Questioning and Cleaning the Body-Fat Data
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Last compiled: February 09, 2023 at 12:36:34 PM
In the article Fitting Percentage of Body Fat to Simple Body Measurements, Johnson (1996) uses the data at http://jse.amstat.org/datasets/fat.dat.txt provided to him by Dr. A. Garth Fischer in a personal communication on October 5, 1994, as a multiple linear regression activity with his students. A subset of the variables at http://jse.amstat.org/datasets/fat.dat.txt is available in the R package mfp by Ambler and Benner (2022) and the data set is used frequently in the text Statistical Regression and Classification by Matloff (2017).
The purpose of this activity is to have the reader critically question, evaluate, and clean the original data provided at http://jse.amstat.org/datasets/fat.dat.txt. The guided activities will reinforce reading data into R using the fread() function from the data.table package written by Dowle and Srinivasan (2022), creating graphs with both packages ggplot2 and plotly written by Wickham, Chang, et al. (2022) and Sievert et al. (2022), respectively, and creating new variables with functions such as mutate() from the dplyr package written by Wickham, François, et al. (2022).
Type complete sentences to answer all questions inside the answer tags provided in the R Markdown document. Round all numeric answers you report inside the answer tags to four decimal places. Use inline R code to report numeric answers inside the answer tags (i.e. do not hard code your numeric answers).
The article by Johnson (1996) defines body-fat determined with the brozek and siri methods as well as fat free weight using Equations (1), (2), and (3), respectively.
\[\begin{equation} \text{bodyfatBrozek} = \frac{457}{\text{density}} - 414.2 \tag{1} \end{equation}\]
\[\begin{equation} \text{bodyfatSiri} = \frac{495}{\text{density}} - 450 \tag{2} \end{equation}\]
\[\begin{equation}
\text{FatFreeWeight} = \left(1 -\frac{\text{brozek}}{100}\times \text{weight_lbs}\right)
\tag{3}
\end{equation}\]
Body Mass Index (BMI) is defined in Equation (4).
\[\begin{equation} \text{BMI} = \frac{\text{kg}}{\text{m}^2} \tag{4} \end{equation}\]
Please use the following conversion factors with this project: 0.453592 kilos per pound and 2.54 centimeters per inch.
For example, a male who weighs 185 pounds and is 71 inches tall has a weight of 83.91 kg and a height of 1.8034 m.
\[\begin{equation} 185 \text{ pounds} \times \frac{0.453592 \text{ kg}}{\text{pounds}} = 83.9145 \text{ kg} \end{equation}\]
\[\begin{equation} 71 \text{ inches} \times \frac{2.54 \text{ cm}}{\text{inches}}\times \frac{1 \text {m}}{100 \text{ cm}} = 1.8034 \text{ m} \end{equation}\]
To read tabular data from the internet, one might use the read.table() function or the fread() function from the data.table package. The general structure to read data from the internet is to provide the URL in the file or input arguments of the read.table() or fread() functions, respectively.
# Example structure
DF1 <- read.table(file = "http://some.url.com")
# load data.table package
library(data.table)
DF2 <- fread(input = "http://some.url.com")- Read the original data from http://jse.amstat.org/datasets/fat.dat.txt into your current R session. Specifically, start by using the
fread()function from thedata.tablepackage to read the data from http://jse.amstat.org/datasets/fat.dat.txt into an object namedbodyfat. The data dictionary for http://jse.amstat.org/datasets/fat.dat.txt is available at http://jse.amstat.org/datasets/fat.txt.
Pass the following vector of names to thecol.namesargument offread():c("case", "brozek", "siri", "density", "age", "weight_lbs", "height_in", "bmi", "fat_free_weight","neck_cm", "chest_cm", "abdomen_cm", "hip_cm", "thigh_cm", "knee_cm", "ankle_cm", "biceps_cm","forearm_cm", "wrist_cm").
# Type your code and comments inside the code chunk
# Obtaining the original data- Use the
read.table()function to read the data from http://jse.amstat.org/datasets/fat.dat.txt into an object namedbodyfat2. Pass the same vector created in Problem 1 to thecol.namesargument ofread.table().
# Type your code and comments inside the code chunk
# Obtaining the original dataThe basic structure of a function in R is
function_name <- function(argument1, argument2, ...){
expression
}The function curve() draws a curve corresponding to a function of the interval [from, to]. For example, to draw the function \(y = x^2\) from -3 to 3 use the following code to obtain Figure 1.
quad <- function(x){x^2}
curve(quad, from = -3, to = 3)
Figure 1: Using the function curve() to draw a quadratic function
To superimpose a function on an open graphics device with curve(), use the argument add = TRUE. For example, to graph \(y = x^2\) with a blue line and \(y = 4 + \sin(8x)\) with a red line from \(-\pi\) to \(\pi\), use the following code:
quad <- function(x){x^2}
curve(quad, from = -pi, to = pi, col = "blue", ylab = "f(x)")
nsin <- function(x){4 + sin(8*x)}
curve(nsin, add = TRUE, col = "red")
- Graph the relationship between
brozekanddensitygiven in Equation (1) with a blue line as well as the relationship betweensirianddensitygiven in Equation (2) with a red line. First, define functionsfbandfsfor Equations (1) and (2), respectively. Use the functioncurve()to draw Equations (1) and (2) on the same graphics device for density values between 1.0 \(\text{gm/cm}^3\) and 1.10 \(\text{gm/cm}^3\).
# Type your code and comments inside the code chunk
# Functions fb and fs
fb <- function(density){(457/density) - 414.2}
fs <- function(density){(495/density) - 450}Code to construct the requested graph using ggplot2 code is given below and the result is visible in Figure 2.
library(ggplot2)
p <- ggplot(data = data.frame(x = c(1, 1.1), y = c(0, 50)), aes(x = x, y = y))
p + stat_function(fun = fb, color = "blue") +
stat_function(fun = fs, color = "red") +
labs(x = expression(paste("density in ", gm/cm^3)), y = "bodyfat") +
theme_bw()
Figure 2: Realtionship between bodyfat and density according to Brozek and Siri definitions
Type your answer here.
Since the relationship between body fat and density for both Equations (1) and (2) is linear, one way to check for unusual observations is to plot the actual values for brozek versus density or siri versus density using the data stored in bodyfat or bodyfat2. Points that do not fall along a straight line should be scrutinized for possible data entry errors or other possible explanations.
The plotly package allows the user to create interactive graphs or to convert ggplot2 graphs to plotly graphs using the ggplotly() function. Given a data frame say DF with variables X and Y, the following template code will create an interactive plotly scatterplot.
library(plotly)
library(ggplot2)
# Create ggplot scatterplot named p
p <- ggplot(data = DF, aes(x = X, y = Y)) +
geom_point()
# Convert ggplot scatterplot to plotly with ggplotly()
p2 <- ggplotly(p)
# Display graph
p2Consider an interactive scatterplot showing the distance required to stop for vehicles traveling at different speeds using the cars data frame.
library(plotly)
library(ggplot2)
# Create ggplot scatterplot named p
p <- ggplot(data = cars, aes(x = speed, y = dist)) +
geom_point()
# Convert ggplot scatterplot to plotly with ggplotly()
p2 <- ggplotly(p)
# Display graph
p2- Create interactive
plotlyscatterplots ofsiriversusdensitywithcasemapped tocolor,weight_lbsversusheight_inwithcasemapped tocolor, andankle_cmversusweight_lbswithcasemapped tocolorto help identify potential outliers. Identify the case numbers for the outliers from your three plots.
# Type your code and comments inside the code chunk
# Creating interactive scatterplot of siri versus density
# Change code below to your plot
ggplot(data = data.frame(x = 1, y = 1), aes(x = x, y = y)) +
geom_point()
Figure 3: Plot of siri versus density
# Type your code and comments inside the code chunk
# Creating interactive scatterplot of weight_lbs versus height_in
# Change code below to your plot
ggplot(data = data.frame(x = 1, y = 1), aes(x = x, y = y)) +
geom_point()
Figure 4: Plot of weight_lbs versus height_in
# Type your code and comments inside the code chunk
# Creating interactive scatterplot of ankle_cm versus weight_lbs
# Change code below to your plot
ggplot(data = data.frame(x = 1, y = 1), aes(x = x, y = y)) +
geom_point()
Figure 5: Interactive scatterplot of ankle_cm versus weight_lbs
Type your complete sentence answer here using inline R code and delete this comment.
- Create a subset of the
bodyfatdata frame namedBFconsisting only of the cases identified as outliers in Problem 5. Use equation (2) to create a new variable inBFnameddensity_C(computed density) based on the reportedsirivalues. Use equation (4) to reverse engineer the computed height in inches based on the values inweight_lbsandbmiusing the conversion factors given at the start of the lab. Store the computed heights in inches in a variable namedheight_in_C. Use the verbmutatefrom the dplyr package to create bothdensity_Candheight_in_C. Show the values of the selected outliers for columnscase,density_C,height_in,height_in_C, andankle_cm. What do you notice about thedensityanddensity_Cvalues inBFfor the scatterplot you created in Figure 3? What do you notice about theheight_invalues inBFfor the scatterplot you created in Figure 4? What do you notice about theankle_cmvalues inBFfor the scatterplot you created in Figure 5?
# Type your code and comments inside the code chunk
# Isolating points of interestType your complete sentence answer here using inline R code and delete this comment.
- Change the reported
densityvalues ofbodyfatbased on your answers from Problem 6. Change theheight_invalue for case 42 to the value you reverse engineered in Problem 6 rounding to the nearest half inch.
# Type your code and comments inside the code chunk
# Updating computed bodyfat values and bmi measurements
# Replacing identified typos of density
# Replacing identified typos in height_in
# Replacing identified typos in ankle_cm- Create variables named
siri_C,brozek_C, andbmi_C, that compute body-fat values rounded to one decimal place according to equations (2), (1), and (4), respectively. Replace the values insiriandbrozekwith the computed values insiri_Candbrozek_Cfor case 182.
# Type your code and comments inside the code chunk- Consider differences between
brozekandbrozek_Cas well as differences betweensiriandsiri_Cgreater than 0.11 to be some type of data entry problem. Use the verbsfilter()andselect()to show the values forcase,siri,siri_C,brozek,brozek_C, anddensitythat are considered data entry problems. If both the computed value for siri and brozek differ from the reported values of siri and brozek, it is likely a data entry problem with the density value. If one of either siri or brozek agrees with its computed value, the one that does not agree with the computed value is likely a data entry problem. Which cases seem to have data entry problems and for what reasons?
# Type your code and comments inside the code chunk
# Identifying bodyfat typos for brozek and siriType your complete sentence answer here using inline R code and delete this comment.
- Fix any data entry errors you identified in Problem 9 in the
bodyfatdata frame. Make sure to update thesiri_C,brozek_C, andbmi_Cvariables inbodyfatusing your corrected values from Problem 9. Write code to verify that there are no differences betweenbrozekandbrozek_Cor any differences betweensiriandsiri_Cgreater than 0.11.
# Type your code and comments inside the code chunk
# Replacing bodyfat typos for brozek and siri
# Updating siri_C, brozek_C, and bmi_C
# Checking for typos according to given criteria- Create interactive plotly scatterplots of
siri_Cversusdensitywithcasemapped tocolorandbrozek_Cversusdensitywithcasemapped tocolorusing the modifiedbodyfatdata frame. Superimpose equations (2) and (1) over their corresponding scatterplots. Comment on the scatterplots.
# interactive plotly scatterplot siri_C vs. density
# Change code below to your plot
ggplot(data = data.frame(x = 1, y = 1), aes(x = x, y = y)) +
geom_point()
# interactive plotly scatterplot brozek_C vs. density
# Change code below to your plot
ggplot(data = data.frame(x = 1, y = 1), aes(x = x, y = y)) +
geom_point()
Type your complete sentence answer here using inline R code and delete this comment.
- How many of the values do you think are potentially rounding errors? Explain your reasoning and show the code you used to identify the errors.
# Type your code and comments inside the code chunk
# Number of rounding discrepancies for siri
# Number of rounding discrepancies for brozek
# Number of rounding discrepancies for bmiType your complete sentence answer here using inline R code and delete this comment.
- What additional variables might you explore to check the quality of the reported data?
Type your complete sentence answer here using inline R code and delete this comment.
# Code for graph suggested in 13
# Change code below to your plot
ggplot(data = data.frame(x = 1, y = 1), aes(x = x, y = y)) +
geom_point()
- Create an object named
bodyfatCleanthat excludes the variablesbrozek,siri, andbmifrom thebodyfatdata frame. Use the functionwrite.csv()orwrite_csv()to store thebodyfatCleanobject as a CSV file.
# writing bodyfat to bodyfatClean.csvThis material is released under an Attribution-NonCommercial-ShareAlike 3.0 United States license. Original author: Alan T. Arnholt