aes(x = Per_Male, y = Bach_Med_Income))

## Example #1.2 - add a layer displaying raw data
ggplot(data = college_majors, mapping = aes(x = Per_Male, y = Bach_Med_Income)) + geom_point()

## Example #1.3 - add another layer (regression line)
ggplot(data = college_majors, mapping = aes(x = Per_Male, y = Bach_Med_Income)) + geom_point() +
  geom_smooth(method = "lm")
## `geom_smooth()` using formula = 'y ~ x'
## Example #1.4 - add another layer (reference text)
ggplot(data = college_majors, mapping = aes(x = Per_Male, y = Bach_Med_Income)) + geom_point() +
  geom_smooth(method = "lm") + 
  annotate("text", x=25, y=80000, label = "Strong, positive relationship")
## `geom_smooth()` using formula = 'y ~ x'

Notice how each layer is added to an existing graph using +. The data frame and variable names from the initial use of ggplot() (the first layer) are passed forward to subsequent layers. This allows + geom_point() to create a scatter plot despite being given no arguments.

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Lab

At this point you should begin working independently with your assigned partner(s). Remember that you should read the lab’s content, not just the questions, and you should all agree with an answer before moving on. Try to let the least experienced member of your group set the pace.

Describing One-Sample Quantitative Data

Last week we introduced the concept of one-sample data, which involves treating all of the observations in our data set as a single sample.

The table below summarizes various things we might include in an analysis of one-sample data depending upon whether the variable of interest is categorical or quantitative:

One-sample Categorical One-sample Quantitative
Descriptive Statistics Frequencies - table()
Proportions - prop.table()		 Mean - mean()
Median - median()
Standard Deviation - sd()
IQR - IQR()
Range - range()
Data Visualizations Bar plot - geom_bar()
Pie chart - geom_bar() + coord_polar()		 Histogram - geom_histogram()
Boxplot - geom_boxplot()
Hypothesis Tests \(H_0: p =\_\_\) \(H_0: \mu =\_\_\)

As you might notice, describing quantitative data can be more complex than describing categorical data. Two of the most important attributes we should describe are a quantitative variable’s center, or where values tend to typically fall, and its spread, or the extent to which values tend to vary relative to the center.

The table below provides a brief description of how each of these is calculated:

Attribute Measure Description
Center Mean Sum of all values divided by the number of values: \(\bar{x} = \frac{1}{n} \sum x_i\)
Median Middle value when data are ordered; if even number of values, average the two middle
Spread Standard Deviation A measure of how spread out values are around the mean: \(s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}\)
IQR Difference between 75th and 25th percentiles: \(\text{IQR} = Q_3 - Q_1\)
Range Maximum value minus minimum value: \(\text{Range} = \max(x) - \min(x)\)

Question #1: For this question you should use the toy data set created below:

example_data = data.frame(x1 = c(1,3,3,4,5,6),
                          x2 = c(1,3,3,4,5,60))
  • Part A: Use R to find the mean and median of the variable x1. Do both of these measures of center seem to accurately describe the “typical value” of this variable’s distribution? Briefly explain.
  • Part B: Now use R to find the mean and median of x2. Do both of these measures of center seem to accurately describe the “typical value” of this variable’s distribution? Briefly explain.
  • Part C: For the variable x2 the value 60 can be considered an outlier, or an extreme observation that is far away from the majority of the observed data. Based upon what you’ve seen in Parts A and B, which measure of center, the mean or median, appears to be more greatly influenced by the presence of outliers?
  • Part D: Use R to find the standard deviations of x1 and x2. How do outliers impact the standard deviation of a variable?

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Visualizing One-Sample Quantitative Data

Descriptive statistics are used to summarize the most important features of a variable, but we’ll also want to know the shape of the variable’s distribution. Visualizations are more effective for assessing shape than numerical summaries, and there are two main types you should know for quantitative data:

  1. Histograms - which group the numerical values into equally spaced intervals called bins, and then use bars to display the frequencies of values falling inside of each bin.
  2. Boxplots - which draw a box connecting the 25th percentile (Q1) and 75th percentile (Q3) with a line inside the box showing the median. Whiskers extend from the box to the most extreme observations that are not considered outliers. Any observations more than 1.5 IQRs beyond the box are displayed individually.

Below is an example of how to create each visualization using ggplot():

## Histogram example
ggplot(data = college_majors, mapping = aes(x = Per_PhD)) + geom_histogram(bins = 20)

## Boxplot example
ggplot(data = college_majors, mapping = aes(x = Per_PhD)) + geom_boxplot()

A few things to note:

  1. Sometimes changing the number of bins in a histogram is necessary to get a clear view of the distributional shape.
  2. You can create a vertical boxplot by mapping the variable of interest to the y aesthetic (rather than x)

Statisticians care about the distributional shape of the data because skew and outliers can influence the reliability of certain hypothesis testing procedures (we’ll talk about them later). For now, you should know the following definitions:

  • A variable is skewed right if its distribution has a long tail of large values.
  • Similarly, a variable is skewed left if its distribution has a long tail of small values.
  • If both tails of the distribution are similar, we’ll describe the distribution as approximately symmetric.
  • Some symmetric distributions follow a bell-curve shape that we’ll soon describe as approximately Normal, in reference to the Normal distribution.

Below are some examples:

Question #2: For this question you should use the Happy Planet data set, which is available at the URL below:

https://remiller1450.github.io/data/HappyPlanet.csv

  • Part A: Create a histogram of the variable HPI, which records each country’s composite happiness index score. Try making histograms with 10, 15, and 25 bins to gauge how the number of bins influences how the histogram looks. After doing so, describe the distribution of “HPI” using one of the four shapes given in this section’s examples.
  • Part B: Repeat the same steps as Part A, but this time using the variable LifeExpectancy. Your final answer should create a histogram displaying the variable and a brief comment describing the shape of the distribution.
  • Part C: Now create a boxplot of the variable LifeExpectancy. Use this boxplot to find approximate values of one measure of center and one measure of spread.

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Hypothesis Testing for One-Sample Quantitative Data

As we’ve discussed, a thorough statistical analysis should include:

  1. An assessment of the sampling procedure
  2. Descriptive statistics and data visualizations that highlight key aspects of the data
  3. Inference (hypothesis testing or other procedures we’ll cover later on)

For one-sample categorical data, we used StatKey simulations to estimate the \(p\)-value in our hypothesis tests. In this section of the lab we’ll do the same for one-sample quantitative data. Next week we’ll learn how to perform similar hypothesis tests in R.

To begin, a key step in any hypothesis test is determining the null distribution, which reflects outcomes that could have occurred had the null hypothesis been true.

  • With one-sample categorical data, we just needed the proportion stated in the null hypothesis (ie: \(p=0.5\)) and the sample size
    • For example, in our helper-hinderer toy-choice study, this allowed us to simulate many possible samples of \(n=16\) infants under the assumption that infants were choosing randomly (\(p=0.5\)).
  • With one-sample quantitative data our hypothesis is about the population-level mean, \(\mu\).
    • The null distribution should be centered at the hypothesized mean, but its spread depends on the amount of sampling variability present in the data.
    • Thus, we need to provide StatKey with both a hypothesized mean and all of the observed numeric values for the variable of interest
    • This is done by uploading the data CSV to StatKey using the “Upload File” button and selecting the column of interest.

After we’ve entered our null hypothesis and data we can use StatKey to simulate outcomes that could have been observed had our null hypothesis been true. We then follow the same procedure to calculate the \(p\)-value. That is, we find the proportion of outcomes at least as extreme as the mean observed in the real sample by checking one of the “left tail”, “two-tail”, or “right tail” boxes then clicking and adjusting the value below the x-axis to match the observed sample mean.

Question #3: Click on this link to download the “Tips” data set. These data were collected by a server working in a chain restaurant in a suburb of New York City. Each row records information about a table they served.

  • Part A: Many people feel that 10% represents the minimum acceptable tip in a restaurant. Perform a one-sided hypothesis test to evaluate whether there is statistical evidence that this server on average receives tip percentages exceeding 10%. Provide the one-sided \(p\)-value you estimate using StatKey along with a one-sentence conclusion.
  • Part B: According to several online sources, 15% is considered a standard tip, reflecting expected/satisfactory service. Perform a two-sided hypothesis test to evaluate whether there is statistical evidence that this server on average receives tip percentages differing from 15%. Provide the two-sided \(p\)-value you estimate using StatKey along with a one-sentence conclusion.

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Practice (required)

In this section you will analyze a few variables from a random sample of \(n=200\) patients admitted to the intensive care unit (ICU) of a research hospital affiliated with Carnegie Mellon University (CMU):

https://remiller1450.github.io/data/ICUAdmissions.csv

The relevant variables for your analyses of these data are:

  • Systolic: The systolic blood pressure reading of the patient when they were admitted
  • Previous: 0 if this is the first time the patient was admitted or 1 if the patient has been re-admitted for the same underlying issue

Question #4:

  • Part A: Name two populations that might be of interest to the researchers who collected these data. One of these populations should be a group that you are highly confident these data provide an unbiased representation of. The other can be a population where the data might present minor-to-moderate sampling bias.
  • Part B: Suppose the researchers wanted to know if these data suggest the readmission rate of ICU patients at CMU hospitals differs from the national average, which is estimated to be 14.67%. Does this scenario involve one-sample categorical or one-sample quantitative data? Briefly explain.
  • Part C: Provide an appropriate set of descriptive statistics and at least one relevant data visualization for the research question described in Part B.
  • Part D: Perform an appropriate hypothesis test, using StatKey to estimate the \(p\)-value, that addresses the research question described in Part B. Your response should clearly state your null and alternative hypotheses using proper statistical notation, provide the \(p\)-value, and make a one-sentence conclusion that uses appropriate context and avoids generic phrases like “reject \(H_0\)”.

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Question #5:

  • Part A: A systolic blood pressure of less than 120 mm Hg is considered healthy. Suppose the researchers want to know if their data suggest ICU patients on average have higher than normal systolic blood pressure when they are admitted. Does this scenario involve one-sample categorical or one-sample quantitative data? Briefly explain.
  • Part B: Provide an appropriate set of descriptive statistics and at least one relevant data visualization for the research question described in Part A.
  • Part C: Perform an appropriate hypothesis test, using StatKey to estimate the \(p\)-value, that addresses the research question described in Part A. Your response should clearly state your null and alternative hypotheses using proper statistical notation, provide the \(p\)-value, and make a one-sentence conclusion that uses appropriate context and avoids generic phrases like “reject \(H_0\)”.