# Links for Binomial Family for m-alternative Forced-choice These provide links for the binomial family for fitting m-alternative forced-choice psychophysical functions. ```r mafc.logit( .m = 2 ) mafc.probit( .m = 2 ) mafc.cloglog( .m = 2 ) mafc.weib( ... ) mafc.cauchit( .m = 2 ) ``` ## Arguments - `.m`: is the integer number (>1) of choices (Default to 2AFC). For m = 1 (Yes/No paradigm), use one of the built-in links for the binomial family. - `...`: just to pass along the formals of `mafc.cloglog`. ## Details These functions provide links for fitting psychometric functions arising from an m-alternative forced-choice experiment. The estimated coefficients of the linear predictor influence both the location and the slope of the psychometric function(s), but provide no means of estimating the upper aymptote which is constrained to approach 1. If the upper asympotote must be estimated, it would be better to maximize directly the likelihood, either with a function like `optim` or `gnlr` from package `gnlm` (available at [https://www.commanster.eu/rcode.html](https://www.commanster.eu/rcode.html)). Alternatively, the function `probit.lambda` can be used with a known upper asymptote, or `glm.lambda` or `glm.WH` to estimate one, with a probit link. `mafc.weib` is just an alias for `mafc.cloglog`. ## Returns Each link returns a list containing functions required for relating the response to the linear predictor in generalized linear models and the name of the link. - **linkfun**: The link function - **linkinv**: The inverse link function - **mu.eta**: The derivative of the inverse link - **valideta**: The domain over which the linear predictor is valid - **link**: A name to be used for the link ## References Williams J, Ramaswamy D and Oulhaj A (2006) 10 Hz flicker improves recognition memory in older people **BMC Neurosci.** 2006 5;7:21 [https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1434755/](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC1434755/) (for an example developed prior to this one, but for m = 2). Klein S. A. (2001) Measuring, estimating, and understanding the psychometric function: a commentary. **Percept Psychophys.**, 63(8) , 1421--1455. Wichmann, F. A. and Hill, N. J. (2001) The psychometric function: I.Fitting, sampling, and goodness of fit. **Percept Psychophys.**, 63(8) , 1293--1313. Yssaad-Fesselier, R. and Knoblauch, K. (2006) Modeling psychometric functions in R. **Behav Res Methods.**, 38(1) , 28--41. (for examples with `gnlr`). ## Author(s) Kenneth Knoblauch ## See Also `family`, `make.link`, `glm`, `optim`, `probit.lambda`, `glm.lambda`, `glm.WH` ## Examples ```r #A toy example, b <- 3.5 g <- 1/3 d <- 0.0 a <- 0.04 p <- c(a, b, g, d) num.tr <- 160 cnt <- 10^seq(-2, -1, length = 6) # contrast levels #simulated observer responses truep <- g + (1 - g - d) * pweibull(cnt, b, a) ny <- rbinom(length(cnt), num.tr, truep) nn <- num.tr - ny phat <- ny/(ny + nn) resp.mat <- matrix(c(ny, nn), ncol = 2) ddprob.glm <- glm(resp.mat ~ cnt, family = binomial(mafc.probit(3))) ddlog.glm <- glm(resp.mat ~ cnt, family = binomial(mafc.logit(3))) # Can fit a Weibull function, but use log contrast as variable ddweib.glm <- glm(resp.mat ~ log(cnt), family = binomial(mafc.cloglog(3))) ddcau.glm <- glm(resp.mat ~ log(cnt), family = binomial(mafc.cauchit(3))) plot(cnt, phat, log = "x", cex = 1.5, ylim = c(0, 1)) pcnt <- seq(0.01, 0.1, len = 100) lines(pcnt, predict(ddprob.glm, data.frame(cnt = pcnt), type = "response"), lwd = 2) lines(pcnt, predict(ddlog.glm, data.frame(cnt = pcnt), type = "response"), lwd = 2, lty = 2) lines(pcnt, predict(ddweib.glm, data.frame(cnt = pcnt), type = "response"), lwd = 3, col = "grey") lines(pcnt, predict(ddcau.glm, data.frame(cnt = pcnt), type = "response"), lwd = 3, col = "grey", lty = 2) # Weibull parameters \alpha and \beta cc <- coef(ddweib.glm) alph <- exp(-cc[1]/cc[2]) bet <- cc[2] #More interesting example with data from Yssaad-Fesselier and Knoblauch data(ecc2) ecc2.glm <- glm(cbind(Correct, Incorrect) ~ Contr * Size * task, family = binomial(mafc.probit(4)), data = ecc2) summary(ecc2.glm) ecc2$fit <- fitted(ecc2.glm) library(lattice) xyplot(Correct/(Correct + Incorrect) ~ Contr | Size * task, data = ecc2, subscripts = TRUE, ID = with(ecc2, Size + as.numeric(task)), scale = list(x = list(log = TRUE), y = list(limits = c(0, 1.05))), xlab = "Contrast", ylab = "Proportion Correct Response", aspect = "xy", panel = function(x, y, subscripts, ID, ...) { which = unique(ID[subscripts]) llines(x, ecc2$fit[which ==ID], col = "black", ...) panel.xyplot(x, y, pch = 16, ...) panel.abline(h = 0.25, lty = 2, ...) } ) ```

mafc function