Figure 1.
Distribution of fitted Weibull values in Table 1.
The fitted values from the suprathreshold (non-starred) conditions of Table 1 were dropped into bins with edges that stepped from 0.8 to 2.4 in jumps of 0.2 (the histogram thus excludes one outlier, the value 6.70 for Henning et al. 's [47] observer GBH at 8.37 cpd). For this histogram, we used the
values that had been fitted using a nonzero lapse rate parameter where available, as this is more likely to reflect the true
. The median of this hybrid population (some including a lapse rate parameter, some not) was 1.43 (indicated by the vertical dashed line).
Table 1.
Fitted Weibull function parameters for 2AFC contrast discrimination.
Figure 2.
Effect of varying Weibull and
on log and linear abscissas.
(A) Varying on a log abscissa: The curve shifts horizontally. (B) Varying
on a log abscissa: The curve is linearly stretched or compressed horizontally. (C) Varying
on a linear abscissa: The curve undergoes a linear horizontal stretch and a change of threshold. (D) Varying
on a linear abscissa: The shape changes in a way that cannot be described as a linear scaling.
Figure 3.
Graphical representation of the probability of a correct response.
The shaded areas in A and B correspond to the integrals in Equations (4) and (5), respectively. The smooth curves trace out the PDF of the noise, , on the internal difference signal,
. As explained in the text,
has to be even-symmetric, and this means that the two integrals in Equations (4) and (5) are equal. The shaded areas correspond to the probability of a correct response. The psychometric function (expressed as a function of
) is the CDF of the noise, increasing from 0.5 to 1 as
increases from 0 to
.
Figure 4.
CDFs and PDFs of four different noise distributions.
The top row shows noise CDFs, , for (A) a Laplace distribution (generalized Gaussian with
), (B) a Gaussian distribution (generalized Gaussian with
), (C) a generalized Gaussian with
, (D) a logistic distribution. Each panel in the bottom row shows the PDF,
, corresponding to the CDF above it. Only the positive halves of the distributions are shown (i.e.
). Note that the use of these colours for the different noise distributions is maintained in Figures 7, 8, 10, 11, 12, 15, and 16.
Figure 5.
Geometrical interpretation of the expression for .
In each panel, the thick, magenta curve represents the transducer function. The horizontal axes represent the transducer input, and the vertical axes represent the transducer output. is the pedestal level, and
is the discrimination threshold. The gradient of the blue line,
, is equal to
, defined in Equation (39). The green line is the tangent to the transducer at point
; its gradient is equal to
, defined in Equation (38). The ratio
is equal to
. For an expansive transducer (panel A),
, so
. For a compressive transducer (panel B),
, so
. For a linear transducer (panel C),
, so
.
Figure 6.
This curve plots the predicted when the Weibull function is fitted to the CDF of generalized Gaussian distributions with a range of different
values. The graph asymptotes to a value of
(see Appendix S1), indicated by the horizontal dashed line. The shape of the generalized Gaussian distribution is determined by
.
-values of 1 and 2 are special cases:
gives a Laplace distribution, and
gives a Gaussian distribution.
Figure 7.
Noise CDFs from Figure 4 plotted against the best-fitting Weibull functions.
The thin, coloured curves shown in (A) to (D) are the CDFs from Figures 4A to 4D, respectively. The thick, black curves are the Weibull functions that give the best (maximum-likelihood) fit across the range of inputs shown on the horizontal axis. This fit was carried out by maximizing the expression
, where
is the noise CDF, and
is the Weibull function whose parameters were being fitted. The Weibull function provides a perfect fit to the Laplace CDF (A), an excellent fit to the Gaussian (B), and logistic (D) CDFs, and an acceptable fit to the generalized Gaussian with
(C); this partly justifies our use of
as an estimate of
in Equation (23). The
values are the
parameters of the fitted Weibull functions. The
values are our analytical estimates of
, given by Equations (52) to (54) for panels (A) to (C), respectively, and Equation (60) for panel (D). In each case,
provides a close match to
. The parameter in brackets in each
term is the shape parameter,
(see Equation (50)). As noted in the text, the CDFs all have a point of inflection at zero. With the exception of panel A, the best-fitting Weibull functions have a point of inflection slightly above zero (
would have to be 1 or less for the steepest point to occur at zero). Nevertheless, the Weibull functions still provide good fits.
Figure 8.
Psychometric functions resulting from power-function transducers and zero pedestal.
The thin, coloured curves show the psychometric function of Equation (65), plotted as a function of . Different rows of panels show psychometric functions with different noise CDFs,
, given by the Laplace distribution (top row of panels), the Gaussian (second row), the generalized Gaussian with
(third row) or logistic (bottom row). Different columns of panels show psychometric functions for different transducer exponents,
, as indicated at the top of the figure. The thick, black curves show the best-fitting (maximum-likelihood) Weibull functions. The curves in the middle column (
, top to bottom) are identical to Figures 7A to 7D, respectively. This is because
gives a linear transducer, and so the psychometric functions for
will have the same shape and same fitted
as the CDF (see Equation (10)). Each panel displays the
value of the best-fitting Weibull function (
) and the estimate,
, where
is given by Equation (52), (53), (54) or (60), as appropriate.
Figure 9.
for the power-function transducer with nonzero pedestal, plotted as a function of Weber fraction.
Each curve gives for a different transducer exponent,
.
asymptotes towards 1 as
decreases, and towards
as
increases. For typical Weber fractions of less than 0.3 (see Table 1),
does not deviate much from 1. The bottom curve, in black, shows the limiting case, as
. All the plotted functions except the one for
are given by Equation (67). In Theorem 4B, we prove that the limiting case as
is identical to the curve corresponding to a logarithmic transducer; this curve is given by Equation (70).
Figure 10.
Psychometric functions resulting from transducer and nonzero pedestal.
The thin, coloured curves show the psychometric function of Equation (69) with . Different rows of panels show psychometric functions with different noise CDFs, as indicated on the right of the figure. Different columns of panels show psychometric functions for different Weber fractions,
. The thick, black curves show the best-fitting (maximum-likelihood) Weibull functions. Each panel displays the
value of the best-fitting Weibull function (
) and the estimate,
, where
is given by Equation (67), and
is given by Equation (52), (53), (54) or (60), as appropriate.
Figure 11.
Psychometric functions resulting from transducer and nonzero pedestal.
All details are the same as in Figure 10, except that the transducer exponent is 0.5.
Figure 12.
Psychometric functions resulting from a logarithmic transducer.
The thin, coloured curves show the psychometric function of Equation (71). Different rows of panels show psychometric functions with different noise CDFs, as indicated on the right of the figure. Different columns of panels show psychometric functions for different Weber fractions, . The thick, black curves show the best-fitting (maximum-likelihood) Weibull functions. Each panel displays the
value of the best-fitting Weibull function (
) and the estimate,
, where
is given by Equation (70), and
is given by Equation (52), (53), (54) or (60), as appropriate.
Figure 13.
for the Legge-Foley transducer with nonzero pedestal.
The curves were generated using Equation (74). Each column of panels has a particular value for , and each row of panels has a particular value for the difference
. Within the panels, the Weber fraction,
, is indicated by the colour of the curve (the legend in the top-left panel applies to all panels). The curves approach horizontal asymptotes on the right (indicated by dotted lines), with vertical position given by Equation (67) with
. This is because, as mentioned earlier, as the input signal increases, the Legge-Foley transducer approaches a power function with exponent
. This asymptote can also be derived from Equation (74) by setting
to 0, which gives the limit as
. On the left, the curves come close to approaching an asymptote with vertical position given by Equation (67) with
because, at low contrasts, the Legge-Foley transducer approximates a power function with exponent
. These near-asymptotes are indicated by dotted lines on the left of each panel. They are not true asymptotes because, even for
, the Legge-Foley transducer is not exactly equal to a power function over a finite range of inputs. The horizontal, dashed lines indicate
. The vertical dashed lines indicate the value of
corresponding to the point of inflection of the Legge-Foley transducer. An expression for this quantity is derived in Appendix S2. For typical values of
and
, including those in this figure, the point of inflection occurs very close to an input of
, giving
. For pedestals above this value, both the target and pedestal will lie in the compressive region of the Legge-Foley transducer, so
must be less than 1. For this reason, none of the curves enter the top-right quadrant in any of the panels.
Figure 14.
Dipper functions for Weibull from Henning and Wichmann's data.
Weibull was fitted to Henning and Wichmann's [40] published data as described in the legend of Table 1. These
values are plotted in black lines and symbols, excluding observer GBH's
value of 13.1 for a pedestal contrast of 0.01, which is obviously an outlier. In each case, the function mapping pedestal contrast to
has a dipper shape. To see whether the dip occurred in the predicted location, we fitted a Legge-Foley transducer model to Henning and Wichmann's data separately for each observer. The model's predicted proportion correct,
, was given by Equation (6) with the transducer function,
, given by the 4-parameter Legge-Foley transducer (Equation (72)), and the noise CDF,
, given by the generalized Gaussian (Equation (41)), which had
as a free parameter, and
set so that
, using Equation (44) (thus we adjusted sensitivity by adjusting the transducer gain, rather than the noise CDF spread). For each pedestal value, Henning and Wichmann reported the contrast differences,
, corresponding to three different performance levels (proportion correct,
= 0.6, 0.75, or 0.9), sampled from their fitted psychometric functions. We performed a maximum-likelihood fit of the Legge-Foley transducer model to the data, by adjusting the parameters to maximize the likelihood,
. Fitted model parameter sets (
,
,
,
,
) were (3.78, 3.38, 0.0322, 15.3, 0.947) for GBH, (3.36, 3.02, 0.00968, 22.7, 2.09) for NAL, and (3.93, 3.51, 0.0102, 18.9, 2.15) for TCC. For each observer and pedestal value, we used a numerical search method to find the threshold,
, corresponding to a proportion correct of
, and then calculated the Weber fraction,
, using Equation (34). We then found
using Equation (74), and
using Equation (50). The analytical prediction of Weibull
is then given by
, and this is plotted in magenta in the figure. Each observer's global minimum in Weibull
was close to that in the analytical prediction.
Figure 15.
Psychometric functions resulting from a Legge-Foley transducer in Meese et al.'s binocular condition.
The thin, green curves show the psychometric functions generated by Meese et al.'s [4] twin-summation model in their binocular condition; in this condition, their transducer is equivalent to the Legge-Foley transducer of Equation (72) with ,
,
,
. Note
is in units of % contrast, as used by Meese et al.; to convert to units of Michelson contrast,
should be divided by 100. The CDF of the noise on the internal difference signal,
, is a cumulative Gaussian with standard deviation given by
. Each panel gives the model's psychometric function for a different pedestal contrast,
, in Meese et al.'s binocular condition. The thick, black curves show the best-fitting (maximum-likelihood) Weibull functions. Each panel displays the
value of the best-fitting Weibull function (
) and the estimate,
, where
is given by Equation (73) for
, and by Equation (74) for the other pedestal levels. The Weber fraction,
, in these equations was calculated from the model's threshold, found by inverting the model's psychometric function using a numerical search method, as explained in the text. Because this model fitted well to Meese et al.'s data, these Weber fractions are close to (but not exactly equal to) the actual Weber fractions obtained in the experiment, given in Table 1.
Figure 16.
Psychometric functions resulting from a Legge-Foley transducer in Meese et al.'s monocular condition.
The same as Figure 15, but for Meese et al.'s monocular condition. In this condition, their twin summation model is equivalent to the Legge-Foley transducer of Equation (72) with the same parameters as those given in the legend to Figure 15, except with and
.
Figure 17.
Psychometric functions for the power-function transducer with nonzero pedestal.
The psychometric function for was generated using Equation (71); the others were generated using Equation (69). In both cases, we assumed Gaussian internal noise (i.e.
is the cumulative Gaussian). All the psychometric functions go through the point
, by definition of the threshold (the abscissa is in threshold units, i.e.
). Each panel shows psychometric functions for a particular Weber fraction. Each curve within a panel shows the psychometric function for a particular transducer exponent,
. The orange curve (
) is the psychometric function plotted in green in the second-to-top row of Figure 10. The blue curve (
) is the psychometric function plotted in green in the second-to-top row of Figure 11. The black line shows the limit as
. As proved in Theorem 4A, this limiting case is identical to the psychometric function for a log transducer. This is the psychometric function plotted in green in the second-to-top row of Figure 12. This figure illustrates two effects. Within each panel, we see how the psychometric function for the power-function transducer converges towards that for a log transducer as the exponent decreases (Theorem 4). Across panels (right-to-left), we see a demonstration of the effect proved in Theorem 3, whereby, with a nonzero pedestal, all psychometric functions converge towards that for a linear transducer as the discrimination threshold decreases (in this case, since we are plotting psychometric functions for Gaussian noise, the functions converge towards the pure noise CDF of Figure 4B as the Weber fraction decreases).
Figure 18.
Effect of a pedestal on the linearity of an expansive power-function transducer.
Each panel in the rightmost column shows the same expansive power-function transducer given by . The panels to the left show parts of this transducer sampled over different ranges of inputs: The width of the range is varied across columns of panels, and the lower limit of the range is varied across rows of panels. The lower limit would correspond to the pedestal value,
, in a discrimination experiment. The abscissa of the curves on the left is the stimulus difference,
, and the ordinate is
, the difference in internal signal values after transduction. The
-value given in each panel is the exponent of the power function that fits best (least squares) to these curves. Each coloured box drawn on a transducer in the right column indicates the part of the transducer that is sampled by the correspondingly coloured curve given in a panel to the left on the same row. It can be seen that, as the pedestal increases, the best-fitting exponent quickly approaches 1, giving an approximately linear mapping from
to
. This linearizing effect is enhanced as the width of the range decreases.
,
, and
are given in arbitrary units: For a given transducer, the best-fitting exponent is determined by the ratio of the pedestal value to the width of the input range. For example, with the transducer shown here, when the pedestal value is equal to the width of the range, the best-fitting exponent is always 1.227; when the pedestal is twice the width of the range, the best-fitting exponent is always 1.126. For a zero pedestal (top row), the best-fitting exponent is always 2, regardless of the width of the input range, and in this sense the power function is not “strongly locally linear” at
.
Figure 19.
(A) The wide, magenta curve shows an expansive power function sampled over a range of inputs from to
, where
. The horizontal blue lines both have length
, and the vertical blue lines have length
and
as indicated. The slope,
, of the secant (the oblique line) across the left half of the curve is given by
, and the slope,
, of the secant across the right half of the curve is given by
. Our index of acceleration,
, is given by
. For the power function, when
,
as
, so the curve is “strongly locally linear” at
. (B) The same as A, but with the bottom of the range of inputs,
, equal to zero. In this case,
depends only on the exponent of the power function, and so it does not approach 1 as
approaches zero. The power function is not “strongly locally linear” at
. (C) The same as A, but for a straight line function. Here,
for all
and
.
Figure 20.
Pairs of noise distribution and transducer exponent consistent with the Weibull parameters for contrast discrimination.
is the generalized Gaussian CDF shape parameter, and
is the power-function transducer exponent. Each curve plots the set of
pairs consistent with one of the fitted psychometric functions for suprathreshold contrast discrimination given in Table 1 (non-starred conditions). Where available, we used the fitted
and
parameters from the Weibull fit that included the lapse rate parameter,
. Note that the contour for Henning et al.'s subject GBH in the 8.37 cpd condition lies out of range of the axes in this figure, and so is not visible. This is because the fitted Weibull β of 6.70 is much higher than usually found – almost certainly an unreliable measurement.