Compute the misclassification rate accuracy rate, sensitivity, precision, and specificity for the following confusion matrix (Round your final answers to 2 decimal places:) Actual Class Class Class Predicted Class 142 Predicted Class 101 240 Misclassification rate Accuracy rate Sensitivity Precision Specificity
The Correct Answer and Explanation is:
To compute the misclassification rate, accuracy rate, sensitivity, precision, and specificity, let’s first define the confusion matrix for a binary classification problem:Confusion Matrix:\text{Confusion Matrix:}Confusion Matrix:Actual \PredictedClass 1Class 2Class 1142101Class 22400\begin{array}{|c|c|c|} \hline \text{Actual \textbackslash Predicted} & \text{Class 1} & \text{Class 2} \\ \hline \text{Class 1} & 142 & 101 \\ \text{Class 2} & 240 & 0 \\ \hline \end{array}Actual \PredictedClass 1Class 2Class 1142240Class 21010
From the matrix, we can extract the following values:
- True Positives (TP): 142 (Class 1 predicted as Class 1)
- False Positives (FP): 101 (Class 2 predicted as Class 1)
- True Negatives (TN): 0 (Class 2 predicted as Class 2)
- False Negatives (FN): 240 (Class 1 predicted as Class 2)
Now, we can compute the following metrics:
1. Misclassification Rate:
This is the fraction of incorrectly predicted instances (either FP or FN) to the total number of instances.Misclassification Rate=FP+FNTP+TN+FP+FN\text{Misclassification Rate} = \frac{FP + FN}{TP + TN + FP + FN}Misclassification Rate=TP+TN+FP+FNFP+FN
Substituting the values:Misclassification Rate=101+240142+0+101+240=341483≈0.71\text{Misclassification Rate} = \frac{101 + 240}{142 + 0 + 101 + 240} = \frac{341}{483} \approx 0.71Misclassification Rate=142+0+101+240101+240=483341≈0.71
So, the misclassification rate is 0.71 or 71%.
2. Accuracy Rate:
Accuracy is the fraction of correctly predicted instances (either TP or TN) to the total number of instances.Accuracy Rate=TP+TNTP+TN+FP+FN\text{Accuracy Rate} = \frac{TP + TN}{TP + TN + FP + FN}Accuracy Rate=TP+TN+FP+FNTP+TN
Substituting the values:Accuracy Rate=142+0142+0+101+240=142483≈0.29\text{Accuracy Rate} = \frac{142 + 0}{142 + 0 + 101 + 240} = \frac{142}{483} \approx 0.29Accuracy Rate=142+0+101+240142+0=483142≈0.29
So, the accuracy rate is 0.29 or 29%.
3. Sensitivity (Recall):
Sensitivity is the fraction of actual positive cases (Class 1) that are correctly identified by the classifier. This is also called the True Positive Rate.Sensitivity=TPTP+FN\text{Sensitivity} = \frac{TP}{TP + FN}Sensitivity=TP+FNTP
Substituting the values:Sensitivity=142142+240=142382≈0.37\text{Sensitivity} = \frac{142}{142 + 240} = \frac{142}{382} \approx 0.37Sensitivity=142+240142=382142≈0.37
So, the sensitivity is 0.37 or 37%.
4. Precision:
Precision is the fraction of predicted positive cases (Class 1) that are actually positive (Class 1). This is also called the Positive Predictive Value.Precision=TPTP+FP\text{Precision} = \frac{TP}{TP + FP}Precision=TP+FPTP
Substituting the values:Precision=142142+101=142243≈0.58\text{Precision} = \frac{142}{142 + 101} = \frac{142}{243} \approx 0.58Precision=142+101142=243142≈0.58
So, the precision is 0.58 or 58%.
5. Specificity:
Specificity is the fraction of actual negative cases (Class 2) that are correctly identified by the classifier. This is also called the True Negative Rate.Specificity=TNTN+FP\text{Specificity} = \frac{TN}{TN + FP}Specificity=TN+FPTN
Substituting the values:Specificity=00+101=0101=0\text{Specificity} = \frac{0}{0 + 101} = \frac{0}{101} = 0Specificity=0+1010=1010=0
So, the specificity is 0 or 0%.
Final Results:
- Misclassification Rate: 71%
- Accuracy Rate: 29%
- Sensitivity: 37%
- Precision: 58%
- Specificity: 0%
These results suggest that the model is not performing well, as it has a high misclassification rate and low accuracy, sensitivity, and specificity.
