20.13 – How to get the distances from a distance tree

Introduction

Extract the patristic distance, the sum of the branch lengths that link two nodes in a tree, for each pair of species.

This distance — see our Chapter 16.6 – Similarity and Distance— is the proportion (p) of amino acid (or nucleotide for DNA or RNA) sites at which the two sequences to be compared are different. It is obtained by dividing the number of amino acid differences by the total number of sites compared. It does not make any correction for multiple substitutions at the same site or differences in evolutionary rates among sites. On a gene tree (Fig. 1), distances are the lengths of the branches connecting the taxa. We want to know how different are two species for the given protein? That’s the distance between them in proportion of amino acid sites that are different by total number compared.

Example

Figure 1. A gene tree of the product (protein HBA1) with five species.

Here’s the Newick format for the tree (HBA1.nwk)

(Mouse:0.0474516,Human:0.104063,((Chicken:0.127652,Alligator:0.202421):0.0616593,Xenopus:0.422801):0.143939);

R code to extract distances and output sorted, pairwise comparisons to a text file

library(ape)

# Create a function
getDis <- function(tree, tips) {
     myTree <- cophenetic(tree)
     myTree <- myTree[,tips]
     xy <- t(combn(colnames(myTree), 2))
     xy <- xy[order(xy[,1], xy[,2]),]
     myOut <- data.frame(xy, myTree[xy])
     colnames(myOut) <- c("Spp1", "Spp2", "Distance")
   return(myOut)
}

# Read a tree file, Newick format
tree5 <-read.tree(text="(Mouse:0.0474516,Human:0.104063,((Chicken:0.127652,Alligator:0.202421):0.0616593,Xenopus:0.422801):0.143939);")

# get taxa names from the tree file
all.tips <- tree5$tip.label; all.tips

# Run the function
myDis <- getDis(tree5, all.tips)

# Check the output
head(myDis)

# Create the results file
write.csv(myDis, file = "my_out.txt")

Example output from head(myDist)

       Spp1    Spp2  Distance
1 Alligator Xenopus 0.6868813
2 Chicken Alligator 0.3300730
3 Chicken   Xenopus 0.6121123
4 Human   Alligator 0.5120823
5 Human     Chicken 0.4373133
6 Human     Xenopus 0.6708030

The function sorts first by Spp1, then by Spp2.

Molecular clock plot

Collect divergence times from timetree.org

A scatterplot of distance HBA protein sequence by log₁₀-transformed millions of years ago divergence time shown in Figure 2. Note, although tempting, calculating the slope from a linear regression to estimate the rate of evolution would not be appropriate without accounting for the lack of independence of the data (see Phylogenetically independent contrasts). Better methods exist, including calculating rate of change after fitting a model that assumes a strict clock vs relaxed clock.

Figure 2. Scatterplot HBA distance by logMYA divergence time

Questions

[pending]

Suggested readings

Bevan, R. B., Lang, B. F., & Bryant, D. (2005). Calculating the evolutionary rates of different genes: a fast, accurate estimator with applications to maximum likelihood phylogenetic analysis. Systematic biology, 54(6), 900-915.

Chapter 20 contents

• Additional topics
• Area under the curve
• Peak detection
• Baseline correction
• Surveys
• Time series
• Dimensional analysis
• Estimating population size
• Diversity indexes
• Survival analysis
• Growth equations and dose response calculations
• Plot a Newick tree
• Phylogenetically independent contrasts
• How to get the distances from a distance tree
• Binary classification
• Meta-analysis

/MD

3.1 – Data types

Understanding data: Types, measurement, and use in analysis.

.

Data? Data refers to collections of facts, information, or statistics about an object. Data are either quantitative (numbers) or qualitative (observed properties that cannot be summarized by numbers).  Data are measured and analyzed for research or reports to be used as evidence in support or against some hypothesis or for some other decision making arena (medicine, policy). Measurement implies a systematic effort to assign a numerical value to the thing that is measured; measurement units are standard quantities used to describe the same kinds of things. Examples of measurement units include kilograms (mass), meter (length), liter (volume), and Celsius (temperature). 

An observation, then, is a single occurrence of something, the variable, we measure or count of gather information about. A unit of observation is the level at which the measure is taken: a single cell, a biopsy containing grams of tissue, one flower among many flowers on a tree, the individual, the family, the community, the population, the ecosystem, … the planet, the solar system, …. If the time period over which the observation may take place varies, then the interval of time may be the unit.

Data also implies a means to code or structure information so that it can be analyzed. Raw data refers to unprocessed collection of information about an object, which then needs to go through data processing in order to be useful in the next steps. If you look more closely, you’ll see that considerable effort is made to standardize data formats for analytical purposes. Good examples of such standards are available in clinical research and genomics.

In statistics, we recognize data which belongs to either of two data types: quantitative or qualitative. We will return to data types repeatedly throughout our statistics journey — knowing which type you directs you to the types of statistical tests that are available to you. In brief, quantitative data types implies estimation of parameters about a population, hence, this data type points the user towards use of parametric statistics; qualitative data types do not lead to estimates of parameters, but provide counting of observations in categories.

Quantitative.

.

Discrete, countable or meristic, example: five Conus shells (Fig 1)

Figure 1. Five Conus shells, example of discrete data type. Click image to view full sized image.

Interval, example: degrees Celsius

Figure 2. Analog thermometer showing office temperature at 23.1 Celsius, example of interval data type. Click image to view full sized image.

Another classical interval scale example is time of day — there’s no meaningful zero time. We can measure amount of time — hours, minutes, seconds, etc, between intervals, but there is no absolute absence of time. 

Ratio, true zero, examples: body mass, capillary blood glucose reading (Fig 3), degrees Kelvin, relative humidity (Fig 4).

Figure 3. Blood glucose reading, 122 mg/dL. Click image to view full sized image.

Figure 4. Analog hygrometer showing office humidity at 65 percent, example of ratio data type. Click image to view full sized image.

I’m a fan of competitive sports and also have interested in measures of performance like running times in lizards! (eg, Beck et al 1998). 

Qualitative.

.

Binomial, yes/no, example: a person either has the condition or they do not; hydrangea petals are sometimes blue (Fig 5).

Figure 5. Flowers (Hydrangea) are blue or they are not, example of binomial data type. Click image to view full sized image.

Nominal, example: names of species. Wolves and dogs are members of Canis lupus and Canis familiaris, respectively, house cats, Felis catus, are not (Fig 6).

Figure 6. Cats are neither dogs or wolves, example of nominal data type. Click image to view full sized image.

Ordinal, ranked, example: Likert scale:

Biologists should know their data types before proceeding with an experiment.

• Strongly disagree
• Disagree
• No opinion
• Agree
• Strongly agree

Although common practice, caution is warranted when converting Likert categories into numerical scale, for example, Strongly agree = 4, Strongly disagree = -4, and so on.  Because it is ordinal, the difference between 4 and -4 can’t be calculated as the difference because it is ranked, not the numerical scale. (See Carifio and Perla 2007, h/t Karen Grace-Martin, The Analysis Factor.)

Examples to try.

.

In R, load the data set diabetic (survival package, which is loaded as part of R Commander), then view the variables. 

For more about R data sets, see Part 6 Working with an included data set in Mike’s Workbook for Biostatistics

R code

data(diabetic, package="survival")

.

R Commander (Fig 7)

Rcmdr: Data → Data in packages → Read data set from an attached package… Double click survival, the list of data sets should appear in the right-hand panel. Select diabetic, then click OK button.

Figure 7. Screenshot Rcmdr Read data from package menu.

View the data by clicking on Rcmdr’s View data set button, or, better, submit the following command in R

head(diabetic)

.

R output

  id laser age   eye trt risk  time status
1  5 argon  28  left   0    9 46.23      0
2  5 argon  28 right   1    9 46.23      0
3 14 xenon  12  left   1    8 42.50      0
4 14 xenon  12 right   0    6 31.30      1
5 16 xenon  9   left   1   11 42.27      0
6 16 xenon  9  right   0   11 42.27      0

.

The command head() displays by default the first six rows of a data frame.

It’s a good idea to read up on the data set. Data sets included with R packages often provide a help page. submit the following command in R to load the help page.

help(diabetic)

.

The data set was subjects with high risk diabetic retinopathy, “each patient had one eye randomized to laser treatment and the other eye received no treatment.”

What are the data types for the variables? I’ll give you the a couple to start. The first column with entries 1 – 6 is called the index variable; it’s row 1, row 2, etc. of the data set and technically is not a data set variable (since it’s assignment is arbitrary) — R adds this for you. Next, the variable labeled id — clearly we see numbers, so we might think meristic, but because these are labels for the subjects, the proper data type is nominal! Try the rest on your own, scroll down to end of the page for best answers.

Questions.

.

1. Assign examples of units of measurement for each kind of measurement. 

1. Darts tossed, Distance from center.
2. Shells, width, length.
3. InfraRed temperature device readings.
4. Body weight.
5. Lung volume.
6. Tomato color morphs (green, yellow).
7. Tomato root length, stem length.
8. Systolic blood pressure.
9. Blood arsenic levels.
10. Body Mass Index.
11. Body Mass Index scale, for example NIH: underweight, normal, overweight, obese.

Assign the data type and provide examples of units of measurement. 

2. What are the data types for

1. Basal 5 hour fasting plasma glucose-to-insulin ratio in mice.
2. Four inbred strains of mice.
3. Height, in inches of six women.
4. Atmospheric carbon dioxide, CO2, levels from Mauna Loa for the month of December, in parts-per-million, ppm.
5. Body temperature readings, deg C, from IR device on forehead or back of the throat. 
6. Six mollusk species.
7. The index variable in the following data set:
      mom  dad
1 67.0 78.5
2 66.5 75.5
3 64.0 75.0
4 58.5 75.0
5 68.0 74.0
6 66.5 74.0

Answers to Examples to try

laser: binomial, there were two types, xenon or argon

age: ratio, years

eye: binomial

trt: binomial, no treatment (0) or laser (1)

risk: ordinal

time: ratio, time to event, number of months

status: binomial

Quiz Chapter 3.1

Data types

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Chapter 3 contents

.

• Introduction.
• Data types.
• Measures of Central Tendency.
• Measures of dispersion.
• Estimating parameters.
• Statistics of error.
• References and suggested readings.

12.4 – ANOVA from “sufficient statistics”

Introduction

By now you should be able to run a one-way ANOVA using R (and R Commander) with ease. As a reminder, You should also be aware that, if you need to, you could use spreadsheet software like Microsoft Excel or LibreOffice Calc to run one way ANOVA on a small data set. Still, there are times when you may need to run a one-way ANOVA on a small data set, and doing so by hand calculator may be just as convenient. What are your available options?

Following the formulas I have given would be one way to calculate ANOVA by hand, but it would be tedious and subject to error. Instead of working with the standard formulas, calculator shortcuts can be derived with a little algebra and this is where I want to draw your attention now. This technique will come in handy in lab classes or other scenarios where you collect some data among a set number of groups and calculate means and standard deviations. The purpose of this posting is to show you how to obtain the necessary statistics to calculate a one-way ANOVA from the available descriptive statistics: means, standard deviations, and sample sizes. In other words, these are the sufficient statistics for one-way ANOVA.

As you recall, a one-way ANOVA yields a single F test of the null hypothesis that all group means are equal. To calculate the F test, you need

• Mean Square Between Groups,
• Mean Squares Within Groups or Error,

F is then calculated as

with degrees of freedom k – 1 for the numerator and N – 1 for the denominator. can also appear as

We can calculate  as

where k is the number of i ^th groups, n_i is the sample size of the ith group,  refers to the overall mean for all of the  sample means.

Next, for the Error Mean Square, , all we need is the average of the sample variances (the square of the sample standard deviation, s).

ANOVA from sufficient statistics

Consider an example data set (Table 1) for which only summary statistics are available (mean and standard deviation, sd). The data set is for life span(days) for several inbred strains of laboratory mice from a study (. Sample size for each group was seven mice.

Table 1. Descriptive statistics female life span (days) for ten different inbred strains of mice (Mus domesticus).

Data from Yuan et al (2021; https://phenome.jax.org).

Spreadsheet calculations.

You have several options at this point, ranging from using your calculator and the formulas above (don’t forget to square the standard deviation to get the variances!), or you could use Microsoft Excel or LibreOffice Calc and enter the necessary formulas by hand (Table 2). You’ll also find many online calculators for one-way ANOVA by sufficient statistics (eg, https://www.danielsoper.com/statcalc/calculator.aspx?id=43).

Table 2. Spreadsheet with formulas for calculating one-way ANOVA from means and standard deviations from statistics presented in Table 1.

For this example, you should get the following

MSB = 219899.5
MSE = 31634.1
F = 6.951
P-value = 4.45E-09

But, R can do it better.

Here’s how. Install the HH package (or RcmdrPlugin.HH for use in Rcmdr) and call the ?aovSufficient function.

Step 1. Install the HH package from a CRAN mirror, eg, cloud.r-project.org, in the usual way.

chooseCRANmirror()
install.packages("HH") 
library(HH)

Step 2. Enter the data. Do this in the usual way (eg, from a text file), or enter directly using the read.table command as follows.

MouseData <- read.table(header=TRUE, sep = ",", text= "
Strain, n.size, average, stdev 
129S1, 32, 787.4, 159.16
A.J, 32, 630.7, 130.20
BALB.c, 32, 734.4, 154.43
BUB.BnJ, 24, 611.3, 218.34
C3H.HeJ, 29, 724.1, 131.48
C57BL.6J, 29, 855.7, 185.34
CBA.J, 30, 622.9, 181.95
FVB.NJ, 26, 750.3, 230.11
P.J, 32, 676.0, 178.82
SWR.J, 31, 831.9, 181.31")
#Check import
head(MouseData)

End of R input

I know, a little hard to read, but from the MouseData to the end bracket ") before the comment line #Check import, that’s all one command.

Of course, you could copy the data and import the data from your computer’s clipboard in Rcmdr: Data → Import data → from text file, clipboard, or URL… 

Once the data set is loaded, proceed to Step 3.

Step 3. In our example, sample size was included for each group. Skip to step 4. If, however, the table lacked the sample size information, you can always add a new variable. For example, if we needed to add sample size to the data frame, we would use the repeat element function, rep().

MouseData$n <- rep(7, 10)

If you check the View data set button in Rcmdr, you will see that the command in Step 3 has added a new variable “n” for each of the eleven rows. The function rep() stands for “replicate elements in vectors” and what it did here was enter a value of 7 for each of the ten rows in the data set. Again, this step is not necessary for this example because sample size is already part of the data frame. Proceed to step 4.

Step 4. Run the one way ANOVA using the sufficient statistics and the HH function aovSufficient

MouseData.aov <- aovSufficient(average ~ Strain, data=MouseData)

Step 5. Get the ANOVA table

summary(MouseData.aov)

Here’s the R output

           Df  Sum Sq Mean Sq F value   Pr(>F) 
Strain      9 1977732  219748   7.123 2.54e-09 ***
Residuals 287 8853743   30849 
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

End R output

To explore other features of the package, type ?aovSufficient at the R prompt (like all R functions, extensive help is generally available for each function in a package).

Meta-analysis and sufficient statistics.

Meta-analysis (see Chapter 20.15 — Meta-analysis), can be performed using sufficient statistics from different tests. Combining sufficient statistics allows for a more precise overall estimate by effectively merging the information from multiple studies, which can be especially useful when individual studies have small sample sizes or lack a unified outcome measure. Common methods include creating summary effect sizes like the standardized mean difference or using p-value combination methods, while ensuring the underlying statistical assumptions are met through goodness-of-fit tests.

Fisher’s method combines the p-values from multiple independent studies into a single overall p-value to test a joint null hypothesis. The method’s test statistic is calculated as the sum of the negative log-transformed p-values. P-values from number of tests are combined; each p-value, , contributes to a variable, , with degrees of freedom. The chi-square distribution is used to evaluate the significance of the Fisher’s method test statistic.

Fisher’s method is available in several R packages (eg,poolr package); here, we’ll just create a simple R code.

Limitations: Methods that only use p-values or test statistics may not adequately address the effects of heterogeneity and potential publication bias among studies. Because many older studies do not report effect sizes and confidence intervals, using test statistics or p-values is often the only available method to combine their results.

Example.

One hypothesis about longevity is that the greater the genetic diversity can lead to longer life. The mice in Table 1 were all from inbred mouse strains. By definition, genetic diversity is restricted in inbred mice, compared to outbred strains. Jackson Labs produced a “Diverse Outbred” line of mice, and a few individuals have lived nearly five years, well past the typical 1 – 2 year lifespan of mice (Cohen 2017, Mullis et al 2025). Thus, a natural comparison is to ask: is there a difference in average lifespan between the DO and inbred mouse strains? We’ll run a series of one-sample t-tests against the mean lifespan for the diverse outbred stock of mice, save the p-values, and apply Fisher’s method. While not strictly a meta-analysis, which would draw from multiple studies, this example outlines the basic steps to take given a series of hypothesis of the same kind.

do_avg= 794.5
# dataset MouseData
tsum.test(average[1], stdev[1], n.size[1], mu=do_avg)

t = -0.25235, df = 31, p-value = 0.8024

p_values <- numeric(length(average))

for (i in 1:length(average)) {
     suppressWarnings({
      test_result <- tsum.test(
      mean.x = average[i],
      s.x = stdev[i],
      n.x = n.size[i],
      mu = do_avg
      )
      p_values[i] <- test_result$p.value
      })
     }

print(p_values)

The p-values from all ten tests are sorted by strain in Table 2.

Table 2. Results of one sample t-tests inbred strains (Table 1) compared to average lifespan of the J:DO diversity outbred mouse strain from Jackson labs

R-code for Fisher’s method

my_pValues <- c(0.08623, 1.601e-05, 0.3368, 0.0007307, 0.2598)
X_squared <- -2 * sum(log(my_pValues),base = exp(1))
myDf <- 2 * length(my_pValues)
combinedP <- pchisq(X_squared, myDf, lower.tail = FALSE)
result_table <- as.data.frame(rbind(X_squared, myDf, combinedP))
rownames(result_table) <- c("chi_sqr", "df", "P-value")
colnames(result_table) <- c("Value")
print(result_table)

and the R output

                Value
chi_sqr 44.7017431931
     df 16.0000000000
P-value 0.0001541831

With the small p-value, we would reject the null hypothesis of no difference in lifespan among the inbred strains compared to the genetically diverse DO strain of mice.

Limitations of ANOVA from sufficient statistics.

This was pretty easy, so it is worth asking — Why go through the bother of analyzing the raw data, why not just go to the summary statistics and run the calculator formula? First, the chief reason against the calculator formula and use of only sufficient statistics loses information about the individual values and therefore you have no access to the residuals. The residual of an observation is the difference between the original observation and the model prediction. The residuals are important for determining whether the model fits the data well and are, therefore, part of the toolkit that statisticians need to do proper data analysis. We will spend considerable time looking at residual patterns and it is an important aspect of doing statistics correctly.

Secondly, while it is possible to extend this approach to more complicated ANOVA problems like the two-way ANOVA (Cohen 2002), the statistical significance of the interaction term(s) calculated in this way are only approximate (the main effects are OK to interpret). Thus, ANOVA from sufficient statistics has its place when all you have is access to descriptive statistics, but its use is limited and not at all the preferred option for data analysis when the original, raw observations are in hand.

Questions

1. Calculate the one-way ANOVA for body weight of 47 female (F) and 97 male (M) cats (kilograms, dataset cats in MASS R package) from the following summary statistics.

2. Bonus: Load the cats data set (package MASS, loaded with Rcmdr) and run a one-way ANOVA using the aov() function via Rcmdr. Are the ANOVA from sufficient statistics the same as results from the direct ANOVA calculation? If not, why not.

Quiz Chapter 12.4

ANOVA from “sufficient statistics”

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Chapter 12 contents

• Introduction
• The need for ANOVA
• One way ANOVA
• Fixed effects, random effects, and agreement
• ANOVA from “sufficient statistics”
• Effect size for ANOVA
• ANOVA posthoc tests
• Many tests one model
• References and suggested readings

4.9 – Heat maps

Introduction

A heat map is a graph of data from a matrix (Wilksonson and Friendly 2009). Heat maps are common in many disciplines in biology, from ecology (e.g., diversity analyses) to genomics (e.g., gene expression profiling) to economics and demographics (Fig. 1). Instead of plotting numbers, color is used to communicate associations between cells in the rows and columns of the matrix.

Heat maps are useful for suggesting trends and typically do not require specialized knowledge to interpret. Provided a color scale is defined, then heat maps do a good job communicating trends. Viewers may rapidly make comparisons as they scan colors, from cold to hot.

Figure 1 provides a classic heat map, counties of USA by percent ethnicity compared to “white” from Census.gov based on the 2010 census. Scale reads shades of blue to white, high percentage(greater than 96.3%) of whites to lower percentage (less than 71%) of whites, respectively. Map generated with mapping tool at United States Census Bureau TIGERweb.

Figure 1. Heat map, USA population by county and percent ethnicity compared to white, graph from census.gov

Figure 2 shows gene expression results from a pilot study we did on metal exposure in cultured rat lung cells compared to cells without metal exposure (i.e., the control group). Genes were selected because of their role in the epithelial-mesenchyme transition, EMT. The color scale is typical for such studies: green represents down-regulation, red indicates up-regulation compared to the controls. Black used to show no difference between treatment and control cells. 

Figure 2. Heat map, gene expression in cultured rat lung cells exposed to metals

Heat maps are good at directing the viewer to areas of strong association between variables, or in the case of comparisons, to draw strong inferences about the association. However, their chief limitations include gradations between colors; like pie charts, it is difficult to interpret the importance of slight changes in color, and the very use of heat map colors does not imply statistical significance (Chapter 8). Some color palettes are poor choices for viewers who may be color blind. A good source about colors is available in the Graphs section of Cookbook for R.

R and heat maps

Lots of specialized packages will do cluster to heat map. Functions include heatmap, heatmap2, heatmap.plus, NeatMap. We’ll step through how to make a heat map with another pilot study data from our lab.

heatmap(). Here’s another heat map, percent DNA in tail from Alkaline Comet Assay (Figure 3). The same cultured cell line, a rat immortalized Type 2-like alveolar lung cell line L2 cells, were grown in media containing witch-hazel tea, a dilute copper solution, or both witch-hazel tea and copper (unpublished data). The hypothesis was that there would be greater DNA damage in cells exposed to copper compared to cells in hazel tea or a combination of copper and hazel tea. Witch hazel is reputed to have antioxidant properties (Pietta et al 1998). A random sample of 10 cells were sampled from each treatment (between 30 and 60 cells counted for each treatment). Within each treatment values were placed in ascending order, so “Cell 1” corresponds to the lowest value for a measured cell in each treatment.

#data arranged in unstacked worksheet
data <- as.matrix(hazelCuUnstack)

#check the import
head(data)

         Copper       Hazel HazelCopper]
[1,] 0.02404672 0.007185706 0.02663191
[2,] 0.06711479 0.027020958 0.03181153
[3,] 0.12196060 0.037725842 0.03743693
[4,] 0.13308991 0.044762867 0.03851548
[5,] 0.13344032 0.045809398 0.18787608
[6,] 0.17537831 0.060942269 0.19494708

#make the heat map
heatmap(data)

Figure 3. A simple heat map generated by heatmap() function, all default options.

The heatmap() function first runs a cluster analysis to group the cells by columns and rows — so similar cells are grouped together. The row and column dendrograms are default; your data are rearranged by the clustering procedure. To generate the heatmap without the dendrograms, add the following to the R code.

heatmap(data, Rowv = NA, Colv = NA)

ggplot2 and aes(). Not straight forward, but ggplot2 (and therefore the Rcmdr plug-in KMggplot2) can be used.  The aes function is part of the “aesthetic mapping” approach (Wickham 2010). The example below takes the same data and introduces use of a custom color palette, brewer.pal. Uses geom_tile, but geom_raster can also be used.

library(RColorBrewer)
#Explore the color profiles available at http://colorbrewer2.org/#type=sequential&scheme=BuGn&n=3
?brewer.pal
hm1.colors <- colorRampPalette(rev(brewer.pal(9, 'RdYlGn')), space='Lab')
#the data set
hazelCu <- read.table("hazelCu.txt", header=TRUE, sep="t", na.strings="NA", dec=".", strip.white=TRUE)
#Confirm the import
head(hazelCu)

Cell Treatment TailPerc
1 1 C 0.02404672
2 2 C 0.06711479
3 3 C 0.12196060
4 4 C 0.13308991
5 5 C 0.13344032
6 6 C 0.17537831

#convert cell number to factor.
hazelCu <- within(hazelCu, {
Cell <- as.factor(Cell)
})

ggplot(hazelCu,aes(x=Treatment,y=Cell,fill=TailPerc)) + geom_tile() + coord_equal() +
scale_fill_gradientn(colours = hm1.colors(100))

Figure 4. ggplot() and aes() functions used to generate a heat map. Colors from brewer.pal

The color scheme used in Figure 3 is common in gene expression studies: green is negative, cooler, while red is positive, hotter. 

Questions

1. What are three advantages of heat map for communicating data. 
2. What are three disadvantages of heat map for communicating data.
3. What color pallet is considered “color-friendly” for accessible visualization?

Quiz Chapter 4.9

Heat maps

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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

4.1 – Bar (column) charts

Introduction

Bar chart, also called a bar graph, or column chart in many spreadsheet apps (e.g., Google Sheets, Microsoft Excel, Libreoffice Calc — “bar charts” is reserved for horizontal bar plots),  are used to compare counts among two or more categories, i.e., an alternative to pie charts (Fig. 1). 

Figure 1. Single nucleotide variants for human gene ACTB by DNA and functional element (data collected 19 May 2022 from NCBI SNP database with Advanced search query). A. Pie chart. Note that slide for “exon – nonsense” is not visible. B. Bar chart – color coded bars to facilitate comparison with pie chart.

Although bar charts are common in the literature (Cumming et al 2007; Streit and Gehlenborg 2014), bar charts may not be a good choice for comparisons of ratio scale data (Streit and Gehlenbor 2014). Bar charts for ratio data are misleading. Parts of the range implied by the bar may never have been observed: the bars of the chart always start at zero. Box (whisker) plots are better for comparisons of ratio scale data and are presented in the next section of this chapter. That said, I will go ahead and present how to create bar charts for both count — generally considered acceptable — and ratio scale data, for which their use is controversial.

Purpose of the bar chart

Like all graphics, a bar chart should tell a story. The purpose of displaying data is to give your readers a quick impression of the general differences among two or more groups of the data. For counts, that’s where the bar chart comes in. The bar chart is preferred over the pie chart because differences are represented by lengths of the bars in the bar chart. Differences among categories in a pie chart are reflected by angles, and it seems that humans are much better at judging lengths than angles.

Bar chart with counts

Figure 2. A simple bar chart

myCombo <- seq(0,10, by=1) 
myCounts <- choose(10, myCombo)    #combinations
barplot(myCounts, names.arg = myCombo, xlab = "Number correct", ylab = "Count",col = "darkblue")

 A stacked bar chart is used to 

myCombo <- seq(0,10, by=1) 
myCounts <- choose(10, myCombo)    #combinations
barplot(myCounts, names.arg = myCombo, xlab = "Number correct", ylab = "Count",col = "darkblue")

Figure 3. The luxury ship RMS Titanic, which sunk 15 April 1912, More than 1500 souls were lost. Public domain image, Wikipedia. Click image to view full sized image.

Stacked bar chart, data set TitanicSurvival in package carData.

Figure 4. A stacked bar chart, survival Titanic

Barplot(passengerClass, by=survived, style="divided", legend.pos="above", xlab="passengerClass", ylab="Frequency")

Bar charts with error bars

Although many data visualization specialists argue against the bar chart — they tend to obscure data — their use is well established. For the familiar bar chart with ratio scale data, the X (horizontal) axis displays the categories of one variable (e.g., location, or treatment group). You plot groups to emphasize comparisons. The Y (vertical) axis then is the mean for each group.

You need error bars with your bar chart to convey variability. A bar chart plus error bars is sometimes called (pejoratively) a dynamite plot. If the mean is displayed, some measure of precision should be (must be?) displayed (Cumming et al 2007). And, as you should recall by now, your choices are standard deviation (Chapter 3.2), standard error of the mean (SEM) (Chapters 3.2, 3.4), or confidence interval (see Chapter 3.4). It is strongly advised, that without a representation of precision, one should not interpret trends or group differences from representations of means (i.e., height of bars) alone.

The bar charts on this page are means plus or minus the standard error of the mean, + SEM. I’m not advocating here for the bar chart — box plots with efforts to show the actual data are better. However, bar charts remain common and may be more familiar to certain audiences. We’ll discuss which choice to make.

Examples

The Copper_rats_PMID3357063 dataset will be used for the next series of graphs (Data set). Refer to Mike’s Workbook for Biostatistics Part07 to review how to import the data.

A portion of the data set is shown below

head(Copper)
         Diet     Body    Heart    Liver
1 Adequate-Cu 320.1381 1.125037 10.259657
2 Adequate-Cu 329.6879 1.158982 9.843295
3 Adequate-Cu 327.9838 1.090374 9.855975
4 Adequate-Cu 334.6669 1.118183 9.942997
5 Adequate-Cu 338.3134 1.172636 9.860971
6 Adequate-Cu 345.4608 1.056183 8.885820

The data set consists of organ weights (heart, liver) from rats fed a diet adequate in copper, deficient in copper, and then a third group who received the adequate diet from perspective of amount of copper, but calorie restricted to match the decreased feeding rates of the rats fed the copper deficient diet. Copper is an essential trace element in our diet. The data set was simulated from descriptive statistics (means, standard deviation, number of subjects) of published data by sampling from a normal distribution. (Table 1, Ovecka et al 1988).

On we go with some graphs.

But, I’m still learning about R graphics, and for bar charts with error bars, I find other packages more straight-forward. So, I’ll use this moment to point out that making a good graph is more about the end product then the particular tools. I have been using other tools for years to make my graphics, so I tend to default on these options first. Graphs presented here are mostly from Veusz software program (pronounced “views”). I like it because the software allows me to edit any of the elements of the graph.

Here’s a typical looking bar chart with ratio scale data: means ± SEM. Let’s look at them more critically.

Figure 5. A bar chart with error bars (standard error of the mean).

Figure 6. Another bar chart. with standard errors of mean

When making comparisons, make sure the axes have the same scale, or consider putting the graphs together.

Figure 7. Bar chart that allows for a comparison among levels of a a factor (organs, liver vs. heart).

Here’s the organ weights again, but taken as ratio of body mass.

Figure 8. Same chart as in Figure 6, but on ratios.

How to make a bar chart with error bars in R

Option 1. First, let’s try a work-around. Instead of an error bar option for bar chart menu, Rcmdr provides a plot of means that allows you to plot with error bars. These are equivalent graphs, the “bar chart” and the “plot of means”, though you should favor the bar chart format for publishing.

Rcmdr: Graphs → Plot of means…

Figure 9. Rcmdr menu popup for Plots Means

Here, Rcmdr takes the data and calculates the mean and your choice of standard errors or deviations, confidence intervals, or no error bars. the resulting graph is below.

Figure 10. Plot of means

That’s an ugly graph (Fig. 10). Functional, good enough for data exploration and preliminary results, and certainly good enough for a Biostatistics homework or report. Additionally, connecting the dots here is a no-no. It implies that if we had measured categories between “adequate copper” and “deficient copper,” then the points would fall on those lines. That would be a complete guess. So, why did I include the connecting lines? That was the default setting for the command, and it makes the point — think before you click. One argument for connecting points in a graph is that it makes it easier for the reader to visualize trends.

This graph (Fig. 10) is fine for exploring data, but you will want to do better for publication.

Let’s make some better graphs with R

Once you are ready to go beyond the default settings available in Rcmdr, there is tremendous functionality in R for graphics. To access R’s potential, you’ll need to get into the commands a bit. I’m going to continue to try and shield you from the programming aspects of R, but from time to time you really need to see what is possible with R. Graphics is one such area. I use the package gplots, with 23 different graphing functions (type at R prompt ?gplots to call up the manual pages).

gplots should be among the packages on your R installation; if not, then install the package and run library(gplots) to complete the installation. We’ll try the barchart2 function.

But first, we need to get means for each of our groups.

At the R command prompt:

hrtWt <- tapply(Dataset$HeartWt, list(Group=Dataset$Group), mean, na.rm=TRUE) 

This code extracts means from our HeartWt variable for each Group, then  stores the three (in this case, because our data set has 3 groups) in the place holder I had called hrtWt. To verify that the three means are there, type “hrtWt” without the quotes then enter.

You should see

hrtWt Group Cu adequate Cu deficient Pair-fed 
               1.200000     1.566667 0.900000

R functions used: tapply, list, mean; na.rm was not needed but would be used to remove all missing values (recall during our import phase we were asked how missing observations were noted in our file; the default is NA).

Next, I want to apply standard error bars

stdDEV <- tapply(Dataset$HeartWt, list(Group=Dataset$Group), sd, na.rm=TRUE)
cil <- hrtWt-(stdDEV/sqrt(3))
ciu <- hrtWt+(stdDEV/sqrt(3))

I used cil and ciu to designate the lower cil and upper ciu values for my ± SEM (standard error of the mean). ciu stands for “confidence interval lower;” ciu stands for “confidence interval upper.”

Finally, here’s the plot command

barplot2(hrtWt, beside = TRUE, main=c("Mice fed different amounts of copper in diet"), col = c("blue", "red", "green"), xlab="Copper in diet", ylab="Heart mass (g)", ylim = c(0, 2), plot.ci=TRUE, ci.l=cil, ci.u=ciu)

Now, draw a box around the graphic

box()

Whew!

What does your new graph look like? My graph is below (Fig. 11).

Figure 11. A bar chart using barplot2.

This works and the point is that once the script is written it easy to make small changes as you need in the future to make nice graphs.

If you are impatient like me, I like a GUI option, at least to start crafting the graph. The Rcmdr plugin KMggplot2 provides a good set of tools to make bar charts with error bars. An even better option I think is to use a software package that is designed for graphics, at least simple graphics like a bar chart. I use SciDAVis and Veusz for simple graphs like pie charts and bar charts; much easier to control.

ggplot2 bar charts with error bars

Nevertheless, here’s how to make a bar chart with error bars using ggplot2 (Fig. 11). First, we need to create a statistics summary. The script printed here was modified from scripts at R Graph Cookbook website.

require(plyr)
summarySE <- function(data=NULL, measurevar, groupvars=NULL, na.rm=FALSE,
     conf.interval=.95, .drop=TRUE) {
length2 <- function (x, na.rm=FALSE) {
     if (na.rm) sum(!is.na(x))
     else length(x)
}

#returns a vector with N, mean, and sd
datac <- ddply(data,groupvars, .drop=.drop,
    .fun = function(xx,col){
      c(N=length2(xx[[col]],na.rm=na.rm),
      mean=mean(xx[[col]],na.rm=na.rm),
      sd=sd(xx[[col]],na.rm=na.rm)
     )
    },
    measurevar
   )

#Rename the "mean" column
datac <- rename(datac,c("mean"=measurevar))

#Calculate the standard error of the mean
datac$se <- datac$sd/sqrt(datac$N)

#Get confidence interval
ciMult <- qt(conf.interval/2 + .5, datac$N-1)
datac$ci <- datac$se*ciMult

return(datac)
}

Applying this function to the BMI dataset yields the following output.

sumBMI <- summarySE(BMI, measurevar="BMI", groupvars=c("Sex", "Smoke"));sumBMI

   Sex Smoke   N      BMI       sd        se       ci 
1    F    No  23 26.83567 7.610271 1.5868511 3.290928 
2    F   Yes  14 25.76133 4.658625 1.2450698 2.689810 
3    M    No  10 26.35731 3.363575 1.0636557 2.406156 
4    M   Yes  27 26.71879 4.675631 0.8998256 1.849618

Now we are ready to make the bar chart with error bars

ggplot(tgc,aes(x=Smoke,y=BMI,fill=Sex)) + 
 geom_bar(position=position_dodge(),stat="identity",color="black") + 
 geom_errorbar(aes(ymin=BMI,ymax=BMI+se),width=.2,position=position_dodge(.9))

Figure 12. A barchart from ggplot2

Questions

1. Why should you use box plots and not bar charts to display comparisons for a ratio scale variable between categories? Obtain a copy of the article by Streit and Gehlenbor 2014 — it’s free! After reading, summarize the pro and cons for box plots over bar charts with error bars.

2. Enter the following data into R. The data are sulfate levels in water, parts per million. 

type = c("Palolo Stream","Chaminade tap water", "Aquafina","Dasani") 
sulfateppm =c(11, 14, 5, 12)
try = data.frame(type,sulfateppm) 
byWater = tapply(try$sulfateppm,list(Group=try$type),mean)

Make a simple bar chart using the boxplot2 function in gplots package. 

3. Change the range of values on the vertical axis to 0, 20

4. Change the color of the bars from gray to blue

5. Add a label to the vertical axis, “Sulfates, ppm” (without the quotes)

6. Add a box around the graph.

Quiz – Chapter 4.1

Bar charts

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Data set

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