Three Ways Classical Test Theory Can Mislead About LLM Judges

Louis Yiven Zhu

cs.LGAug 31, 2026

Three Ways Classical Test Theory Can Mislead About LLM Judges

Authors: Louis Yiven Zhu

Organizations: University of Oxford

Abstract

Evaluations that use a large language model (LLM) as a judge have begun to borrow reliability statistics from classical test theory and its extensions. We examine three such statistics that need one administration and no gold labels. None of them can isolate the judge, because one judge under one prompt supplies no variance component of its own. Claude Haiku 4.5 judged 210 constructed short answers against ten-element checklists. On the 180 with parsed verdicts, the Kuder-Richardson coefficient (KR-20) came out at 0.5223 on the judge's verdicts and 0.5231 on error-free gold verdicts. In simulation, bank design alone moves KR-20 from 0.01 to 0.68 at the judge's measured 4.72% error rate. The dependability index $Φ(λ)$ , a ratio of mean squared distances from the pass mark, sits 0.22 to 0.38 below the judge's accuracy against gold and returns 0.54 to 0.68 on error-free gold verdicts. Livingston-Lewis accuracy treats the rubric elements as a sample, and at a pass mark of five elements it credits error-free gold scores with 0.78, close to the judge's 0.81. A statement about the judge therefore needs gold labels or a varied scorer facet, and a reliability ratio needs the bank's spread beside it. One of the four closest judge-evaluation papers varies the prompt and still reads a reliability below 0.7 as a sign that a model cannot serve as a judge, although that reliability moves with the spread of the samples scored. We derive a decision table and four reporting lines from these two rules.

Figures & tables

Figure 1: Why KR-20 cannot identify the judge’s reliability. A fixed LLM judge scores

n

responses against

K

binary rubric elements and fills the verdict matrix

X_{ij}\in\{0,1\}

. KR-20 partitions the matrix’s variance into components for responses, elements and a residual. One judge under one prompt phrasing supplies no variance along a scorer axis. Errors that the judge repeats within a response merge with the response component, errors that it repeats across responses on one element merge with the element component, and the rest merge with the residual. Bank construction and judge error can thus produce the same coefficient, one that characterises the measurement system as a whole.

Figure 2: KR-20 responds to bank design and to judge error, and bank design dominates at realistic error rates. (a) Full grid, 60 replicates per cell; outlined cells have near-identical coefficients (

0.451

and

0.457

) at twice the judge error. (b) At the measured

4.72\%

error rate, bank design alone moves the coefficient across most of its range (

\pm 1

SD shaded). (c) At the widest bank, judge error lowers the coefficient by

0.04

5\%

and by

0.46

30\%

Question	Quantity that answers it	Requires	Tempting substitute
How often does the judge’s pass or fail match gold?	Accuracy against gold, with cluster intervals	Gold labels	$\Phi(\lambda)$ , a ratio of squared distances, or Livingston–Lewis accuracy, which counts element sampling as error
Where does the judge err, and in which direction?	False-accept and false-reject rates per checklist line	Gold labels	A whole-rubric reliability coefficient
Do the judge’s decisions depend on the wording of its instruction?	Phrasing variance components from a person-by-phrasing study (Appendix A.5 )	Varied phrasings	Agreement across repeated samples at one prompt, which measures decode-time noise
Do the checklist elements rank responses consistently?	KR-20	Neither	KR-20 on judge verdicts read as a property of the judge
Would pass or fail survive another sample of checklist elements?	Livingston–Lewis consistency and accuracy	Neither	Accuracy against gold, which holds the elements fixed

Table 1: Questions that a practitioner asks, and the quantities that answer them. The first three questions concern the judge and require gold labels or varied phrasings. The last two concern the bank and the checklist, and the label-free statistics answer them.

Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

Artefact	File	Contents
Bank	judge_item_bank.csv	210 responses with question id, question and answer text, element text, element-level gold ( $G_{ij}$ ) and judge ( $X_{ij}$ ) verdicts and totals; judge fields blank for the 30 unparsed responses
Estimators	estimators.py	KR-20, KR-21, inter-element correlation, beta-binomial MLE and goodness of fit, Livingston–Lewis DC and CA, variance components and $\Phi(\lambda)$ , cluster bootstrap
Bank statistics	bank_stats.py	Every quantity measured on the real bank, with cluster intervals and the independence check of Section 3
Simulations	sweeps.py	Two-way sweep, measured-rate run, one-directional sweep, $\Phi(\lambda)$ control, Livingston–Lewis estimand control, phrasing identification check
Published grids	sweep_grid.json	Grid values as reported here, including the 60-replicate run at the measured $4.72\%$ error rate
Figures	make_figures.py	Regenerates Figures 2 , A.1 and A.2 from the released data; Figure 1 is a schematic

Appendix

Table A.1: Contents of the released bundle. Running python bank_stats.py , python sweeps.py and python make_figures.py regenerates every quantity and data figure with numpy , scipy , pandas and matplotlib .

Question. How does a refrigerator move heat out of its interior?

Answer as presented to the judge. A refrigerant fluid circulates through a closed loop of coils. Evaporation releases heat into the interior, which is then vented outside separately. Expansion causes the refrigerant to cool to a temperature below the interior. The cycle repeats continuously, net moving heat from cold interior to warm exterior against the natural gradient. Heat release at the condenser causes the refrigerant to condense into a liquid. The cold refrigerant absorbs heat from the interior at the evaporator coils.

Appendix

Table A.2: One response from the bank ( item_id 30, question 3), with gold verdicts

G_{ij}

and judge verdicts

X_{ij}

for all ten elements, quoted verbatim from the released file. An element with

G_{ij}=0

is either absent or represented by its distractor. Here the second sentence of the answer is the distractor for element 9. It inverts the direction of heat flow, and the judge rejects element 9. The judge errs once, accepting element 6, whose slot is empty.

Element position	1	2	3	4	5	6	7	8	9	10
Gold rate	0.506	0.583	0.561	0.533	0.550	0.567	0.561	0.528	0.544	0.578
Judge error	0.144	0.044	0.072	0.061	0.044	0.033	0.033	0.000	0.022	0.017

Appendix

Table A.3: Gold presence rates and judge error rates by element position over the 180 scored responses. Each question has its own checklist, and each position pools 13 different lines. Presence rates stay close to balance by construction, whereas judge error varies by an order of magnitude across positions.

Figure A.1: Bank and judge diagnostics over the 180 scored responses. (a) Gold and judge total-score distributions, with 6 gold totals and 9 judge totals at the ceiling of 10. (b) Judge error rate by element position, pooled over the 13 scored questions, with the aggregate

4.72\%

dashed. (c) Judge against gold totals with the identity line, where the displacement above the line is the judge’s leniency.

	Judge per-element error
Bank spread	0%	4.72%	5%	10%	20%	30%
0.0	-0.000	0.007	-0.014	-0.011	0.015	0.003
0.2	0.111	0.098	0.092	0.062	0.068	-0.009
0.4	0.366	0.317	0.328	0.253	0.174	0.045
0.6	0.575	0.522	0.519	0.451	0.325	0.168
0.8	0.722	0.679	0.682	0.608	0.457	0.266

Appendix

Table A.4: Mean KR-20 over 60 replicates per cell. Reading down any column shows bank design moving the coefficient, and reading across any row shows judge error moving it. The bold column comes from a separate 60-replicate run at the measured

4.72\%

error rate and is the column Section 3 quotes. The

5\%

column belongs to the two-way grid and comes from its own random stream, and the two columns differ by up to

0.021

Bank spread	0%	4.72%	5%	10%	20%	30%
0.0	0.087	0.085	0.111	0.106	0.091	0.112
0.2	0.097	0.083	0.097	0.078	0.093	0.100
0.4	0.061	0.068	0.063	0.085	0.088	0.111
0.6	0.044	0.047	0.048	0.041	0.065	0.084
0.8	0.021	0.030	0.028	0.043	0.059	0.078

Appendix

Table A.5: Per-cell standard deviations for Table A.4 . Precision is lowest at low bank spread, where the coefficient’s true value lies near zero and its sampling variance is largest. We leave the apparent non-monotonicity in the two lowest-spread rows uninterpreted for that reason.

Figure A.2: Two quantities that resemble accuracies without being them. (a)

\Phi(\lambda)

against the judge’s classification accuracy against gold at four cut scores on the 180 scored responses, with arrows marking gaps of

0.22

0.38

. (b) In a simulation without element sampling, where the judge’s true score departs from gold, accuracy against the judge’s own true score exceeds accuracy against gold at every cut. On the real bank, element sampling reverses the sign of this difference (Table A.6 ).

	Accuracy	$95\%$	LL CA	LL CA	Accuracy	$1-$ LL CA	LL DC
Cut	vs gold	interval	judge	gold	$-$ LL CA	(gold)	judge
4	0.961	0.922–0.994	0.882	0.846	0.079	0.154	0.836
5	0.928	0.893–0.962	0.812	0.781	0.116	0.219	0.755
6	0.900	0.851–0.945	0.759	0.750	0.141	0.250	0.697
7	0.917	0.856–0.973	0.750	0.769	0.167	0.231	0.689

Appendix

Table A.6: Livingston–Lewis (LL) estimates on the 180 scored responses. Applied to the error-free gold totals, the estimator falls short of perfect accuracy by more than the judge’s own difference at every cut. The bank’s element sampling dominates the difference column, and the judge’s gate errors offset part of it.

Scenario	$\sigma^{2}(\text{responses})$	$\sigma^{2}(\text{phrasing})$	$\sigma^{2}(\text{resid})$	$\Phi$
Phrasing irrelevant	0.994	0.000	0.090	0.916
Mild phrasing effect	0.994	0.039	0.090	0.886
Large phrasing effect	0.994	0.365	0.090	0.721
Deterministic given phrasing	0.996	0.177	0.000	0.865

Appendix

Table A.7: Identification check over 200 simulations,

N=210

R=3

, by two-way random-effects analysis of variance.

\Phi

refers to a single phrasing and averages the per-simulation values.

Line	Worked wording
Bank	Reliability computed over $K=10$ binary checklist elements on 180 responses from 13 question clusters. SD of gold totals $2.16$ on a $0$ – $10$ scale. Mean inter-element correlation $0.098$ .
Gold comparison	KR-20 is $0.5223$ on judge verdicts and $0.5231$ on gold verdicts. Livingston–Lewis accuracy at a pass mark of five elements is $0.812$ on judge totals and $0.781$ on gold totals. Because error-free scoring lands in the same region, neither value describes the judge.
$\Phi(\lambda)$	Dependability coefficient $\Phi(\lambda=0.5)=0.600$ on judge verdicts and $0.543$ on gold verdicts. This is a ratio of squared distances from the pass mark and is not comparable to the classification accuracy of $0.928$ at the same pass mark.
Judge error	Against gold, the judge accepts $10.4\%$ of absent elements ( $95\%$ cluster interval $5.5\%$ – $15.5\%$ ) and rejects $0.1\%$ of present ones ( $0.0\%$ – $0.3\%$ ). Five of the 130 checklist lines hold 25 of the 85 errors. Gate accuracy at a pass mark of five elements is $0.928$ ( $0.893$ – $0.962$ ) on the 180 scored responses and $0.795$ on all 210 when unparsed outputs count as errors.

Appendix

Table A.8: Worked wording for the four reporting lines of Section 5 , with this study’s values.

Term	Meaning
Response	One constructed short answer, the unit the judge scores; the released file calls it an item
Bank	The set of responses a judge scores, here 210 constructed short answers
Element	One binary checklist criterion, with $K=10$ per question and a separate checklist for each question
Element position	The index $j$ of an element within its question’s checklist
Gold verdict $G_{ij}$	Whether element $j$ is present in response $i$ , fixed by construction
Judge verdict $X_{ij}$	The judge’s YES ( $1$ ) or NO ( $0$ ) for element $j$ of response $i$

Appendix

Table A.9: Terms and notation.

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