Confusion Matrices and Error Taxonomy

A screening system predicts either “review” or “do not review,” while the reference later records whether the event was truly urgent. Before averaging anything, the evaluation contains four distinct outcomes. Those cells describe different mistakes, different affected cases, and often different remedies. A single scalar can hide which cell changed.

Michigan Medicine tested the Epic Sepsis Model against 38,455 hospitalizations recorded between December 2018 and October 2019. Sepsis occurred in 2,552 of them, about 7%. The external validation, published by Wong and colleagues in JAMA Internal Medicine in 2021, reported “a hospitalization-level area under the receiver operating characteristic curve of 0.63 (95% CI, 0.62-0.64)”. That is the number the model is usually remembered by. It shows none of the four counts underneath it.

The same paper gives the counts. The model “did not identify 1709 patients with sepsis (67%) despite generating alerts for an ESM score of 6 or higher for 6971 of all 38 455 hospitalized patients (18%), thus creating a large burden of alert fatigue”. Put those two sentences into one matrix and the four cells fall out. Of the 6,971 alerts, 843 had sepsis behind them and 6,128 did not, so 12.1% of alerts were true — about one in eight. Of the 2,552 patients who developed sepsis, 843 were identified and 1,709 were not, a sensitivity of 33.0%. The fourth cell is the remaining 29,775 hospitalizations with neither an alert nor sepsis. Two cells moved in opposite directions: a third of cases caught, seven alerts in eight wasted. The 0.63 reports neither.

Rows and columns must be labeled. Conventions differ between libraries and between reports, and an unlabeled matrix can be read in two incompatible ways.

One axis is what the system predicted. The other is what the reference process later recorded. The predicted-positive column holds the true positives (reference positive) and the false positives (reference negative). This is where workload enters, and the column a threshold change moves first. The predicted-negative column holds the false negatives (reference positive) and the true negatives (reference negative). This is where missed cases accumulate. It is also where the fallback policy decides the consequence — what happens to a case the system says nothing about.

The Michigan Medicine evaluation fills all four cells at the alert threshold of an ESM score of 6 or higher: 843 true positives, 6,128 false positives, 1,709 false negatives, 29,775 true negatives. They sum to the 38,455 hospitalizations. Every ratio in the rest of this lesson is built from those four numbers and nothing else.

A real four-cell split is more instructive than a hypothetical one, because the cells rarely fail together. ProPublica analysed COMPAS risk scores from Broward County in May 2016. Its contingency tables cover the 7,214 defendants scored in 2013–2014 for whom at least two years of recidivism information was available, with criminal records collected through 1 April 2016. ProPublica summarised one cell of that table: “Our analysis found that black defendants who did not recidivate over a two-year period were nearly twice as likely to be misclassified as higher risk compared to their white counterparts (45 percent vs. 23 percent).” The other error cell ran the other way.

Two families of metrics answer different questions. Reference-conditioned metrics — recall or sensitivity, specificity, the false-negative rate, the false-positive rate — ask: among the cases the reference calls positive, or negative, how often does the system respond correctly? Prediction-conditioned metrics — precision or positive predictive value, and negative predictive value — ask: among the cases the system calls positive, or negative, how often is it right? The second family depends strongly on prevalence and speaks directly to review yield. The first does not.

Within the reference-conditioned family alone, the two error rates need not travel together. NIST's Face Recognition Vendor Test Part 3 (NISTIR 8280), published in December 2019, ran 18.27 million images of 8.49 million people through 189 mostly commercial algorithms from 99 developers. Its executive summary reports: “Across demographics, false positives rates often vary by factors of 10 to beyond 100 times.” False negatives behave differently in the same evaluation. They “tend to be more algorithm-specific, and vary often by factors below 3”. One cell varies by two orders of magnitude across groups. The other varies by less than three-fold.

An independent university study found the two cells moving in opposite directions across the same groups. Krishnapriya and colleagues measured face-recognition accuracy on the MORPH dataset in 2019. At a fixed decision threshold, the African-American image cohort has the higher false match rate and the lower false non-match rate. A report that names only one of the two families, or only one of its rates, can show that cohort as better served or worse served at will — without changing a single decision the system made.

For multiple classes, the diagonal contains correct predictions and off-diagonal cells show which classes are confused. A global accuracy can conceal a systematic failure on one cell, and the identity of that cell is usually what a deployment cares about. Inspect counts, row-normalized recall, column-normalized precision, and support for each class. Rare classes need uncertainty estimates, because a handful of errors can move their rates substantially.

The standard demonstration is Gender Shades, published in 2018 by Joy Buolamwini and Timnit Gebru. Their abstract states: “We evaluate 3 commercial gender classification systems using our dataset and show that darker-skinned females are the most misclassified group (with error rates of up to 34.7%). The maximum error rate for lighter-skinned males is 0.8%.” The dataset is the Pilot Parliaments Benchmark, 1,270 subjects. The paper reports true positive rate, error rate, positive predictive value and false positive rate for each of the four intersectional groups rather than one headline number. The spread runs from 0.8% for lighter-skinned males to 34.7% for darker-skinned females, and no single overall accuracy figure for any of the three classifiers displays it.

The finding held when other people rebuilt the test. Later in 2018, a team from IBM Research and UC Berkeley measured the same subgroup ordering on their own reconstruction of the benchmark, PPB*. The IBM Watson classifier scored 82.5% on dark females against 99.3–99.5% on the other three groups. They argued at the same time that skin type itself is not the cause. The ordering of the cells is robust. The explanation of it is a separate question, and the matrix does not answer it.

Reading it reliably means going through the cases before the ratios: verify definitions, inspect raw counts, normalize deliberately, attach consequences, compare slices.

Step 1 — name the positive class, reference process, unit and threshold — is not paperwork. A 2016 reanalysis in Federal Probation took the same Broward County COMPAS data and changed only the binning of the three-level risk score. For Black defendants, binning low against moderate-plus-high gives a false positive rate of 0.42 and a false negative rate of 0.28. Binning low-plus-moderate against high gives 0.14 and 0.62 on the identical defendants. Same people, same instrument, same two-year window, and a false positive rate of either 0.42 or 0.14 depending on where the cut is drawn. Flores, Bechtel and Lowenkamp drew the conclusion: “Regardless, what should be taken away from these tables is the fact that when recommended practices are followed for calculating performance indicators of predictive instruments, a somewhat different pattern of results and conclusions is drawn.” A matrix without its threshold stated is not a result.

Step 2 — check support, missing outcomes and impossible cells — extends to distrusting the reference itself. An audit of the test sets of ten widely used benchmarks, presented at NeurIPS in 2021, put a number on the problem. Northcutt and colleagues write: “we estimate an average of at least 3.3% errors across the 10 datasets, where for example label errors comprise at least 6% of the ImageNet validation set.” Vasudevan and colleagues at Google reached a compatible conclusion a year later: on expert multi-label review, nearly half of the remaining “mistakes” made by top ImageNet models were not mistakes at all. Where the reference is wrong at a few percent, a share of every off-diagonal cell is the label failing rather than the model. From the counts alone, the two are indistinguishable.

Steps 3 to 5 follow from those two. Choose row, column or total normalization for the question actually being asked. Map each error cell to workload, delay, harm or cost: 6,128 false alerts and 1,709 missed sepsis cases are not the same kind of loss. Then repeat the whole matrix for time, subgroup, site and severity. The Broward tables and the Pilot Parliaments groups both show that the aggregate cell is an average over cells that disagree.