# MixDelta: PSE/JND from GLMM Estimates using Delta Method In MixedPsy: Statistical Tools for the Analysis of Psychophysical Data

## Description

Estimate Points of Subjective Equivalence (PSE), Just Noticeable Differences (JND) and the related Standard Errors from a GLMM by means of delta method. The method applies to models with a probit link function, one continuous predictor, and one (optional) factorial predictor.

## Usage

 `1` ```MixDelta(xplode.obj, alpha = 0.05, p = 0.75) ```

## Arguments

 `xplode.obj` an object of class `xplode.obj`. The fitted model (object of class `merMod`, specifically of subclass `glmerMod`) includes one continuous predictor and one (optional) factorial predictor. `alpha` significance level of the confidence intervals. Default is 0.05 (value for 95% confidence interval). `p` probability value relative to the JND upper limit. Default is 0.75 (value for 50% JND).

## Details

When the model includes a factorial predictor, the function is based on a recursive use of `glmer` and re-order of levels of the factorial predictor. The JND estimate assumes a probit link function.

## Value

A matrix including estimate, standard error, inferior and superior bounds of the confidence interval of PSE and JND. If a factorial predictor is included in the model, the function returns a list, each item containing a matrix for the estimates relative to a level of the predictor.

## Note

The delta method is based on the assumption of asymptotic normal distribution of the parameters estimates. This may result in an incorrect variance estimation. For a more reliable (but more time-consuming) estimation based on bootstrap method, use `pseMer`.

## References

Moscatelli, A., Mezzetti, M., & Lacquaniti, F. (2012). Modeling psychophysical data at the population-level: The generalized linear mixed model. Journal of Vision, 12(11):26, 1-17. doi:10.1167/12.11.26

Casella, G., & Berger, R. L. (2002). Statistical inference (2nd ed.). Pacific Grove, CA: Duxbury Press

`glmer` for fitting Generalized Linear Mixed Models. `xplode` for interfacing values from a fitted GLMM to `MixedPsy` functions. `pseMer` for bootstrap-based confidence intervals of psychometric parameters.
 ``` 1 2 3 4 5 6 7 8 9 10 11 12 13``` ```library(lme4) #univariable GLMM (one continuous predictor) mod.uni = glmer(formula = cbind(Longer, Total - Longer) ~ X + (1 | Subject), family = binomial(link = "probit"), data = simul_data) xplode.uni = xplode(model = mod.uni, name.cont = "X") MixDelta(xplode.uni) #multivariable GLMM (one continuous and one factorial predictor) mod.multi <- glmer(cbind(faster, slower) ~ speed * vibration + (1 + speed| subject), family = binomial(link = "probit"), data = vibro_exp3) xplode.multi <- xplode(model = mod.multi, name.cont = "speed", name.factor = "vibration") MixDelta(xplode.multi) ```