BIOS 4120 Lab 3

Objectives

In today’s lab we will:

Use RStudio to create graphics from a dataset
Discuss the relationship between hypothesis testing and confidence intervals

Common Plot Types and Their Functions

Creating a Bar Chart

For a bar chart, each column represents a group defined by a categorical variable. If you have data that is categorical (like we see in the ‘titanic’ dataset), you will want to use a bar chart to display information.

Bar Plots – ‘barplot()’

A simple way to create barplots is to use the barplot function.

counts <- c(8, 15, 11, 16)
barplot(height = counts, 
        names.arg = c("I", "II", "III", "IV"), 
        col = "blue", 
        main = "Distribution of Cancer Stage", 
        xlab = "Stage",
        ylab = "Frequency")

Determining Parameters for Plotting Functions

You will notice that there are a lot of parameters here (height, names.arg, etc.). The best way to determine how each parameter is used is to use the “?” command in the console.

?barplot

Notice that in the documentation, a lot of the parameters have default values. Therefore, for the barplot function, “height” is the only required parameter; using all other parameters is optional.

There is no real limit on how many parameters you can put in one of these functions, but it is easiest to read if you separate them onto different lines.

Common Graphical Parameters for Plot Functions

xlab, ylab – the label on the x-axis and y-axis, respectively
xlim, ylim – a vector representing the x and y limits, respectively
pch – an integer representing the type of plot points. You can also create your own plot points with quotes. e.g. “|”
lty – an integer representing the type of line
main – sets a title for the plot
col – sets the color for points, lines, or graphics for your plot
names.arg – the names of the bars

See http://www.statmethods.net/advgraphs/parameters.html for point, color, and line options and there corresponding codings

Note: Not all plots have the same arguments. The help function is a great way to see what arguments are part of a function

Adding Lines or Text to Plots

Once you run your plotting command and the figure pops up, you can still add either lines or text to the plot using these functions:

# a: intercept
# b: slope
# h: the y-value(s) for horizontal line(s)
# v: the x-value(s) for vertical line(s)
abline(a, b, h, v) 

# x: x coordinate for the text
# y: y coordinate for the text
# "Text": The text to be written
text(x, y, "Text")

These functions are called \(\bf{after}\) you the graph function is called.

Note that to clear the previous lines/text, run the plot function again. All previous lines and text will remain on the graph until a new plot is made.

Creating Legends

The legend function in R can be used to add a legend to a plot. There are a variety of arguments that this legend function can take. Some of the most commonly used are described below.

you can set x equal to a specified keyword (“bottomright”, “bottom”, “bottomleft”, “left”, “topleft”, “top”, “topright”, “right”) to indicate placement of the legend or you can use the x and y arguments to specify x and y coordinates for legend position.
the legend argument takes a character expression to be included in the legend
lwd takes an integer to indicate the line width
the col argument dictates the color of the points or lines that appear in the legend

counts <- c(8, 15, 11, 16)
barplot(height = counts, 
        names.arg = c("I", "II", "III", "IV"), 
        col = "pink", 
        main = "Distribution of Cancer Stage", 
        xlab = "Stage",
        ylab = "Frequency")
abline(h = c(8, 15), lty = 2, lwd = 4, col = c("blue", "red"))

stages <- c("Stage 1 Freq", "Stage 2 Freq")
legend(x = "topleft", 
       legend = stages,
       lty = 2,
       lwd = 4,
       col = c("blue", "red"))

(The usage of a legend and lines on this specific plot isn’t particularly useful to us, but we do it anyway to show how you can add these elements to a plot).

Storing Plots

Usually you can right-click on a plot in RStudio or R to copy them and then easily paste them onto a Google document. If this is giving you trouble, you can always save a plot to a pdf using the form below (after you have set a working directory):

pdf("Name_of_Plot.pdf", height = 3.5, width = 5.25)

## PLOT CODE HERE

dev.off()

This will save a pdf of the plot you set up in the folder you are working.

Bar Plots Using ggplot

Lastly we’ll go over how to use ggplot to create bar plots. Using ggplot is currently a popular way to create graphics. We’ll be using ggplot to illustrate relationships between variables in the titanic dataset we went over last lab. It isn’t necessary that you know ggplot, but its a tool available to you to make more complex and prettier plots.

To use the ggplot function you will need to install and load the ggplot2 package which can be done with the following code.

#install.packages(ggplot2)
library(ggplot2)

titanic <- read.delim("http://myweb.uiowa.edu/pbreheny/data/titanic.txt")

ggplot(titanic, aes(x = Class)) +
  geom_bar(position = "stack", fill="forest green") +
  ylab("Number of Individuals in each Class") +
  xlab("Class")

ggplot(titanic, aes(x = Survived)) +
  geom_bar(position = "stack", fill="forest green") +
  ggtitle("Death and Survival Counts for Individuals on the Titanic") +
  theme(plot.title = element_text(hjust = 0.5))

ggplot(titanic, aes(x = Class)) +
  geom_bar(position = "stack", aes(fill = Survived))+
  xlab("Class")

ggplot(titanic, aes(x = Class)) +
  geom_bar(position = "stack", aes(fill = Survived)) +
  facet_wrap(c("Age", "Sex")) +
  xlab("Class")

ggplot(titanic, aes(x = Class)) +
  geom_bar(position = "stack", aes(fill = Survived)) +
  facet_wrap(c("Age", "Sex"), scales = "free") +
  xlab("Class")

Group Bar Plot Exercise - we will use the barplot() to work the following problem

We are interested in the amount of people who eat at Iowa City restaurants downtown. On the night of this study, there were 16 people at Shorts, 21 at Donnelly’s Pub, 10 at Blue Moose, and 13 at Joe’s Place.

Step One

Create a vector named “people” of the counts.

people <- c(16, 21, 10, 13)

Step Two

Create a basic bar plot of “people”.

barplot(people)

What improvements could be made?

Main title
Label the bars
Label the x axis
Color
Make the y axis longer than the tallest bar

Step Three

Give the bar plot a main title.

barplot(people, 
        main = "Iowa City Night Life")

Step Four

Adjust the y axis to be high enough.

barplot(people, 
        main = "Iowa City Night Life",
        ylim = c(0,25))

Step Five

Give the axes appropriate names and labels.

To do this first create a vector named “restaurants” that contains the names of the restaurants.

## Create a vector named "bars" that contains the names of the bars.

restaurants <- c("Shorts", "Donnelly's Pub", "Blue Moose", "Joe's Place")

## Add the bar categories to the bar plot

barplot(people, 
        main = "Iowa City Night Life",
        ylim = c(0,25),
        names.arg = restaurants)

## Label the y-axis

barplot(people, 
        main = "Iowa City Night Life",
        ylim = c(0,25),
        names.arg = restaurants,
        ylab = "Number of People",
        xlab = "Restaurants in Iowa City")

Step Six

Let’s say that the threshold of having fun is 10 people.

Add a line at this threshold and label it “Fun Threshold”

barplot(people, 
        main = "Iowa City Night Life",
        ylim = c(0,25),
        names.arg = restaurants,
        ylab = "Number of People",
        col = "orange")

abline(a = 10, b = 0)

# Or 

abline(h = 10)

# Make the line green, dotted line
barplot(people, 
        main = "Iowa City Night Life",
        ylim = c(0,25),
        names.arg = restaurants,
        ylab = "Number of People",
        col = "orange")

abline(h = 10, col = "green", lty = 2, lwd = 2)

## Label the line

text(x = 3, y = 11, labels = "Fun Threshold")

Step Seven

Create a legend describing the line instead of the text function.

barplot(people, 
        main = "Iowa City Night Life",
        ylim = c(0,25),
        names.arg = restaurants,
        ylab = "Number of People",
        col = "orange")

abline(h = 10, col = "green", lty = 2, lwd = 2)

legend(x = "topright",
       legend = "Fun Threshold",
       lty = 2,
       lwd = 2,
       col = "green")

Step Eight

Save the plot.

Hint: Use the “pdf” and “dev.off” functions.

pdf("Bar-Plot-01.pdf", height = 3.5, width = 7.25)

barplot(people, 
        main = "Iowa City Night Life",
        ylim = c(0,25),
        names.arg = restaurants,
        ylab = "Number of People",
        col = "orange")

abline(h = 10, col = "green", lty = 2)

legend(x = "topright",
       legend = "Fun Threshold",
       lty = 2,
       col = "green")

dev.off()

## png 
##   2

Hypothesis Testing and Confidence Intervals

There is close relationship between confidence intervals and hypothesis testing. All values within a constructed 95% interval are considered “plausible” values for the parameter that we are estimating. Values outside the interval are rejected as unlikely and improbable.

Confidence Intervals Interpretation:

If you were to repeat the process of creating a confidence interval an infinite number of times, 95% of the interval estimates for \(\mu\) will contain the true parameter value, \(\mu\). We treat the population mean \(\mu\) as being fixed. Any particular interval may or may not contain the true population mean \(\mu\).

We say that we are “95% confident” that the interval contains the true population \(\mu\) because the procedure used to construct this interval produces a correct interval estimate 95% of the time.
We DO NOT say there is a 95% probability that \(\mu\) lies between these two values. (\(\mu\) is fixed)

Visualization

Hypothesis Testing Analogy - “Null Until Proven Otherwise”

In class, you learned that there are a lot of wrong ways to think about the hypothesis testing process. The courtroom is a helpful example that illustrates the correct usage of p-values and hypothesis tests. Let’s look at it in terms of “innocent until proven guilty”: As the person analyzing data, you are the judge. The hypothesis test is the trial, and the null hypothesis is the defendant.

If the evidence presented doesn’t prove the defendant is guilty beyond a reasonable doubt, you still have not proved that the defendant is innocent. (We never say that we accept the null hypothesis)

So how would that verdict be announced? It enters the court record as “Not guilty.” That phrase is perfect: “Not guilty” doesn’t mean the defendant is innocent, because that has not been proven. It just means the prosecution couldn’t prove its case to the necessary, “beyond a reasonable doubt” standard. It failed to convince the judge to abandon the assumption of innocence.

If you follow that rationale, then you can see that “failure to reject the null” is just the statistical equivalent of “not guilty.” In a trial, the burden of proof falls to the prosecution. When analyzing data, the entire burden of proof falls to the sample data you’ve collected. This is why our sampling procedure is so important. Just as “not guilty” is not the same thing as “innocent,” neither is “failing to reject” the same as “accepting” the null hypothesis.

This method of thinking about hypothesis tests will come in handy when we start formally testing our own hypotheses.

Source: http://blog.minitab.com/blog/understanding-statistics/things-statisticians-say-failure-to-reject-the-null-hypothesis

Relationship between Confidence Intervals & Hypothesis Testing

If the value of the parameter specified by the null hypothesis (for instance Ho = 0) is contained within the 95% interval, then the null hypothesis cannot be rejected at the 0.05 level. If the value specified by the null hypothesis is not in the interval, then the null hypothesis can be rejected at the 0.05 level. Likewise, for a 99% confidence interval, if the value specified by the null hypothesis is in the interval, then the null hypothesis cannot be rejected at the 0.01 level.

*Disclaimer: though these concepts have a strong relationship, one method is not a substitute for the other. On future homework and quizzes, you will need to know how to do both methods.

Practice Problems

In lab last week we worked with the titanic dataset. Today we are wanting to know whether sex played a significant role in the survival rates of the passengers on-board. Therefore, we want to compare survival rates between males and females.

Define the null hypothesis for this study on the ‘titanic’ dataset?
Say for example that we have the following null hypothesis \(H_o:\mu_{female}\) = 0.5. We obtain a 95% confidence interval (0.415, 0.481). Remember that interpretation of this confidence interval states that we are 95% confident that the true population \(\mu\) lies within this interval. Would we reject or retain the null hypothesis?

True or False

Suppose we have a test with an alpha level of 0.05. If we find a p-value of 0.03, we can reject the null hypothesis.
From our results in question 3, the 95% confidence interval would contain the specified null hypothesis.
Suppose we have a test with an alpha level of 0.01. If we find a p-value of 0.03, we can reject the null hypothesis.
From our results in question 5, the 99% confidence interval would contain the specified null hypothesis.

Answers

Ho: average survival rate for females = average survival rate for males
Reject
True
False
False
True