5  Types of Data

5.1 Setup: Packages Used Here

knitr::opts_chunk$set(comment = NA)

library(janitor)
library(gtsummary)
library(kableExtra)
library(tidyverse)

theme_set(theme_bw())

We’ll also use the describe() function from the psych package in what follows, but I won’t load the whole psych package here.

5.2 Data require structure and context

Descriptive statistics are concerned with the presentation, organization and summary of data, as suggested in Norman and Streiner (2014). This includes various methods of organizing and graphing data to get an idea of what those data can tell us.

As Vittinghoff et al. (2012) suggest, the nature of the measurement determines how best to describe it statistically, and the main distinction is between numerical and categorical variables. Even this is a little tricky - plenty of data can have values that look like numerical values, but are just numerals serving as labels.

As Bock, Velleman, and De Veaux (2004) point out, the truly critical notion, of course, is that data values, no matter what kind, are useless without their contexts. The Five W’s (Who, What [and in what units], When, Where, Why, and often How) are just as useful for establishing the context of data as they are in journalism. If you can’t answer Who and What, in particular, you don’t have any useful information.

In general, each row of a data frame corresponds to an individual (respondent, experimental unit, record, or observation) about whom some characteristics are gathered in columns (and these characteristics may be called variables, factors or data elements.) Every column / variable should have a name that indicates what it is measuring, and every row / observation should have a name that indicates who is being measured.

5.3 Reading in the “Complete Cases” Sample

Let’s begin by loading into the nh_500cc tibble the information from the nh_adult500cc.Rds file we created in Section 4.5. Notice that I am simplifying the name of the tibble, to save me some typing.

nh_500cc <- read_rds("data/nh_adult500cc.Rds")

One obvious hurdle we’ll avoid for the moment is what to do about missing data, since the nh_500cc data are specifically drawn from complete responses. Working with complete cases only can introduce bias to our estimates and visualizations, so it will be necessary in time to address what we should do when a complete-case analysis isn’t a good choice. We’ll return to this issue later.

5.4 Quantitative Variables

Variables recorded in numbers that we use as numbers are called quantitative. Familiar examples include incomes, heights, weights, ages, distances, times, and counts. All quantitative variables have measurement units, which tell you how the quantitative variable was measured. Without units (like miles per hour, angstroms, yen or degrees Celsius) the values of a quantitative variable have no meaning.

• It does little good to be told the price of something if you don’t know the currency being used.

• You might be surprised to see someone whose age is 72 listed in a database on childhood diseases until you find out that age is measured in months.

• Often just seeking the units can reveal a variable whose definition is challenging - just how do we measure “friendliness”, or “success,” for example.

• Quantitative variables may also be classified by whether they are continuous or can only take on a discrete set of values. Continuous data may take on any value, within a defined range. Suppose we are measuring height. While height is really continuous, our measuring stick usually only lets us measure with a certain degree of precision. If our measurements are only trustworthy to the nearest centimeter with the ruler we have, we might describe them as discrete measures. But we could always get a more precise ruler. The measurement divisions we make in moving from a continuous concept to a discrete measurement are usually fairly arbitrary. Another way to think of this, if you enjoy music, is that, as suggested in Norman and Streiner (2014), a piano is a discrete instrument, but a violin is a continuous one, enabling finer distinctions between notes than the piano is capable of making. Sometimes the distinction between continuous and discrete is important, but usually, it’s not.

5.5 Quantitative Variables in nh_500cc

Here’s a list of the variables contained in our nh_500cc tibble.

names(nh_500cc)

 [1] "ID"            "Sex"           "Age"           "Height"       
 [5] "Weight"        "Race"          "Education"     "BMI"          
 [9] "SBP"           "DBP"           "Pulse"         "PhysActive"   
[13] "Smoke100"      "SleepTrouble"  "MaritalStatus" "HealthGen"    

The nh_500cc data includes seven quantitative variables, including Age, Height, Weight, BMI, SBP, DBP and Pulse.

• We know these are quantitative variables because they have units:
  • Age in years, Height in centimeters, Weight in kilograms,
  • BMI in kg/m², the BP measurements in mm Hg, and Pulse in beats per minute.

Let’s summarize them with the describe() function from the psych package.

nh_500cc |>
  select(Age, Height, Weight, BMI, SBP, DBP, Pulse) |>
  psych::describe() |>
  kbl() |>
  kable_styling()

	vars	n	mean	sd	median	trimmed	mad	min	max	range	skew	kurtosis	se
Age	1	500	41.6060	12.803853	42.00	41.48000	16.30860	21.0	64.0	43.0	0.0469923	-1.2377866	0.5726057
Height	2	500	169.3726	9.935372	169.05	169.19025	10.52646	144.8	200.4	55.6	0.1618323	-0.2539869	0.4443233
Weight	3	500	82.7094	20.884652	80.00	81.05650	19.94097	41.9	184.5	142.6	0.9835819	1.7024261	0.9339900
BMI	4	500	28.7574	6.567995	27.70	28.11625	6.07866	17.0	63.3	46.3	1.1522475	2.2838638	0.2937297
SBP	5	500	119.3320	15.049344	119.00	118.41250	13.34340	84.0	221.0	137.0	1.1872427	4.7556753	0.6730271
DBP	6	500	72.1580	11.245709	72.00	72.24500	8.89560	0.0	110.0	110.0	-0.5339975	4.5794674	0.5029234
Pulse	7	500	74.1640	11.500505	74.00	73.76500	11.86080	48.0	114.0	66.0	0.3414617	-0.0027802	0.5143182

As an alternative, we could use tbl_summary() from the gtsummary package, as well. The approach below works nicely for producing a mean, standard deviation, and five-number summary for each of the variables we’ve identified as quantitative.

• Quantitative variables lend themselves to many of the summaries we will discuss, like means, quantiles, and our various measures of spread, like the standard deviation or inter-quartile range. They also have at least a chance to follow the Normal distribution.

nh_500cc |>
  select(Age, Height, Weight, BMI, SBP, DBP, Pulse) |>
  tbl_summary( statistic = list(all_continuous() ~ "{mean} ({sd}): 
                     [ {min}, {p25}, {median}, {p75}, {max} ]")) 

Characteristic	N = 5001
Age	42 (13): [ 21, 30, 42, 52, 64 ]
Height	169 (10): [ 145, 162, 169, 176, 200 ]
Weight	83 (21): [ 42, 68, 80, 94, 185 ]
BMI	29 (7): [ 17, 24, 28, 32, 63 ]
SBP	119 (15): [ 84, 110, 119, 127, 221 ]
DBP	72 (11): [ 0, 66, 72, 79, 110 ]
Pulse	74 (12): [ 48, 66, 74, 82, 114 ]
1 Mean (SD): [ Minimum, 25%, Median, 75%, Maximum ]

5.5.1 A look at BMI (Body-Mass Index)

The definition of BMI (body-mass index) for adult subjects (which is expressed in units of kg/m²) is:

\[ \mbox{Body Mass Index} = \frac{\mbox{weight in kg}}{(\mbox{height in meters})^2} = 703 \times \frac{\mbox{weight in pounds}}{(\mbox{height in inches})^2} \]

[BMI is essentially] … a measure of a person’s thinness or thickness… BMI was designed for use as a simple means of classifying average sedentary (physically inactive) populations, with an average body composition. For these individuals, the current value recommendations are as follow: a BMI from 18.5 up to 25 may indicate optimal weight, a BMI lower than 18.5 suggests the person is underweight, a number from 25 up to 30 may indicate the person is overweight, and a number from 30 upwards suggests the person is obese.

Wikipedia, https://en.wikipedia.org/wiki/Body_mass_index

5.5.2 Types of Quantitative Variables

Depending on the context, we would likely treat most of these quantitative variables as discrete given that are measurements are fairly crude (this is certainly true for Age, measured in years) although BMI is probably continuous in most settings, even though it is a function of two other measures (Height and Weight) which are rounded off to integer numbers of centimeters and kilograms, respectively.

It is also possible to separate out quantitative variables into ratio variables or interval variables.

• An interval variable has equal distances between values, but the zero point is arbitrary.
• A ratio variable has equal intervals between values, and a meaningful zero point.

For example, weight is an example of a ratio variable, while IQ is an example of an interval variable. We all know what zero weight is. An intelligence score like IQ is a different matter. We say that the average IQ is 100, but that’s only by convention. We could just as easily have decided to add 400 to every IQ value and make the average 500 instead. Because IQ’s intervals are equal, the difference between and IQ of 70 and an IQ of 80 is the same as the difference between 120 and 130. However, an IQ of 100 is not twice as high as an IQ of 50. The point is that if the zero point is artificial and movable, then the differences between numbers are meaningful but the ratios between them are not.

On the other hand, most lab test values are ratio variables, as are physical characteristics like height and weight. Each of the quantitative variables in our nh_500cc data can be thought of as a ratio variable.A person who weighs 100 kg is twice as heavy as one who weighs 50 kg; even when we convert kg to pounds, this is still true. For the most part, we can treat and analyze interval or ratio variables the same way.

5.6 Qualitative (Categorical) Variables

Qualitative or categorical variables consist of names of categories. These names may be numerical, but the numbers (or names) are simply codes to identify the groups or categories into which the individuals are divided. Categorical variables with two categories, like yes or no, up or down, or, more generally, 1 and 0, are called binary variables. Those with more than two-categories are sometimes called multi-categorical variables.

In the nh_500cc data, we have eight categorical variables, four binary and four with multiple categories.

nh_500cc |>
  select(Sex, PhysActive, Smoke100, SleepTrouble,
         Race, Education, MaritalStatus, HealthGen) |>
  summary()

     Sex      PhysActive Smoke100  SleepTrouble       Race    
 female:236   No :216    No :291   No :380      Asian   : 51  
 male  :264   Yes:284    Yes:209   Yes:120      Black   : 81  
                                                Hispanic: 37  
                                                Mexican : 48  
                                                White   :262  
                                                Other   : 21  
          Education        MaritalStatus     HealthGen  
 8th Grade     : 26   Divorced    : 47   Excellent: 52  
 9 - 11th Grade: 59   LivePartner : 46   Vgood    :167  
 High School   : 89   Married     :256   Good     :204  
 Some College  :153   NeverMarried:125   Fair     : 65  
 College Grad  :173   Separated   : 17   Poor     : 12  
                      Widowed     :  9                  

5.6.1 Nominal vs. Ordinal Categories

• When the categories included in a variable are merely names, and come in no particular order, we sometimes call them nominal variables. The most important summary of such a variable is usually a table of frequencies, and the mode becomes an important single summary, while the mean and median are essentially useless.

In the nh_500cc data, Race is a nominal variable with multiple unordered categories. So is MaritalStatus.

• The alternative categorical variable (where order matters) is called ordinal, and includes variables that are sometimes thought of as falling right in between quantitative and qualitative variables.

Examples of ordinal multi-categorical variables in the nh_500cc data include the Education and HealthGen variables.

• Answers to questions like “How is your overall physical health?” with available responses Excellent, Very Good, Good, Fair or Poor, which are often coded as 1-5, certainly provide a perceived order, but a group of people with average health status 4 (Very Good) is not necessarily twice as healthy as a group with average health status of 2 (Fair).

• Sometimes we treat the values from ordinal variables as sufficiently scaled to permit us to use quantitative approaches like means, quantiles, and standard deviations to summarize and model the results, and at other times, we’ll treat ordinal variables as if they were nominal, with tables and percentages our primary tools.

• Note that all binary variables may be treated as either ordinal, or nominal.

Binary variables in the nh_500cc data include Sex, PhysActive, Smoke100, SleepTrouble. Each can be thought of as either ordinal or nominal.

Lots of variables may be treated as either quantitative or qualitative, depending on how we use them. For instance, we usually think of age as a quantitative variable, but if we simply use age to make the distinction between “child” and “adult” then we are using it to describe categorical information. Just because your variable’s values are numbers, don’t assume that the information provided is quantitative.

5.7 Tabulating Binary Variables

Note how the tbl_summary() approach works with binary variables, and in particular with variables coded Yes and No, like PhysActive, Smoke100 and SleepTrouble.

nh_500cc |>
  select(Sex, PhysActive, Smoke100, SleepTrouble) |>
  tbl_summary()

Characteristic	N = 5001
Sex
female	236 (47%)
male	264 (53%)
PhysActive	284 (57%)
Smoke100	209 (42%)
SleepTrouble	120 (24%)
1 n (%)

We can also summarize any particular variable with the tabyl() function from the janitor package.

nh_500cc |>
  tabyl(Sex)

    Sex   n percent
 female 236   0.472
   male 264   0.528

Or, we can make a basic cross-tabulation of two binary variables, like this:

nh_500cc |>
  tabyl(PhysActive, Smoke100) |>
  adorn_title()

            Smoke100    
 PhysActive       No Yes
         No      111 105
        Yes      180 104

5.8 Tabulating Multi-Categorical Variables

nh_500cc |>
  select(Race, Education, MaritalStatus, HealthGen) |>
  tbl_summary()

Characteristic	N = 5001
Race
Asian	51 (10%)
Black	81 (16%)
Hispanic	37 (7.4%)
Mexican	48 (9.6%)
White	262 (52%)
Other	21 (4.2%)
Education
8th Grade	26 (5.2%)
9 - 11th Grade	59 (12%)
High School	89 (18%)
Some College	153 (31%)
College Grad	173 (35%)
MaritalStatus
Divorced	47 (9.4%)
LivePartner	46 (9.2%)
Married	256 (51%)
NeverMarried	125 (25%)
Separated	17 (3.4%)
Widowed	9 (1.8%)
HealthGen
Excellent	52 (10%)
Vgood	167 (33%)
Good	204 (41%)
Fair	65 (13%)
Poor	12 (2.4%)
1 n (%)

We can also use tabyl() to look at combinations of multi-categorical variables, whether they are ordinal or nominal.

nh_500cc |>
  tabyl(Education, HealthGen) |>
  adorn_totals(where = c("row", "col")) |>
  adorn_title() |>
  kbl(align = 'lrrrrrc') |>
  kable_styling(full_width = FALSE)

	HealthGen
Education	Excellent	Vgood	Good	Fair	Poor	Total
8th Grade	2	3	9	9	3	26
9 - 11th Grade	4	12	28	12	3	59
High School	9	26	36	18	0	89
Some College	11	46	75	17	4	153
College Grad	26	80	56	9	2	173
Total	52	167	204	65	12	500

5.9 Coming Up

Next, we’ll look at several additional approaches to building tabular and graphical summaries for these data, beyond the ideas provided here.

Bock, David E., Paul F. Velleman, and Richard D. De Veaux. 2004. Stats: Modelling the World. Boston MA: Pearson Addison-Wesley.

Norman, Geoffrey R., and David L. Streiner. 2014. Biostatistics: The Bare Essentials. Fourth. People’s Medical Publishing House.

Vittinghoff, Eric, David V. Glidden, Stephen C. Shiboski, and Charles E. McCulloch. 2012. Regression Methods in Biostatistics: Linear, Logistic, Survival, and Repeated Measures Models. Second. Springer-Verlag, Inc. http://www.biostat.ucsf.edu/vgsm/.