**Abstract**:

**Abstract**
**To find out whether there are separate visual mechanisms for size, density, and numerosity computation in textures, we investigated the ability of human observers both to discriminate differences in numerosity between approximately circular textures and to decide whether the differences were due to a change in circle size or dot density. The standard texture always contained 64 irregularly spaced fuzzy dots of random contrast polarity. Dots were added or subtracted from the test pattern either by changing the dot density with the pattern size constant or the pattern size with density constant. Observers had to decide whether size or density had changed and whether the change was an increase or a decrease (mixed task condition). In a separate condition, they reported differences in numerosity (numerosity condition). Numerosity changes were more accurately reported when they were correlated with changes in size than with changes in density, arguing against a single mechanism for numerosity. Observers showed a bias toward reporting larger patterns as denser and vice versa. The data were consistent with a mechanism in which observers compute numerosity from size and density signals and decide whether size or density had changed by the signal detection theory MAX rule after transforming signals into z-values. Because thresholds were not significantly lower in the numerosity condition than in the mixed task condition, we conclude that the direct numerosity mechanism, if it exists, must be noisier than the mechanisms that respond to changes in size and density.**

^{23}) represents the number of elementary entities of a substance in 1 mole of that substance. Clearly, this very large number was not obtained by enumeration. Instead, it was established in the first instance as a ratio of the charge of 1 mole of electrons to the elementary charge. By suitable choice of two measurements having the same dimension, a dimensionless number can always be obtained as a ratio.

^{1}

^{−2}) by the area (dimension m

^{2}) and thus obtain a dimensionless quantity. It is known that both density (Barlow, 1978; Durgin, 2008) and area (Morgan, 2005; Nachmias, 2011) can be measured by human observers, so this is a plausible way to compute number. The fact that Weber fractions for number discrimination are no lower than for density (Ross & Burr, 2010) might seem to support this argument. However, Ross and Burr contradicted the density explanation with an experiment that compared density discrimination and number discrimination using the same kind of stimuli. The stimuli were dot clusters varying in size and dot density (Figure 1). Conditions of constant area, constant density, and constant number were randomly intermingled so that the observer had no way of knowing on a given trial whether to base a number decision on density or area. They argue that this would have made it hard for the observer to use density as a proxy for number. However, it is still the case that the observer could report a greater number when they see a greater density than the standard or when they see a greater area. Ross and Burr argue that this decision process would introduce greater noise into the computation of number. However, to determine what loss of accuracy this would produce, if any, requires a signal-detection model of the choice process, and that is the main purpose of the present article.

^{2}. Stimulus presentation was controlled by MATLAB and the PTB3 version of the Psychtoolbox (Brainard, 1997). On each trial, subjects saw consecutively two stimuli, which they were required to compare for number, density, or size. Each stimulus contained a number of fuzzy dots with a diameter of 10 arcmin and a Gaussian decline with a space constant of 2.5 arcmin. The dots were black (0.4 cd/m

^{2}) or white (300 cd/m

^{2}) with equal probability and were randomly scattered within a notional circle with a standard radius of 3.1° (150 pixels). The standard stimulus, which was always presented first for 0.5 s, contained 64 dots. The variable stimulus, which was presented second for 0.5 s, contained 64 ± 64·W dots, where W is a fraction between 0% and 100% in steps of 5%. The value of W on each trial was chosen by an adaptive procedure (Watt & Andrews, 1981) designed to obtain the 50% point (

*μ*) and the standard deviation (

*σ*) of the psychometric function efficiently by concentrating values of W at

*μ*±

*σ*. There was a 0.75 s blank period before each stimulus, during which only a fixation point was presented in the center of the screen. The test and reference positions were separately offset from the fixation point to avoid interference by afterimages and to prevent the observer from using landmarks on the screen for size judgments. The offset was randomly selected in both x and y direction from a uniform distribution with a width equal to ±0.75 of the circle radius.

*μ*(50% point) and

*σ*(standard deviation) using maximum-likelihood methods (Morgan, 2012). Confidence limits (95%) for the individual points on the psychometric functions and those for the fitted parameters of the psychometric functions were obtained by a bootstrapping procedure. The maximum likelihood values were used to generate 640 new psychometric functions by simulation of the exact experimental procedure. The central 95% of the fitted values were taken to define the confidence limits. The optimal parameters of the psychometric functions for all subjects and conditions are given in Table 1.

Obs | Single stair | Mixed task | Numerosity task | |||

Dens | Size | Dens | Size | Dens | Size | |

1 | ||||||

μ | 3.82 | 2.13 | −0.33 | 0.18 | −1.48 | −0.86 |

σ | 15.81 | 9.35 | 15.3 | 7.45 | 14.03 | 9.19 |

log(L) | 242.66 | 182.44 | 225.77 | 154.30 | 302.30 | 255.28 |

χ^{2} | 11.68*** | 19.52*** | 7.33*** | |||

2 | ||||||

μ | 0.16 | 2.65 | 2.96 | 1.90 | 2.47 | 1.49 |

σ | 19.17 | 10.97 | 22.16 | 10.89 | 35.82 | 13.23 |

log(L) | 257.65 | 192.77 | 277.41 | 202.06 | 280.38 | 239.73 |

χ^{2} | 13.06*** | 18.08*** | 30.86*** | |||

3 | ||||||

μ | 4.30 | −0.81 | 6.18 | 2.76 | 4.93 | 2.90 |

σ | 14.91 | 5.93 | 15.32 | 6.27 | 12.43 | 8.95 |

log(L) | 246.69 | 160.44 | 243.58 | 144.33 | 205.70 | 178.59 |

χ^{2} | 40.92*** | 33.39*** | 5.93** | |||

4 | ||||||

μ | −2.41 | −4.78 | 0.34 | 0.46 | −2.34 | −2.74 |

σ | 14.19 | 9.52 | 14.00 | 9.45 | 11.41 | 15.15 |

log(L) | 421.32 | 354.45 | 398.21 | 313.75 | 387.55 | 450.80 |

χ^{2} | 18.89*** | 11.79*** | 6.71** |

*μ*s and one

*σ*) to four-parameter fits (two

*μ*s and two

*σ*s). In all conditions but one, Weber fractions for density-varying trials were significantly higher than Weber fractions for size-varying trials (

*df*= 1, see Table 1). Only naïve Observer 4 showed, in the number discrimination task, a significantly higher Weber fraction on size-varying trials than on density-varying trials.

*μ*and standard deviation

*σ*. Having chosen the source, the observer decides the response (which stimulus was larger/denser/more numerous) on the basis of the sign of the deviation. Details of the calculation are given in the Appendix. The blue curves in Figure 3 show the fits based on the four-parameter MAX model. It should be noted that this model for two channels gives identical results for the described experiment to a model in which the two signals are summed before the response on the direction of the change is made (Morgan & Solomon, 2006).

Model A | ||||||||

Obs | μ _{Dens} | σ _{Dens} | μ _{Size} | Ratio | log(L) | |||

1 | 1.39 | 7.66 | 1.28 | 1.70 | 1973.5 | |||

2 | 0.50 | 10.00 | 1.08 | 2.03 | 2087.1 | |||

3 | 1.12 | 6.21 | 0.07 | 2.09 | 1795.9 | |||

4 | −3.84 | 10.59 | −3.48 | 0.95 | 3416.2 | |||

Model B | ||||||||

Obs | μ _{Dens} | σ _{Dens} | μ _{Size} | Ratio | Crossover | log(L) | ||

1 | 0.92 | 8.05 | 0.77 | 1.72 | 0.20 | 1958.5 | ||

2 | 0.60 | 9.89 | 1.18 | 1.97 | −0.04 | 2086.6 | ||

3 | 0.98 | 6.34 | −0.14 | 2.21 | 0.11 | 1789.6 | ||

4 | −3.40 | 11.69 | −3.18 | 0.92 | 0.23 | 3398.8 | ||

Model C | ||||||||

Obs | μ _{Dens} | σ _{Dens} | μ _{Size} | Ratio | Crossover | Bias | log(L) | |

1 | 0.91 | 8.03 | 0.77 | 1.73 | 0.20 | 0.01 | 1958.6 | |

2 | 0.59 | 9.69 | 1.21 | 2.08 | −0.03 | 0.09 | 2084.7 | |

3 | 0.86 | 6.12 | −0.19 | 2.43 | 0.13 | 0.14 | 1786.1 | |

4 | −2.53 | 9.62 | −3.11 | 1.33 | 0.22 | 0.59 | 3277.7 | |

Model Comparison | ||||||||

Obs 1 | Obs 2 | Obs 3 | Obs 4 | |||||

χ ^{2} | Sig | χ ^{2} | Sig | χ ^{2} | Sig | χ ^{2} | Sig | |

A vs. B | 29.94 | *** | 0.87 | — | 12.50 | *** | 34.90 | *** |

B vs. C | −0.17 | — | 3.91 | * | 7.00 | ** | 242.20 | *** |

*p*< 0.001) for Observer 4, Observer 3 (

*p*< 0.01), and Observer 2 (

*p*< 0.05), but not for Observer 1 (

*p*> 0.1).

^{2}Our additional findings are (a) thresholds for explicit size discrimination are lower than those for density, when expressed as changes in numerosity; (b) thresholds for size and density are no higher when observers have to distinguish them explicitly, rather than reporting numerosity; (c) thresholds in the interleaved conditions (mixed task and numerosity task) are similar to those predicted from a MAX model from signal detection theory; and (d) there is a bias toward reporting larger patterns as denser and vice versa.

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^{1}Possibly to avoid the implication that numbers are established by enumeration, the SI system of units prefers to replace numbers by constants. Thus, Avogadro's number has been replaced by Avogadro's constant with units of moles

^{−1}. Another subtlety is that some dimensional analysts prefer to consider numbers as constants having the dimension “one.”

^{2}The exception is Observer 2's threshold on density-varying trials in the numerosity condition, which is abnormally elevated with low confidence (see Figure 2).

*m*is assumed to be the signed difference between the test and a comparatively noiseless internal standard. The probability of a larger/denser response to a signal

*m*is computed as the integral from –inf to

*m*of a normal distribution with variance

*σ*

^{2}and mean (

*m*+

*μ*), where

*μ*is the observer's bias. There is no late noise in the model.

*σ*

_{A}

^{2}and

*σ*

_{B}

^{2}. Any bias in deciding on the direction of change (first or second interval larger/denser/more numerous) is denoted as

*μ*

_{A}and

*μ*

_{B}. The model assumes that the observer decides whether size or density is different from the standard by determining which of the two noisy signals (in A and B) on a given trial has the larger absolute value (the MAX rule). At the same time, the sign of the Max signal is used to determine whether the difference is positive or negative.

*μ*and

*σ*being the midpoint and the standard deviation of the probability density function. The external signal size is given as

*m*, which can take on negative or positive values.

*σ*

_{A}

^{2}and

*σ*

_{B}

^{2}and means

*μ*

_{A}and

*μ*

_{B}, the first of which carries a signed signal

*m*. Following Green and Swets (1966), we assume that each sensory event can be mapped on to a signed internal variable

_{A}*x*. The probability that a sample of the source distribution carrying the signal

*m*takes on the internal value

_{A}*x*and that this has a larger absolute value than the sample from source B carrying no signal (

*m*= 0) is given by

_{B}*m*> 0, this would be the probability of deciding that the test is denser than the standard, when this is in fact the case.

_{A}*x*than the sample from the other source, meaning the signal source is correctly identified as A, and that

*x*< 0 (for example, if

*m*> 0 deciding that the stimulus is less dense when it is in fact denser) is found by integrating as follows over all negative values of

_{A}*x*:

*m*of source A is wrongly perceived as coming from source B. For example,

_{A}*p*

_{3}could be the probability that a denser test was perceived as larger than the standard.

*p*

_{4}is the probability that the signal

*m*is perceived as coming from source B and as being smaller than 0 (e.g., perceived smaller when it was denser).

_{A}*p*

_{1}+

*p*

_{3}, where source A is density. The probability of correctly identifying the change as density in the mixed task condition is given by (

*p*

_{1}+

*p*

_{2}) on density-varying trials.

*p*

_{1}+

*p*

_{3}(with density on the left and size on the right).

*p*

_{1}/(

*p*

_{1}+

*p*

_{2}).

*p*

_{1}+

*p*

_{2}.