# Modeling and correction of sensitivity thresholds determined by best EstimateThreshold (BET)

Caroline Peltier, Alix Rollinat, Christophe Martin

<figure><img src="image_1.png" /></figure>

## To cite this version:

Caroline Peltier, Alix Rollinat, Christophe Martin. Modeling and correction of sensitivity thresholds determined by best EstimateThreshold (BET). Food Quality and Preference, 2026, 135, pp.105690. 10.1016/j.foodqual.2025.105690. hal-05273662

HAL Id: hal-05273662

https://hal.inrae.fr/hal-05273662v1

Submitted on 17 Oct 2025

HAL is a multi-disciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

<figure><img src="image_2.png" /></figure>
<figure><img src="image_3.png" /></figure>

# Modeling and correction of sensitivity thresholds determined by best EstimateThreshold (BET)☆

Caroline Peltier<sup>a,b,*</sup>, Alix Rollinat<sup>a,b</sup>, Christophe Martin<sup>a,b</sup>

## ARTICLE INFO

**Keywords:**

BET

Sensitivity threshold

## ABSTRACT

The r-Alternative Forced Choice (r-AFC) test is a test of discrimination in which the subject is presented with three samples, one of which is a test sample containing a nominated stimulus (test sample), the other being references. The subject is instructed to indicate the test sample. Taste and odor sensitivity thresholds are frequently determined using successive r-AFC tests with stimuli in increasing concentrations. The Best Estimate Threshold (BET) method consists in using successive 3-AFC with increasing concentrations to estimate sensitivity threshold. Then, the threshold is estimated using the geometrical mean of the highest concentration that caused an error and the concentration directly below it. However, a subject who feels no difference between the samples may give a correct answer by chance. It leads to consequent potential bias in the determination of the sensitivity thresholds.

This paper aims to formalize and model the thresholds obtained in successive r-AFC in order to quantify the errors inherent in such protocols. It establishes that, when you assumed that the distribution of the true sensitivity threshold is known in the population, the threshold obtained by r-AFC can be modelled with a variable following a specific probability law.

This paper presents the theory of this model, then illustrate it with simulations and application on a real dataset. An R package dedicated to these analyses, AFCR, was also created and is available on github (http://github.com/ChemoSens/AFCR). Therefore, sensory scientists could use the package as a help to set up their sensory protocol or-to analyze their own data.

## 1. Introduction

The detection threshold (DT) is defined as the smallest amount of a substance or the smallest intensity of a stimulus that can be detected with a 50 % probability (ASTM, 2019). For ISO 13301:2018, the detection threshold is the lowest concentration of a substance, or the lowest intensity of a stimulus, that a subject can perceive in at least 50 % of tests. Widely speaking, the detection threshold can be seen as a level below which no sensation would be produced by a stimulus and above which a sensation would reach consciousness (Lawless & Heymann, 1999).

The notion of DT serves as an informative benchmark of when a taste or aroma material will have an effect on consumer perception of that taste or flavor (Hough et al., 2013). It can be related to several factors including physiological factors such as age (Methven et al., 2012; Mojet

et al., 2005), sex (Martin & Neyraud, 2021), genetics (Zhu et al., 2023), microbiota (Licandro et al., 2023), but also other factors such as food habits like vegetarianism (Nuvoli et al., 2023 and Jalil Mozdehi et al., 2021).

DT can be determined with the up-down staircase technique, a procedure in which the next stimulus intensity level to be tested depends on detection or non-detection at the previous interval. Thus, the subject will track around the threshold level, ascending the intensity of the stimuli when performance is incorrect (non-detection is the response) and descending in intensity when performance is correct (or detection is indicated) (Lawless & Heymann, 1999). This procedure has several issues, such as the fatigue of the subject and a bias of anticipation. Furthermore, it is a time-consuming procedure that is impossible to be conducted on a large number of subjects. To get around this problem, later workers introduced a forced-choice element to the trials at each

<footnote><sup>a</sup> Centre des Sciences du Goût et de l'Alimentation, CNRS, INRAE, L'Institut Agro, Université Bourgogne, F-21000 Dijon, France</footnote>

<footnote><sup>b</sup> CNRS, INRAE, PROBE Research Infrastructure, Chemosens Facility, F-21000 Dijon, France</footnote>

<footnote>* This article is part of a Special issue entitled: 'Sensometrics Conference 2024 (Invite Only)' published in Food Quality and Preference.</footnote>

<footnote>* Corresponding author at: Centre des Sciences du Goût et de l'Alimentation, CNRS, INRAE, L'Institut Agro, Université Bourgogne, F-21000 Dijon, France.
E-mail address: caroline.peltier@inrae.fr (C. Peltier).</footnote>
concentration step. A typical example is the ascending series of 3-Alternated Forced Choice tests, recognized as standard for fast threshold testing (ASTM, 2019).

The 3-Alternative Forced Choice (3-AFC) test is a test of discrimination in which the subject is presented with three samples, one of which is a test sample containing a nominated stimulus, the other being references, and where the subject is instructed to indicate the test sample.

According to the Best Estimate Threshold (BET) method, successive 3-AFC tests with stimuli in increasing concentrations are conducted, then the sensory threshold is estimated using the geometrical mean of the highest concentration that caused an error and the concentration directly below it as presented in ASTM standard E679-19 (last reversal rule). This ASTM standard also indicates that samples are prepared in 5-8 concentration steps, each differing by a multiplicative factor (e.g., for a factor of 3: x/27, x/9, x/3, x, 3x, 9x, 27x). Today, the BET is the most recommended method in methodological manuals.

However, there exist some drawbacks of the 3-AFC data treatment that were analyzed by Lawless (2010). A first drawback could be that there is no consideration for the fact that a subject might detect on early trials (at low concentration in the series) but become adapted, fatigued or overwhelmed at the higher concentration (not much above their individual detection threshold) and miss the target. A major second drawback is no formal consideration of the fact that a subject can be correct by guessing. To account for the guessing issue, Lawless (2010) presented an alternative way to estimate group thresholds. At each concentration step, the percent correct for the group as a whole is known, so a plot of the percent correct versus log concentration can be used to interpolate the 50 % detection level. Because the task is forced-choice, a subject can be correct by guessing so a correction for guessing must be employed:

$$p_D = \frac{p_{\text{obs}} - p_{\text{chance}}}{1 - p_{\text{chance}}} \qquad (1)$$

where $p_D$ is the proportion of discriminators, $p_{\text{obs}}$ is the proportion of correct answers and $p_{\text{chance}}$ is the probability of getting a correct answer by chance (0.33 for 3-AFC test). The second step was to calculate the proportion of discriminators for each concentration, then to fit a logistic model on it $\ln(\frac{p_D}{1-p_D}) = a + b \ln(\text{Concentration})$ for finally identifying the group threshold as the log-concentration where $p_D=0.5$.

Hough et al. (2013) pointed out a limitation of this approach: in the regression, only the proportion of correct responses is considered. As a result, the responses of a given subject are no longer treated as a connected sequence, even though this continuity is central to threshold estimation. In other words, the fact that a subject consistently gives correct (or incorrect) responses at higher concentrations is not taken into account. They proposed an improved method in which a first step consists in adjusting responses considering they may have been obtained by chance, using an algorithm with random replacements. This algorithm was not based on an explicit model of the BET variable and can be difficult to conduct in practice.

To our knowledge, the individual BET threshold was not modelled as a random variable yet. This paper aims to present a new theoretical framework for analyzing BET results based on a probabilistic model and is structured as follows. First, the models are presented, including a probabilistic model of the BET distribution given the 'true threshold' distribution, and a correction procedure for the BET distribution based on this model. Then, these methods are applied and illustrated through practical cases involving both simulated and real data.

## 2. Mathematical background

### 2.1. Preliminary remarks and notations

This paper is dedicated to BET results but can be extended to K

successive r-AFC tests where K is the number of concentrations to be tested and r is the number of samples tested at each AFC test ($r = 3$ for 3-AFC and BET). In this paper, we consider that K concentrations $c_1 < ... < c_K$ were successively tested. In BET, these concentrations are chosen such as $c_{k+1} = A^*c_k$ where A is a multiplicative constant. We consider also two virtual concentrations that are not presented to the subjects but used in the threshold calculations: $c_0 = \frac{c_1}{A}$ and $c_{K+1} = A^*c_K$. Fig. 1a represents such an experimentation with A = 2 and $c_1 = 1$.

As a first remark, the assumption of this paper is that, at a given time, there is a 'true sensitivity threshold'. The existence of a concentration limit where the savor/aroma is not perceived anymore can be discussed. Indeed, a sensory threshold can vary with time for the same subject (Lawless & Heymann, 1999). However, the same authors claim that the measured threshold remains an appealing and useful concept to many workers involved in sensory assessments. This hypothesis of the existence of a 'true threshold' is the basis of threshold research when conducting BET. In this paper, we differentiate what we called the "true sensory threshold", noted $T_{\text{true}}$, and the threshold obtained in a BET experimentation, noted $T_{\text{BET}}$. Both of them can be expressed as a concentration.

As a second remark, we would like to highlight that the individual BET does not allow to reach continuous values but returns only a small number of discrete values that poorly estimate the threshold of a subject. For example, if a subject missed the $k^{th}$ test, but succeeds all higher concentrations, it could be supposed that their true threshold is in $|c_k; c_{k+1}| = I_k$. The BET consists in calculating $t_k$ as the geometric mean between $c_k$ and $c_{k+1}$: a numeric value that seems very precise (especially more precise that the interval $I_k$). However, this value only is a representative of this interval, but in the facts, the position of the 'true' sensory threshold of the subject is in this interval is unknown: the 'true' threshold could as well be $c_k + 0.001$ or $c_{k+1}$. In practice, almost nothing in this paper would change by replacing the BET estimation by any other estimation in $I_k$ (even by a qualitative ordinal one). Therefore, these representations of concentration intervals are noted $t_{c_k c_{k+1}}$ in the paper. They correspond to $t_{c_k c_{k+1}} = \sqrt{c_k c_{k+1}}$ for BET. This remark justifies that the BET is related to a discrete random variable and why discrete probability distributions were used in the remainder of this text. Fig. 1b illustrates the threshold obtained with BET with 6 concentrations.

### 2.2. A probabilistic model for the BET distribution knowing the 'true threshold' distribution

Let's assume that a subject had a true sensitivity threshold higher than the highest concentration tested in the experiment. In other terms, he does not feel the aroma/taste within the range of concentrations presented. In this case, he would randomly answer all the r-AFC tests with a probability to give the good answer of $q = 1/r$.

Therefore, the distribution of probability of $T_{\text{BET}}$ would be:

$$\forall k \in 1, ..., K, P(T_{\text{BET}} = t_{c_k c_{k+1}}) = q^{k-k} p$$

$$P(T_{\text{BET}} = t_{c_0 c_1}) = q^K$$

Fig. 1c. illustrates the probabilities of $T_{\text{BET}}$ when a subject answers randomly. As expected, the sum of these probabilities is 1.

$$q^K + \sum_{k=1}^{K} q^{K-k} p = q^K + p \sum_{0}^{K-1} q^i = q^K + p \frac{1-q^K}{1-q} = q^K + 1 - q^K = 1$$

Therefore, a table of the probability of $T_{\text{BET}}$ knowing $T_{\text{true}}$ can be obtained (table 1).

This table can be stored in a matrix noted M, such as:
* The second one (2) assuming that the true detection thresholds are normally distributed. This situation illustrates a common assumption of sensory sensitivity heterogeneity within the population: responses are typically centered around a mean value, with a few individuals being either hypo- or hypersensitive.

In both situations, the true detection threshold and the obtained BET distributions were plotted for illustrating the risks associated with random response bias.

## 3.2. Application on astringency dataset

A first dataset used to illustrate our modeling on real data concerned astringency thresholds. 193 subjects participated in a BET experimentation on eight increasing astringency concentrations ($c_1 = 0.015$ g/L; $c_2 = 0.027$ g/L; $c_3 = 0.05$ g/L; $c_4 = 0.091$ g/L; $c_5 = 0.166$ g/L; $c_6 = 0.302$ g/L; $c_7 = 0.55$ g/L, $c_8 = 1.001$ g/L). The concentrations were chosen according to ASTM (ASTM, 2019) assessing that the increase in concentration of the test substance per scale step should be by a constant factor (≈1.82 here). It means that the differences between two successive log-concentrations are constant. Therefore, we consider that $c_0 = 0.008$ g/L and $c_9 = 1.822$ g/L. Here, the difference between two successive log-concentrations is 0.26 (log in base 10). This number will be useful to compare the difference between two-log concentrations and the difference between true and BET thresholds. The distribution of sensitivity threshold and the group thresholds were calculated for our method with Eq. 4, then with Lawless's method and Hough's method.

## 3.3. Application on TOM dataset

A second dataset used to illustrate our modeling on real data concerned taste thresholds. In a research project called Taste and Oral Microbiota (TOM, Martin & Neyraud, 2021), 82 participants took part in a study on sensitivity to the five basic tastes. Fructose for sweet taste (CAS: 57–48-7, Cooper, France), sodium chloride for salty taste (NaCl, CAS: 7647-14-5, Cooper, France), citric acid for sour taste (CAS: 5949-29-1, Cooper, France), quinine hydrochloride for bitter taste (Quinine HCl, CAS: 6119-47-7, Merck KGaA, Germany), and monosodium glutamate for umami taste (MSG, CAS: 6106-04-3, Merck KGaA, Germany) were used as the prototypical taste compounds.

The sensitivity test, called T@sty, involved edible test sheets, each designed to measure sensitivity to a specific taste. Each sheet contained six sets of three pre-cut, detachable discs, which were placed on the tongue to perform the measurements. The final surface concentrations of the discs were as follows: 2.1, 2.7, 4.1, 7.1, 16.1, and 35.4 µg/cm² (fructose); 4.9, 5.2, 5.6, 6.7, 8.8, and 12.6 µg/cm² (NaCl); 3.6, 4.1, 5.6, 9.5, 17.1, and 24.7 µg/cm² (citric acid); 2.9, 9.5, 19.2, 48.8, 98.3, and 133.5 ng/cm² (quinine HCl); and 1.1, 1.6, 3.7, 9, 19.6 and 29.6 µg/cm² (MSG). We used here the first replicate of the dataset to estimate the distribution of 'true' sensitivity threshold and the group thresholds for our method (Eq. 4), Lawless's method and Hough's method.

## 3.4. Computational aspects

### 3.4.1. Computational environment and parameters

Computations were made with R (4.3.0). The R package AFCR contains all the functions described in this paper and is freely available on github (https://github.com/ChemoSens/AFCR). This package contains all the simulation functions presented in this paper, but also allows to automatically compute group thresholds according to Lawless (2010) or Hough et al. (2013).

The code used to reproduce the results of the paper is available on (https://github.com/ChemoSens/ExternalCode/AFCR_Results).

### 3.4.2. Implementation of algorithms for group thresholds

The Lawless formula (Eq. 1) returns the discriminators' proportion and was implemented in the R package with the logistic regression that follows (see Introduction). Hough's correction was computed as presented in algorithm 1 (see below). For a given subject, we consider the first concentration after which the subject gives only correct answers (among increasing concentrations). This answer can be correct by chance (1/3 in 3-AFC case). In this case, a Bernoulli trial ($p = p_D$) is conducted and replaces the original obtained value. The same process is conducted considering the following concentration, with another Bernoulli trial ($p = p_D$). Finally, the averaged threshold is estimated with a linear model of the resulting proportion of discriminators according to the log-concentrations (base 10). These thresholds are estimated taking a proportion of discriminators of 0.5.

**Algorithm 1.** Algorithm for Hough correction

houghCorrection = function (triangularMatrix)
  I <- number of rows of triangularMatrix
  J <- number of columns of triangularMatrix
  continue=TRUE
  p_D <- the proportion of discriminators for each concentration according to Lawless formula
  for each line i of triangularMatrix (corresponding to one subject), do
    while j<J and continue=TRUE, do
      if triangularMatrix[i,j] == 1
        if all tests for the concentrations higher than j are successful
          triangularMatrix[i,j] <- Bernoulli(p_D[j])
          continue=FALSE
        if (j<J)
          triangularMatrix[i,j+1] <- Bernoulli(p_D[j]^2)
        else
          triangularMatrix[i,j] <- Bernoulli(p_D[j])
    return(triangularMatrix)
<figure><img src="image_4.png" /><figcaption>**Fig. 2.** Distribution of true threshold when everybody has the same threshold in $[c_3; c_4]$ (Fig. 2a) and related estimation of BET distribution (Fig. 2b). Fig. 2c represents a normal true threshold distribution while Fig. 2d represents the related estimation of BET distribution in this situation.</figcaption></figure>

# 4. Results

## 4.1. Simulation of the BET distributions in two situations

As presented in Fig. 2a and b, the thresholds of around a third of subjects were underestimated in the first situation. For almost 20 %, the threshold was underestimated by one concentration. Finally, for about 10 % of subjects, the threshold was underestimated by two or more concentrations. This bias is particularly problematic when the distribution of individual thresholds is used to select subjects who are more sensitive than others. Indeed, all the subjects whose BET results are lower than $c_3$ are false positives: they just answer correctly by chance. Regarding the second situation (Fig. 2c and d), if the panel leader decides to select a ‘more sensitive group’ with subjects who have a BET strictly lower than $c_2$, this group would contain 25 % of the panel (0.11 + 0.14 = 0.25). However, considering the true sensitivity scores, only 13 % (0.04 + 0.09) of the population have a BET strictly lower than $c_2$. This implies that if the original panel contains 100 subjects, the selected group will contain 25 subjects, but only 13 of them will have a true sensitivity threshold lower than $c_2$, which is almost half of the selected group. Similarly, if the panel leader selects only the group with a BET strictly lower than $c_1$ (11 subjects), only 4 of them will have a true threshold strictly lower than $c_1$.

## 4.2. Application on the astringency dataset

In accordance with ASTM E679-19, the average panel threshold was determined by taking the geometric mean of all individual BETs and

returned a concentration of 0.11 g/L for the BET and 0.15 g/L for the estimated corrected threshold. To illustrate this amplitude of difference in a multiplicative context (geometrical mean), the difference of the logarithm of these two quantities was calculated as 0.13 while the difference between two consecutive log-concentrations was 0.26.

The obtained BET distribution was plotted on Fig. 3a. The distribution of true thresholds was inferred in the population according to the formula presented in 2.3 (Eq. 3) and was plotted on Fig. 3b. In this case, three intervals of BET have a high and similar number of subjects: $[c_3; c_4]$, $[c_4; c_5]$, $[c_5; c_6]$, indicating variability in the sensitivity of astringency in the panel. Indeed, as seen in the previous section, if there was no variability in the sensitivity, around 66 % of the subjects would have the same BET as presented in the application of 4.1.

The analysis of the same data returns 0.12 g/L with Lawless technique. Regarding the Hough analysis, including random variations, 100 simulations were conducted, giving values comprised between 0.1569 g/L and 0.1769 g/L with an average of 0.165 g/L and a standard deviation of 0.004 (estimation of the 50 % of discriminators by linear interpolation). Our estimation of averaged threshold is between the Lawless and the Hough ones.

The objective of the presented study was not to calculate average threshold but to identify groups of sensitivity to astringency. Unfortunately, the left extreme group: $[c_0; c_1]$, $[c_1; c_2]$ is more likely to be due to false positive than to true threshold lower than $c_2$. Indeed, this group of subjects contains only 3 % subjects of the panel (193 subjects) after correction instead of 8 % as induced by the Fig. 3. The panel leader should be therefore be cautious in the interpretation of 'group of higher sensitivity than the panel'.

<figure><img src="image_5.png" /><figcaption>**Fig. 3.** Observed BET threshold distribution in proportion (a) and estimated corrected threshold distribution (b).</figcaption></figure>
<caption>**Table 2**</caption>

Distribution of the observed and corrected BET for each taste. The group threshold was calculated with geometrical average for both cases (GM), with Lawless approach (L) and Hough estimate (H).

<table><thead><tr><th></th><th></th><th>c0: c1</th><th>c1: c2</th><th>c2: c3</th><th>c3: c4</th><th>c4: c5</th><th>c5: c6</th><th>c6: c7</th><th>GM</th><th>L</th><th>H</th></tr></thead><tbody><tr><td rowspan="2">Bitter</td><td>observed</td><td>8</td><td>7</td><td>13</td><td>22</td><td>20</td><td>15</td><td>15</td><td>36.44</td><td>24.06</td><td>31.88</td></tr><tr><td>corrected</td><td>4.5</td><td>4</td><td>8.5</td><td>23</td><td>22.5</td><td>15</td><td>22.5</td><td>50.19</td><td></td><td></td></tr><tr><td rowspan="2">Sweet</td><td>observed</td><td>25</td><td>18</td><td>15</td><td>15</td><td>16</td><td>6</td><td>5</td><td>4.24</td><td>7.06</td><td>8.69</td></tr><tr><td>corrected</td><td>16</td><td>19.5</td><td>15</td><td>14.5</td><td>21</td><td>6.5</td><td>7.5</td><td>5.21</td><td></td><td></td></tr><tr><td rowspan="2">Sour</td><td>observed</td><td>12</td><td>16</td><td>15</td><td>15</td><td>17</td><td>10</td><td>15</td><td>8.36</td><td>8.49</td><td>9.81</td></tr><tr><td>corrected</td><td>4</td><td>16.5</td><td>15</td><td>14</td><td>20.5</td><td>7.5</td><td>22.5</td><td>9.88</td><td></td><td></td></tr><tr><td rowspan="2">Salty</td><td>observed</td><td>18</td><td>20</td><td>19</td><td>20</td><td>18</td><td>3</td><td>2</td><td>5.77</td><td>7.07</td><td>7.78</td></tr><tr><td>corrected</td><td>8</td><td>20.5</td><td>18.5</td><td>21</td><td>25</td><td>3.5</td><td>3</td><td>6.17</td><td></td><td></td></tr><tr><td rowspan="2">Umami</td><td>observed</td><td>18</td><td>14</td><td>19</td><td>19</td><td>8</td><td>18</td><td>4</td><td>4.45</td><td>5.60</td><td>7.11</td></tr><tr><td>corrected</td><td>11</td><td>11.5</td><td>19</td><td>24.5</td><td>3</td><td>25</td><td>6</td><td>5.92</td><td></td><td></td></tr></tbody></table>

### 4.3. Application on the TOM dataset

Table 2 shows that the distributions for all tastes was affected by the BET correction. For example, among the subjects whose thresholds fell within the interval $|c_0: c_1|$ for Bitter, nearly half were potential false positive (3.5/8). This tendency was even more pronounced for other tastes such as Sour (8/12 false positives). Regarding the estimation of group threshold, as expected, the average concentration was consistently higher for corrected values. Similarly, the Hough thresholds were systematically higher than Lawless thresholds. However, no clear trend was observed between our approach and Hough's results: in some cases, they were similar; in others, they differed, and not in a consistent direction.

## 5. Discussion

### 5.1. Relations with other sensitivity thresholds

The approach of this paper was to correct the distribution of the BET threshold using BET as a random variable. This approach is different from the ones proposed by the literature (Hough et al., 2013; Lawless, 2010). Regarding the Lawless approach, it considers only the proportion of correct answers without considering the individual BET results. Regarding the Hough approach, it consists in randomly modifying a result (success or failure) for some concentration just below the BET based on an iterative algorithm. This approach does not consider potential two successive successes by chance which occurs 1 on 9 and considers the proportion of discriminators in the imputation. This algorithm was not available for the community and was implemented in AFCR package. Finally, this algorithm is mainly empirical (based on the proportion of real discriminators) while our approach is mainly theoretical. In practice, the Hough approach and the BET correction presented in this paper are not equivalent. It was expected since they do not rely on the same rationale.

Including both approaches in the package will allow further comparisons on other datasets and help to understand the differences. Piornos et al. (2019) previously compared the Hough approach with BET methods for calculating group thresholds. Similar comparisons including our approach could be conducted in the future. It would also be relevant to design a study comparing the corrected threshold distributions with those obtained using the staircase method. Theoretically, this would reduce false positives and produce results closer to the corrected distribution than the original BET distribution.

### 5.2. Interest of simulations

This model also allows different protocols to be theoretically compared with simulations. In the paper, we presented the case of homogeneous true threshold, but an infinity of other a priori distributions could be tested. This theoretical a priori distribution $P(T_{true} \in [c_i; c_{i+1}])$ is based on two elements (i) the continuous distribution of the true

threshold in the population (ii) the chosen test concentrations.

The question of the continuous distribution of the true threshold in the population is a difficult question but is always underlying while selecting the concentrations in a BET. By the recommendations presented in ASTM, the true concentration threshold is supposed to be distributed following a log-normal distribution whose characteristics (average and standard deviations) are to be determined with the literature and/or pre-tests.

In practice, these distributions are often unknown and poorly estimated. If no prior information is available, we recommend conducting pre-tests, for example using the staircase method. These pre-tests would allow both average and standard deviations to be estimated and would also be an opportunity to determine appropriate initial and final concentrations for the BET.

As the trend towards data FAIRification continues, an increasing amount of sensory threshold data may become accessible. Ideally, a framework should be established to integrate all these data sources and facilitate comprehensive querying. Achieving this integration would require standardizing experimental practices.

Regarding the second element, and assuming known the first one, the choice of the number of concentrations to be used, the minimal and maximal concentration and the multiplicative factor directly impact the resulting BET distribution. The simulation tool could give theoretical answers to very practical questions such as “If the theoretical distribution of threshold follows this log-normal law, what BET results would be expected with my choices of concentrations? In this purpose, the following steps should be followed (i) Formalizing one theoretical true distribution and a set of concentrations (ii) inferring $P(T_{true})$ from the supposed distribution and concentrations (iii) use the tool with these parameters (iv) observe the obtained distribution $P(T_{BET})$ without any experiment.

### 5.3. Adding a possibility to fail when deconcentrated

The model presented in this paper is based on the strong assumption that a subject with a true threshold lower than a given concentration will always give the good answer. Alternative distributions could also be considered by including noise in the model. In this case, the probability of the threshold obtained with BET knowing the sensory threshold is not systematically null when the sensory threshold is lower than the observed one. Many options of modeling the probability to fail for a concentration higher than the true threshold could be considered. The matrix below illustrates an example of distribution considering that an unconcentrated subject can fail to the concentration higher than their sensitivity threshold with a probability of $\epsilon$. For $\epsilon = 0$, this matrix is exactly equivalent to the model presented in this paper. Consequently, this presents a potential generalization of the model, that was not tested in this paper.
$$M = \begin{pmatrix} 1 - \varepsilon & \varepsilon & 0 & \dots & 0 & 0 \\ q(1 - \varepsilon) & p(1 - \varepsilon) & \varepsilon & \dots & 0 & 0 \\ q^2(1 - \varepsilon) & pq(1 - \varepsilon) & p(1 - \varepsilon) & \dots & 0 & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots & \vdots \\ q^{K-1}(1 - \varepsilon) & pq^{K-2}(1 - \varepsilon) & \dots & \dots & p & \varepsilon \\ q^K & pq^{K-1} & \dots & \dots & pq & p \end{pmatrix}$$

Indeed, the estimation of $\varepsilon$ would require further assumptions (based on psychological knowledge) or experimentations (knowing a true threshold, quantifying the number of mistakes when the concentration is higher than the true threshold, previously determined by staircase method).

Other models of the probability of the threshold obtained with BET knowing the sensory threshold could also be implemented: with a constant probability to fail whatever the concentration with a probability to fail that decreases when the concentration increases. We hope that this paper and the related available package will lead to further studies in this domain.

## 5.4. Experimental perspectives for sensitivity thresholds

The BET correction presented in this paper does not adjust individual thresholds: it only corrects the distribution probabilistically. If the panel leader is interested in individual BETs, other options should be considered, such as adding replicates or using more sophisticated protocols. For example, during the BET experiment, the subject can indicate the confidence in their choice (Kim et al., 2020, Kim et al., 2011; Lawless et al., 2000) or the intensity of their perception on a scale. Further works will focus on these perspectives (Martin et al., 2025).

# 6. Conclusion

This paper presents a new tool to help the sensory analyst to model a

# Appendix A. Relations with the geometric truncated law

The geometric distribution is a discrete probability indicating the probability distribution of the number $X$ of Bernoulli trials (with $p_{Ber}$ parameter) required to get one success. It is defined for a potential infinity of trials. Besides, the truncated geometric distribution of parameter $K$ and $p_{Ber}$ is close to this distribution except that a limited of number of trials ($K$) is considered. In this case the probability of $X = k$ is:

$$\forall k \in 1, \dots, K, P(X = k) = (1 - p_{Ber})^{k-1} p_{Ber} = (q_{Ber})^{k-1} p_{Ber}$$

Conventionally, if no success is observed, $X = 0$. and so, $P(X = 0) = (1 - p_{Ber})^K = q_{Ber}^K$.

The related expectation of this law is

$$E(X) = \frac{1}{p_{Ber}} (1 - (Kp_{Ber} + 1)(1 - p_{Ber})^K)$$

This truncated probability law seems to be very close to the distribution obtained in 2.3.

$$\forall k \in 1, \dots, K, P(T_{BET} = t_{c_k c_{k+1}}) = q^{K-k} p$$

$$P(T_{BET} = t_{c_0 c_1}) = q^K$$

Indeed, let's consider the function $f: t_{c_k c_{k+1}} \to K - k + 1$ and $U = f(T_{BET})$. Thus, considering the decreasing $c_K > \dots > c_1$ concentrations U is the rank of the first mistake and takes values in $\{1, \dots, K + 1\}$.

In this case,

$$\forall k \in 1, \dots, K, P(U = K - k + 1) = q^{K-k} p$$

$$P(U = K + 1) = q^K$$

Making a variable change ($j = K - k + 1$) gives:

$$\forall j \in 1, \dots, K, P(U = j) = q^{j-1} p$$

$$P(U = K + 1) = q^K$$

Consequently, $U = f(T_{BET})$ almost follows a geometrical truncated law. The only difference is that

$$P(U = K + 1) = q^K$$

BET experiment. It allows an estimation of the distribution of the true threshold to be obtained from BET results. This model could be extended to include inattention or fatigue mistakes. Therefore, the panel leader can quantify the errors produced induced by any BET protocol and react accordingly. The R package AFCR containing the code is available online.

# Credit authorship contribution statement

Caroline Peltier: Writing – original draft, Software, Methodology, Formal analysis, Conceptualization. Alix Rollinat: Writing – review & editing, Methodology, Investigation, Conceptualization. Christophe Martin: Writing – review & editing, Methodology, Conceptualization.

# Ethical statements

As this study is based on statistical methodology, no ethical approval was required.

Note that the real dataset used for illustration received the approval of the Ethics Committee for Research (CPP Ile de France I 23.01785.000230–2023-A00291–44) and was conducted according to the French law (loi Jardé) and Declaration of Helsinki.”

# Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Instead of the conventional

$$P(U = 0) = q^K$$

Consequently, this convention of truncated geometrical law is modified: instead of having X with values in {0, ..., K} with the convention $P(X = 0) = q^K$; a variable U that has values in {1, ..., K + 1} is considered, with the convention $P(U=K + 1) = q^K$.

Therefore, considering the Bernoulli trial ‘fail to an AFC-test’ with a probability $p = 1-1/p$; the variable U follows a modified truncated geometric law with parameter p and K.

This changes the value of expectation of U to

$$E(U) = \frac{1}{p} (1 - (Kp + 1)q^K) + q^K(K + 1)$$

$$E(U) = \frac{1}{p} - \left( K + \frac{1}{p} \right) q^K + q^K(K + 1) = \frac{1}{p} (1 - q^K) + q^K$$

Consequently, considering the change of variable $U = f(T_{BET})$, the probabilistic law of U is a modified geometric truncated law whose support is {1, ..., K + 1} instead of {0, ..., K}.

## Data availability

Data will be made available on request.

## References

Hough, G., Methven, L., & Lawless, H. T. (2013). Survival analysis statistics applied to threshold data obtained from the ascending forced-choice method of Limits. *Journal of Sensory Studies*, 28, 414–421. https://doi.org/10.1111/joss.12067

Jalil Mozhdehi, F., Abeywickrema, S., Bremer, P. J., & Peng, M. (2021). Comparing taste detection thresholds across individuals following vegan, vegetarian, or omnivore diets. *Foods*, 10, 2704. https://doi.org/10.3390/foods10112704

Kim, M. K., Drake, S. L., & Drake, M. A. (2011). Evaluation of key flavor compounds in reduced- and full-fat cheddar cheeses using sensory studies on model systems: Key flavour compounds in cheddar cheeses. *Journal of Sensory Studies*, 26, 278–290. https://doi.org/10.1111/j.1745-459X.2011.00343.x

Kim, M. K., Jang, H. W., & Lee, K.-G. (2020). Characterization of key aroma-active compounds isolated from Omija fruit treated differently based on odor activity values and descriptive sensory analysis. *Foods*, 9, 638. https://doi.org/10.3390/foods9050638

Lawless, H. T. (2010). A simple alternative analysis for threshold data determined by ascending forced-choice methods of LIMITS. *Journal of Sensory Studies*, 25, 332–346. https://doi.org/10.1111/j.1745-459X.2009.00262.x

Lawless, H. T., Hartono, C., & Hernandez, S. (2000). Thresholds and suprathreshold intensity functions for casaicin in oil and aqueous based carriers. *Journal of Sensory Studies*, 15, 437–477. https://doi.org/10.1111/j.1745-459X.2000.tb00281.x

Lawless, H. T., & Heymann, H. (1999). Sensory evaluation of food. Springer US, Boston, MA. https://doi.org/10.1007/978-1-4615-7843-7

Licandro, H., Truntzer, C., Fromentin, S., Morabito, C., Quinquis, B., Pons, N., Martin, C., Blottière, H. M., & Neyraud, E. (2023). The bacterial species profiles of the lingual and salivary microbiota differ with basic tastes sensitivity in human. *Scientific Reports*, 13, 20339. https://doi.org/10.1038/s41598-023-47636-1

Martin, C., & Neyraud, E. (2021). Impact of very hot drink consumption habits, age, and sex, on taste sensitivity. *Foods*, 10. https://doi.org/10.3390/foods10051139

Martin, C., Rollinat, A., Tournier, C., & Peltier, C. (2025). A simple procedure to assess the reliability of the best estimate thresholds (BETs): A case study on the detection threshold of tannic acid astringency. *Food Quality and Preference*, 127, 10547. https://doi.org/10.1016/j.foodqual.2025.105437

Methven, L., Allen, V. J., Withers, C. A., & Gosney, M. A. (2012). Ageing and taste. *The Proceedings of the Nutrition Society*, 71, 556–565. https://doi.org/10.1017/S0029665112000742

Mojet, J., Christ-Hazelhof, E., & Heidema, J. (2005). Taste perception with age: Pleasantness and its relationships with threshold sensitivity and supra-threshold intensity of five taste qualities. *Food Quality and Preference*, 16, 413–423. https://doi.org/10.1016/j.foodqual.2004.08.001

Nuvoli, C., Fillion, L., Lacoste Gregorutti, C., & Labbe, D. (2023). Comparison of sensitivity to taste and astringency stimuli among vegans and omnivores. *Physiology & Behavior*, 262, Article 114092. https://doi.org/10.1016/j.physbeh.2023.114092

Piornos, J. A., Delgado, A., De La Burgade, R. C. J., Methven, L., Balagiannis, D. P., Koussissi, E., Brouwer, E., & Parker, J. K. (2019). Orthonasal and retronasal detection thresholds of 26 aroma compounds in a model alcohol-free beer: Effect of threshold calculation method. *Food Research International*, 123, 317–326. https://doi.org/10.1016/j.foodres.2019.04.034

Zhu, Y., Liu, J., & Liu, Y. (2023). Understanding the relationship between umami taste sensitivity and genetics, food-related behavior, and nutrition. *Current Opinion in Food Science*, 50, Article 100980. https://doi.org/10.1016/j.cofs.2022.100980