Fit Psychometric Functions (PFs) to data using a Maximum Likelihood (ML) criterion. Estimate the standard errors of the free parameters, and determine the 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. 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?). 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 condition²) where 'a' and 'b' are free parameters? We understand. See examples on our Model Comparison page. 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.

Palamedes can also be used to fit Psychometric Functions (PFs) to data using a Bayesian criterion (this requires installation of third-party 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, Palamedes will automatically fit a hierarchical model that includes posterior distributions across the hyper-parameters specifying the distribution of parameter values (e.g., mean, standard deviation) across observers. The Bayesian psychometric function fitting procedures are demonstrated in the PAL_PFHB_xxx_Demo files located in the PalamedesDemos folder. For more information see Prins, Behavior Research Methods, 2023.

Palamedes can statistically compare custom models defined across multiple conditions in order 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 instead. Finally, Palamedes can compare models that use custom-parametrization of the parameters of a PF (for examples, see our Model Comparison page). 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. To learn more about statistical model comparisons see Prins and Kingdom (2018).

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 (see Prins, 2013). 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.

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.

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 gs and ps 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.

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.