3.1 – Data types

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

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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.

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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.

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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.

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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")

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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)

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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

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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)

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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.

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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

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

4.3 – Box plot

Introduction

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

Purpose and design criteria

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

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

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

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

R Code

Command line

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

boxplot(OliveMoment~Treatment)

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

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

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

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

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

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

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

Rcmdr: Graph → Boxplot…

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

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

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

Figure 4. Select the group variable

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

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

And here is the resulting box plot (Fig 6)

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

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

Apply Tidyverse-view to enhance look of boxplot graphic

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

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

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

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

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

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

Figure 10. Default box plot from KMggplot.

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

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

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

Conclusions

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

Questions

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

Quiz Chapter 4.3

Box plots

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Chapter 4 contents

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