Data

The data used for this project is displayed in Table 1 shown in the Appendix section at the end of this report, which includes measurements for Concho Water Snakes. This dataset contains one categorical variable, X1, which represents genders male and female, and two quantitative variables: X2 and X3. The variable X2 represents tail length in millimeters and X3 represents snout to vent length in millimeters. The dataset includes a total of 66 observations for both genders. Out of the 66 observations, there are \(n_1=37\) observations for both tail length and snout to vent length for female Concho Water Snakes and \(n_2=29\) observations for male Concho Water Snakes. The observations in this dataset can be denoted as \(X_{\ell j}\) where \(\ell=1,\cdots,g\) and \(j=1,\cdots,n_{\ell}\) such that

\[Female: X_{11}, X_{12}, \cdots, X_{1n_1}\] \[Male: X_{21}, X_{22}, \cdots, X_{2n_2} \] \[ \begin{aligned} p&\text{: total number of features in a single observation = 2}\\ g&\text{: total number of populations = 2}\\ n_\ell&\text{: number of observation in the $\ell$th population} \end{aligned} \]

Methods

I. Summary Statistics

The following chart displays the general summary statistics of the data being explored for this project. All two quantitative variables, \(X2\) ( tail length ) and \(X3\) ( snout to vent length ), contain integer values and \(X1\) (gender) is a categorical variable with two factors. In this dataset, there are a total 37 observations for female and 29 observations for Male Concho Water Snakes. Therefore, the sample sizes are not equal across two genders. In the graph column of this chart, the bar graph for each variable shows the type of distributions each variable have. Both X2 and X3 are approximately normally distributed. The chart also shows that the data does not contain any missing values for all three features. The second column in the chart shows the computed mean, standard deviation, minimum, median, maximum, and interquartile range (IQR), and coefficient of variation (CV) for each quantitative variable. The number of distinct values in each quantitative variable are also included, showing that there are slightly more distinct values in X3 than X2. The upper and lower quartiles are also shown in Table 2 below. Since X2 and X3 are two different measures, it is not necessary to compare them. However, the summary statistics of these two variables can be compared across two genders. These two variables can also be used to classify Female and Male Concho Water Snakes.

Data Frame Summary

data

Dimensions: 66 x 3
Duplicates: 0

Variable	Stats / Values	Freqs (% of Valid)	Graph	Missing
X1 [factor]	1. Female 2. Male	37 ( 56.1% ) 29 ( 43.9% )		0 (0.0%)
X2 [integer]	Mean (sd) : 167.6 (21.1) min ≤ med ≤ max: 115 ≤ 170 ≤ 211 IQR (CV) : 23.5 (0.1)	41 distinct values		0 (0.0%)
X3 [integer]	Mean (sd) : 517.3 (68.6) min ≤ med ≤ max: 361 ≤ 498 ≤ 683 IQR (CV) : 97.2 (0.1)	58 distinct values		0 (0.0%)

Generated by summarytools 1.0.1 (R version 4.2.0)
2022-08-25

The summary statistic for each gender category is displayed in Table 3 below. The mean column shows that the average values for each variable X2 and X3 between two genders slightly differ. Female Concho Water Snakes has higher average tail length (X2) compare to the male Concho Water Snakes. Male Concho Water Snakes has higher snout to vent length (X3) mean compare to the Female Concho Water Snakes. This observation can be helpful in answering the question of whether Female and Male Concho Water Snakes can be classified using these two measurements. Based on the standard deviation column included for each gender, it appears that X2 and X3 have the higher standard deviation in Male than Female so these variables are more dispersed relative to their mean in Male. Additionally, the standard deviation in X2 for Male and Female are approximately equivalent with a very small difference.

The variance-covariance matrix for Female Concho Water Snakes is displayed in Table 4 below. The variance of X2 (tail length) in this population is 413.0030 while the variance of X3 (snout to vent length) is 872.6877 The covariance between X2 and X3 is 2417.9805, which is positive so this suggests that the two variables tend to increase or decrease in tandem.

The variance-covariance matrix for Male Concho Water Snakes is displayed in Table 5 below. The variance of X2 (tail length) in this population is 420.1675 while the variance of X3 (snout to vent length) is 5355.741. The covariance between X2 and X3 is 1401.844, which is positive so this suggests that the two variables also increase or decrease in tandem in this population, which is similar to the female population. The variance-covariance values for X2 and X3 in Male appears to be generally slightly higher than the values in the variance-covariance matrix for Female above.

II. Data Visualizations

Based on the pairwise plot below (Figure 1), the correlations between X2 and X3 for two genders are almost equivalent. There appears to be an overall significantly high and positive correlation between X2 and X3 across two populations (genders) and within each population. In this figure, the density plots also show that the two variables are approximately normally distributed in two populations and do not significantly deviate from the normality assumption. The exception is X2 in Female and X3 in Male, which appears to have a somewhat bimodal distribution.

Figure 1: Pairwise scatter plot by gender

The side-by-side boxplots of X2 and X3 for two genders are also attached below (Figure 2). Based on this plot, there are two outliers that deviate from other points, 127 mm and 133 mm tail lengths in Female. The plot also further supports that previous finding that the mean of X2 are higher in Female than in Male. Meanwhile, Male has significantly higher mean in X3 than Female Concho Water Snakes. However, this finding can be further investigated using other quantitative methods to verify if X2 and X3 are indeed different across two genders.

Based on the 95% confidence ellipse for Male and Female Concho Water Snakes shown below, there are some points (\(x_2\), \(x_3\)) in Female population that is included in the Male cluster. Therefore, we can infer that there might be some misclassification present.

III. Minimum ECM Rule

Assuming multivariate normal distributions with common variance covariance matrix, equal prior probabilities (\(p_1=p_2=0.5\)), and equal costs of misclassification, the Minimum Expected Cost of Misclassification (ECM) Rule for two genders of Concho Water Snakes using the outputs below is defined as follows. This result is also obtained using the methods specified in Part III of the Material and Methods section.

Allocate \((X_1, X_2)\) to \(\pi_1\) if: \(0.3564042X_2 - 0.1238378X_3 \ge -5.07174\)

Allocate \((X_1, X_2)\) to \(\pi_2\) if: \(0.3564042X_2- 0.1238378X_3 < -5.07174\)

##          [,1]
## X2  0.3564042
## X3 -0.1238378

##          [,1]
## [1,] -5.07174

IV. Confusion Matrix & APER

Confusion Matrix

The confusion matrix of the data is

where

\(34\) is the number of \(\pi_1\) items correctly classified

\(27\) is the number of \(\pi_2\) items correctly classified

\(3\) is the number of \(\pi_1\) items correctly misclassified

\(2\) is the number of \(\pi_2\) items correctly misclassified

##         p
##          Female Male
##   Female     34    3
##   Male        2   27

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction Female Male
##     Female     34    2
##     Male        3   27
##                                          
##                Accuracy : 0.9242         
##                  95% CI : (0.832, 0.9749)
##     No Information Rate : 0.5606         
##     P-Value [Acc > NIR] : 7.579e-11      
##                                          
##                   Kappa : 0.8468         
##                                          
##  Mcnemar's Test P-Value : 1              
##                                          
##             Sensitivity : 0.9189         
##             Specificity : 0.9310         
##          Pos Pred Value : 0.9444         
##          Neg Pred Value : 0.9000         
##              Prevalence : 0.5606         
##          Detection Rate : 0.5152         
##    Detection Prevalence : 0.5455         
##       Balanced Accuracy : 0.9250         
##                                          
##        'Positive' Class : Female         
## 

Apparent Error Rate (APER)

The Apparent Error Rate (APER), which is calculated using the formula provided in the Methods section is displayed below.

\[APER = \frac{3+2}{66} = 0.07575758\]

## [1] 0.07575758

V. Confusion Matrix (Holdout Procedure)

The confusion matrix obtained using the holdout procedure appears to be equivalent to the confusion matrix obtained above in the previous section. Figure 5 below, which shows the partition plot, also shows that there are three misclassified observations from the Female population and two misclassified observations from the Male population, which verifies the misclassifications shown in Figure 4 above. The estimated expected actual error rate (AER) is also 0.07575758.

##         
##          Female Male
##   Female     34    3
##   Male        2   27

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction Female Male
##     Female     34    2
##     Male        3   27
##                                          
##                Accuracy : 0.9242         
##                  95% CI : (0.832, 0.9749)
##     No Information Rate : 0.5606         
##     P-Value [Acc > NIR] : 7.579e-11      
##                                          
##                   Kappa : 0.8468         
##                                          
##  Mcnemar's Test P-Value : 1              
##                                          
##             Sensitivity : 0.9189         
##             Specificity : 0.9310         
##          Pos Pred Value : 0.9444         
##          Neg Pred Value : 0.9000         
##              Prevalence : 0.5606         
##          Detection Rate : 0.5152         
##    Detection Prevalence : 0.5455         
##       Balanced Accuracy : 0.9250         
##                                          
##        'Positive' Class : Female         
## 

## Actual Error Rate (AER): 0.07575758

Figure 5: Partition Plot