16.6 – Similarity and Distance

Introduction

A measure of dependence between two random variables. Unlike Pearson Product Moment correlation, distance correlation measures strength of association between the variables whether or not the relationship is linear. Distance correlation is a recent addition to the literature, first reported by Gábor J. Székely (e.g., Székely et al. 2007). The package correlation (Makowski et al 2019) offers distance correlation and significance test.

Example, fly wing dataset introduced 16.1 – Product moment correlation

library(correlation)
Area <- c(0.446, 0.876, 0.390, 0.510, 0.736, 0.453, 0.882, 0.394, 0.503, 0.535, 0.441, 0.889, 0.391, 0.514, 0.546, 0.453, 0.882, 0.394, 0.503, 0.535)
Length <- c(1.524, 2.202, 1.520, 1.620, 1.710, 1.551, 2.228, 1.460, 1.659, 1.719, 1.534, 2.223, 1.490, 1.633, 1.704, 1.551, 2.228, 1.468, 1.659, 1.719)
FlyWings <- data.frame(Area, Length)
correlation(FlyWings,method="distance")

Output from R

# Correlation Matrix (distance-method)
Parameter1 | Parameter2 |    r |       95% CI | t(169) |      p
------------------------------------------------------------------
      Area |     Length | 0.92 | [0.80, 0.97] |  30.47 | < .001***

p-value adjustment method: Holm (1979)
Observations: 20

The product-moment correlation was 0.97 with 95% confidence interval (0.92, 0.99). The note about “p-value adjustment method: Holm (1979)” refers to the algorithm used to mitigate the multicomparison problem, which we first introduced in Chapter 12.1. The correction is necessary in this context because of how the algorithm conducts the test of the distance correlation. Please see Székely and Rizzo (2013) for more details.

Which should you report? For cases where it makes sense to test for a linear association, then the Product-moment correlation is the one to use. For other cases where no inference of linearity is expected, then the distance correlation makes sense.

Similarity and Distance

Similarity and distance are related mathematically. When two things are similar, the distance between them is small; When two things are dissimilar, the distance between them is great. Whether similarity (sometimes dissimilarity) or distance, the estimate is a statistic. The difference between the two is that the typical distance measures one sees in biostatistics all obey the triangle inequality rule while similarity (dissimilarity) indices do not necessarily obey the triangle inequality rule.

Distance measures

Distance is a way to talk about how far (or how close) two objects are from each other (Fig. 1). The distance may be relate to physical distance (map distance), or in mathematics, distance is a metric or statistic. Euclidean distance is the distance between two points in either the X-Y plane or 3-dimensional space measures the length of a segment connecting the two points (e.g., x₁,y₁ = 1, 4 and x₂,y₂ = 1,4).

Figure 1. Cartesian plot of two points, the first at x₁ = 1 and y₁ = 1 and the second at x₂ = 4 and y₂ = 4.

For two points (x₁, y₁ and x₂, y₂) described in two dimensions (e.g., an xy plane), the distance d is given by

For two points described in three (e.g., an xyz-space), or more dimensions, the distance d is given by

Distances of this form are Euclidean distances and can be directly obtained by use of the Pythagorean Theorem. The triangle inequality rule then applies (i.e., the sum of any two sides must be less than the length of the remaining side). Euclidean distance measures also include

• Manhattan distance: the sum of absolute difference between the measures in all dimensions of two points.

• Chebyshev distance: also called the maximum value distance, the distance between two points is the greatest of their differences along any coordinate dimension.

Note: We first met Chebyshev in Chapter 3.5.

Example

There are a number of distance measures. Let’s begin discussion of distance with geographic distance as an example. Consider the distances between cities (Table 1).

Table 1. Distances (miles) among cities.

This table is a distance matrix — note that along the diagonal are “zeros,” which should make sense — the distance between an object and itself is, well zero. Above and below the diagonal you see the distance between one city and another. This is a special kind of matrix called a symmetric matrix. Enter the distance in miles (1 mile = 1.609344) between 2 cities (this is “pairwise“). There are many resources “out there,” to help you with this. For example, I found a web site called mapcrow that allowed me to enter the cities and calculate distances between them.

To get the distance matrix, use this online resource, the Geographic Distance Matrix Calculator.

Real world problem, use geodist package. Provide latitude and longitude coordinates

The dist() function in R base will return Manhattan distance and others () on matrices.

R script

HNL <- c(0, 2667.57, 5323.37, 3849.99, 3891.82)
SEA <- c(2667.57, 0, 6590.23, 4776.81, 1888.06)
MAN <- c(5323.37, 6590.23, 0, 1835.1, 8471.48)
TOK <- c(3849.99, 4776.81, 1835.1, 0, 6664.82)
HOU <- c(3891.82, 1888.06, 8471.48, 6664.82, 0)
mat.city <- rbind(HNL, SEA, MAN, TOK, HOU)
colnames(mat.city) <- c("HNL", "SEA", "MAN", "TOK", "HOU")
mat.city
dist(mat.city, method = "manhattan")

R output

        HNL       SEA     MAN      TOK
SEA 9532.58 
MAN 21163.95 25361.39 
TOK 16070.49 20267.93 8763.66 
HOU 14526.09 8769.63 27906.40 22896.60

Other distances, replace “manhattan” with

For example,

dist(mat.city, method = "euclidean")

R output

         HNL       SEA     MAN       TOK
SEA 4550.917 
MAN 9853.774 12079.347 
TOK 7345.155 9615.750 3931.736 
HOU 6980.976 3966.360 13820.922 11883.931

The Chebyshev distance doesn’t seem to be part of base R, but is available in other packages (philentropy, function distance).

R script

require(philentropy)
distance(mat.city, method = "chebyshev", diag=FALSE, upper=FALSE)

R output

     v1         v2      v3      v4      v5
v1    0.00 2667.57 5323.37 3849.99 3891.82
v2 2667.57    0.00 6590.23 4776.81 1888.06
v3 5323.37 6590.23    0.00 1835.10 8471.48
v4 3849.99 4776.81 1835.10    0.00 6664.82
v5 3891.82 1888.06 8471.48 6664.82    0.00

Distance measures used in biology

It is easy to see how the concept of genetic distance between a group of species (or populations within a species) could be used to help build a network, with genetically similar species grouped together and genetically distant species represented in a way to represent how far removed they are from each other. Here, we speak of distance as in similarity: two species (populations) that are similar are close together, and the distance between them is short. In contrast, two species (populations) that are not similar would be represented by a great distance between them. Genetic distance is the amount of divergence of species from each other. Smaller genetic distances reflects close genetic relationship. Hamming distance is a common choice, good for categorical data.

Here’s an example (Fig. 2), RAPD gel for five kinds of beans. RAPD stands for random amplified polymorphic DNA.

Figure 2. RAPD gel (simulated) five kinds of beans.

Samples were small red beans (SRB), garbanzo beans (GB), split green pea (SGP), baby lima beans (BLB), and black eye peas (BEP). RAPD primer 1 was applied to samples in lanes 1 – 6; RAPD primer 2 was applied to samples in lane 7 – 12. Lane 1 & 7 = SRB; Lane 2 & 8 = GB; Lanes 3 & 9 = SGP; Lane 4 & 10 = BLB; Lane 5 & 11 = BB; Lane 6 & 12 = BEP.

Here’s how to go from gel documentation to the information needed for genetic distance calculations (see below). I’ll use “1” to indicate presence of a band, “0” to indicate absence of a band, and “?” to indicate no information. For simplicity, I ignored the RF value, but ranked the bands by order of largest (= 1) to smallest (=8) fragment.

We need three pieces of information from the gel to calculate genetic distance.

N_A = the number of markers for taxon A

N_B = the number of markers for taxon B

N_AB = the number of markers in common between A and B (this is the pairwise part — we are comparing taxa two at a time.

First, compare the beans against the same primer. My results for primer 1 are in Table 2; results for primer 2 are in Table 3.

Table 2. Bands for Primer 1

Table 3 for Primer 2

From Table 2 and Table 3, count N_A (= N_B ) for each taxon. Results are in Table 4.

Table 4. Bands for each taxon.

To get Hamming distance between SRB and BB (from Table 2 and Table 3), R script as follows.

SRB <- c(1, 1, 1, 0, 0, 1, 0, 1)
GB <- c(0, 0, 1, 1, 0, 1, 1, 1)
# sum the inequality between the two vectors 
sum(SRB != GB)

R output

> sum(SRB != GB)
[1] 4

Hamming distance between small red beans and black beans for the markers is 4.

As you can see, there is no simple relationship among the taxa; there is no obvious combination of markers that readily group the taxa by similarity. So, I need a computer to help me. I need a measure of genetic distance, a measure that indicates how (dis)similar the different varieties are for our genetic markers. I’ll use a distance calculation that counts only the “present” markers, not the “absent” markers, which is more appropriate for RAPD. I need to get the N_AB values, the number of shared markers between pairs of taxa.

Table 5. N_AB

The equation for calculating Nei’s distance is:

where N_A = number of bands in taxon “A”, N_B = number of bands in taxon “B”, and N_AB is the number of bands in common between A and B (Nei and Li 1979). Here’s an example calculation.

Let A = SRB and B = GB, then

The R package adegenet includes Nei’s distance (and others), but the data set assumes more information than I provided here (e.g., ploidy, alleles).

Which distance measure to use?

Unfortunately, not a simple answer. A quick search of journals will likely return “Euclidean distance” the most common (Evolution = 328, Ecology = 298). The different distance (similarity) measures were developed to solve sometimes related, but often different problems, and so a review of the statistical properties would be warranted before making a choice. Helpful reviews

Questions

1. Write up three learning outcomes for this page. Hint: Point your favorite generative AI to this page and ask for help.

2. Review all of the different correlation estimates we introduced in Chapter 16 and construct a table to help you learn. Product moment correlation is presented as example.

3. Be able to define and distinguish between similarity measures, dissimilarity measures quantify, and distance.

4. Return to our counts of colored candies from mini bags of Skittles, M&M plain chocolate, and M&M peanut. Which two are most similar? Here’s a data sample. Calculate the Euclidean pairwise distances, then the Manhattan distances among the bags of candies. Do you get the same answer?

The data

Example R code

mySites <- data.frame(
Site <- c("A","B","C"),
Spp1 <- c(10, 5, 12),
Spp2 <- c(7, 15, 8),
Spp3 <- c(3, 2, 9)
)

# View the matrix
print(mySites)

# Exclude the "Site" column
myAbundances = mySites[,-1]

# Calculate the Euclidean distance matrix
dist(myAbundances,method="euclidean")

The output should be

          1          2
2  9.486833  
3  6.403124  12.124356

where “1” refers to site A, “2” refers to site B, and “3” refers to site C. We would conclude that sites A and C are most similar, or have similar species abundance, because they have the smallest Euclidean distance. As you can imagine, there’s more to interpreting these statistics, and in particular, Euclidean distance can lead to some confusing interpretations about abundance and species composition between sites, eg, Orlóci paradox. Interested readers should see Ricotta 2021.

Why then teach Euclidean distance? The Euclidean distance is a standard measure, it helps lay the foundation for understanding geometric concepts. Its limitations are a reminder that students need to be aware that alternative methods are sometimes needed.

Quiz Chapter 16.6

Similarity and Distance

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Chapter 16 contents

• Introduction
• Product moment correlation
• Causation and Partial correlation
• Data aggregation and correlation
• Spearman and other correlations
• Instrument reliability and validity
• Similarity and Distance
• References and suggested readings

10.1 – Compare two independent sample means

Introduction

We introduced the concept of comparing a sample statistic (mean) against a population parameter (Chapter 6.7, Normal deviate) or one-sample t-test against a specified mean (eg, from published data or from theory, Chapter 8.5).

Consider now a basic experimental design, the randomized control trial, or RCT (Fig 1), introduced in Chapter 2.4.

Figure 1. A two group Randomized Control Trial.

Subjects randomly selected from population of interest, then again — random assignment — once recruited into one of two treatment groups. Importantly, subjects belong to one treatment arm only: no subject simultaneously receives the treatment and the control. This is in contrast to the paired design, in which subjects receive both treatments (see Chapter 10.3).

In inferential statistics about an experiment, we are more likely trying to test if sample means are different. For example

• two species grown in common garden, do they differ in growth rate?
• human subjects given a new treatment have better outcomes compare to those receiving a control treatment (eg, placebo).

The equivalent null hypothesis is that two samples are pulled from the same population. We write the null hypothesis as

and the corresponding alternative hypothesis, H_A, then must be

Question: Is this a one tailed or two-tailed hypothesis?

Answer: Two-tailed (review Chapter 8.4)

Worked example

We introduce the two-sample t-test, or better, the independent sample t-test.

where the numerator is the difference between the two sample means and the denominator is the standard error of the differences between the two groups standard errors. The formula for

The choice of independent sample over two-sample is best because it emphasizes that the two groups (the two samples), must be comprise of independent sampling units. This is a pretty straight-forward requirement; you have randomly assigned twenty individuals to two groups, a control group (n = 10) and a treatment group (n = 10). Individuals are either in the control group or they are in the treatment group — they cannot simultaneously appear in both groups.

We will work our way through this test by example. For starters, let’s use the same lizard dataset (see Example data set, below), four body mass recordings (grams) each for house geckos (Hemidactylus frenatus, Fig 2) and the Carolina anole (Anolis carolensis, Fig 3), two of many lizard species introduced to Hawaiʻi.

Figure 2. Male Hemidactylus frenatus, central Oahu, M. Dohm.

Figure 3. Male Anolis carolinensis, ʻAkaka Falls, Big Island of Hawaiʻi, M. Dohm.

Example data set

Geckos: 3.186, 2.427, 4.031, 1.995

Anoles: 5.515, 5.659, 6.739, 3.184

Question: How would you go about creating a data frame with the values in long form (stacked worksheet), including a label variable and the body mass?

Answer: At the R prompt, type

Geckos <- c(3.186, 2.427, 4.031, 1.995); Anoles = c(5.515, 5.659, 6.739, 3.184) #create two vectors
bmass <- c(Geckos, Anoles)    #combine the two vectors into a single vector holding all of the body mass records
species <- c("gecko", "gecko", "gecko", "gecko", "anole", "anole", "anole", "anole")   #create your label variable
lizards <- data.frame(species, bmass)  #create your data frame
lizards   #print your data frame  species bmass
1   gecko 3.186
2   gecko 2.427
3   gecko 4.031
4   gecko 1.995
5   anole 5.515
6   anole 5.659
7   anole 6.739
8   anole 3.184

Note also that you can enter data into the Data editor by creating the data frame first then adding values. To edit the data frame “lizards” type fix(lizards) at the R prompt, then close the data frame when you have added or changed values as needed.

As always, begin with an exploration of the data, including a graph (Fig 4).

Figure 4. Box plot of lizard body mass.

We can see already that there’s greater spread of data for the Anoles compared to the Geckos, but the median values differ. Small sample sizes can be a problem for analyses as we can only have reduced confidence in our conclusions. However, we press on for the sake of demonstration.

Let’s test the null hypothesis, H₀, ie, the two species of lizards have the same mean body mass.

Rcmdr: Statistics → Means → Independent-samples t-test…

In this next image I posted the Rcmdr menu popup for the Independent Samples t-test (Fig 5). Later versions of Rcmdr split the settings for this command into two tabs; the first tab allows for the selection of the variables and setting the hypotheses whereas the second tab, labeled Options, permits additional choices. The default selections need your attention: to actually conduct the t-test you need to answer “No” to the question, “Assume equal variance?” (Fig 6).

Figure 5. Rcmdr menu for Independent sample t-test.

Select the Options tab (Fig 6) to select null hypothesis and to select the t-test and not the Welch-test (which is the default, ie, No to the prompt “Assume equal variances?”).

Figure 6. Rcmdr Options menu for Independent sample t-test.

Let’s look at the results and break down the parts of the test.

t.test(Body.mass~Lizard, alternative='two.sided', conf.level=.95, 
+ var.equal=TRUE, data=lizards)

Two Sample t-test

data: Body.mass by Lizard 
t = 2.7117, df = 6, p-value = 0.03503
alternative hypothesis: true difference in means is not equal to 0 
95 percent confidence interval:
0.2308685 4.4981315 
sample estimates:
mean in group Anolis mean in group Gecko 
5.27425 2.90975

Consider the R session output above and answer the following questions.

Questions for the worked example

1. Which lizard group had the greater mean value, Anolis or Gecko?
2. What are the assumptions necessary for you to use the independent sample t-test?
3. What does “two-sided” mean?
4. What was the null hypothesis?
5. Was this a one-tailed or two-tailed test of the null hypothesis?
6. What is the value of the test statistic?
7. How many degrees of freedom?
8. What is the critical value for this test?
9. What is the value of the lower limit of the 95% confidence interval?
10. What is the value of the lower limit of the 99% confidence interval?
11. True or False. If the null hypothesis is accepted, then zero is a value included in the 95% confidence interval.
12. Do you accept the null hypothesis? Explain your selection.

Try another example

DNA damage, changes in the chemical structure of nucleotide bases or breakage of the DNA chains, occurs in cells under many circumstances. The comet assay, or single-cell gel electrophoresis, is one method for visualizing and measuring DNA strand breaks in cells. Exposed cells are mixed with a low-melting temperature agarose and placed onto a microscope slide. The cells are then lysed with an alkaline detergent and high salts. When current is applied across the slide, undamaged DNA remains in the nucleus, whereas damaged DNA extends towards the anode to form a comet-like tail, with imaging assisted by including a fluorescent dye like Sybr-Green. Examples of comets are shown below (Fig 7).

Figure 7. Comet examples. A Intact cell, no DNA damage, B Cell with some DNA damage, a slight tail to the right is evident, C  Cell with significant DNA damage, a large tail is evident. M. Dohm.

In an experiment, immortalized lung epithelial cells were exposed to dilute copper solutions for 30 minutes then washed with PBS. The comet assay was applied to these cells and for comparison, to cells without copper exposure but otherwise treated the same way (controls). The data are available at the bottom of this page (scroll down or click here).

Again, you should begin all analyses with an exploration of the data, including a graph (Fig 8).

Figure 8. Boxplot of comet tail lengths for cells with and without (control) exposure to copper in the cell medium for 30 minutes.

Let’s look at the R output for the t-test analysis.

    Two Sample t-test

data:  CometTail by Treatment
t = -5.8502, df = 38, p-value = 9.139e-07
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
 -22.39865 -10.88213
sample estimates:
mean in group Control  mean in group Copper 
             11.14533              27.78571

Consider the R session output above and answer the following questions.

Questions for Comet assay data set

1. Which cell group had the greater mean value, Copper-exposed or Control-exposed cells?
2. What are the assumptions necessary for you to use the independent sample t-test?
3. What does “two-sided” mean?
4. What was the null hypothesis?
5. Was this a one-tailed or two-tailed test of the null hypothesis?
6. What is the value of the test statistic?
7. How many degrees of freedom?
8. What is the critical value for this test?
9. What is the value of the lower limit of the 95% confidence interval?
10. What is the value of the lower limit of the 99% confidence interval?
11. True or False. If the null hypothesis is accepted, then zero is a value included in the 95% confidence interval.
12. Do you accept the null hypothesis? Explain your selection.

T test from summary statistics

In some cases you may only have to summary statistics for data, eg, the means and the standard deviations. We can use the equations of the t test to write a simple formula, where the user provides the known means, standard deviations, and sample size. For example, create a simple function with readline for user input.

myTtest <- function() {
   mnx <- as.numeric(readline(prompt="Enter mean of x: "))
   stdevx <- as.numeric(readline(prompt="Enter sd of x: "))
   nx <- as.numeric(readline(prompt="Enter n of x: "))
   mny <- as.numeric(readline(prompt="Enter mean of y: "))
   stdevy <- as.numeric(readline("Enter sd of y: "))
   ny <- as.numeric(readline(prompt="Enter n of y: "))
   myTvalue <- abs(((mnx-mny)-0)/sqrt(((stdevx^2)/nx)+(stdevy^2)/ny))
   myDF <- as.integer(nx+ny-2)
   myPvalue <- pt(myTvalue,myDF,lower.tail=FALSE)*2
   myResults <- c(myTvalue, myDF, myPvalue)
   report <- c("T-test: ", "df: ", "two-tailed p-value: ")
   cat(sprintf("%s %3.3f, ", report, myResults))
}

then run the function by typing myTest() at the R prompt and entering the means, standard deviations, and sample size when prompted.

myTtest()
Enter mean of x: 2.91
Enter sd of x: .895
Enter n of x: 4
Enter mean of y: 5.27
Enter sd of y: 1.497
Enter n of y: 4
T-test: 2.706, df: 6.000, two-tailed p-value: 0.035

P-values from confidence intervals

While we expect certain reporting criteria for published, it is not uncommon that one or more elements are missing. For example, while means plus or minus standard deviations for two groups may be reported, the confidence interval of the difference may not be reported, or, an inexact p-value is reported like “< 0.05,” but we “need” an exact p-value for our meta-analysis.  A little math, and these missing statistics can be calculated.

We can use the lizard study as an example; while all of the expected elements were reported, including the p-value (0.03503), let’s say the p value was reported as “statistically significant at p < 0.05.”

We need

lower and upper confidence intervals: 0.2309 and 4.498, respectively.

critical value from t distribution, df = 6 (2 tailed, therefore 0.05/2). A little R code we have:

the standard error, which can be calculated as

and then the p-value,

which is pretty close to the result from R (p = 0.03503). The difference is likely due to the small sample size.

Questions

1. Don’t forget to work through the Questions for the Comet tail data set (scroll up or click here).
2. Microsoft Excel, LibreOffice Calc, and Google sheets spreadsheet software all include t-test functions and return the p-value. Consider two variables big (100, 110, 120, 100, 110, 210, 200) and small (0,1,1,2,0,1,0). (Note — these two groups are obviously very different, calculating a t-test on their difference is silly, just for this question.)  If formatting is set to the default two decimal places for Number cell category, the p-value will return as “0.00.” How should you report the p-value in this case?

Quiz Chapter 10.1

Compare two independent sample means

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Comet assay data set

Chapter 10 contents

• Introduction
• Compare two independent sample means
• Digging deeper into t-test Plus the Welch test
• Paired t-test
• References and suggested reading