Twenty-two listeners participated in the study, with ten in Experiment 1 (six male and four female, ages 23–28 years), four in Experiment 2 (two male and two female, ages 25–29 years), and eight in Experiment 3 (four male and four female, ages 21–29 years). Listeners were informed about the purpose of the study, gave their written consent to participate, and were remunerated on an hourly basis. Twelve listeners had participated in other experiments in our lab during which detection and discrimination thresholds were measured using forced-choice procedures, but none had participated in experiments using the method of adjustment. For each listener, audiograms were measured prior to the experiments by means of standard audiometry (Otometrics Madsen Itera II) and found to be normal for both ears. The study was approved by the ethics committee of the Otto von Guericke University, Magdeburg.

Each acoustic stimulus consisted of an onset during which the amplitude of the stimulus rose linearly from zero to the maximum, a plateau during which the amplitude remained constant, and an offset during which the amplitude decayed linearly from the maximum to zero. Depending on the experiment, stimuli could differ in four attributes: onset duration, total duration, carrier, and sound level. They were generated in MATLAB R2010a (The Mathworks, Natick, MA, USA) and stored in the Waveform Audio File Format (WAVE) for replay by custom-made stimulus-presentation software. The system was calibrated using defined sound sources (Microtech Gefell MG 4000 and Brüel and Kjaer pistonphone 4228) and an artificial ear (Brüel and Kjaer 4153) equipped with a condenser microphone (Brüel and Kjaer 4133) connected to a conditioning amplifier (Brüel and Kjaer Nexus) and a multimeter (Agilent 34401A). During presentation, stimulus levels were controlled by a programmable attenuator (PA5, Tucker-Davis Technologies) linked downstream to a 24-bit sound card (Juli@, ESI Audiotechnik GmbH) of a desktop computer.

All data analyses were performed using MATLAB R2010a (The Mathworks, Natick, MA, USA) with the Curve Fitting Toolbox, the Optimization Toolbox, the Parallel Computing Toolbox, and the Statistics Toolbox. In all statistical tests the α-criterion was 0.05.

According to our theoretical considerations, onset durations should be treated as elements of the multiplicative Euclidean vector space ℝ_>0 rather than of the additive Euclidean vector space ℝ. To investigate empirically whether the matches of onset durations are consistent with an additive or with a multiplicative statistical model, we used the maximum-likelihood method to fit additive normal and multiplicative normal distributions to the distributions of matches obtained in Experiments 1 and 2, separately for each condition and listener (576 distributions in total, each based on between 36 and 68 matches). Even if other distributions might fit a particular dataset better than the additive normal and multiplicative normal distributions do, these two represent the simplest distributions describing data that behave according to an additive or a multiplicative Gaussian error model (Limpert et al., 2001; Limpert and Stahel, 2011). In fact, they are the maximum-entropy distributions when only the mean and the variance are known (Park and Bera, 2009). Fitting these two distributions therefore suffices to distinguish between additive and multiplicative models as long as no other constraints are imposed. The results of the fits are summarized in Figure 3.

Figure 3A shows, for one example listener in Experiment 1, the matches (blue dots; n_mat = 62 per St) for each of the eight standard onset durations (gray vertical lines) of tone stimuli presented at 34 dB SL. Superimposed are the cumulative distribution functions (CDFs) of the fitted multiplicative normal distributions (black lines). They fit the data well.

Figure 3B is a scatterplot of the negative log-likelihoods from the fits to the multiplicative normal distribution, ℓ_mult, and to the additive normal distribution, ℓ_add, for all conditions and listeners in Experiments 1 and 2. The data from the example listener and conditions in Figure 3A are indicated separately. The smaller ℓ, the better the fit. In 471 of 576 cases (82 %), the multiplicative normal distribution fits the data better than the additive normal distribution. This is illustrated more clearly in the inset of Figure 3B, which shows the cumulative distribution of the differences between the negative log-likelihoods, Δℓ = ℓ_mult − ℓ_add. Negative values of Δℓ support the multiplicative normal distribution, whereas positive values support the additive normal distribution. Dashed lines indicate the fraction of negative differences (0.82). The median of Δℓ (−11.47) is significantly different from zero (Wilcoxon signed rank test; p = 5 · 10⁻⁶¹; two-tailed).

Because previous studies have obtained only few matches per listener (e.g., two matches in van Heuven and van den Broecke, 1979; van den Broecke and van Heuven, 1983), we also tested how the number of matches included in the fits influences the support for each model. From the distribution of matches for each condition and listener, we randomly drew a number of matches (n_mat ∈ {6, 12, 18, 24, …} or all), fitted them with additive normal and multiplicative normal distributions using the maximum likelihood method, and computed Δℓ. To reduce the noise, we arithmetically averaged the Δℓ across all conditions for a given n_mat and listener. Figure 3C plots these averages of Δℓ (dots) as a function of n_mat, separately for each of the 14 listeners in Experiments 1 and 2. For all listeners but one, the multiplicative normal distribution is increasingly favored as additional matches are included. This shows that a sufficient number of matches per listener and condition is required for a clear empirical differentiation between the additive and the multiplicative models. Superimposed in Figure 3C (solid lines) are least-squares fits of the data from each listener to the straight-line equation Δℓ = α · (n_mat − 2) with n_mat ≥ 2. The offset was fixed to take into account that Δℓ = 0 for n_mat = 2. The slope α, which represents the change in Δℓ per match included in the fit, was a free parameter. For each listener, the slope is plotted in Figure 3D against the geometric average across all conditions of the geometric Weber fraction, W_G (see derivation in next section). The slopes are shallow for listeners with small average W_G (narrow distributions; high precision) and steep for listeners with large average W_G (broad distributions; low precision). The black line shows the relationship predicted under the assumption that all matches follow perfect multiplicative normal distributions. The data points fall near, though mostly above or to the right of, the prediction. Simulations (not shown) revealed that this is partly explained by averaging Δℓ and W_G from different distributions. In addition, the experimental data do not follow perfect multiplicative normal distributions.

In summary, when a sufficient number of matches is included, clear empirical support for the multiplicative model is obtained. This corroborates our theoretical considerations and the conclusion that onset-duration adjustment data should be described using geometric statistics and novel psychophysical measures, rather than arithmetic statistics and conventional psychophysical measures used in previous studies (e.g., van Heuven and van den Broecke, 1979; van den Broecke and van Heuven, 1983; Smurzyński and Houtsma, 1989). The novel measures are derived in the next section.

Here, we derive probabilistic definitions of the psychophysical performance measures for the method of adjustment when the physical stimuli used can have only positive real magnitudes. We base our derivation on the (one-dimensional) Euclidean vector space ℝ_>0 in which multiplication and exponentiation take on the role of vector addition and scalar multiplication and distances between any two elements x, y ∈ ℝ_>0 are measured with the log-ratio distance function distℝ>0(x,y)=| ln(x)-ln(y) |=|ln(xy)|. To obtain the operations and properties of ℝ_>0, one can exploit the fact that ℝ and ℝ_>0 are isometrically isomorphic, meaning that there are bijective functions from one space to the other that preserve the algebraic and metric structure. The most natural choice of bijections is the exponential function exp:ℝ→ℝ>0,x↦ex and the logarithmic function ln:ℝ_>0 → ℝ, x ↦ log_e(x), where e ≈ 2.71828 is the Euler-Napier number. The Euclidean vector spaces ℝ and ℝ_>0 are compared in Table 2 of the Supplementary Material.

For the method of adjustment, we are interested in calculating, for a given standard magnitude St, the constant and variable errors of a distribution of matches with positive-real magnitudes. To simplify matters, we assume that X_St approximately follows the canonical Gaussian distribution on ℝ_>0, namely the multiplicative normal distribution, for which the relationships between parameters and statistical measures are well known (see Table 3 of the Supplementary Material). Let μ₀ and σ² be the parameters representing a potential constant error and the variable error, respectively, and let μ = ln(St). Note that when X_St follows a multiplicative normal distribution, the logarithm of X_St follows an additive normal distribution with the same parameters, i.e., ln(XSt)~N(μ+μ0,σ2). Conceptually, the point of subjective equality should be the center of X_St, such that (5)PSEG= eμ+μ0=St · eμ0= GM(XSt) .

We propose to define the constant error as the directed log-ratio distance between the PSE and the St: (6)CEG=ln(PSEGSt)=ln(PSEG)−ln(St)=μ0 .

This directed distance differs from dist_ℝ>0 (PSE_G, St) in that the sign of the error is retained. It is a dimensionless measure with the logarithmic auxiliary unit neper (Np), which can be converted to the more familiar decibel (dB), where 1 Np = (20/ln(10)) dB ≈ 8.686 dB. By analogy with the Euclidean space ℝ, where the magnitude of the comparison stimulus that is just-noticeably different from the St is typically chosen to be Co = St ± AS(X_St), let us define (7)Co=St ⋇ GS(XSt) , where the symbol ⋇ denotes “times or divided by”. Precision can then be quantified by a dimensionless difference limen defined as the geometric standard deviation of X_St: (8)DLG={Co/St,Co≥StSt/Co,Co<St= GS(XSt) .

Unlike the conventional difference limen, DL_A = AS(X_St), which must be divided by St to yield a relative measure of sensitivity, the proposed difference limen could be used directly as such a measure, because it is already such a ratio. For this reason it would be more correct to refer to it as a ratio limen, but by analogy to the conventional measure, we use the term geometric difference limen.

Note, however, that DL_G ∈ ℝ_>1, whereas we want Weber fractions to be elements of ℝ_>0, so that they are comparable to conventional Weber fractions. One might be tempted to define a novel Weber fraction as W_G = |DL_G − 1| = |GS(X_St) − 1| = |GCV(X_St)|, which is the geometric coefficient of variation proposed by Kirkwood (1979). However, this would be inconsistent with the operations allowed on ℝ_>0. Instead, we propose to use the logarithm of DL_G, (9)WG=ln(DLG)= distℝ>0 (Co,St)=σ , because σ ∈ ℝ_>0 alone satisfies the properties required of the Weber fraction as a measure of relative sensitivity. Again, the log-ratio distance between Co and St is a dimensionless quantity with the auxiliary unit neper, which can be converted to decibels. Due to their common auxiliary units, CE_G and W_G can be directly compared. The proposed geometric measures are contrasted with the conventional measures in Table 1, and we will use our experimental results to compare both types of measures in the sections that follow.

The purpose of Experiment 1 was to examine how accuracy and precision of onset-duration matching depend on the onset duration, the sound level, and the carrier of the standard and to explore inter-listener variability. We compare the results when accuracy and precision are quantified by the conventional and the proposed measures. We find quantitative and qualitative differences which may even lead to opposite conclusions.

The purpose of Experiment 2 was to explore whether listeners use differences in the plateau duration, rather than in the onset duration, of the standard and comparison stimuli for matching. In Experiment 1, the total duration (400 ms) and the offset duration (50 ms) were fixed, such that plateau duration covaried with onset duration. If listeners actually matched plateau duration, and if plateau-duration matching approximately followed Weber's law, then the performance of the apparent onset-duration matching would improve by shortening, and worsen by prolonging, the plateau duration of the standard. To test this idea, we employed standards with total durations of 400 ms (as in Experiment 1), 200 ms, and 800 ms. The offset duration was fixed at 50 ms (as in Experiment 1). Again, tone and noise carriers were used but with only two sound levels (10 dB SL and 34 dB SL) and only two onset-durations of the standard (2 and 16 ms).

Neither accuracy nor precision have a monotonic dependence on plateau duration (Figure 7). For the majority of combinations of sound level, carrier, and onset duration (7/8 for CE_A; 5/8 for CE_G; 8/8 for W_A and 6/8 for W_G), the null hypothesis of equal medians can be maintained (Friedman test; p > 0.105). For the remaining combinations, p-values were 0.0498 (4/6) or 0.0388 (2/6). In most cases, however, accuracy and precision changed nonmonotonically. In addition, precision changed much less than predicted if listeners matched plateau durations in accordance with Weber's law (represented by the yellow dashed lines in Figures 7D,H). It is therefore unlikely that listeners performed Experiments 1 and 2 by matching plateau durations. It is likely that listeners instead use onset cues, specifically differences in onset duration per se or associated spectral differences (see Discussion).

The results of Experiment 2, obtained with four listeners who did not take part in Experiment 1, also corroborate the differences in the arithmetic and geometric measures of accuracy and precision described in the previous section (Figure 7). They also corroborate the major findings of Experiment 1: accuracy and precision are higher for the tone than for the noise carrier, accuracy and precision increase with increasing sound level for both carriers, estimates of the constant error are more positive for short (2 ms) than for long (16 ms) onset durations of the standard, and precision increases with increasing onset duration of the standard (Figure 7).

In Experiment 3, we examined the effects of restricting the range of adjustable comparison stimuli on the measures of accuracy and precision. The experiment was motivated by our observation in Experiments 1 and 2 that some of the distributions of matches for standards with short onset durations, low sound levels, and noise carriers, might have been subject to boundary effects due to restriction of the range of adjustable comparison stimuli. In Experiment 3, a single standard stimulus with an onset duration of 10.24 ms was used, and the range of adjustable comparison stimuli was increasingly restricted from values between 0.64 and 163.84 ms (first degree of restriction) to values between 6.64 and 15.79 ms (seventh degree of restriction; see Table 2).

Figure 8 shows the results of Experiment 3. The arithmetic mean of CE_A decreases with increasing range restriction from about 0.5 to about 0. The arithmetic mean of CE_G increases from about −0.15 and approaches 0 with increasing range restriction (Figure 8A). The absolute value of both measures of the constant error therefore diminishes as the range of adjustable comparison stimuli is restricted. The associated arithmetic standard deviation of CE_A also decreases with increasing range restriction, whereas that of CE_G shows no systematic change (Figure 8B). The geometric means of both W_A and W_G decrease monotonically with increasing range restriction, both from about 0.6 to 0.25 (Figure 8C). The decrease is expected because W_A and W_G are equal to 0 when only a single value of the comparison stimulus can be chosen. Precision therefore appears to improve with increasing range restriction. The geometric standard deviation of W_A decreases with increasing range restriction whereas that of W_G changes little over the seven degrees of range restriction we tested. Ultimately, however, the geometric standard deviations of W_A and W_G must approach 1 as the range of adjustable comparison stimuli is increasingly restricted. They are equal to 1 when only a single value of the comparison stimulus can be chosen. Boundary effects imposed on the distributions of matches by the restricted range of comparison stimuli might therefore underlie the positive values of CE_G for short onset durations of the standard, as illustrated in Figure 5B, and the initial increase in W_G with increasing onset duration of the standard for some combinations of tone level and carrier, as illustrated in Figure 5D.

We derived novel definitions of the psychophysical measures of the point of subjective equality, the constant error, the difference limen, and the Weber fraction for stimuli with positive-real magnitudes. Our aim was consistency: When two positive-real quantities are combined, then the result should also be positive real. This requirement is not always met when positive-real quantities are treated as if they were elements of the additive vector space of real numbers. The multiplicative Euclidean vector space of positive-real numbers endowed with the log-ratio distance function, however, fully satisfies this requirement. In a recent note, Graff (2014) promotes the use of the log-ratio distance function, arguing that relative differences between physical quantities are often more informative than absolute differences. Moreover, he makes the point that the determination of psychophysical laws first requires a clear-cut, objective expression of the physics of stimuli and responses. He shows that the log-ratio distance function exhibits all properties desired of a measure of relative differences, such as symmetry in selecting a reference, agreement between inverted units, and additivity. We build upon these ideas by linking the algebraic properties of the multiplicative Euclidean vector space with descriptive statistics. The geometric mean, unlike the arithmetic mean, is consistent with the structure of the multiplicative Euclidean space. Our Gedanken experiment (Figure 2) demonstrates that the conventional psychophysical measures possess a convexity bias of purely mathematical origin. As summarized in Figure 2C, the conventional arithmetic constant error would imply the existence of a positive bias or perceptual illusion (i.e., when CE_A > 0), whose magnitude increases with the width of the distribution of matches, when no bias or illusion exists (i.e., when CE_G = 0). A true bias or illusion (i.e., when CE_G ≠ 0) would be overestimated, missed, or underestimated, depending on its sign and magnitude and on the width of the distribution of matches. It is even possible, as in the present study, that CE_A > 0 while CE_G < 0 (see quadrant II in Figure 2C).

It may be criticized that the Gedanken experiment rests on the assumption that matches follow a multiplicative normal distribution and that the outcome may differ if the matches followed other distributions. However, the multiplicative normal distribution is the simplest model describing data that behave according to a multiplicative model (Limpert et al., 2001; Limpert and Stahel, 2011), and it is the maximum-entropy distribution on ℝ_>0 when only the mean and the variance are known (Park and Bera, 2009). Furthermore, the multiplicative normal distribution may be difficult to distinguish from other multiplicative distributions, such as the Pareto distribution, a power-law distribution (Newman, 2005; Clauset et al., 2009). We chose the multiplicative normal distribution because it allowed us to exploit the known relationships between its parameters and the statistical descriptive measures (Table 3 of the Supplementary Material) and to analytically determine the magnitude of the convexity bias. The particular distribution on ℝ_>0 is, however, not so relevant for the main argument: the convexity bias is a consequence of the AM-GM inequality, a special case of Jensen's inequality, and its existence is independent of the underlying distribution. The Gedanken experiment applies not only to the matching of onset durations, but also challenges matching studies in which the existence of perceptual illusions was inferred from, or their magnitudes were quantified by, the conventional constant error. This includes reports of the auditory time-stretching illusion (e.g., Sasaki et al., 2010) and also certain optical illusions such as the famous Müller-Lyer illusion (e.g., Restle and Decker, 1977; Bulatov et al., 1997; Bulatov and Bertulis, 1999; Woloszyn, 2010), the vertical–horizontal illusion (e.g., Higashiyama, 1996), and Titchener and Delboeuf illusions (e.g., Pressey, 1977). The impact of the convexity bias on measures of such perceptual illusions remains to be investigated. Another issue is how the theoretical considerations can be applied to other psychophysical methods, such as forced-choice procedures, where performance is quantified differently.

It may be argued that the problems we have identified above can be avoided if the conventional arithmetic analysis is performed on log-transformed data, as is often the case in the literature. Indeed, the geometric mean and standard deviation on which our proposed measures of accuracy and precision are based can be obtained by exp-transforming the arithmetic mean and standard deviation of the log-transformed random variable (see Table 1 of the Supplementary Material). However, log-transformation is often used only to make skewed distributions more symmetric (e.g., bell shaped), a requirement of many statistical tests. In hearing sciences, log-transformation of the stimulus amplitude (into decibels) serves mainly to compress the wide range of sound pressures (> six orders of magnitude) or sound intensities (> twelve orders of magnitude) that are processed by auditory systems into a more manageable range of values and to respect the observation that equal increments in level correspond to roughly equal increments in sensation (Hassall and Zaveri 1988, p. 31; Moore 2013, p. 10; Pickles 2013, p. 3). When to log-transform data, when to back-transform measures obtained in log-space, or how to proceed with the back-transformed measures may often be only vaguely known. Because, historically, concepts in descriptive statistics have usually been developed based on the additive Euclidean vector space of real numbers, their mathematical relationships to the multiplicative Euclidean vector space of positive-real numbers may be little known. To avoid biases that are of purely mathematical origin, it is essential to respect that corresponding mathematical operations differ between the vector space of real numbers and that of positive-real numbers. For example, addition corresponds to multiplication, multiplication corresponds to exponentiation, and the Euclidean distance function dist_ℝ (x, y) = |x − y| corresponds to the Euclidean distance function dist_ℝ>0 (x, y) = |ln(x) − ln(y)|. Our study may be the first in the field of perception science to demonstrate the mathematical necessity for geometric measures of performance when stimulus magnitudes can only be positive real. Our approach extends beyond simply log-transforming data and using arithmetic statistics.

We reexamined onset-duration matching with respect to stimulus factors affecting the accuracy and precision of adjustments, as well as with respect to inter-listener variability measured with the conventional and the proposed psychophysical measures. We demonstrated that the multiplicative normal distribution usually fits the distributions of matches better than the additive normal distribution (Figure 3). This is in line with our theoretical considerations according to which matches of onset durations, or of other stimulus attributes that can attain only positive real magnitudes, should comply with a multiplicative rather than an additive model. We also showed that the superiority of the fit of the multiplicative normal distribution over that of the additive normal distribution depends on the width of the empirical distribution and on the number of matches. Unlike previous studies which are based on only few matches per listener and condition (e.g., van Heuven and van den Broecke, 1979; van den Broecke and van Heuven, 1983), the listeners in our study made many (up to 68) matches per condition. This is clearly sufficient to demonstrate the superiority of the multiplicative model and the use of geometric rather than additive descriptive statistical measures to characterize the distributions of matches and to quantify performance.

It may be argued that the superiority of the multiplicative model, demonstrated here (Figure 3), results from the logarithmic spacing of the adjustable onset durations of the comparison stimuli and that opposite results would have been obtained with linear spacing. For the fitting procedure, however, it makes no difference how the samples are spaced. Nonetheless, it is conceivable that the spacing of the stimuli might affect the way listeners perform the tasks. For example, incrementing the onset duration by some amount above the standard constitutes a smaller relative change than decrementing it by the same amount below the standard. Such an asymmetry might bias the distribution of matches.

When comparing the behavior of the conventional and the proposed measures of accuracy and precision across listeners and conditions, we observed that the conventional measures display a high degree of heteroscedasticity and a strong dependence on the onset duration and the sound level of the standard. This would mean that the generalization gradients of performance differ markedly between listeners. In contrast, the proposed measures display a high degree of homoscedasticity and a weak dependence on the onset duration and the sound level of the standard (Figures 4–6), meaning that the generalization gradients of performance are similar for different listeners.

Furthermore, we found that the carrier of the standard has a significant influence on accuracy and precision of onset-duration matching, an effect detected when using the proposed geometric measures but missed with the conventional measures (Figure 6). We found that both accuracy and precision were better for the narrowband (tone) than the broadband (noise) carrier. This is consistent with the trend observed in the data of van Heuven and van den Broecke (1979) and van den Broecke and van Heuven (1983) and observed when comparing the findings of Kewley-Port and Pisoni (1984) with those of Smurzyński and Houtsma (1989) (see Introduction and Figure 1B). The better performance with narrowband than broadband carriers can be explained by assuming that listeners make use of additional spectral cues available when using the former. The spectral splatter induced by the onset of a tone increases with decreasing onset duration. These spectral differences might be exploited by the listeners, in addition to the temporal differences, when matching the onset durations of tones. With broadband carriers, the spectral differences are less pronounced or absent.

We found no systematic effect of total duration (and hence plateau duration) on accuracy or precision (Figure 7), suggesting that differences in plateau duration can be excluded as a dominant cue for onset-duration matching in our experiments. This is in line with a conclusion drawn by van Heuven and van den Broecke (1979). This leaves the temporal differences at the onset (and, as discussed above, their associated spectral differences) as potential cues. One crucial parameter may be the rate of rise of the amplitude at stimulus onset. This rate doubles when onset duration is halved and sound level is held constant. Under these conditions, accuracy and precision slightly decrease with increasing rate of rise, even with the geometric measures. On the other hand, the rate of rise of the amplitude at stimulus onset also increases when sound level is increased and onset duration is held constant. The rate of rise increases by a factor of four for a 12-dB increment in sound level. Under these conditions, however, accuracy and precision improve with increasing rate of rise. The rate of rise therefore cannot be the only factor. The onset responses of cortical and subcortical neurons are very sensitive to the rate of rise of the stimulus amplitude, as observed in many studies that have varied onset duration or sound level (Hall and Feng, 1988; Phillips, 1988; Gooler and Feng, 1992; Heil, 1997a,b; Heil and Irvine, 1998; Lee et al., 2016), but onset responses are not a unique function of this rate. Based on the effects of varying onset duration, onset shape, and sound level on the onset responses of cat auditory cortical neurons, Heil and colleagues suggested that the first spikes of the responsive population of neurons track the onset envelope and the initial steady-state portion of the stimulus (for review, see e.g., Heil, 2003). They also concluded that each temporal envelope will evoke a unique spatiotemporal response pattern across the tonotopic and isofrequency axes of auditory cortical maps. Such unique patterns may be the basis for (or a physiological correlate of) the onset-matching capabilities and their dependence on onset duration, level, and carrier of the standard reported here.

Notably, the sign of the constant error in our data was affected by the type of measure used. When using the conventional measure, the relative CE_A, values were typically positive (Figures 4–8), implying that listeners overestimated onset durations. Our analyses, however, show that in our data positive values of CE_A are likely the result of the convexity bias. The proposed measure of accuracy, the geometric constant error CE_G, was negative for most listeners and conditions. Values of CE_G tended to be positive only for the shortest onset duration of the standard, which might be due to a boundary effect imposed by the restricted range of comparison stimuli. The presence of predominantly negative values of CE_G may be due to presentation-order effects (Gescheider, 1997), given that the standard was always presented before the comparison stimulus. Whether this is a viable explanation could be assessed, if feasible, in future studies that also employ reversed and balanced (e.g., randomized) sequences of standard and comparison stimuli.

The results of Experiment 3 show that constant errors and Weber fractions converge toward zero when the range of values from which the comparison stimuli can be drawn is increasingly restricted (Figure 8). Thus, when the range of adjustable comparison stimuli is too narrow, the presence of systematic biases or illusions—no matter if positive or negative—will be missed, and precision will be overestimated. This supports a conclusion drawn by van Heuven and van den Broecke (1979), who argued that ceiling effects reduced the standard deviations computed from the distribution of matches to standards with the longest onset durations. A similar reduction of the standard deviations will occur for matches to standards with the shortest onset durations. Range effects therefore cannot be the origin of the rapid increase of conventional Weber fractions with decreasing onset duration at short onset durations of the standard.

Arithmetic descriptive statistics are appropriate for random variables over the additive Euclidean space ℝ, whereas geometric descriptive statistics should be used for random variables over the multiplicative Euclidean space ℝ_>0, in which relative differences are measured with the log-ratio distance function. Our results imply that psychophysical studies should be reevaluated if arithmetic descriptive statistics were used to quantify performance when geometric statistics should have been used instead. This applies when the magnitudes of the stimuli can be only positive real, because then the measures of performance will be affected by the convexity bias. Unfortunately, it is not possible to simply compute the proposed geometric Weber fractions and geometric constant errors from the conventional arithmetic Weber fractions and arithmetic constant errors, because W_A and CE_A depend on both CE_G and W_G (lines in Figure 2C; prediction surfaces in Figures 4C,F). It will therefore be necessary to reanalyze the raw data. The size of the convexity bias will differ across studies because it depends on the width of the distribution, which will differ across conditions, tasks, and subjects.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.