# Capacity Coefficient for Exhaustive (AND) Processing Calculates the Capacity Coefficient for Exhaustive (AND) Processing ```r capacity.and(RT, CR=NULL, ratio=TRUE) ``` ## Arguments - `RT`: A list of response time arrays. The first array in the list is assumed to be the exhaustive condition. - `CR`: A list of correct/incorrect indicator arrays. If NULL, assumes all are correct. - `ratio`: Indicates whether to return the standard ratio capacity coefficient or, if FALSE, the difference form. ## Details The AND capacity coefficient compares performance on task to an unlimited-capacity, independent, parallel (UCIP) model using cumulative reverse hazard functions. Suppose $Ki(t)$ is the cumulative reverse hazard function for response times when process $i$ is completed in isolation and $Ki(t)$ is the cumulative reverse hazard function for response times when all processes must completed together. Then the AND capacity coefficient is given by, $$ C_{\rm AND}(t)=\frac{\sum_i K_i(t)}{K_{\rm AND}(t)}.Cand(t)=sum[Ki(t)]/[Kand(t)]. $$ The numerator is the estimated cumulative reverse hazard function for the UCIP model, based on the response times for each process in isolation and the denominator is the actual performance. $Cand(t)<1$ implies worse performance than the UCIP model. This indicates that either there are limited processing resources, there is inhibition among the subprocesses, or the items are not processed in parallel (e.g., the items may be processed serially). $Cand(t)>1$ implies better performance than the UCIP model. This indicates that either there are more processing resources available per process when there are more processes, that there is facilitation among the subprocesses, or the items are not processed in parallel (e.g., the items may be processed coactively). The difference form of the capacity coefficient (returned if ratio=FALSE) is given by, $$ C_{\rm AND}(t)=K_{\rm AND}(t) - \sum_i K_i(t).Cand(t)=Kand(t) - sum[Ki(t)]. $$ Negative values indicate worse than UCIP performance and positive values indicate better than UCIP performance. ## Returns - **Ct**: An object of class approxfun representing the estimated AND capacity coefficient. - **Var**: An object of class approxfun representing the variance of the estimated AND capacity coefficient. Only returned if ratio=FALSE. - **Ctest**: A list with class "htest" that is returned from `ucip.test` and contains the statistic and p-value. ## References Townsend, J.T. & Wenger, M.J. (2004). A theory of interactive parallel processing: New capacity measures and predictions for a response time inequality series. **Psychological Review, 111**, 1003--1035. Townsend, J.T. & Nozawa, G. (1995). Spatio-temporal properties of elementary perception: An investigation of parallel, serial and coactive theories. **Journal of Mathematical Psychology, 39**, 321-360. Houpt, J.W. & Townsend, J.T. (2012). Statistical Measures for Workload Capacity Analysis. **Journal of Mathematical Psychology, 56**, 341-355. Houpt, J.W., Blaha, L.M., McIntire, J.P., Havig, P.R. and Townsend, J.T. (2013). Systems Factorial Technology with R. **Behavior Research Methods**. ## Author(s) Joe Houpt ## See Also `ucip.test` `capacityGroup` `capacity.or` `estimateUCIPand` `estimateNAK` `approxfun` ## Examples ```r rate1 <- .35 rate2 <- .3 RT.pa <- rexp(100, rate1) RT.ap <- rexp(100, rate2) RT.pp.limited <- pmax( rexp(100, .5*rate1), rexp(100, .5*rate2)) RT.pp.unlimited <- pmax( rexp(100, rate1), rexp(100, rate2)) RT.pp.super <- pmax( rexp(100, 2*rate1), rexp(100, 2*rate2)) tvec <- sort(unique(c(RT.pa, RT.ap, RT.pp.limited, RT.pp.unlimited, RT.pp.super))) cap.limited <- capacity.and(RT=list(RT.pp.limited, RT.pa, RT.ap)) print(cap.limited$Ctest) cap.unlimited <- capacity.and(RT=list(RT.pp.unlimited, RT.pa, RT.ap)) cap.super <- capacity.and(RT=list(RT.pp.super, RT.pa, RT.ap)) matplot(tvec, cbind(cap.limited$Ct(tvec), cap.unlimited$Ct(tvec), cap.super$Ct(tvec)), type='l', lty=1, ylim=c(0,3), col=2:4, main="Example Capacity Functions", xlab="Time", ylab="C(t)") abline(1,0) legend('topright', c("Limited", "Unlimited", "Super"), lty=1, col=2:4) ```

capacity.and function