Predicting Height from Metacarpal Bone Length

This study aims to estimate a model that would predict the height of an individual from the metacarpal bone length. The study utilizes a data set of nine individuals and employs a simple linear regression model to establish the relationship between height and metacarpal bone length. The results show a positive linear relationship between the two variables, with a unit increase in metacarpal bone length resulting in an increase in height by 1.70 cm.

Data Science
Student Name:
Instructor Name:
Course Number:
1 June 2019

Aim
The aim of this study is to estimate a model that would predict the Height of an individual from
Metacarpal Bone Length.
Introduction
Development – the essential process is estimated by estimating an individual's height, which itself is a
combination of the length of specific appendages and bones. Whenever anthropologists want to
analyze human skeletal remains, they look for important piece of information which is the living
stature. This study therefore seeks to come up with an appropriate model that would help predict the
height of an individual from the Metacarpal bone length.
Method
The aim of this study was to come up with a predictive model that would help predict the height of an
individual based on the metacarpal bone length (Long, 2009) . A simple linear regression model is
employed because there is only one explanatory (independent) variable that is being considered to
predict the dependent variable (Tofallis, 2009) . This technique is useful in determining the
relationship between two variables (dependent and independent variable).
The following regression model was to be estimated;
y=β0 + β1 x1 +ε
Where
y=dependent variable ( height ) , x1=independent variable ( length ) , β0 =intercept ( constant ) coefficient , β1=coeffic
ε ¿ error term
Data

For purposes of this study, the study utilized a data comprising of nine individuals. For a random
sample of nine humans, the length of the metacarpal bone (mm) and the height (cm) were measured.
Data set is given as follows:
Table 1: Data set
Metacarpal bone
length
(cm)
Height
(cm)
45 171
51 178
39 157
41 163
48 172
49 183
46 173
43 175
47 173
Source: Musgrave, J., and Harneja, N. (1978), ``The estimation of adult stature from metacarpal bone length'',
American
Journal of Physical Anthropology, 48, 113‐120.
Results
Exploratory data analysis (EDA)
In this section, we present the summary statistics for the two variables under investigation. The results
shows that the average metacarpal bone length is 45.44 cm with the median metacarpal bone length
being 46 cm. The highest and the lowest metacarpal length were 51 cm and 39 cm. Considering the
median and the mean, we can say that the distribution is close to normal distribution since the mean
and the median values are almost equal. However, this will be confirmed by the boxplot below. For
the height, the average height is given as 171.7 cm with the median height being 173.0 cm. The
minimum and the maximum height of the individuals included in the study is 157 cm and 183 cm
respectively. These values on height suggest that the distribution is close to normal distribution.

Boxplot
We plotted two boxplots for the metacarpal bone length and the
height of individual to try and check on the distribution of the two
variables. The first plot is the boxplot of the metacarpal bone length while the second plot is the
boxplot for the height of individual.
Considering figure 1 below, we can see that the distribution of the metacarpal bone length is not
normally distributed but is rather skewed to the left (negatively skewed). However, there are no
outliers in the dataset as can be seen from the boxplot.
Figure 1: Boxplot on length
Figure 2 below presents the boxplot for height. As can be seen, the plot shows that the distribution of
height is close to normal distribution, however, the dataset has some outliers.
Metacarpal.bone.length
Height
Min. :39.00
Min. :157.0
1st Qu.:43.00 1st
Qu.:171.0
Median :46.00
Median :173.0
Mean :45.44
Mean :171.7
3rd Qu.:48.00 3rd
Qu.:175.0
Max. :51.00 Max.
:183.0
Table 1: Summary Statistics

Figure 2: Box plot on height
Relationship between the two variables (length and height)
A scatter plot is presented below to try and establish the relationship that exists between height of an
individual and the metacarpal bone length. From the figure below, it can be seen that a positive linear
relationship exists between the two variables (Emerson, et al., 2013) .
Figure 3: Scatter plot of height versus metacarpal bone length
Interpretation of results

In this section, we present the interpretation of the results of the study. Table 2 below presents the
results of the regression analysis.
First, we consider the goodness of fit of the estimated model. The
overall model was found to be significant and appropriate to predict
the height of an individual using the metacarpal bone length [F(1, 7)
= 19.19, p = .000]. The value of R-squared was found to be 0.7327,
which implies that 73.27% of the variation in the dependent variable
(height) is explained by the metacarpal bone length (independent
variable) in the model.
Both the intercept and the metacarpal bone length were found to be
significant in the model (p < 0.05).
> model<-
lm(Height~Metacarpal.bo
ne.length)
> summary(model)
Call:
lm(formula = Height ~
Metacarpal.bone.length)
Residuals:
Min 1Q Median
3Q Max
-4.0102 -3.1091 -1.1128
0.3891 7.4880
Coefficients:
Estimate
Std. Error t value
(Intercept)
94.428 17.691 5.338
Metacarpal.bone.length
1.700 0.388 4.380
Pr(>|t|)
(Intercept)
0.00108 **
Metacarpal.bone.length
0.00323 **
---
Signif. codes:
0 ‘***’ 0.001 ‘**’ 0.01 ‘*’
0.05 ‘.’ 0.1 ‘ ’ 1
Residual standard error:
4.255 on 7 degrees of
freedom
Multiple R-squared:
0.7327, Adjusted R-
squared: 0.6945
F-statistic: 19.19 on 1
and 7 DF, p-value:
0.003234
Table 2: Regression results

The coefficient of the metacarpal bone length was found to be 1.70 which shows that a positive
relationship exists between the metacarpal bone length and the height of an individual. The coefficient
further shows that a unit increase in the metacarpal bone length is expected to result in an increase in
the height of an individual by 1.70 cm. Similarly, a unit decrease in the metacarpal bone length is
expected to result in a decrease in the height of an individual by 1.70 cm.
The intercept coefficient is given as 94.428. This value means that holding metacarpal bone length
constant (zero value for metacarpal bone length), we would expect the height of an individual to be
94.428 cm.
Based on the above findings, the final model for predicting the height of an individual is given as
follows;
y=94.428+1.70 x1
Where y=dependent variable ( height ) , x1=independent variable ( length )
Conclusion
This study aimed to estimate a model that would predict the Height of an individual from
Metacarpal Bone Length. A sample of nine individuals was utilized in the study. Results showed
that a positive linear relationship exists between metacarpal bone length and the height of an
individual. It was established that a unit increase in metacarpal bone length is likely to result in
an increase in the height of an individual by 1.70 cm.

References
Emerson, W. J., Green, A. W., Schoerke, B. & Crowley, J., 2013. The Generalized Pairs Plot.
Journal of Computational and Graphical Statistics, 22(1), p. 79–91.
Long, Y., 2009. Human age estimation by metric learning for regression problems. International
Conference on Computer Analysis of Images and Patterns, 5(2), p. 74–82.
Tofallis, C., 2009. Least Squares Percentage Regression. Journal of Modern Applied Statistical
Methods, 7(5), p. 526–534.

References
Emerson, W. J., Green, A. W., Schoerke, B. & Crowley, J., 2013. The Generalized Pairs Plot.
Journal of Computational and Graphical Statistics, 22(1), p. 79–91.
Long, Y., 2009. Human age estimation by metric learning for regression problems. International
Conference on Computer Analysis of Images and Patterns, 5(2), p. 74–82.
Musgrave, J. & Harneja, N., 1978. The estimation of adult stature from metacarpal bone length.
American Journal of Physical Anthropology, Volume 48, p. 113‐120.
Tofallis, C., 2009. Least Squares Percentage Regression. Journal of Modern Applied Statistical
Methods, 7(5), p. 526–534.

Appendix
R codes
data<-read.csv("C:\\Users\\310187796\\Documents\\data2.csv")
str(data)
attach(data)
summary(data)
boxplot(Metacarpal.bone.length, main="Boxplot on length",
cex.lab=1.0, cex.axis=1.0, cex.main=1.0, cex.sub=1.0)
boxplot(Height, main="Boxplot on height",
cex.lab=1.0, cex.axis=1.0, cex.main=1.0, cex.sub=1.0)
model<-lm(Height~Metacarpal.bone.length)
summary(model)
plot(Height, Metacarpal.bone.length, main="Scatterplot of height vs length",
xlab="Metacarpal bone length ", ylab="Height ", cex.lab=0.75,
cex.axis=0.75, cex.main=0.75, cex.sub=0.75, pch=19)