Psychometric
Function Fitting

Palamedes can be used to fit Psychometric Functions (PFs) to data using a Maximum Likelihood (ML) criterion. Various
forms of PF are supported (Weibull, Logistic, Gumbel, Cumulative Normal, Hyperbolic Secant, Quick). Make any combination of
the PF's parameters (threshold, slope, guess rate, lapse rate) free parameters. Estimate the standard errors of free parameters
using a parametric or non-parametric bootstrap. Determine Goodness-of-Fit of the fit. Palamedes can also use a Bayesian criterion
to determine the best-fitting PF to your data and the standard errors of the parameters. The psychometric function fitting
procedures are demonstrated in PAL_PFML_Demo, PAL_PFML_SearchGrid_Demo, PAL_PFML_lapseFit_Demo, PAL_gammaEQlambda_Demo (all
use ML) and PAL_PFBA_Demo (Bayesian). All demo files are located in the PalamedesDemos folder.

Multi-condition Model
Fitting

Palamedes can also fit PFs to several conditions simultaneously while allowing flexibility in constraining the free
parameters between conditions in order to define a 'model'. For example: you wish to fit separate PFs to more than one condition
in your experiment but you wish to estimate a single, shared lapse rate across all conditions (and why wouldn't you?). Palamedes
can do this. Another example: you wish to constrain thresholds to increase linearly with condition but you wish to fit a single
shared slope to all conditions. Palamedes can do this too. With the introduction of custom-reparametrization of parameters
in Palamedes version 1.1.0 you can constrain any of the PFs parameters *any* *which way* you wish (minor assembly
required). Need to fix the threshold values in conditions 1, 3, and 6 to equal 1, 2, and pi respectively but have thresholds
in conditions 2, 4, 5, and 7 adhere to 'threshold(condition) = a x sin(b x condition^2) where 'a' and 'b' are free parameters?
We understand. Click here for examples of what you can do. Palamedes can perform a bootstrap analysis to estimate standard errors on
the parameters of your custom-defined model, and determine its Goodness-of-Fit. The model fitting procedures are demonstrated
in PAL_PFLR_Demo, PAL_PFLR_FourGroup_Demo, PAL_PFLR_CustomDefine_Demo, and PAL_PFML_lapseFit_Demo in the PalamedesDemos folder.

Model Comparisons

Palamedes can statistically compare models defined across multiple
conditions (see above: Multi-condition Model Fitting) to test, for example, whether thresholds differ between conditions,
whether slopes differ between conditions, whether the lapse rate equals zero, etcetera, etcetera. Users savvy with the General
Linear Model may use contrasts to test, for example, whether thresholds increase linearly with, say, adaptation duration
or whether the data warrant a quadratic trend. Finally, Palamedes can compare models that use custom-parametrization
of the parameters of a PF (for examples, click here). The model comparison procedures are demonstrated in PAL_PFLR_Demo, PAL_PFLR_FourGroup_Demo, PAL_PFLR_CustomDefine_Demo,
and PAL_PFML_lapseFit_Demo in the PalamedesDemos folder.

Adaptive Procedures

Palamedes can be used to guide stimulus selection in your experiments. Palamedes can implement up/down procedures,
‘running fit’ procedures (best PEST, QUEST), and the psi method. As of Palamedes 1.6.0 the Psi-method can treat
any of the PF's four parameters either as a parameter of primary interest whose estimation should be optimized,
as a nuisance parameter whose estimation should be subservient to the estimation of primary interest or as
fixed. This avoids bias in parameter estimates (Prins, 2013; http://www.journalofvision.org/content/13/7/3). All adaptive methods offer considerable flexibility regarding the specifics. The up/down, running fit, and psi method procedures
are demonstrated in PAL_AMUD_Demo, PAL_AMRF_Demo, and PAL_AMPM_Demo respectively. All are located in the PalamedesDemos folder.

Signal Detection Measures

Palamedes can calculate the Signal-Detection-Theory (SDT) measure *d*' (d-prime) and criterion values from proportion
hits and false alarms, or proportion correct, (and vice versa) for a large variety of psychophysical tasks (e.g. 1AFC, 2AFC,
MAFC, Same-Different, Match-to-Sample, Oddity) and under different models (Independent Observation, Differencing).
Palamedes can also fit ROC (receiver operating characteristic) curves to single-interval (1AFC) rating-scale data in
order to obtain estimates of both *d*’ and the ratio of standard deviations (SD ratio) of the two underlying
distributions (e.g. noise and signal-plus-noise), and furthermore determine whether the SD ratio is significantly different
from unity. The signal detection procedures are demonstrated in PAL_SDT_1AFC_DPtoPHF_Demo, PAL_SDT_1AFC_PHFtoDP_Demo,
PAL_SDT_DPtoPCcomparison_Demo, PAL_SDT_1AFC_PCtoDPcomparison_Demo, PAL_SDT_DPtoPCacrossM_Demo and PAL_SDT_ROCML_Demo in the
PalamedesDemos folder. Palamedes can fit SDT models to psychometric functions using PAL_SDT_PFML_Fit, in order to estimate
the stimulus scaling factor *g* and transducer
exponent *p, *based on the relation *d*' = *(gx)*^*p,** *where *x *is stimulus intensity.
The use of the SDT psychometric function fitting routines is demonstrated in PAL_SDT_PF_Demo.

Summation Modeling

Palamedes can calculate a variety of measures for modeling detection tasks
involving multiple stimuli. Palamedes can calculate proportion correct from stimulus amplitude (or level), and vice-versa,
for detecting multiple stimuli assuming either probability or additive summation under the assumptions of Signal-Detection-Theory
(SDT). The routines use novel formulae that are solved by numerical integration. For probability summation, PAL_SDT_PS_SLtoPC,
and for additive summation, PAL_SDT_PS_SLtoPC, calculate
proportion correct for any *x, g*, *p*, *M*, *Q* and *n, *where *x* is
stimulus level, *g* the stimulus level scaling factor, *p* the exponent on the transducer function,
*M* the number of alternatives in the forced-choice task, *Q* the number of monitored channels
and *n* the number of stimuli/signals (the relationship between *x* and the conventional Signal-Detection-Theory
measure *d*' is *d*' = *(gx)*^*p*)*. *PAL_SDT_PS_PCtoSL and PAL_SDT_AS_PCtoSL perform the inverse of this function,
i.e. calculate *x* from proportion correct. For unequal stimulus intensities there is PAL_SDT_PS_uneqSLtoPS, PAL_SDT_PS_2uneqSLtoPS and PAL_SDT_PS_PCto2uneqSL
for probability, and PAL_SDT_AS_uneqSLtoPS, PAL_SDT_AS_2uneqSLtoPS and PAL_SDT_AS_PCto2uneqSL for additive summation. For verification
purposes MonteCarlo simulations of the probability summation formulae can be performed using PAL_SDT_PS_MonteCarlo_SLtoPC
and PAL_SDT_PS_Montcarlo_uneqSLtoPC. Using PAL_SDT_Summ_MultiplePFML_Fit, Palamedes can fit multiple psychometric functions (PFs) obtained from different combinations of stimuli
with both additive and probability summation models in order to estimate *g*s* *and *p*s and determine which model is better. PAL_SDT_PSvAS_3PFmultipleFit_Demo
and PAL_SDT_PSvAS_SummSquare_Demo demonstrates the usage of the fitting routine
for 3-PF and 5-PF cases, the latter for data displayed in a conventional summation square.

Maximum Likelihood Difference Scaling

Palamedes will derive parameter estimates describing the transducer function based on an observer’s judgments
about the perceived differences between stimuli, for stimuli presented as pairs, triads or quadruples (double pairs). A
bootstrap procedure may be used to determine standard errors. The maximum likelihood difference scaling procedures are demonstrated
for simulated data sets in PAL_MLDS_Demo in the PalamedesDemos folder.