All about Confusion Matrix— Preparing for Interview Questions

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Index

1. What is confusion Matrix?
2. Different Metrics derived from Confusion Matrix
3. Connection with Type I and Type II errors in detail
4. Power of test (1 — β)
5. Use Cases
6. Case Study
7. List of Interview Questions

What is Confusion Matrix?

A confusion matrix is a table that is often used to describe the performance of a classification model on a set of data for which the true values are known. It is a 2x2 matrix for binary classification (though it can be expanded for multi-class problems). The four outcomes are:

• True Positives (TP): The cases in which the model predicted yes (or the positive class), and the truth is also yes.
• True Negatives (TN): The cases in which the model predicted no (or the negative class), and the truth is also no.
• False Positives (FP), Type I error: The cases in which the model predicted yes, but the truth is no.
• False Negatives (FN), Type II error: The cases in which the model predicted no, but the truth is yes.

The confusion matrix looks like this:

Source — Wikipedia

Detailed Version

Source — Wikipedia

Different Metrics derived from Confusion Matrix

Precision: Precision is the ratio of correctly predicted positive observations to the total predicted positive observations.

Formula: Precision = TP / (TP + FP)

Recall (Sensitivity): Recall is the ratio of correctly predicted positive observations to all observations in actual class.

Formula: Recall = TP / (TP + FN)

F1-Score: The F1 Score is the weighted average of Precision and Recall. It tries to find the balance between precision and recall.

Formula: F1 Score = 2*(Recall * Precision) / (Recall + Precision)

Accuracy: Accuracy is the most intuitive performance measure. It is simply a ratio of correctly predicted observation to the total observations.

Formula: Accuracy = (TP+TN) / (TP+FP+FN+TN)

Type I error (False Positive rate): This is the situation where you reject the null hypothesis when it is actually true. In terms of the confusion matrix, it’s when you wrongly predict the positive class.

Formula: Type I error = FP / (FP + TN)

Type II error (False Negative rate): This is the situation where you fail to reject the null hypothesis when it is actually false. In terms of the confusion matrix, it’s when you wrongly predict the negative class.

Formula: Type II error = FN / (FN + TP)

ROC-AUC Curve:

Receiver Operating Characteristic (ROC) is a probability curve that plots the true positive rate (sensitivity or recall) against the false positive rate (1 — specificity) at various threshold settings.

Area Under the Curve (AUC) is the area under the ROC curve. If the AUC is high (close to 1), the model is better at distinguishing between positive and negative classes. An AUC of 0.5 represents a model that is no better than random.

Source — Wikipedia

Thresholding:

Thresholding is used to create binary classification outcomes. For example, you might predict a class “1” if the probability of the class is above a certain threshold, say 0.5, and “0” otherwise. By changing this threshold, you can increase or decrease the recall or precision of a classifier, and this trade-off is often visualized with an ROC curve.

Connection with Type I and Type II errors in detail

Type I and Type II errors are terms used in statistical hypothesis testing, while False Positives and False Negatives are terms used more commonly in machine learning and binary classification tests. However, they refer to similar concepts and can be connected as follows:

Type I Error: This occurs when we reject the null hypothesis when it is actually true. In terms of binary classification, this is analogous to a False Positive. A False Positive is when we predict an event will happen, but it does not. For example, if we predict a patient has a disease (when in reality they do not), we have committed a Type I error or produced a False Positive.

Type II Error: This occurs when we accept the null hypothesis when it is actually false. In terms of binary classification, this is analogous to a False Negative. A False Negative is when we predict an event will not happen, but it does. For example, if we predict a patient does not have a disease (when in reality they do), we have committed a Type II error or produced a False Negative.

These errors are inversely related: reducing Type I error rate often increases Type II error rate, and vice versa. This is known as the trade-off between sensitivity and specificity in binary classification tasks. The balance between these two types of errors depends on the specific problem and the costs associated with each type of error.

Power of Test (1 — β)

The power of a statistical test is the probability that the test will reject the null hypothesis when the null hypothesis is actually false (i.e., it correctly identifies a true effect). The power of a test is represented as 1 — β, where β is the probability of making a Type II error.

A Type II error occurs when the null hypothesis is false, but we fail to reject it. In other words, it’s when we miss a real effect, also known as a “false negative”. The rate of these Type II errors is represented by the term “beta” (β).

So, if β is the probability of making a Type II error, then 1 — β is the probability of not making a Type II error, i.e., correctly identifying a true effect when it exists. This is why 1 — β is called the power of a test.

The power of a test depends on three factors:

1. The alpha level (α), or the probability of rejecting the null hypothesis when it is true (Type I error rate).
2. The effect size, or the magnitude of the difference or relationship that exists in the population.
3. The sample size, or the number of observations in the study.

Increasing your sample size or your effect size (if you have control over this, such as in an experimental design) can increase the power of your test. Conversely, setting a lower alpha level (being more stringent about what you consider statistically significant) can decrease the power of your test.

Use Cases

F1 Score:

The F1 Score is preferred in situations where we want a balance between Precision and Recall.

1. Imbalanced Classes: When you have data that is highly skewed or imbalanced, accuracy becomes a less informative measure of the performance of a model. This is where the F1 score can be more useful since it considers both precision and recall. For example, in fraud detection or cancer prediction, the positive class (fraud/cancer) might be a very small proportion.
2. Both False Positives and False Negatives are Important: There are some cases where both types of errors are equally important. For instance, in the case of a spam detection model, you want to avoid non-spam (good) emails being classified as spam (False Positives) and also spam emails getting into the inbox (False Negatives).

ROC-AUC Curve:

The ROC-AUC score is important in the following cases:

1. Comparing Different Models: ROC-AUC is a good metric for comparing different models’ performances because it measures how well the model can distinguish between classes. The model with a higher AUC is generally the better one.
2. Threshold Selection: ROC curve also helps in choosing the optimal threshold value for classification, which balances True Positive and False Positive rates.

When Precision is More Important than Recall:

In certain situations, we care more about minimizing false positives, even at the cost of having more false negatives. This is when Precision becomes more important.

1. Email Spam Detection: You wouldn’t want to have important emails wrongly classified as spam. Here, Precision is preferred over Recall because false positives (non-spam emails predicted as spam) are more problematic than false negatives (spam emails predicted as non-spam).
2. Safe Content Filtering (like Kids YouTube): When filtering content for children, it’s very important that all the unsafe content is restricted (high precision). It’s less of a concern if some safe content is also restricted (lower recall).
3. Legal Cases (Innocent until proven guilty): The principle that every person is considered innocent until proven guilty focuses on high precision. We want to ensure that every convicted person is guilty (even if we miss some guilty people), rather than convicting innocent people.

When Recall is More Important than Precision:

In other situations, we want to minimize false negatives, even if it means having more false positives. This is when Recall becomes more important.

1. Cancer Diagnosis: When diagnosing cancer, doctors wouldn’t want to miss anyone who has the disease. Here, Recall is preferred over Precision because false negatives (patients with cancer predicted as non-cancerous) are more problematic than false positives (patients without cancer predicted as having cancer).
2. Fraud Detection: It’s more important to capture all potentially fraudulent activities even if it means some normal activities are flagged for review.

Case Study

Suppose a machine learning model is used to predict whether a patient has cancer (1) or not (0). Let’s consider we have 1000 patients for the test.

The model’s outcomes versus the actual outcomes are as follows:

• True Positives (TP): The model predicted 100 patients have cancer and they actually do.
• False Positives (FP): The model predicted 30 patients have cancer, but they do not.
• True Negatives (TN): The model predicted 800 patients do not have cancer, and they actually do not.
• False Negatives (FN): The model predicted 70 patients do not have cancer, but they actually do.

Now, let’s calculate the metrics using these values.

Precision: Precision is the ratio of correctly predicted positive observations to the total predicted positive observations.

Precision = TP / (TP + FP) = 100 / (100 + 30) = 0.769 ~ 76.9%

Recall (Sensitivity): Recall is the ratio of correctly predicted positive observations to all observations in actual class.

Recall = TP / (TP + FN) = 100 / (100 + 70) = 0.588 ~ 58.8%

F1-Score: The F1 Score is the weighted average of Precision and Recall.

F1 Score = 2*(Recall * Precision) / (Recall + Precision) = 2*(0.769 * 0.588) / (0.769 + 0.588) = 0.666 ~ 66.6%

Accuracy: Accuracy is the ratio of correctly predicted observation to the total observations.

Accuracy = (TP+TN) / (TP+FP+FN+TN) = (100 + 800) / (100 + 30 + 70 + 800) = 0.9 ~ 90%

Type I error (False Positive rate): This is when you wrongly predict the positive class.

Type I error = FP / (FP + TN) = 30 / (30 + 800) = 0.036 ~ 3.6%

Type II error (False Negative rate): This is when you wrongly predict the negative class.

Type II error = FN / (FN + TP) = 70 / (70 + 100) = 0.412 ~ 41.2%

In this case, the Type II error is quite high, which means the model often predicts “no cancer” when the patient actually has cancer.

Given the importance of early detection in cancer treatment, this model’s performance might be considered inadequate, despite its high accuracy, due to the high number of false negatives.

The F1 score, which is lower than the accuracy, gives a better picture of the model’s performance in this case.

List of Sample Interview Questions

1. How can a model have a high accuracy but a low F1 score? Can you provide an example where this might happen?
2. How can you choose the threshold value in a classification problem? Can you explain the concept of ROC curve and how it helps in selecting the optimal threshold?
3. Given a case where both false positives and false negatives are costly, how would you evaluate your model? Which metric would you choose to optimize?
4. How can a model with a lower accuracy have a higher ROC-AUC score than a model with higher accuracy? Explain with a hypothetical example.
5. Can you explain the concept of Precision-Recall curve? How is it different from the ROC-AUC curve and when should we use it?
6. How can we use the ROC-AUC curve for multi-class classification problems?
7. Suppose we have an extremely imbalanced dataset. Which metrics would you rely on to evaluate your model and why?
8. Explain the difference between micro-averaging and macro-averaging in the context of a multi-class classification problem. How would these impact Precision, Recall, and F1-score?
9. What does it mean when your model’s ROC-AUC score is less than 0.5? What might be happening and what steps could you take to diagnose or fix the issue?
10. Discuss the trade-off between sensitivity and specificity in the context of the ROC-AUC curve. Can you provide a real-world example where you might prefer a high specificity over sensitivity, and vice versa?
11. How would you handle a situation where your false positive and false negative costs are not the same? How would this change your choice of model or evaluation metric?
12. If your model’s ROC curve is a diagonal line from (0, 0) to (1, 1), what does this indicate about your model? Can you describe a situation where you might observe this?
13. Explain the concept of class imbalance and how it affects Precision, Recall, and F1 score. How would you adapt the calculation of these metrics to better handle class imbalance?
14. What is the relationship between the Gini coefficient and the ROC-AUC score? How can you derive one from the other?
15. How would you interpret an ROC-AUC score of 1.0? What could this indicate about your data or model?
16. How would you use a confusion matrix to evaluate a multi-class classification problem? What additional challenges does multi-class classification introduce?
17. How do Precision and Recall change as you move along the ROC curve? How would you find the optimal point on the ROC curve that balances these metrics?
18. Explain the concept of the “No Free Lunch” theorem in the context of model evaluation. How does this influence your choice of evaluation metric?
19. Suppose you have a model with high precision but low recall. What strategies could you employ to try to increase recall, and what might be the potential impacts on precision?
20. Explain the relationship between the F1 score and the harmonic mean. Why do we use the harmonic mean in the F1 score instead of the arithmetic mean?