# Probability of faking. The function gives the conditional probability of replacement $p(f=k|d=h,\theta_F)$ for discrete values in the range $1, \ldots, Q$. ```r pfake(k, h = k, p = c(0,0), Q = 5, gam = c(1,1), del = c(1,1), fake.model = c("uninformative", "average", "slight", "extreme")) ``` ## Arguments - `k`: A fake value. - `h`: An observed original value. - `p`: Overall probability of replacement: `p[1]` indicates the faking good probability, `p[2]` indicates the faking bad probability. - `Q`: Max value in the discrete r.v. range: $1, \ldots, Q$. - `gam`: Gamma parameter: `gam[,1]` indicates the faking good parameter $\gamma_{+}$, `gam[,2]` indicates the faking bad parameter $\gamma_{-}$. - `del`: Delta parameter: `del[,1]` indicates the faking good parameter $\delta_{+}$, `del[,2]` indicates the faking bad parameter $\delta_{-}$. - `fake.model`: A character string indicating the model for the conditional replacement distribution. The options are: `uninformative` (default option) [`gam = c(1,1)` and `del = c(1,1)`]; `average` [`gam = c(3,3)` and `del = c(3,3)`]; `slight` [`gam = c(1.5,4)` and `del = c(4,1.5)`]; `extreme` [`gam = c(4,1.5)` and `del = c(1.5,4)`]. ## Returns Gives the conditional probability distribution based on the following equation $$ p(f=k|d=h,\theta_F)=\left\{\begin{array}{cl}DG(k;h+1,Q,\gamma_{+},\delta_{+}) \pi_{+} & 1 \leq h < k \leq Q \\DG(k;q,h-1,\gamma_{-},\delta_{-}) \pi_{-} & 1 \leq k < h \leq Q \\1-(\pi_{+}+\pi{-}) & 1 < h=k < Q \\1- \pi_{+} & k=h=1 \\1- \pi_{-} & k=h=Q\end{array}\right. $$ with $\theta_F$ and $DG$ being the parameter vector $(\gamma_{+},\gamma_{-},\delta_{+},\delta_{-},\pi_{+},\pi_{-})$ and the generalized Beta distribution for discrete variables (`dgBetaD`) with bounds $a=h+1$ (resp. $a=1$) and $b=Q$ (resp $b=h-1$). The parameter $\pi_{+}$ denotes the probability of faking good, $\pi_{-}$ indicates the probability of faking bad. Note that the faking probabilities must satisfy the following condition: $\pi_{+}+\pi_{-} \leq 1$. ## References Lombardi, L., Pastore, M. (2014). sgr: A Package for Simulating Conditional Fake Ordinal Data. **The R Journal**, 6, 164-177. Pastore, M., Lombardi, L. (2014). The impact of faking on Cronbach's Alpha for dichotomous and ordered rating scores. **Quality & Quantity**, 48, 1191-1211. ## Author(s) Massimiliano Pastore & Luigi Lombardi ## Examples ```r x <- 1:7 GA <- c(1,3,1.5,8); DE <- c(1,3,4,2.5) ### fake good par(mfrow=c(2,2)) for (j in 1:4) { y <- NULL for (i in x) y <- c(y,pfake(x[i],h=4,Q=7, gam=c(GA[j],GA[j]),del=c(DE[j],DE[j]),p=c(.4,0))) plot(x,y,type="h",panel.first=points(x,y,pch=19), main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.7), ylab="Replacement probability") } ### fake bad for (j in 1:4) { y <- NULL for (i in x) y <- c(y,pfake(x[i],h=4,Q=7, gam=c(GA[j],GA[j]),del=c(DE[j],DE[j]),p=c(0,.4))) plot(x,y,type="h",panel.first=points(x,y,pch=19), main=paste("gamma=",GA[j]," delta=",DE[j],sep=""),ylim=c(0,.7), ylab="Replacement probability") } ### fake good and fake bad P = c(.4,.4) par(mfrow=c(2,4)) for (j in x) { y <- NULL for (i in x) { y <- c(y,pfake(x[i],h=x[j],Q=max(x),gam=c(GA[1],GA[1]),del=c(DE[1],DE[1]),p=P)) } plot(x,y,type="h",panel.first=points(x,y,pch=19), main=paste("h=",x[j],sep=""),ylim=c(0,1), ylab="Replacement probability") print(sum(y,na.rm=TRUE)) } ### using the fake.model argument x <- 1:5 models <- c("uninformative","average","slight","extreme") par(mfrow=c(2,2)) for (j in 1:4) { y <- NULL for (i in x) y <- c(y,pfake(x[i],h=2,Q=max(x), fake.model=models[j],p=c(.45,0))) # fake good plot(x,y,type="h",panel.first=points(x,y,pch=19), main=paste(models[j],"model"),ylim=c(0,1), ylab="Replacement probability") } par(mfrow=c(2,2)) for (j in 1:4) { y <- NULL for (i in x) y <- c(y,pfake(x[i],h=4,Q=max(x), fake.model=models[j],p=c(0,.45))) # fake bad plot(x,y,type="h",panel.first=points(x,y,pch=19), main=paste(models[j],"model"),ylim=c(0,1), ylab="Replacement probability") } ```

pfake function