8.5 – One sample t-test

Introduction.

We’re now talking about the traditional, classical two group comparison involving continuous data types. Thus begins your introduction to parametric statistics. One sample tests involve questions like, how many — what proportion of — people would we expect are shorter or taller than two standard deviations from the mean? This type of question assumes a population and we use properties of the normal distribution and, hence, these are called parametric tests because the assumption is that the data has been sampled from a particular probability distribution.

However, when we start asking questions about a sample statistic (eg, the sample mean), we cannot use the normal distribution directly, ie, we cannot use Z and the normal table as we did before (Chapter 6.7). This is because we do not know the population standard deviation and therefore must use an estimate of the variation (s) to calculate the standard error of the mean.

With the introduction of the t-statistic, we’re now into full inferential statistics-mode. What we do have are estimates of these parameters. The t-test — aka Student’s t-test — was developed for the purpose of testing sample means when the true population parameters are not known.

The equation of the one sample t-test. Note the resemblance in form with the Z-score!

where is the sample standard error of the sample mean (SEM).

For example, weight change of mice given a hormone (leptin) or placebo. The , but under the null hypothesis, the mean change is “really” zero (). How unlikely is our value of 5 g?

Speaking of null hypotheses, can you say (or write) the null and alternative hypotheses in this example? How about in symbolic form?

We want to know if our sample mean could have been obtained by chance alone from a population where the true change in weight was zero.

$s = 3, n = 20$

and

$s_{\overline{X}} = \frac{s}{\sqrt{n}} = \frac{3}{\sqrt{20}} = 0.6708$

and we take these values and plug them into our equation of the t-test

$t = \frac{\left(5 – 0\right)}{0.67} = 7.45$

Then recall that Degrees of Freedom are DF = n – 1 so we have DF = 20 – 1 = 19 for the one sample t-test. And the Critical Value is found in the appropriate table of critical values for the t distribution (Fig 1)

Figure 1. Table of a portion of the Critical values of the t distribution. Red selections highlight critical value for t-test at α = 5% and df = 19.

Or, and better, use R

qt(c(0.025), df=19, lower.tail=FALSE)

where qt() is function call to find t-score of the p^th percentile (cf 3.3 – Measures of dispersion) of the Student t distribution. For a two tailed test, we recall that 0.025 is lower tail and 0.025 is upper tail.

In this example we would be willing to reject the Null Hypothesis if there was a positive OR a negative change in weight.

This was an example of a “two-tailed test” which is “2-tail” or α₍₂₎ in Table of critical values of the t distribution.

Critical Value for α₍₂₎ = 0.05, df = 19, = 2.093

Do we accept or reject the Null Hypothesis?

A typical inference workflow.

Note the general form of how the statistical test is processed, a form which actually applies to any statistical inference test.

1. Identify the type of data
2. State the null hypothesis (2 tailed? 1 tailed?)
3. Select the test statistic (t-test) and determine its properties
4. Calculate the test statistic (the value of the result of the t-test)
5. Find degrees of freedom
6. For the DF, get the critical value
7. Compare critical value to test statistic
8. Do we accept or reject the null hypothesis?

And then we ask, given the results of the test of inference, What is the biological interpretation? Statistical significance is not necessarily evidence of biological importance. In addition to statistical significance, the magnitude of the difference — the effect size — is important as part of interpreting results from an experiment. Statistical significance is at least in part because of sample size — the large the sample size, the smaller the standard error of the mean, therefore even small differences may be statistically significant, yet biologically unimportant. Effect size is discussed in Ch9.1 – Chi-square test: Goodness of fit, Ch11.4 – Two sample effect size and Ch12.5 – Effect size for ANOVA.

R Code.

Let’s try a one-sample t-test. Consider the following data set: body mass of four geckos and four Anoles lizards (Dohm unpublished data).

For starters, let’s say that you have reason to believe that the true mean for all small lizards is 5 grams (g).

Geckos: 3.186, 2.427, 4.031, 1.995
Anoles: 5.515, 5.659, 6.739, 3.184

Get the data into R (Rcmdr)

By now you should be able to load this data in one of several ways. If you haven’t already entered the data, check out Part 07. Working with your own data in Mike’s Workbook for Biostatistics.

Once we have our data.frame, proceed to carry out the statistical test.

To get the one-sample t-test in Rcmdr, click on Statistics → Means → Single-sample t-test… Because there is only one numerical variable, Body.mass, that is the only one that shows up in the Variable (pick one) window (Fig 2).

Figure 2. Screenshot Rcmdr single-sample t-test menu.

Type in the value 5.0 in the Null hypothesis: m = u box.

Question 1: Quick! Can you write, in plain old English, the statistical null hypothesis???

Answer 1: For example: No difference between gecko and Anolis lizard mean body mass.

Click OK

The results go to the Output Window.

t.test(lizards$Body.mass, alternative='two.sided', mu=5.0, conf.level=.95)
One Sample t-test
data: lizards$Body.mass
t = -1.5079, df = 7, p-value = 0.1753
alternative hypothesis: true mean is not equal to 5
95 percent confidence interval:
2.668108 5.515892
sample estimates:
mean of x 4.092

end of R output

Let’s identify the parts of the R output from the one sample t-test. R reports the name of the test and identifies

1. The dataset$variable used (lizards$Body.mass). The data set was called “lizards” and the variable was “Body.mass”. R uses the dollar sign ($) to denote the dataset and variable within the data set.
2. The value of the t test statistic was (t = -1.5079). It is negative because the sample mean was less than the population mean — you should be able to verify this!
3. The degrees of freedom, df = 7
4. The p-value = 0.1753
5. 95% confidence interval of the population mean; lower limit = 2.668108, upper limit = 5.515892
6. The sample mean = 4.092

Take a step back and review.

Let’s make sure we “get” the logic of the hypothesis testing we have just completed.

Consider the one-sample t-test.

Step 1. Define H_O and H_A. The null hypothesis might be that a sample mean, , is equal to μ = 5.

$H_0 : \mu = 5$

The alternate is that the sample mean is not equal to 20.

$H_A : \mu \ne 5$

Where did the value 5 come from? It could be a value from the literature (does the new sample differ from values obtained in another lab?). The point is that the value is known in advance, before the experiment is conducted, and that makes it a one-sample t-test.

One tailed hypothesis or two?

We introduced you to the idea of “tails of a test” (Ch08.4). As you should recall, a null/alternative hypothesis for a two-tailed test may be written as

Null hypothesis

versus the alternative hypothesis

where  is the sample mean and is the population mean.

Alternatively, we can write one-tailed tests of null/alternative hypothesis

for the null hypothesis versus the alternative hypothesis

Question 2: Are all possible outcomes of the one-tailed test covered by these two hypotheses?

Answer 2: Yes

Question 3: What was the SEM for this problem?

Answer 3: It would be the sample standard deviation divided by the square root of the sample size.

Step 2. Decide how certain you wish to be (with what probability) that the sample mean is different from μ. As stated previously, in biology, we say that we are willing to be incorrect 5% of the time (Cowles and Davis 1982; Cohen 1994). This means we are likely to correctly reject the null hypothesis 100% – 5% = 95% of the time, which is the definition of statistical power. We do this by setting the Type I error to be 5% (alpha, α = 0.05). The Type I error is the chance that we will reject a null hypothesis, but the true condition in the population we sampled was actually “no difference.”

Step 3. Carry out the calculation of the test statistic. In other words, get the value of t from the equation above by hand, or, if using R (yes!) simply identify the test statistic value from the R output after conducting the one sample t test.

Step 4. Evaluate the result of the test. If the value of the test statistic is greater than the critical value for the test, then you conclude that the chance (the P-value) that the result could be from that population is not likely and you therefore reject the null hypothesis.

Question 4: What is the critical value for a one-sample t-test with df = 7?

Answer 4: From R, we get + 2.365 for the two-tailed test. R code was qt(c(.025), df=7, lower.tail=FALSE)

Hint; you need the table or better, use R

Rcmdr: Distributions → Continuous distributions → t distributions → t quantiles

You also need to know three additional things to answer this question.

1. You need to know alpha (α), which we have said generally is set at 5%.
2. You also need to know the degrees of freedom (DF) for the test. For a one sample t-test, DF = n – 1, where n is the sample size.
3. You also must know whether your test is one or two-tailed.
4. You then use the t-distribution (the tables of the t-distribution at the back of your book) to obtain the critical value. Note that if you use R, the actual p-value is returned.

Why learn the equations when I can just do this in R?

Rcmdr does this for you as soon as you click OK. Rcmdr returns the value of the test statistic and the p-value. R does not show you the critical value, but instead returns the probability that your test statistic is as large as it is AND the null hypothesis is true. From our one-sample t-test example, the Rcmdr output. The simple answer is that in order to understand the R output properly you need to know where each item of the output for a particual test comes from and how to interpret it. Thus, the best way is to have the equations available and to understand the algorithmic approach to statistical inference.

And, this is as good of time as any to show you how to skip the RCmdr GUI and go straight to R.

First, create your variables. At the R prompt enter the first variable

liz <- c("G","G","G","G","A","A","A","A")

and then create the second variable

bm <- c(3.186,2.427,4.031,1.995,5.515,5.659,6.739,3.184)

Next, create a data frame. Think of a data frame as another word for worksheet.

lizz <- data.frame(liz,bm)

Verify that entries are correct. At the R prompt type “lizz” wthout the quotes and you should see

lizz
 liz bm
1 G 3.186
2 G 2.427
3 G 4.031
4 G 1.995
5 A 5.515
6 A 5.659
7 A 6.739
8 A 3.184

End of R output

Carry out the t-test by typing at the R prompt the following

t.test(lizz$bm, alternative='two.sided', mu=5, conf.level=.95)

And, like the Rcmdr output we have for the one-sample t-test the following R output

One Sample t-test
data: lizards$Body.mass
t = -1.5079, df = 7, p-value = 0.1753 alternative hypothesis: true mean is not equal to 5
95 percent confidence interval:
2.668108 5.515892
sample estimates:
mean of x
4.092

End of R output

which, as you probably guessed, is the same as what we got from RCmdr.

Question 5: From the R output of the one sample t-test, what was the value of the test statistic?

1. -1.5079
2. 7
3. 0.1753
4. 2.668108
5. 5.515892
6. 4.092

Answer 5: -1.5079

Why fall back on statistical tables? Think of this as a basic skill. In statistics and for some statistical tests, Rcmdr and other software may not provide the information needed to decide that your test statistic is large, and a table in a statistics book is the best way to evaluate the test.

For now, double check Rcmdr by looking up the critical value from the t-table.

Check critical value against our test statistic

Df = 8 – 1 = 7

The test is two-tailed, therefore α(2)

α = 0.05 (note that two-tailed critical value is 2.365. T was equal to 1.51 (since t-distribution is symmetrical, we can ignore the negative sign), which is smaller than 2.365 and so we would agree with Rcmdr — we cannot reject the null hypothesis.

Question 6: From the R output of the one sample t-test, what was the P-value?

1. -1.5079
2. 7
3. 0.1753
4. 2.668108
5. 5.515892

Answer 6: 0.1753

Question 7: We would reject the null hypothesis

1. False
2. True

Answer 7: False — p-value, 17.5%, is greater than Type I error of 5%.

Questions

Seven questions, with answers, were provided for you within the text in this chapter. Here’s one more, but without answers.

8. Here’s a small data set for you to try your hand at the one-sample t-test and Rcmdr. The dataset contains cell counts, five counts of the numbers of beads in a liquid with an automated cell counter (Scepter, Millipore USA). The true value is 200,000 beads per milliliter fluid; the manufacturer claims that the Scepter is accurate within 15%. Does the data conform to the expectations of the manufacturer? Write a hypothesis then test your hypothesis with the one-sample t-test. Here’s the data.

Quiz Chapter 8.5

One sample t-test

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Chapter 8 contents

• Introduction
• The null and alternative hypotheses
• The controversy over proper hypothesis testing
• Sampling distribution and hypothesis testing
• Tails of a test
• One sample t-test
• Confidence limits for the estimate of population mean
• References and suggested readings

4.10 – Graph software

Introduction

You may already have experience with use of spreadsheet programs to create bar charts and scatter plots. Microsoft Office Excel, Google Sheets, Numbers for Mac, and LibreOffice Calc are good at these kinds of graphs — although arguably, even the finished graphics from these products are not suitable for most journal publications. 

For bar charts, pie charts, and scatter plots, if a spreadsheet app is your preference, go for it, at least for your statistics class. This choice will work for you, at least it will meet the minimum requirements asked of you.

However, you will find spreadsheet apps are typically inadequate for generating the kinds of graphics one would use in even routine statistical analyses (e.g., box plots, dot plots, histograms, scatter plots with trend lines and confidence intervals, etc.). And, without considerable effort, most of the interesting graphics (e.g., box plots, heat maps, mosaic plots, ternary plots, violin plots), are impossible to make with spreadsheet programs.

At this point, you can probably discern that, while I’m not a fan of spreadsheet graphics, I’m also not a purist — you’ll find spreadsheet graphics scattered throughout Mike’s Biostatistics Book. Beyond my personal bias, I can make the positive case for switching from spreadsheet app to R for graphics is that the learning curve for making good graphs with Excel and other spreadsheet apps is as steep as learning how to make graphs in R (see Why do we use R Software?). In fact, for the common graphs, R and graphics packages like lattice, ggplot2, make it easier to create publishing-quality graphics.

Alternatives to base R plot

This is a good point to discuss your choice of graphic software — I will show you how to generate simple graphs in R and R Commander which primarily rely on plotting functions available in the base R package. These will do for most of the homework. R provides many ways to produce elegant, publication-quality graphs. However, because of its power, R graphics requires lots of process iterations in order to get the graph just right. Thus, while R is our software of choice, other apps may be worth looking at for special graphics work. 

My list emphasizes open source and or free software available both on Windows and macOS personal computers. Data set used for comparison from Veusz (Table).

1. GrapheR — R package that provides a basic GUI (Fig 1) that relies on Tcl/Tk — like R Commander — that helps you generate good scatter plots, histograms, and bar charts.  Box plot with confidence intervals of medians (Fig 2).

Figure 1. Screenshot of GrapheR GUI menu, box plot options.

Figure 2. Box plot made with GrapheR.

2. RcmdrPlugin.KMggplot2 — a plugin for R Commander that provides extensive graph manipulation via the ggplot2 package, part of the Tidyverse environment (Fig 3). Box plot with data point, jitter (Fig 4).

Figure 3. Screenshot of KMggplot2 GUI menu, box plot options.

Figure 4. Box plot graph made with GrapheR with jitter applied to avoid overplotting of points.

Figure 5. Box plot graph made with GrapheR with beeswarm applied to avoid overplotting of points.

3. A bit more work, but worth a look. Use plotly library to create interactive web application to display your data. 

install.packages("plotly")
library(plotly)
fig <- plot_ly(y = Bees, type = "box", name="Bees")
fig <- fig %>% add_trace(y = Butterflies, name="Butterflies")
fig

code modified from example at https://plotly.com/r/box-plots/

Figure 6. Screenshot of plotly box plot. Live version, data points visible when mouse pointer hover.

4. Veusz, at https://veusz.github.io/. Includes a tutorial to get started. Mac users will need to download the dmg file with the curl command in the terminal app instead of via browser, as explained here.

Figure 7. Screenshot of box plot example in Veusz GUI.

5. SciDAVis, “Scientific Data Analysis and Visualization,” is a package capable of generating lots of kinds of graphs along with curve fitting routines and other mathematical processing, https://scidavis.sourceforge.net/. SciDAVis is very similar to QtiPlot and OriginLab, and runs on both macOS and WinPC.

Figure 8. Screenshot SciDAVis app with default box plot.

More sophisticated graphics can, and when you gain confidence in R, you’ll find that there are many more sophisticated packages that you could add to R to make really impressive graphs. However, the point is to get the best graph, and there are many tools out there that can serve this end.

Chapter 4 contents

• Graphs and tables
• Bar (column) charts
• Histograms
• Box plot
• Mosaic plots
• Scatter plots
• Adding a second Y axis
• Q-Q plot
• Ternary plots
• Heat maps
• Graph software
• References and suggested readings

4.6 – Adding a second Y axis

Introduction

Scatter plots are used to show association between two continuous variables. However, it is not uncommon to have a third variable for which association between the same X variable is expected. Thus, a common scatter plot includes a second Y axis. These graphs are a bit more involved to make regardless of which application used. The purpose of this short section is to provide a way to create a plot with two Y axes against a common X axis. Additional plotting options are also provided and explained.

The data set

The first draft of this post was written in September 4, 2019, peak time for hurricanes. Hurricane Dorian passed the Bahamas as a category 4 storm heading for Florida. For context I include a screenshot of imagery from NOAA; you can see Dorian along the Florida coastline as well as four additional storms lined up from the coast of Africa across the Atlantic Ocean (Fig 1).

Figure 1. Screenshot from NOAA GOES-East – Sector view: Tropical Atlantic – GeoColor, 4 September 2019. Click image to view full sized image.

The data set here is the number of Atlantic Ocean Category 1, 2, 3, 4, or 5 storms since 1900, tabulated by decade. Storms are categorized by wind speed according to the Saffir-Simpson Hurricane Wind Scale, or SSHWS.

Additional data, average levels of carbon dioxide (CO₂), a greenhouse gas, measured at Mauna Loa and average global temperature index from NOAA were also acquired for the same period. (The temperature index is called anomaly data as it is the difference of temperature between the average temperatures between 1950 and 1980). 

#Create the time data, 12 decades, with 2010 based on 8 years only -- up to 2018
decade <- seq(1900,2010, by = 10)
#Saffir-Simpson Hurricane Wind Scale; compiled from Wikipedia, checked against NOAA
cat01 <- c(2,0,1,3,0,9,4,11,18,11,27,46)
cat02 <- c(2,0,1,3,10,4,5,4,3,13,9,8)
cat03 <- c(7,2,3,1,4,10,9,5,4,12,11,7)
cat04 <- c(1,6,5,8,9,10,9,5,5,10,15,8)
cat05 <- c(0,0,2,6,0,2,4,3,3,2,9,4)
#land ocean anomaly temperature index, NOAA
tempIndex <- c(-0.317,-0.329,-0.241,-0.123,0.042,-0.048,-0.028,0.034,0.247,0.387,0.591,0.791)
#mauna kea CO2, mean by decade, record starts 1958
co2 <- c(NA,NA,NA,NA,NA,316.0,320.3,330.9,345.5,360.5,378.6,399.0)

Combine all of the variables into a single data frame.

storms <- data.frame(decade,tempIndex,cat01,cat02,cat03,cat04,cat05,co2)
head(storms)

Output from R should look like

  decade tempIndex cat01 cat02 cat03 cat04 cat05 co2
1   1900    -0.317     2     2     7     1     0  NA
2   1910    -0.329     0     0     2     6     0  NA
3   1920    -0.241     1     1     3     5     2  NA
4   1930    -0.123     3     3     1     8     6  NA
5   1940     0.042     0    10     4     9     0  NA
6   1950    -0.048     9     4    10    10     2 316

Create a new variable with all categories of storms summed by decade. 

#Sum all of the storms
allHur <- cat01+cat02+cat03+cat04+cat05
#Add new variable to the data frame
storms$allHur <- allHur
#Attach the data frame so you don't have to keep referring to the data frame when you call variables.
attach(storms)

Now, create a simple scatter plot, number of hurricanes by decade (Fig 2).

plot(decade,allHur,pch=16)

Figure 2. Plot of hurricanes from 1900 to present by decade.

A note on data management

This note is not needed for the plots. However, it’s as good a place as any to comment about how to work with your data in R. For the hurricane data, I set the data frame storms as an unstacked (wide) worksheet. The code that follows shows how to use the reshape2 package, part of the tidyverse set of packages for data manipulation, to convert the data frame from unstacked (wide) to stacked (long).

#code modified from examples presented by Thomas Eisfeld
library(reshape2)
library(dplyr)
tryMe <- melt(storms, id.vars=c("decade", "tempIndex", "co2"), variable.name = "SSHWS", value.name = "hurricanes")
head(tryMe)

You can also take a stacked (long) worksheet and convert it to unstacked (wide) with dcast().

tryAgain <- dcast(tryMe, decade + tempIndex + co2 ~ SSHWS)

Make the plots

At last, here’s the code for adding a second Y axis. Code modified from https://www.r-bloggers.com/r-single-plot-with-two-different-y-axes/. The work flow begins by setting parameters of the plotting window, creating the first plot, followed up by adding the second plot (par=TRUE), setting graphing elements, then adding labels and a legend.

First plot example: Total number of hurricanes by decades, with Temperature Index by decades. Number of hurricanes represented on first (left) axis and Temperature Index represented on second (right) axis (Fig 3).

par(mar = c(5,5,2,5))
#Create the first plot
plot(tryAgain$decade, tryAgain$allHur, pch=16, cex=1.5, col="black", xlab="Decades", ylab="Hurricanes", main="Atlantic Hurricanes by Decade")

the syntax par(mar = c(bottom, left, top, right))

#Add the second plot
par(new=T)
plot(tryAgain$decade, tryAgain$tempIndex, type="l", lty="dashed", lwd=2, col="red3", axes=F, xlab=NA, ylab=NA)
axis(side=4)
mtext(side=4,line=3,"Temperature Index")

#Add a legend
legend("topleft",legend=c("Hurricanes", "Temperature Index"),lty=c(0,2), pch=c(16,NA), col=c("black", "red3"))

Figure 3. Total number of hurricanes by decades, with Temperature Index by decades. Number of hurricanes represented on first (left) axis and Temperature Index represented on second (right) axis.

Next plot example: Total number of hurricanes by decades, with Atmospheric CO₂ measured at Mauna Loa by decades. Number of hurricanes represented on first (left) axis and Atmospheric CO₂ represented on second (right) axis (Fig 4).

par(mar = c(5,5,2,5))
plot(tryAgain$decade, tryAgain$allHur, pch=16, cex=1.5, col="black", xlab="Decades", ylab="Hurricanes", main="Atlantic Hurricanes by Decade")

Create space for the second plot in the same frame (use T or TRUE, both work).

par(new=T)

Code for the second plot. Note that axes=F (“F” or “FALSE” works) is used to suppress printing axes — Recall that lines were already created by the first plot; if you do not add this, then additional lines are added to the plot.”NA” is used to suppress adding labels; if you do not add this code, then labels will be printed over the existing labels created for the first plot. The code axis=4 sets the right-hand Y axis to active. The next code, mtext() is used to place the axis label for the second Y axis.

plot(tryAgain$decade, tryAgain$co2, type="l", lty="dashed", lwd=2, col="red3", axes=F, xlab=NA, ylab=NA)
axis(side=4)
mtext(side=4,line=3,"Mean CO2, ppm")

Add the legend.

legend("topleft",legend=c("Hurricanes", "CO2"),lty=c(0,2), pch=c(16,NA), col=c("black", "blue"))

Figure 4. Total number of hurricanes by decades, with Atmospheric CO₂ measured at Mauna Kea by decades. Number of hurricanes represented on first (left) axis and Atmospheric CO₂ represented on second (right) axis.

Go ahead and try these plots. Change the settings (e.g., lty, pch, cex, col), and note how the graphs look.

Bubble (chart) plot

From the basic scatter plot a variety of variants have been developed to convey additional information about ratio scale variables. The bubble chart is one common example and can be used as an alternative to adding a second Y-axis in some cases. A bubble plot uses the area of a bubble to represent a third variable, while a second y-axis is used to display a second set of data with its own scale on the same chart.

In brief, a third variable is added, mapped to circle size. Spreadsheet apps also include options to make bubble charts and the wonderful plot.ly library provides a vehicle to produce sophisticated graphs including bubble charts online or as a package in R — and it works well with the ggplot2 package. A nice bubble chart introduction is available at Atlassian. 

Examples of bubble charts with R code are provided in Ch16.3. That said, what’s one more?  Here’s example R code:

ggplot(storms, aes(x = decade, y = tempIndex, size = total_storms)) +
  geom_point(alpha = 0.7, color = "lightgreen") + # Added color = "lightgreen"
  scale_size_continuous(range = c(3, 15)) +
  labs(title = "Temperature Index vs. Decade with Total Storms",
       x = "Decade",
       y = "Land Ocean Anomaly Temperature Index",
       size = "Total Storms") +
  theme_classic()

and the plot, Figure 4.

Figure 4. Bubble plot of association between total hurricanes and Land-Ocean Temperature Anomalies by Decade.

If you are interested in trying yourself, a bubble chart yourself, CO₂ datas with the hurricane data set is a good candidate for a bubble plot.

Questions

1. Using the plot() command, make the following graphs
   • One scatter plot for each category of storm by decade
   • Explore the kinds of graph elements available by changing pch values. Create your own 
   • Change point size by changing valued for cex
2. Create a new plot with Decade on X axis, Temperature Index on first Y axis, and CO₂ on the second Y axis.
   • Include a legend for the plot.
3. What is a Q-Q plot used for in statistics?
4. Looking at the plot in Figure 5, explain why the confidence lines get further and further away from the straight line.
5. What are some advantages and disadvantages of heat map for communicating data.
6. What color pallet is considered “color-friendly” for accessible visualization?
7. Create a table with three columns.

• In column 1, list the name for each graph type discussed in Chapter 5.
• In column 2, state the kind of data appropriate for each graph.
• In column 3, enter the name of a function in the base R package you can use to make the graph
• In column 4, enter the name of a function from the ggplot2 package you can use to make the graph

Quiz Chapter 4.6

Adding a second Y axis

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Chapter 4 contents

• Graphs and tables (How to report statistics)
• Bar (column) charts
• Histograms
• Box plot
• Mosaic plots
• Scatter plots
• Add a second Y axis
• Q-Q plot
• Ternary plots
• Heat maps
• Graph software
• References and suggested readingsh

4.3 – Box plot

Introduction

Box plots, also called whisker plots, should be your routine choice for exploring ratio scale data. Like bar charts, box plots are used to compare ratio scale data collected for two or more groups. Box plots serve the same purpose as bar charts with error bars, but box plots provide more information.

Purpose and design criteria

Box plots are useful tool for getting a sense of central tendency and spread of data. These types of plots are useful diagnostic plots. Use them during initial stages of data analyses. All summary features of box plots are based on ranks (not sums). So, they are less sensitive to extreme values (outliers). Box plots reveal asymmetry. Standard deviations are symmetric.

The median splits each batch of numbers in half (center line). The “hinge” (median value) splits the remaining halves in half again (the quartiles). The first, second (median), and third quartiles describes the interquartile range, or IQR, 75% of the data (Fig 1). Outlier points can be identified, for example, with an asterisk or by id number (Fig 1).

Figure 1. A box plot. Elements of box plot labelled. 

We’ll use the data set described in the previous section, so if you have not already done so, get the data from Table 1, Chapter 4.2 into your R software.

R Code

Command line

We’ll provide code for the base graph shown in Figure 2A. At the R prompt, type

boxplot(OliveMoment~Treatment)

Figure 2A. Box plot, default graph in base package

Boxplot is a common function offered in several packages. In the base installation of R, the function is boxplot(). The car package, which is installed as part of R Commander installation, includes Boxplot(), which is a “wrapper function” for boxplot(). Note the difference: base package is all lower case, car package the “B” is uppercase. One difference, base boxplot() permits horizontal orientation of the plot (Fig 2B).

boxplot(OliveMoment ~ Treatment,  horizontal=TRUE, col="steelblue")

Figure 2B. Same graph, but with color and made horizontal; boxplot(), default graph in base package

Base package boxplot() has additional features and options compared to Note 3: Boxplot() in the car package. i.e., not all barcode() options are wrapped. For example, I had more success adding original points to boxplot() graph (Fig 2C) following the function call with stripchart().

stripchart(OliveMoment ~ Treatment, method = "overplot", pch = 19, add = TRUE)

Figure 2C. Same graph, added original points; boxplot(), default graph in base package.

Rcmdr: Graph → Boxplot…

Select the response variable, then click on the Plot by: button

Figure 3. Popup menu in R Commander: Select the response variable and set the Plot by: option.

Next, select the Groups (Factor) variables (Fig 4). Click OK to proceed

Figure 4. Select the group variable

Back to the Box Plot menu, click “Options” tab to add details to the plot, including a graph title and how outliers are noted (Fig 5), 

Figure 5. Options tab, enter labels for axes and a title.

And here is the resulting box plot (Fig 6)

Figure 6. Resulting box plot from car package implemented in R Commander. Outliers are identified by row id number.

The graph is functional, if not particularly compelling. The data set was “olive moments” from Comet Assays of an immortalized rat lung cell line exposed to dilute copper solution (Cu), Hazel tea (Hazel), or Hazel & Copper solution. 

Apply Tidyverse-view to enhance look of boxplot graphic

Load the ggplot2 package via the Rcmdr plugin to add options to your graph. As a reminder, to install Rcmdr plugins you must first download and install them from an R mirror like any other package, then load the plugin via Rcmdr Tools → Load Rcmdr plug-in(s)… (Fig 7, Fig 8).

Figure 7. Screen shot of Load Rcmdr plug-ins menu, Click OK to proceed (see Fig 8).

Figure 8. To complete installation of the plug-in, restart R Commander.

Significant improvement, albeit with an “eye of the beholder” caveat, can be made over the base package. For example, ggplot2 provides additional themes to improve on the basic box plot. Figure 8 shows the options available in the Rcmdr plugin KMggplot2, and the default box plot is shown in Fig 9.

Figure 9. Menu of KMggplot2. A title was added, all else remained set to defaults.

The next series of plots, Fig 10 – 12,  explore available formats for the charts.

Figure 10. Default box plot from KMggplot.

Figure 11. “Economist” theme box plot from KMggplot2.

And finally, since the box plot is often used to explore data sets, some recommend including the actual data points on a box plot to facilitate pattern recognition. This can be accomplished in the KMggplot2 plugin by checking “Jitter” under the Add data points option (see Fig 8). Jitter helps to visualize overlapping points at the expense of accurate representation. I also selected the Tufte theme, which results in the image displayed in Figure 12. 

Figure 12. Tufte theme and data points added to the box plot.

Conclusions

As part of your move from the world of Microsoft Excel graphics to recommended graphs by statisticians, the box plot is used to replace the bar charts plus error bars that you may have learned in previous classes. The second conclusion? I presented a number of versions of the same graph, differing only by style. Pick a style of graphics and be consistent.

Questions

1. Why is a box plot preferred over a bar chart for ratio scale data, even if an appropriate error bar is included?
2. With your comet data (Table 1, Chapter 4.2), explore the different themes available in the box plot commands available to you in Rcmdr. Which theme do you prefer and why?

Quiz Chapter 4.3

Box plots

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Chapter 4 contents

• Graphs and tables (How to report statistics)
• Bar (column) charts
• Histograms
• Box plot
• Mosaic plots
• Scatter plots
• Add a second Y axis
• Q-Q plot
• Ternary plots
• Heat maps
• Graph software
• References and suggested readings