Accuracy and Misclassification Rate (when randomly predicting a binary outcome)
When randomly predicting a binary outcome, the accuracy and the error rate (or misclassification rate) depends on the probability distribution of the classes and the specific method used for random prediction.
Let’s consider the following:
Case 1: Equal Probability of Each Class
If we are randomly predicting a binary outcome, and the two classes are equally likely (i.e., the probability of class 0 or class 1 is both 50%), the accuracy can be estimated by the probability of making a correct prediction, and the error rate can be estimated by the probability of making an incorrect prediction.
- Since the predictions are random, each prediction has a 50% chance of being true/wrong.
- Hence, both the accuracy and the error rate (misclassification rate) in this case would be 50% because there is no information being used to make a better prediction, and thus half of the predictions would be correct/incorrect.
Case 2: Unequal Class Probabilities
If the classes are not equally likely, say one class has a higher probability than the other, the error rate will depend on how the random predictor chooses its class. For example, if class 0 occurs with 70% probability and class 1 occurs with 30%, and the random predictor chooses class 0 with a probability of 70% and class 1 with a probability of 30%, the error rate can be calculated by considering the following:
- If the model predicts class 0 70% of the time, the error rate would be the probability that class 1 occurs (which is 30%).
- Similarly, if the model predicts class 1 30% of the time, the error rate would be the probability that class 0 occurs (which is 70%).
Thus, in this case, the error rate would be a weighted sum of these misclassifications.
Likewise, the accuracy can be calculated as:
- If the model predicts class 0 70% of the time, the accuracy rate would be the probability that class 0 occurs (which is 70%).
- Similarly, if the model predicts class 1 30% of the time, the accuracy rate would be the probability that class 1 occurs (which is 30%).
And, the accuracy would be a weighted sum of these classifications.
General Formula for Error Rate
For random prediction, where the probabilities of class 0 and class 1 are p0 and p1, the error rate can be approximated as:
E = p0 × (1 − p0) + p1 × (1 − p1)
where:
- p0 is the probability of the model predicting class 0.
- p1 is the probability of the model predicting class 1.
If we take the sample above, the calculation would be:
Error Rate = 0.7 x 0.3 + 0.3 x 0.7 = 0.42
In a case where the classes are imbalanced, the error rate will vary accordingly, but it will generally be higher than the misclassification rate for a more informed model that uses the feature set to make predictions.
In summary:
- If equal probabilities (50% chance for each class), the error rate is 50%.
- If unequal probabilities, the error rate depends on how often each class is chosen, but it will still be higher than a model that actually learns from the data.
General Formula for Accuracy
We can calculate the accuracy as the sum of the probability of correctly predicting class 0 and the probability of correctly predicting class 1:
Accuracy = (p0 × p0) + (p1 × p1)
where:
- p0 × p0 is the probability of correctly predicting class 0.
- p1 × p1 is the probability of correctly predicting class 1.
If we take the sample above, the calculation would be:
Accuracy = 0.7 x 0.7 + 0.3 x 0.3 = 0.58
This also justifies that Error Rate = 1 – Accuracy.
In the sample above: 0.42 = 1 – 0.58
Possible error scenarios: Round-off and Individual Probability
Depending on the precision we take, sometimes we may end up with a round-off error or floating-point precision during the calculations.
But some other times such issues can arise due to a deeper conceptual issue: A slight difference may arise due to the fact that we’re not simply calculating the total misclassification, but rather the individual probability of making an error based on class probabilities.
Each term in the error rate calculation adjusts the error for each class, considering the probability distribution across both classes.
Limitations of Accuracy and Error Rate
Lack of Insight into Class Imbalance
- The misclassification rate does not account for class imbalances in the dataset. For example, if we have a dataset where 95% of instances belong to class 0 and 5% belong to class 1, a model that always predicts class 0 will have a low misclassification rate of 5%. However, it will fail completely to predict the minority class (class 1).
- Metrics like precision, recall, and F1 score provide more detailed insight into how well the model is performing, especially when the data is imbalanced.
No Differentiation Between False Positives and False Negatives
- The misclassification rate simply measures the fraction of incorrect predictions, without differentiating between false positives (FP) and false negatives (FN).
- In many applications (e.g., medical diagnostics, fraud detection), it’s important to distinguish between these types of errors, as they have different consequences. Metrics like precision, recall, and F1 score are much more informative because they specifically address these concerns.
No Performance Granularity
- Misclassification rate provides a very basic view of performance. It doesn’t give detailed insights into the underlying reasons for the errors. On the other hand, the confusion matrix offers a more granular breakdown of the model’s errors, allowing you to calculate the precision, recall, and other metrics. This granular information can be much more helpful when analyzing model performance.
F1 Score and Other Metrics Better Capture Model Trade-offs
- The F1 score, for example, balances precision and recall, providing a more comprehensive view of a model’s performance when handling imbalanced datasets. Misclassification rate does not provide such a balanced trade-off between the different types of classification errors (false positives and false negatives).