Psychometric
Function Fitting, Maximum-Likelihood Criterion

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. The maximum-likelihood psychometric function fitting procedures are demonstrated
in PAL_PFML_Demo, PAL_PFML_SearchGrid_Demo, PAL_PFML_lapseFit_Demo, PAL_gammaEQlambda_Demo. All demo files are located
in the PalamedesDemos folder. 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.

Psychometric Function Fitting, Bayesian Criterion

Palamedes
can also be used to fit Psychometric Functions (PFs) to data using a Bayesian criterion (this requires installation of third-part
software: JAGS or Stan, see here for more information). 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. Much of the multi-condition fitting capabilities described above are also available
when you use the Bayesian Criterion to fit PFs. You can even fit data from multiple observers (as well as multiple conditions)
when you use the Bayesian criterion. When there are multiple subjects represented in your data file, Palamedes will automatically
fit a model that includes posterior distributions across parameters that correpond to the average parameter values across
observers. This is called Hierarchical Modeling. The Bayesian psychometric function fitting procedures are demonstrated
in the PAL_PFHB_xxx_Demo files located 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.