12.6 – ANOVA posthoc tests

Introduction

Tests of the null hypothesis in a one-way ANOVA yields one answer: either you reject the null, or you do not reject the null hypothesis — with the usual caveats about “provisionally reject,” etc (Ch8.2 and Ch8.3). But while there was only one factor (population, drug treatment, etc) in a one-way ANOVA, there are usually many treatments (eg, multiple levels, four different populations, three doses of a drug plus a placebo).

ANOVA plus posthoc tests solves the multiple comparison problem we discussed: you still get your tests of all group differences, but with adjustments to the procedures so that these tests are conducted without suffering the increase in Type I error = the multiple comparison problem. If the null hypothesis is rejected, you may then proceed to posthoc tests among the groups to identify differences without increasing the chance of rejecting the null hypothesis too often.

Consider the following example of four populations scored for some outcome, sim.ch12 (scroll down the page, or click here to get the R code).

Bring the data frame, sim.ch12, into current memory in Rcmdr by selecting the data set.  Next, run the one-way ANOVA.

Rcmdr: Statistics → Means → One-way ANOVA…

which brings up the following menu (Fig 1).

Figure 1. One-way ANOVA menu in R Commander.

Notice that Rcmdr menu correctly identifies the Factor variable, which contains text labels for each group, and the Response variable, which contains the numerical observations.

newName <- as.factor(oldVariable)

If your data set contains more variables, then you would need to sort through these and select the correct model (Fig 1).

To get the default Tukey posthoc tests simply check the Pairwise comparisons box and then click OK.

For a test of the null that four groups have the same mean, a publishable ANOVA table would look like Table 1.

Table 1. The ANOVA table

pvalue(1.108357e-15, accuracy=0.0001, decimal.mark = ".", add_p=TRUE)

[1] "p<0.0001"

Here’s the R output for ANOVA

AnovaModel.1 <- aov(Values ~ Label, data = sim.Ch12)
summary(AnovaModel.1)
          Df  Sum Sq Mean Sq  F value     Pr(>F) 
Label      3  389627  129876    76.44   1.11e-15 ***
Residuals 36   61167    1699 
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

with(sim.Ch12, numSummary(Values, groups = Label, statistics = c("mean", "sd")))
      mean       sd data:n
Pop1 150.3 35.66838 10
Pop2  99.0 43.25634 10
Pop3 130.9 42.36469 10
Pop4 350.7 43.10723 10

End of R output.

When to conduct posthoc tests.

Recall that all we can say is that a difference has been found and we reject the null hypothesis. However, we do not know if group 1 = group 2, but both are different from group 3, or some other combination. So we may want additional tools. We can conduct posthoc tests (also called multiple comparisons tests).

Once a difference has been detected (F test statistic > F critical value, therefore P < 0.05), then posteriori tests, also called unplanned comparisons, can be used to tell which means differ.

There are also cases for which some comparisons were planned ahead of time and these are called a priori or planned comparisons; even though you conduct the tests after the ANOVA, you were always interested in particular comparisons. This is an important distinction: planned comparisons are more powerful, more aligned with what we understand to be the scientific method. An argument — transparency, avoids Type II errors — can be made that if comparisons are planned, and only a few tests are conducted, no correction for “multiple tests” is warranted. Moreover, it is defensible to report all p-values from a hundred tests without correction, acknowledging that if all null hypotheses were “true,” then 5 would be expected to be less than Type I of 5% by chance alone.

Posthoc tests.

ANOVA was introduced by Fisher in 1918, and, as readers know by know, returns whether or not significant differences are present among groups, but no which groups may differ.  Unsurprisingly, given the needs of researchers to compare groups, many tests were developed since Fisher’s ANOVA to conduct multiple comparison tests. Collectively, they are often referred to as posthoc tests (Ruxton and Beauchamp 2008).

There are many different flavors of these tests, and R offers several, but I will hold you responsible only for three such comparisons: Tukey’s, Dunnett’s, and Bonferroni (Dunn). These named tests are among the common ones (Table 2, Midway et al 2020), but you should be aware that the problem of multiple comparisons and inflated error rates has received quite a lot of recent attention because the size of data sets has increased in many fields, eg, genome wide-association studies in genetics or data mining in economics or business analytics. A related topic then is the issue of “false positives.” New approaches include Holm-Bonferroni. There are others — it is a regular “cottage industry” in applied statistics to a problem that, while recognized, has not achieved a universal agreed solution. Best we can do is be aware and deal with it and know that the problem is one mostly of big data (eg, microarray and other high-throughput approaches). Equally important, some of these tests are better than others, and some may not be consistent with best practices.

Let’s take a look at these procedures.

Performing multiple comparisons and the one-way ANOVA

a. Tukey’s: “honestly (wholly) significant difference test”

Tests versus where A and B can be any pairwise combination of two means you wish to compare. Thus, this is likely a case of unplanned comparisons. There are comparisons.

where

and n is the harmonic mean of the sample sizes of the two groups being compared. If the sample sizes are equal, then the simple arithmetic mean is the same as the harmonic mean.

• The significance level is the probability of encountering at least one Type I error (probability of rejecting H_o when it is true). This is called the experimentwise (familywise) error rate whereas before we talked about the comparisonwise (individual) error rate.

Three options to get the posthoc test Tukey:

(1) TukeyHSD() function is included in base statistics.

(2) use a function called mcp or in Rcmdr, Tukey is the default option in the one-way ANOVA command. The function mcp is part of the multcomp package, which is included in the Rcmdr package.

(3) specify method = "HSD" in PostHocTest function in DescTools package.

In brief, run the anova first, save the output to an object, then call an appropriate command from a package to conduct the posthoc test.

Rcmdr: Statistics → Means → One-way ANOVA
Check “Pairwise comparisons of means” to get the Tukey’s HSD test (Fig 2).

Figure 2. Screenshot Rcmdr menu: Select Tukey posthoc tests with the one-way ANOVA.

Tukey’s honestly significant difference is available without installing additional packages. Simply run

AnovaModel.1 <- aov(formula = Values ~ Label, data = sim.Ch12)
TukeyHSD(AnovaModel.1)

and R output returned is

Tukey multiple comparisons of means
95% family-wise confidence level

Fit: aov(formula = Value ~ Label, data = sim.Ch12)

$Label
           diff        lwr        upr     p adj
Pop2-Pop1 -51.3 -100.94731  -1.652695 0.0406116
Pop3-Pop1 -19.4  -69.04731  30.247305 0.7200796
Pop4-Pop1 200.4  150.75269 250.047305 0.0000000
Pop3-Pop2 31.9   -17.74731  81.547305 0.3232616
Pop4-Pop2 251.7  202.05269 301.347305 0.0000000
Pop4-Pop3 219.8  170.15269 269.447305 0.0000000

Installation of R Commander also includes installation of the multcomp package, which contains several post hoc tests, including Tukey and Dunnett’s tests.

R output follows. There’s a lot, but much of it is repeat information. Take your time, here we go.

.Pairs <- glht(AnovaModel.1, linfct = mcp(Label = "Tukey"))
summary(.Pairs) # pairwise tests
     Simultaneous Tests for General Linear Hypotheses
Multiple Comparisons of Means: Tukey Contrasts
 Fit: aov(formula = Values ~ Label, data = sim.Ch12) 

 Linear Hypotheses: 
                  Estimate Std. Error t value Pr(>|t|)     
 Pop2 - Pop1 == 0   -51.30      18.43  -2.783   0.0405 *   
 Pop3 - Pop1 == 0   -19.40      18.43  -1.052   0.7201     
 Pop4 - Pop1 == 0   200.40      18.43  10.871   <0.001 *** 
 Pop3 - Pop2 == 0    31.90      18.43   1.730   0.3233     
 Pop4 - Pop2 == 0   251.70      18.43  13.654   <0.001 *** 
 Pop4 - Pop3 == 0   219.80      18.43  11.924   <0.001 *** 
 --- 
 Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
 (Adjusted p values reported -- single-step method) 

 confint(.Pairs) # confidence intervals 
      Simultaneous Confidence Intervals 

 Multiple Comparisons of Means: Tukey Contrasts 

 Fit: aov(formula = Values ~ Label, data = sim.Ch12) 

 Quantile = 2.6927 
 95% family-wise confidence level 
   
 Linear Hypotheses: 
                  Estimate  lwr       upr       
 Pop2 - Pop1 == 0  -51.3000 -100.9382   -1.6618 
 Pop3 - Pop1 == 0  -19.4000  -69.0382   30.2382 
 Pop4 - Pop1 == 0  200.4000  150.7618  250.0382 
 Pop3 - Pop2 == 0   31.9000  -17.7382   81.5382 
 Pop4 - Pop2 == 0  251.7000  202.0618  301.3382 
 Pop4 - Pop3 == 0  219.8000  170.1618  269.4382 

Figure 3. Plot of confidence intervals of Tukey HSD.

R Commander includes a default 95% CI plot (Fig 3). From this graph, you can quickly identify the pairwise comparisons for which 0 (zero, dotted vertical line) is included in the interval, ie, there is no difference between the means (eg, Pop1 is different from Pop4, but Pop1 is not different from Pop3).

b. Dunnett’s Test for comparisons against a control group

• There are situations where we might want to compare our experimental Populations to one control Population or group. Thus, this is likely a case of planned comparisons.
• This is common in medical research where there is a placebo (control pill with no drug) or sham operations (operations where everything but the critical operation is done).
• This is also a common research design in ecological or agricultural research where some animal or plant populations are exposed to an environmental factor (eg, fertilizer, pesticide, pollutant, competitors, herbivores) and other animal or plant populations are not exposed to these environmental factor.
• The difference in the statistical procedure for analyzing this type of research design is that the experimental groups may only be compared to the control group.
• This results in fewer comparisons.
• The formula is the same as for the Tukey’s Multiple Comparison test, except for the calculation of the SE.

Standard Error is changed by multiplying the MS_Error by 2.

And n is the harmonic mean of the sample sizes of the two groups being compared.

where

R Commander doesn’t provide a simple way to get Dunnett, but we can get it simply enough if we are willing to write some script. Fortunately (OK, by design!), Rcmdr prints commands.

Look at the Output window from the one-way ANOVA with pairwise comparisons: it provides clues as to how we can modify the mcp command (mcp stands for multiple comparisons).

First, I had run the one-way ANOVA command and noted the model (AnovaModel.3). Second, I wrote the following script, modified from above.

Pairs <- glht(AnovaModel.1, linfct = mcp(Label = c("Pop2 - Pop1 = 0", "Pop3 - Pop1 = 0", "Pop4 - Pop1 = 0")))

where Label was my name for the Factor variable. Note that I specified the comparisons I wanted R to make. When I submit the script, nothing shows up in the Output window because the results are stored in  Pairs.

Next, I then need to ask R to provide confidence intervals

confint(Pairs)

R output window

Pairs <- glht(AnovaModel.1, linfct = mcp(Label = c("Pop2 - Pop1 = 0", "Pop3 - Pop1 = 0", "Pop4 - Pop1 = 0"))) 
confint(Pairs) 

Simultaneous Confidence Intervals 

 Multiple Comparisons of Means: User-defined Contrasts 

 Fit: aov(formula = Values ~ Label, data = sim.Ch12) 

 Estimated Quantile = 2.4524 
 95% family-wise confidence level 

 Linear Hypotheses: 
  ...... lwr ....... upr 
 Pop2 - Pop1 == 0  .. -51.3000  . -96.5080  ... -6.0920 
 Pop3 - Pop1 == 0  .. -19.4000  . -64.6080  ... 25.8080 
 Pop4 - Pop1 == 0  .. 200.4000  . 155.1920  .. 245.6080

Look for intervals that include zero, therefore, the group does not differ from the Control group (Pop1). How many groups differed from the Control group?

Alternatively, I may write

Tryme <- glht(AnovaModel.1, linfct = mcp(Label = "Dunnett"))
confint(Tryme)

It’s the same — in fact, the default mcp test is Dunnett’s test.

Tryme <- glht(AnovaModel.1, linfct = mcp(Label = "Dunnett")) 
confint(Tryme) 

Simultaneous Confidence Intervals 

Multiple Comparisons of Means: Dunnett Contrasts 

Fit: aov(formula = Values ~ Label, data = sim.Ch12) 

Estimated Quantile = 2.4514 
95% family-wise confidence level 

Linear Hypotheses: 
Estimate lwr upr 
Pop2 - Pop1 == 0 -51.3000 -96.4895 -6.1105 
Pop3 - Pop1 == 0 -19.4000 -64.5895 25.7895 
Pop4 - Pop1 == 0 200.4000 155.2105 245.5895                                   

Level 
[1] Pop1 Pop2 Pop3 Pop4
Levels: Pop1 Pop2 Pop3 Pop4

Level <- relevel(Level, ref = "Pop2"); Level
[1] Pop1 Pop2 Pop3 Pop4
Levels: Pop2 Pop1 Pop3 Pop4

is.factor(Level)
[1] TRUE

Level <- as.factor(Level)

Level <- factor(Level, levels = c("Pop2", "Pop4", "Pop1", "Pop3")); Level

[1] Pop1 Pop2 Pop3 Pop4
Levels: Pop2 Pop4 Pop1 Pop3

c. Bonferroni t

The Bonferroni t test is a popular tool for conducting multiple comparisons. This, too, is consistent with an unplanned comparisons approach. The rationale for this particular test is that the MS_error is a good estimate of the pooled variances for all groups in the ANOVA.

and DF = N – k.

Thus, for k = 4 groups,

Here’s a simplified version if you prefer to get all pairwise tests

Use this information then to determine how many total comparisons will be made, then if necessary, use to adjust Type I error rate for one test (the exerimentwise error rate).

For our example, is 0.05/6 = 0.00833. Thus, for a difference between two means to be statistically significant, the P-value must be less than 0.00833.

For Bonferroni, we will use the following script.

1. Set up one-way ANOVA model (ours has been saved as AnovaModel.1),
2. Collect all pairwise comparisons with the mcp(~"Tukey") stored in a vector (I called mine Whynot),
3. and finally, get the Bonferroni adjusted test of the comparisons with the summary command, but addtest = adjusted("bonferroni").

It’s a bit much, but we end up with a very nice output to work with.

Whynot <- glht(AnovaModel.1, linfct = mcp(Label = "Tukey")) 
summary(Whynot, test = adjusted("bonferroni")) 

 Simultaneous Tests for General Linear Hypotheses

Multiple Comparisons of Means: Tukey Contrasts

Fit: aov(formula = Values ~ Label, data = sim.Ch12)

Linear Hypotheses:
Estimate Std. Error t value Pr(>|t|) 
Pop2 - Pop1 == 0 -51.30 18.43 -2.783 0.0512 . 
Pop3 - Pop1 == 0 -19.40 18.43 -1.052 1.0000 
Pop4 - Pop1 == 0 200.40 18.43 10.871 3.82e-12 ***
Pop3 - Pop2 == 0 31.90 18.43 1.730 0.5527 
Pop4 - Pop2 == 0 251.70 18.43 13.654 5.33e-15 ***
Pop4 - Pop3 == 0 219.80 18.43 11.924 2.77e-13 ***
  --- 
 Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1 
 (Adjusted p values reported -- bonferroni method)        

Questions

1. Be able to define and contrast experiment-wise and family-wise error rates.
2. Read and interpret R output
   1. Refer back to the Tukey HSD output. Among the four populations, which pairwise groups were considered “statistically significant” following use of the Tukey HSD?
   2. Refer back to the Dunnett’s output. Among the four populations, which population was taken as the control group for comparison?
   3. Refer back to the Dunnett’s output. Which pairwise groups were considered “statistically significant” from the control group?
   4. Refer back to the Bonferroni output. Among the four populations, which pairwise groups were considered “statistically significant” following use of the Bonferroni correction?
   5. Compare and contrast interpretation of results for posthoc comparisons among the four populations based on the three different posthoc methods
3. Be able to distinguish when Tukey HSD and Dunnet’s post hoc tests are appropriate.
4. Some microarray researchers object to use of Bonferroni correction because it is too “conservative.” In the context of statistical testing, what errors are the researchers talking about when they say the correction is “conservative.”

Quiz Chapter 12.6

ANOVA posthoc tests

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Data set used in this page

R code # copy the data from the full table to clipboard and paste between the double quotes in text=””)

sim.ch12 <- read.table(header=TRUE, sep=",",text="")

#check the dataframe
head(sim.ch12)

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

3.2 – Measures of Central Tendency

Introduction to summary statistics.

For a sample of observations we can begin the summary by identifying the “typical” value. Various statistics are used to describe the middle and collectively these are referred to as measures of central tendency. The mean, the median, and the mode are the most common measures of central tendency. In the situation in which we work with data from a population census, we would calculate population descriptive statistics — not inferential statistics; because we more often work with samples from populations, we report sample descriptive statistics.

First we review the familiar arithmetic mean and introduce the weighted mean. Next we introduce “other means,” which you may not be as familiar with.

Means.

There are several means beyond the simple arithmetic average or mean. Here we review a few. We use this topic also to start introducing standard notation we will use throughout the book.

The population mean, μ (pronounced “mu”) is given by

X_i, an observation on the i-th individual

Σ, or “sigma,” which instructs you to add them up (the X’s), from i = 1 (the first observation) to i = N (the last observation).

The sample mean, , pronounced “eks bar,” of a collection of observations is given by 

Weighted arithmetic mean.

In some cases you may have several samples from the same population. If the sample sizes are the same, you can calculate the average of averages without any fuss — just take all of the sample means and add them up, then divide by the total number of samples. If the sample sizes differ, then you needs to weight (W) each sample mean by its sample size. Simply divide each sample mean by its appropriate sample size, then add all of these up. That is the weighted average.

More generally, we can write

For example, consider a variable containing the following observations, for example, over the course of watching birds gathering at a drinking source (eg, recorded over 20 days, a single 5 minute interval per day), we can count how many times zero, one, two, etc birds are observed. 

Table 1. A sample of counts of observations, eg, how many birds

The observation “4” was observed four times; the observation “5” was observed twice, and so on for a total of 15 observations.

What is the arithmetic mean of these 15 observations? To solve this, well, you have a couple of choices. You could copy the numbers down as often as they appear and then calculate the mean in the usual way.

Other “means”.

The arithmetic average (illustrated above) is not the only way to estimate the mean.

The trimmed mean, also called the truncated mean, is a useful approach when data is widely dispersed — data spread away from the middle (see Chapter 3.3). Thus, the trimmed mean will be less influenced compared to the arithmetic mean by outlier data points, ie, data far from other data points in the set.

You would use the trimmed mean to describe the middle of a data set in which a plot shows most of the values are clumped together around a middle – and yet you see a few values that are much smaller or much greater. A specified percentage of the smallest and largest values are removed from the data set and then the simple arithmetic mean is calculated for the trimmed data set. For example, given a data set of daily rainfall for different cities, you might wish to remove the driest 5% and wettest 5% of the days in order to better compare the rainfall trends for the cities.

Calculating the trimmed mean is straight-forward in R: use the same built-in function, mean(), but add some options.

Note 2: This is a good time to remind you how to get help with R commands. Do you recall how to get help in R? 

At the R prompt type

help(mean) 

or

?mean

The R Documentation page for mean() will pop up (assuming you allowed R to install help pages as html). Figure 1 shows a screenshot of a portion of the help page for mean()

Figure 1. A portion of the R help page about the function mean.

From the help page (Fig 1) we can see that we can specify a trimmed mean by adding options to the mean(x, ...) command. For our x variable defined above, get the trimmed mean after 25% of the data are removed.

mean(x, trim=0.25)

[1] 6

From the help page we see that the only required option you need to feed the mean command is the name of the variable, in this case, “x” (it can be, of course, any name provided the data are attached). In this case we removed 25% of the values – 12.5% of the smallest values and 12.5% of the largest values – that’s also called the interquartile mean. We use a dot plot to show these trends (Fig 2).

Figure 2. Dot plot of our x variable with locations of the mean (blue) and the trimmed mean (red).  The Dotplot(x) function in package RcmdrMisc was used in Rcmdr to make this graphic. Arrows were added by hand. Dotplot() example code presented in Chapter 3.4.

If we recalculate a trimmed mean after dropping 10% of the points, or even 40% of the points, we get the same mean value of 6. The trimmed mean is an example of a robust estimator; it’s resistant to the influence of outliers.

Another useful descriptor of the middle is the geometric mean. The geometric mean is useful for calculating the average of ratios.  Geometric mean would be used when you want to compare central tendency for different variables, each differing in scale. For example, gene expression results, reported as fold-changes, for different genes often shows tremendous differences among genes and would be best described by logarithmic scale, not arithmetic scale. Geometric mean expression values would be better choice for central tendency. Other examples are found in economics: for example, calculating compound interest or interest. The geometric mean applies whenever the scale is multiplicative and not additive

The geometric mean is given by the equation

The geometric mean (gm) is equivalent to log-transforming your data, then calculating the arithmetic mean, and transforming the result back (with the antilog exponent.) As you recall, for our simple data set the arithmetic mean was 6.2. The geometric mean for this data was 5.977. Taking the natural log for each of the values from our simple data set, then calculating the arithmetic mean we have 1.788. 

The antilog of this value is

exp(1.788)

[1] 5.977

Another frequently encountered mean is the harmonic mean, which is defined by the equation

Harmonic mean is appropriate for averaging rates. For example, what is the average speed traveled if you travel 30 miles per hour (mph) between point A and B, then on the return trip, your speed was 40 mph? If you think (30 + 40)/2 = 35 mph, then this would be incorrect — after all, the distance covered has not changed, just the time. The harmonic mean returns 34.2 mph. Let

y = c(30,40)

The harmonic mean returns 34.2 mph ( see below “How to calculate these other means”)

Both harmonic and geometric means apply for values greater than zero.

How to calculate these “other” means.

In Microsoft Excel, calculate geometric mean via the function GEOMEAN(); calculate harmonic mean via the function HARMEAN().

The base R (and Rcmdr) doesn’t have built in functions for these, although you could download and install some R packages which do (eg, package psych, geometric.mean(variable), harmonic.mean(variable)). It is quicker to just to calculate these by submitting a snippet of code into the script window

For geometric mean of variable “x” at the R prompt type

exp(mean(log(x)))

For harmonic mean of variable “x” at the R prompt type

1/mean(1/x)

where is the base of the natural logarithm, Euler’s number, and log is the natural logarithm (in R, to get log to other bases you can use log10 for base 10 logarithm or log2 for base 2 logarithm, or  log(x, base = n) for any base n of the variable x, and variable is the name of the variable you wish to do the calculations on.

R code: Do try on your own!

Here’s some numbers to try your hand. For example, create a variable containing a few numbers, any numbers. and write it to the variable named z

myZ = c(3,4,6,7,9,11,4)

Now, calculate the arithmetic mean, the geometric mean, and the harmonic mean for the variable myZ. You should get Table 4.2)}

Table 2. Comparison of different means for myZ.

Try three more. In R (or R Commander script window), create three new variables.

varA = c(3,3,3,3)

varB = c(1,2,3,12)

varC = c(-3,0,1,3)

Now, calculate the arithmetic mean, geometric mean, and harmonic mean for each variable.

For the simple arithmetic mean

mean(varA)

For the geometric mean, use the formula above

exp(mean(log(varA)))

For the harmonic mean, use the formula above

1/mean(1/varA)

What did you get?

Other measures of central tendency

Median 

The median, which lacks an accepted notation — we’ll go with , divides a set of observed numbers into two equal halves. Half the observations are above the median, half of the observations are below the median. Arrange data from lowest to highest, take the middle measurement:

odd number of measurements minus the middle value minus even number of measurements minus average of the 2 middle values

Or, more succinctly, we have

To get the median in R type at the R prompt

median(variable)

and of course, replace variable with the name of the variable containing the numbers. For our x variable created earlier, the function median returns in R

[1] 6

[Note that the median for x was the same as the trimmed mean for x, which is consistent with with our view that the trimmed mean is a robust estimator of the middle of a data set.

Mode

Mode is another way to express the middle and it refers to the most frequent occurring measurement. Use of mode makes most sense for discrete or countable numbers. For a normal distribution, the mean, median and mode will be the same value. Note that a data set may have more than one mode. For example, what is the mode for the variable  we created earlier?

myX = c(4,4,4,4,5,5,6,6,6,7,7,8,8,8,15)

For this small data set we see that “4” is the most frequent with a count of four occurrences in the set.

Mode would seem like a straightforward function in R. However, it turns out there is not a mode function in the base package.

A little explanation is in order. In R, typing mode at the R prompt like so

mode(myX)

returns

[1] "numeric" 

Not the answer we were expecting. In R, mode command is used to tell you what the mode (ie, way or manner in which some task is accomplished) of storage is for the variables.

In order to get the statistical mode we want, we either hunt down a package that contains mode estimation (eg, install the package DescTools use the Mode function — see below), or we can write a little code.

A quick Google search found a number of answers at stackoverflow.com (eg, question 2547402). The simplest response was to use names and max commands like so

temp = table(as.vector(myX))

names (temp)[temp==max(temp)]

Why the median may be a better middle than the mean

Comparing the two measures of central tendency can tell you without plotting how your data are distributed about the middle. Sample distributions are discussed in Chapter 6. 

• When the distribution of the data is symmetric or normally distributed (discussed in Chapter 6.7) then the mean and the median will be about the same value
• When data are right-skewed (a few large values), then the mean will be greater than the median.
• When the data are left-skewed (a few small values), then the mean will be less than the median.

Here’s an illustration (Fig 3). I sampled 100 points from a random normal distribution with mean zero and standard deviation 0ne and another 100 points from a log-normal distribution also with mean zero and standard deviation one. In Figure 3, the histograms (see Chapter 4.2 – Histograms), along with summary statistics (see Note 4). Means are indicated with red arrows and medians are indicated with blue arrows.

Figure 3. Normal and lognormal distributions with mean (red) and median (blue) noted for comparison.

So the median is a better descriptor or the central tendency of a sample distribution when the distribution is NOT normally distributed.

Scaling and transformation of data

Sometimes it is useful to standardize your data so that the variables all have the same scale. One algorithm for standardization is called normalization. Normalization implies that you correct the data so that data has a mean, μ, of zero, and a standard deviation, σ, of 1 (unit variance). There are several ways to standardize, each with strengths and limitations. To normalize we use the Z-score equation (see Chapter 6.7 for other uses of Z score).

where X_i is each observation in your data set.

Normalization will make outliers, the few points in a data set that are noticeably different from the central tendency of the rest of the data, smaller and less influential. When you normalize multiple sets of data, then each will have the same mean (0) and variance (unit variance), but the ranges will differ. An example of this is the simple product moment correlation — by standardizing you change the variances for the different variables to have the same unit variance.

As we will see later in class it is also useful to expand or contract the variability of the data or to change the shape of the distribution (if the data is not normally distributed). For example, if you compare individuals of a population for many morphological traits (eg body size, growth rate), the spread of points (called a distribution) will look more like a Poisson distribution (not symmetrical about the mean, a few individuals may be much larger…). This is partly due to the way in which morphological traits are measured. We normally measure body size on a linear scale (inches or centimeters). However, body size is affected by physiological processes that are more related to volume. Therefore, the more appropriate scale of measurement is on a log scale. We can transform the data measured on a linear scale to a log scale. For morphological traits this can produce a distribution that is normally distributed (bell shape). There are many more statistical procedures for data that is normally distributed than there are statistical procedures for Poisson distributions or any other type of distribution. Additional discussion about data transformation is introduced in Chapter 13.3.

You can always uncode or unstandardize your data after performing the statistical procedures and return to the original scale. In fact, when reporting descriptive statistics you should report the untransformed, uncoded data. Moreover, you will find it useful to report means adjusted for other variables (eg, from ANOVA or regression); if the ANOVA or regression equations are performed on transformed or coded data you would want to back calculate to the original scale after applying the ANOVA or regression adjustments. This advise will make more sense after we’ve discussed ANOVA (Chapter 12) and linear regression (Chapter 17).

R operators

The names command can be used to retrieve the names contained in the variable (if text types) or to set the names of the observations, which is what we are using it for here. We set the numbers to text names “4”, “5”, etc. then find the maximum count of named items in the temp table. The double equals operator (==) is used to tell R to find the object that is “equal to” something we specify, in this case, the max value (R Language Definition 2014). Table 3 shows common operators in R.

Table 3. Common arithmetic* and comparison** operators

help(Arithmetic)

help(Comparison)

The R package modeest has a number of algorithms for calculating the mode, depending on the kind of data you are working with. After installing the package and its dependencies, type at the R prompt 

require(modeest)
mfv(myX)
[1] 4

Creating objects in R

Everything in R is an object (Chambers 2008). We provide many worked examples in Part 07. Working with your own data – Mike’s Workbook for Biostatistics. In brief, create a variable in R by assigning the vector x, either directly at the R prompt or in a script window (Rcmdr, RStudio), like so

myX = c(4,4,4,4,5,5,6,6,6,7,7,8,8,8,15)

The function c(), which stands for combine  is used to combine the set of numbers into the object, myX. For small sets like this you may find it convenient to enter the values one by one and let R store it into the vector for you. For R installed on your computer, use the scan() function and your keyboard — scan() doesn’t work consistently in serverless R instances (eg., Google CoLab). Careful! Make sure that you remember to assign the results from scan to a vector.

I’ll create the object tryScan just to distinguish it from x, although I will enter the same values. Until R receives an interrupt signal from you, it will prompt you to enter numbers one row at a time. When you’ve reached the end, use the keyboard shortcut Ctrl + q (Windows) or Cmd + q (macOS) to interrupt keyboard input. 

tryScan = scan()
1: 4
2: 4
3: 4
4: 4
5: 5
6: 5
7: 6
8: 6
9: 6
10: 7
11: 7
12: 8
13: 8
14: 8
15: 15
Read 15 items

tryScan
 [1] 4 4 4 4 5 5 6 6 6 7 7 8 8 8 15

The function tryScan is a very useful command, with many options, and can be used for more than keyboard entry. For example, you can paste from your computer’s clipboard a column of numbers from your spreadsheet.

R code calculate central tendency

Once we have the vectormyX, calculate the mean by entering at the R prompt

mean(myX)

and you should get the answer of 6.466667

And of course, you don’t type in the R prompt >, right?

Or, for the better option, create two variables, one containing the list of observed numbers and the second that contains the frequency for each observed number in the series. You would then use the command for weighted mean.

myY = c(4,5,6,7,8,15)

myW = c(4/15, 1/15, 3/15, 2/15, 4/15, 1/15)

Note — you can check that the frequencies sum to 1 by using the sum command like so

sum(myW)

For the weighted mean, the command is

weighted.mean(myY,myW)

and, the answer returned is 6.466667, the same as before.

There’s an R package for that.

We list code examples for all of the statistics described on this page, but as you can imagine, there are packages that include these and other useful functions, or “wrapper functions” that make writing code easier in R. A good example is the DescTools package. One advantage of using a package like DescTools is that it cuts through the need to remember specific commands for basic data exploration. For example, just call the Desc() function to yield several measures of central tendency and dispersion (see Chapter 3.3). 

The R code and output follows, along with a plot (Fig 4). The mean and median are highlighted in orange.

Desc(myX)
──────────────────────────────────────────────────────────────────────────────── 
myX (numeric)

length      n   NAs unique   0s  mean meanCI'
    15     15     0      6    0  6.47    4.92
       100.0%  0.0%        0.0%          8.02

   .05    .10   .25 median  .75   .90     .95
  4.00   4.00  4.50   6.00 7.50  8.00   10.10

range      sd vcoef    mad  IQR  skew    kurt
11.00    2.80  0.43   2.97 3.00  1.72    2.94

  value freq   perc cumfreq cumperc
1     4    4  26.7%       4   26.7%
2     5    2  13.3%       6   40.0%
3     6    3  20.0%       9   60.0%
4     7    2  13.3%      11   73.3%
5     8    3  20.0%      14   93.3%
6    15    1   6.7%      15  100.0%

' 95%-CI (classic)

Figure 4. Histogram, box plot, and cumulative distribution function plot generated by default by Desc() call.

Lots of output! This is a good example of how to explore data — describe the middle (eg, mean and median), but also the variability about the middle (eg, range, standard deviation — sd, mean absolute deviation — mad, coefficient of variation — vcoef), and distribution shape (quantiles — 0.05, 0.25 etc, kurt, skew). The plots — histogram, box plot, and cumulative distribution plots — are explained in Chapter 4.

Use aggregate() to calculate summary statistics by group.

So far, all code presumes need to calculate descriptive statistics one vector at a time. In general, we need summary statistics for more than one group or strata. For example, we may ask, what are the average body temperature readings measured by IR device on different body areas from several subjects (question 8a, Homework 2A: Descriptive statistics in Mike’s Workbook for Biostatistics).

The aggregate() function in R simplifies calculating means by groups by providing a concise and efficient way to apply a summary function (like mean()) to a variable, grouped by one or more other variables. The format for aggregate is

 aggregate(x = variable_to_aggregate ~ grouping_variable, FUN = mean, data = your_dataframe)

Questio8a had two levels of body regions, “Forehead” and “Throat.” Assuming the variable is named bodyTemp and the grouping variable is called bodyGroup and stored in the attached data.frame question08a, then the following R code provides the requested summary statistics by groups.

aggregate(bodyTemp ~ bodyGroup, data = question08a, 
function(x) c(mean = mean(x)))

Additional statistics can be requested by adding one after the other following mean(x), eg, median = median(x).

The advantage of use of R Commander, it’s easy to get multiple statistics by groups by following the drop down menus, beginning with Statistics > Summaries > Table of statistics… (Fig 5).

Figure 5. Screenshot R Commander, summary statistics by group.

Questions

1. Find the help page in R for the median function. How does the function handle missing values?
2. For a simple data set like the following myY <- c(1,1,3,6) you should now be able to calculate, by hand, the
   • mean • median • mode
3. If the observations for a ratio scale variable are normally (symmetrically) distributed, which statistic of central tendency is best (eg, less sensitive to outlier values)?
4. In the names() command, what do you think the result will be if you replace max in the command with min?
5. If data are right skewed, what will be the order of the mean, median, and mode?
6. Body mass of Rhinella marina (formerly Bufo marinus, Fig 6),
   bufo <- c(71.3, 71.4, 74.1, 85.4, 85.4, 86.6, 97.4, 99.6, 107, 115.7, 135.7, 156.2)

Figure 6. Female Rhinella marina (formerly Bufo marinus), Chaminade University campus. Body length 23.5 cm. Click image to view full sized image.

7. Under what circumstances is it appropriate to transform raw data?
8. What does “standardize” or “normalize” imply as actions to take with data?

Quiz Chapter 3.2

Measures of Central Tendency

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Chapter 3 contents

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