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Don't Let Your Likert Scales Grow Up To Be Visual Analog Scales: Understanding the Relationship Between Number of Response Categories and Measurement Error

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read If measurement error rises with the number of response options, the optimal Likert scale has 4-7 categories, and converting to a 100-point visual analog scale can sharply reduce reliability.

desk verdict A useful formal framework for a classic scale-design question, but the headline VAS warning rests on assumed error-growth slopes not estimated from data. read the letter →

arxiv 2502.02846 v1 pith:KPYTVCUN submitted 2025-02-05 stat.ME

Siqi Sun , Karen M. Schmidt , Teague R. Henry This is my paper · ORCID
classification stat.ME MSC 62P15
keywords LikertscaleVisualAnalogresponsecategoriesmeasurementerrorreliabilityGradedModelitemvariableconstructionsimulation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper asks whether there is an optimal number of response categories for survey items, and where it falls. Using simulations built on an item variable construction of the graded response model, it finds that if measurement error is independent of category count, reliability rises and then plateaus with no true optimum. Under the more realistic assumption that measurement error grows as more fine-grained options are added, reliability peaks at 7, 5, or 4 categories for small, medium, or large error-growth rates, then declines. The authors conclude that converting a validated Likert item to a 100-point visual analog scale can sharply reduce reliability, and that any response-format change requires fresh validation.

What carries the argument

The item variable construction of the Graded Response Model (GRM) is the mechanism that carries the argument. Each item has a continuous latent response variable $\gamma_{ij} \sim N(\theta_i, \sigma_j)$, where $\sigma_j$ is the item's measurement error on the latent scale; observed category choices are obtained by thresholding $\gamma_{ij}$. This reparameterization of Samejima's GRM lets the authors define measurement error independently of the response format, then impose different linear dependencies between $\sigma_j$ and the number of categories and trace how the optimal category count shifts.

What would settle it

Estimate item-level measurement error directly by administering the same item text with 2, 5, 7, 11, and 101 response options to the same respondents, then check whether item discrimination or test-retest reliability declines as categories increase; if reliability stays flat or keeps improving through 100 options, the predicted reliability collapse and the VAS caution are falsified for that construct.

Watch

Extended reading notes

Core claim

The paper's central claim is that the optimal number of response options is governed by how item measurement error depends on category count. Using the item variable construction of the graded response model, the authors simulate responses across 2 to 100 categories. When error is independent of the number of categories, recovery of the true score improves and then plateaus after roughly 5 to 10 categories, so there is no single optimum. When error increases linearly with the number of categories, a clear optimum emerges at 7, 5, and 4 response options for small, medium, and large dependency structures, with reliability declining past that point; the standard error of a regression coefficient follows the same pattern. The paper therefore argues that a VAS, considered as a 101-point Likert scale, will typically have lower reliability than a shorter Likert version of the same item, and that a format change should trigger re-validation.

Load-bearing premise

The argument rests on the assumption that an item's measurement error grows linearly with the number of response categories at the specific rates used in the simulation; if the true relationship is flat, nonlinear, or varies by construct, the predicted optima and the warning against VAS conversion do not follow.

Editorial extensions

If this is right

  • If measurement error grows with category count, reliability peaks at 7, 5, or 4 response options depending on the growth rate, then declines beyond that point.
  • Converting an existing Likert item to a 100-point visual analog scale will decrease reliability when such error growth is present.
  • When measurement error is independent of category count, there is no true optimum; reliability just increases and then plateaus.
  • Changing the response format of a validated measure requires re-validation because the measurement error of the scale is likely to change.
  • Adding more items (three instead of one) improves true-score recovery but does not move the optimal number of response categories.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The specific optima of 7, 5, and 4 are direct consequences of the chosen linear error-growth slopes; real-world error could grow nonlinearly, so the robust qualitative prediction is that an optimum exists well below VAS-like counts.
  • A direct empirical test would estimate item discrimination or test-retest reliability for the same item text under 2, 5, 7, 11, and 101 response options, which would map the true dependency structure and identify construct-specific optima.
  • For ecological momentary assessment, where single-item measures dominate, the reliability penalty of VAS conversion may be largest, but momentary-state constructs might exhibit flatter error growth than trait measures, making construct type a moderator worth testing.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper simulates Likert- and VAS-style response data using the graded response model with an item-variable parameterization, and examines how the number of response categories affects recovery of the latent trait (Spearman correlation) and the standard error of a regression coefficient. In the independent-error condition, reliability increases and then plateaus. In the dependent-error condition, measurement error is assumed to increase linearly with the number of categories at three slopes (small, medium, large), yielding optima at 7, 5, and 4 categories, respectively. The authors conclude that converting Likert items to VAS (0-100) will drastically reduce reliability and that any such conversion requires re-validation.

Significance. If taken as a conditional simulation study, the paper is a useful proof of concept: it demonstrates that under a monotonically increasing error-category relationship, a finite optimum exists and the location of that optimum shifts with the error-growth rate. The item-variable construction of the GRM is a clean way to separate categorization error from item-level measurement error. However, the practical conclusions are not empirically supported because the linear error-growth slopes in Section 2.2 are chosen by fiat, not estimated from data, and the reported optima are direct consequences of those slopes. The paper is also transparent about this dependence in Section 5, but the abstract and conclusion overstate the generalizability. With appropriate reframing and additional simulation detail, the paper could be a worthwhile contribution to the response-format literature.

major comments (3)
  1. [§2.2, §3.2, §5] The optima of 7, 5, and 4 categories are direct outputs of the three linear error-dependency sequences specified in Section 2.2 (σ_j from 0.05-0.5, 0.1-1.0, and 0.2-2.0 across k=2..20). These slopes are not estimated from any empirical data, and the independent-error condition in Section 3.1 shows that no optimum arises without them. The abstract's claim that 'conversion of any Likert scale item to VAS will result in a drastic decrease in reliability' is therefore not supported by the simulations alone. The authors should reframe the contribution as a conditional demonstration, explicitly state that the optima are functions of the assumed error-growth function, and ideally include a sensitivity analysis across a range of functional forms (e.g., sublinear, concave, or stepwise) to show how the conclusion depends on the shape of σ(k).
  2. [§2.2, §3.2, Figures 5-6] The dependent-error simulations only vary the number of response categories from 2 to 20, yet the abstract and conclusion extrapolate the 'drastic decrease' to VAS scales with 0-100 response points. This extrapolation assumes that linear error growth continues unchanged from k=20 to k=100, which is an untested assumption. The authors should either simulate the full 2-100 range under their dependency rules, or explicitly restrict the practical claim to the range of categories actually simulated and discuss what additional evidence would be needed to extend it to VAS.
  3. [§2.2, all figures] The number of Monte Carlo replications is never reported, and none of the figures include error bars or confidence bands. For a study whose central claim is the location of an optimum and the existence of a decline beyond it, the reader cannot distinguish a real turning point from Monte Carlo noise. The authors should report the number of replications and add uncertainty estimates (e.g., standard errors or confidence bands across replications) to Figures 1-6, or at least report the Monte Carlo standard error for the reported optima.
minor comments (4)
  1. [§1.1] The text states there is a one-to-one relationship between σ_j and the GRM discrimination parameter but does not provide the analytic mapping; giving the formula would improve reproducibility and help readers interpret the error sequences.
  2. [§2.2] The independent-error condition sets σ_j to values from 0.1 to 1.0, but Figure 1's legend shows only 0.25, 0.50, 0.75, 1.00; clarify whether these are the full set of levels or a selected subset, and if the latter, why the subset was chosen.
  3. [§3.2, Figure 6] The text says the SE for large dependency 'rises again after approximately five response options,' but Figure 6 appears to show the increase beginning around four categories; please align the description with the plotted results.
  4. [General] The manuscript references Supplementary Materials for regression-coefficient bias figures but does not state how to access them; please include a data/code availability statement.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the reported optima are conditional simulation outcomes, and the paper explicitly disclaims design-dependence.

full rationale

The paper's central 'optimum' results are obtained by Monte Carlo simulation under an explicitly stated data-generating process. Section 2.2 fixes sigma_j as a linear function of the number of categories (small 0.05-0.5, medium 0.1-1.0, large 0.2-2.0). That is an input assumption, not a fitted parameter, and the reliability optima (7, 5, and 4) are legitimate derived outcomes of that model rather than a renaming of the input. The conclusion that VAS conversion would decrease reliability is explicitly conditional: the abstract says 'If measurement error increases with the number of response categories,' and Section 5 states 'these optima are highly dependent on our simulation design.' The question of whether the assumed error-growth relationship is realistic or empirically calibrated is a validity/external-evidence concern, not a circularity concern under the rubric. The only self-citation (Schmidt, 2010) is used as supporting literature for the general point that reducing response categories can improve measurement; it is not load-bearing for the simulation or the central derivation. No self-definitional equivalence, fitted-input-as-prediction, imported uniqueness, or ansatz-via-citation was found.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests on standard GRM assumptions plus the authors' chosen linear error-dependency mappings. No new entities are introduced; the item variable is a standard Lord-Novick construction. The main burden is that the headline optima are direct outputs of the assumed dependency slopes, with no independent empirical calibration.

free parameters (4)
  • Error-dependency slopes (small/medium/large) = 0.025, 0.05, and 0.1 per additional category over categories 2 to 20
    Chosen by authors to define small, medium, and large dependency; they directly determine the reported optima of 7, 5, and 4 categories. No empirical calibration is provided.
  • Item measurement error sigma_j grid = 0.1 to 1.0 in increments of 0.1
    Design values used for the independent-error condition; they affect plateau height but not the qualitative shape of results.
  • Threshold span and spacing = Evenly spaced thresholds in [-2, 2]; binary threshold 0
    Choice affects how many categories are needed to approximate the latent trait and therefore where the plateau appears.
  • Predictor relation constants = Regression slope 0.5, residual SD 0.2
    Fixed simulation constants for the third-variable regression; they influence SE magnitudes but not the main ordinal-category conclusions.
assumptions (6)
  • domain assumption Latent trait theta_i follows N(0,1) and item variable gamma_ij follows N(theta_i, sigma_j)
    Foundation of item variable construction from Lord and Novick, used to generate data in Section 2.1.
  • standard math Response category is determined by thresholds on gamma_ij with probability 1, producing an ogive graded response model
    Samejima's GRM assumption, used in the ICC equation in Section 1.1.
  • domain assumption Measurement error is independent of the number of response categories in the first simulation set
    Baseline condition; the authors acknowledge this is a theoretical case and question its realism.
  • ad hoc to paper Measurement error increases linearly with the number of response categories in the second simulation set
    Central premise; the linear mapping is constructed in Section 2.2 with no empirical estimate, yet it drives all dependent-error findings.
  • domain assumption Unidimensionality and a stable trait within a measurement occasion
    Assumed by the simulation model; the limitations section admits trait fluctuation is not modeled.
  • domain assumption Composite or mean scoring of items is used rather than model-based scores
    Items are averaged before regression; the authors assert CFA or IRT would not change conclusions but do not demonstrate this.

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Cite this review

Pith. "Pith review of Don't Let Your Likert Scales Grow Up To Be Visual Analog Scales: Understanding the Relationship Between Number of Response Categories and Measurement Error." pith.science (2026). https://pith.science/paper/KPYTVCUN

@misc{pith2026250202846,
  author       = {Pith},
  title        = {Pith review of: Don't Let Your Likert Scales Grow Up To Be Visual Analog Scales: Understanding the Relationship Between Number of Response Categories and Measurement Error},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KPYTVCUN}},
  note         = {Machine review of arXiv:2502.02846}
}
read the original abstract

The use of Visual Analog Scales (VAS), which can be broadly conceptualized as items where the response scale is 0-100, has surged recently due to the convenience of digital assessments. However, there is no consensus as to whether the use of VAS scales is optimal in a measurement sense. Put differently, in the 90+ years since Likert introduced his eponymous scale, the field does not know how to determine the optimal number of response options for a given item. In the current work, we investigate the optimal number of response categories using a series of simulations. We find that when the measurement error of an item is not dependent on the number of response categories, there is no true optimum; rather, reliability increases with number of response options and then plateaus. However, under the more realistic assumption that the measurement error of an item increases with the number of response categories, we find a clear optimum that depends on the rate of that increase. If measurement error increases with the number of response categories, then conversion of any Likert scale item to VAS will result in a drastic decrease in reliability. Finally, if researchers do want to change the response scale of a validated measure, they must re-validate the new measure as the measurement error of the scale is likely to change.

Figures

Figures reproduced from arXiv: 2502.02846 by the authors.

Figure 1
Figure 1. True - Estimated Spearman’s Correlations. This plot illustrates the estimated Spearman’s [PITH_FULL_IMAGE:figures/full_fig_p007_1.png] view at source ↗
Figure 2
Figure 2. Delta - Spearman’s Correlations with True Score. This plot illustrates the changes in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Standard Error of The Estimated Regression Coefficient. This plot shows the standard [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Delta - Standard Errors. This plot illustrates the change in the standard error (SE) of the [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: True - Estimated Spearman’s Correlations When Measurement Errors Are Dependent On [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: Standard Error of The Estimated Regression Coefficient When Measurement Errors Are De [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

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Pith tools

Reviewed August 9, 2026 · model on record in the stance chip above.