Evaluation of binary classifiers

Source: Wikipedia, the free encyclopedia.
From the confusion matrix you can derive four basic measures.

The evaluation of binary classifiers compares two methods of assigning a binary attribute, one of which is usually a standard method and the other is being investigated. There are many metrics that can be used to measure the performance of a classifier or predictor; different fields have different preferences for specific metrics due to different goals. For example, in medicine sensitivity and specificity are often used, while in computer science precision and recall are preferred. An important distinction is between metrics that are independent on the prevalence (how often each category occurs in the population), and metrics that depend on the prevalence – both types are useful, but they have very different properties.

Probabilistic classification models go beyond providing binary outputs and instead produce probability scores for each class. These models are designed to assess the likelihood or probability of an instance belonging to different classes. In the context of evaluating probabilistic classifiers, alternative evaluation metrics have been developed to properly assess the performance of these models. These metrics take into account the probabilistic nature of the classifier's output and provide a more comprehensive assessment of its effectiveness in assigning accurate probabilities to different classes. These evaluation metrics aim to capture the degree of calibration, discrimination, and overall accuracy of the probabilistic classifier's predictions.

Contingency table

Given a data set, a classification (the output of a classifier on that set) gives two numbers: the number of positives and the number of negatives, which add up to the total size of the set. To evaluate a classifier, one compares its output to another reference classification – ideally a perfect classification, but in practice the output of another

scale invariant (scaling all the numbers by the same factor does not change the output), to make them independent of population size, which is achieved by using ratios of homogeneous functions, most simply homogeneous linear or homogeneous quadratic
functions.

Say we test some people for the presence of a disease. Some of these people have the disease, and our test correctly says they are positive. They are called

false positives (FP). These can be arranged into a 2×2 contingency table (confusion matrix
), conventionally with the test result on the vertical axis and the actual condition on the horizontal axis.

These numbers can then be totaled, yielding both a

marginal totals
. Totaling the entire table, the number of true positives, false negatives, true negatives, and false positives add up to 100% of the set. Totaling the columns (adding vertically) the number of true positives and false positives add up to 100% of the test positives, and likewise for negatives. Totaling the rows (adding horizontally), the number of true positives and false negatives add up to 100% of the condition positives (conversely for negatives). The basic marginal ratio statistics are obtained by dividing the 2×2=4 values in the table by the marginal totals (either rows or columns), yielding 2 auxiliary 2×2 tables, for a total of 8 ratios. These ratios come in 4 complementary pairs, each pair summing to 1, and so each of these derived 2×2 tables can be summarized as a pair of 2 numbers, together with their complements. Further statistics can be obtained by taking ratios of these ratios, ratios of ratios, or more complicated functions.

The contingency table and the most common derived ratios are summarized below; see sequel for details.

Predicted condition Sources: [1][2][3][4][5][6][7][8][9]
Total population
= P + N
Predicted Positive (PP) Predicted Negative (PN) Informedness, bookmaker informedness (BM)
= TPR + TNR − 1
Prevalence threshold
(PT)
= TPR × FPR - FPR/TPR - FPR
Actual condition
Positive (P) [a]
True positive (TP),
hit[b]
False negative
(FN),
miss, underestimation
power

= TP/P = 1 − FNR
type II error [c]

= FN/P = 1 − TPR
Negative (N)[d]
False positive (FP),
false alarm, overestimation
True negative (TN),
correct rejection[e]
type I error [f]

= FP/N = 1 − TNR
specificity
(SPC), selectivity
= TN/N = 1 − FPR
Prevalence
= P/P + N
precision

= TP/PP = 1 − FDR
False omission rate
(FOR)
= FN/PN = 1 − NPV
Positive likelihood ratio
(LR+)
= TPR/FPR
Negative likelihood ratio
(LR−)
= FNR/TNR
Accuracy (ACC)
= TP + TN/P + N
False discovery rate (FDR)
= FP/PP = 1 − PPV
Negative predictive value
(NPV)
= TN/PN = 1 − FOR
Markedness (MK), deltaP (Δp)
= PPV + NPV − 1
Diagnostic odds ratio (DOR)
= LR+/LR−
Balanced accuracy (BA)
= TPR + TNR/2
F1 score

= 2 PPV × TPR/PPV + TPR = 2 TP/2 TP + FP + FN
Fowlkes–Mallows index (FM)
= PPV × TPR
Matthews correlation coefficient
(MCC)
= TPR × TNR × PPV × NPV - FNR × FPR × FOR × FDR
Threat score (TS), critical success index (CSI), Jaccard index
= TP/TP + FN + FP
  1. ^ the number of real positive cases in the data
  2. ^ A test result that correctly indicates the presence of a condition or characteristic
  3. ^ Type II error: A test result which wrongly indicates that a particular condition or attribute is absent
  4. ^ the number of real negative cases in the data
  5. ^ A test result that correctly indicates the absence of a condition or characteristic
  6. ^ Type I error: A test result which wrongly indicates that a particular condition or attribute is present


Note that the rows correspond to the condition actually being positive or negative (or classified as such by the gold standard), as indicated by the color-coding, and the associated statistics are prevalence-independent, while the columns correspond to the test being positive or negative, and the associated statistics are prevalence-dependent. There are analogous likelihood ratios for prediction values, but these are less commonly used, and not depicted above.

Sensitivity and specificity

The fundamental prevalence-independent statistics are sensitivity and specificity.

True Positive Rate (TPR), also known as recall
, is the proportion of people that tested positive and are positive (True Positive, TP) of all the people that actually are positive (Condition Positive, CP = TP + FN). It can be seen as the probability that the test is positive given that the patient is sick. With higher sensitivity, fewer actual cases of disease go undetected (or, in the case of the factory quality control, fewer faulty products go to the market).

True Negative Rate
(TNR) is the proportion of people that tested negative and are negative (True Negative, TN) of all the people that actually are negative (Condition Negative, CN = TN + FP). As with sensitivity, it can be looked at as the probability that the test result is negative given that the patient is not sick. With higher specificity, fewer healthy people are labeled as sick (or, in the factory case, fewer good products are discarded).

The relationship between sensitivity and specificity, as well as the performance of the classifier, can be visualized and studied using the

Receiver Operating Characteristic
(ROC) curve.

In theory, sensitivity and specificity are independent in the sense that it is possible to achieve 100% in both (such as in the red/blue ball example given above). In more practical, less contrived instances, however, there is usually a trade-off, such that they are inversely proportional to one another to some extent. This is because we rarely measure the actual thing we would like to classify; rather, we generally measure an indicator of the thing we would like to classify, referred to as a surrogate marker. The reason why 100% is achievable in the ball example is because redness and blueness is determined by directly detecting redness and blueness. However, indicators are sometimes compromised, such as when non-indicators mimic indicators or when indicators are time-dependent, only becoming evident after a certain lag time. The following example of a pregnancy test will make use of such an indicator.

Modern pregnancy tests do not use the pregnancy itself to determine pregnancy status; rather,

embryogenesis
, the sensitivity of modern pregnancy tests cannot be 100% (because false negatives are possible).

Likelihood ratios

Positive and negative predictive values

In addition to sensitivity and specificity, the performance of a binary classification test can be measured with

negative predictive value
(NPV). The positive prediction value answers the question "If the test result is positive, how well does that predict an actual presence of disease?". It is calculated as TP/(TP + FP); that is, it is the proportion of true positives out of all positive results. The negative prediction value is the same, but for negatives, naturally.

Impact of prevalence on prediction values

Prevalence has a significant impact on prediction values. As an example, suppose there is a test for a disease with 99% sensitivity and 99% specificity. If 2000 people are tested and the prevalence (in the sample) is 50%, 1000 of them are sick and 1000 of them are healthy. Thus about 990 true positives and 990 true negatives are likely, with 10 false positives and 10 false negatives. The positive and negative prediction values would be 99%, so there can be high confidence in the result.

However, if the prevalence is only 5%, so of the 2000 people only 100 are really sick, then the prediction values change significantly. The likely result is 99 true positives, 1 false negative, 1881 true negatives and 19 false positives. Of the 19+99 people tested positive, only 99 really have the disease – that means, intuitively, that given that a patient's test result is positive, there is only 84% chance that they really have the disease. On the other hand, given that the patient's test result is negative, there is only 1 chance in 1882, or 0.05% probability, that the patient has the disease despite the test result.

Likelihood ratios

Precision and recall

Precision and recall can be interpreted as (estimated) conditional probabilities: Precision is given by while recall is given by ,[10] where is the predicted class and is the actual class. Both quantities are therefore connected by Bayes' theorem.

Relationships

There are various relationships between these ratios.

If the prevalence, sensitivity, and specificity are known, the positive predictive value can be obtained from the following identity:

If the prevalence, sensitivity, and specificity are known, the negative predictive value can be obtained from the following identity:

Single metrics

In addition to the paired metrics, there are also single metrics that give a single number to evaluate the test.

Perhaps the simplest statistic is accuracy or fraction correct (FC), which measures the fraction of all instances that are correctly categorized; it is the ratio of the number of correct classifications to the total number of correct or incorrect classifications: (TP + TN)/total population = (TP + TN)/(TP + TN + FP + FN). As such, it compares estimates of pre- and post-test probability. In total ignorance, one can compare a rule to flipping a coin (p0=0.5). This measure is prevalence-dependent. If 90% of people with COVID symptoms don't have COVID, the prior probability P(-) is 0.9, and the simple rule "Classify all such patients as COVID-free." would be 90% accurate. Diagnosis should be better than that. One can construct a "One-proportion z-test" with p0 as max(priors) = max(P(-),P(+)) for a diagnostic method hoping to beat a simple rule using the most likely outcome. Here, the hypotheses are "Ho: p ≤ 0.9 vs. Ha: p > 0.9", rejecting Ho for large values of z. One diagnostic rule could be compared to another if the other's accuracy is known and substituted for p0 in calculating the z statistic. If not known and calculated from data, an accuracy comparison test could be made using "Two-proportion z-test, pooled for Ho: p1 = p2".

Not used very much is the complementary statistic, the fraction incorrect (FiC): FC + FiC = 1, or (FP + FN)/(TP + TN + FP + FN) – this is the sum of the

antidiagonal, divided by the total population. Cost-weighted fractions incorrect could compare expected
costs of misclassification for different methods.

The

.

Alternative metrics

An

of precision and recall:

.

F-scores do not take the true negative rate into account and, therefore, are more suited to

dual. The component regression coefficients of the Matthews correlation coefficient are markedness (deltap) and informedness (Youden's J statistic or deltap').[13]

See also

References

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    PMID 31898477
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  8. ^ Chicco D, Toetsch N, Jurman G (February 2021). "The Matthews correlation coefficient (MCC) is more reliable than balanced accuracy, bookmaker informedness, and markedness in two-class confusion matrix evaluation". BioData Mining. 14 (13): 13.
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  10. .
  11. .
  12. ^ Powers, David M. W. (2012). "The Problem with Kappa" (PDF). Conference of the European Chapter of the Association for Computational Linguistics (EACL2012) Joint ROBUS-UNSUP Workshop. Archived from the original (PDF) on 2016-05-18. Retrieved 2012-07-20.
  13. S2CID 17104364
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External links