# Compute score density Computes the cumulated density for distribution function, the probability density function, and the log probability density function as fd objects by spline smoothing of the score values `indexdens` using the basis object `logdensbasis`. The norming constant `C` is also output. The score values may score index values `index`, expected test score values `mu`, or arc length locations on the test information or scale curve. The argument functional data object `logdensfd` should have a range that is appropriate for the score values being represented: For score indices, [0,100], for expected test scores, the range of observed or expected scores; and for test information curve locations in the interval [0,`infoSurp`]. ```r index_distn(indexdens, logdensbasis, pvec=c(0.05, 0.25, 0.50, 0.75, 0.95), nfine = 101) ``` ## Arguments - `indexdens`: A vector of score index, test score, or arc length values. In the score index case, these are usually only the values in the interior of the interval [0,100]. - `logdensbasis`: A functional basis object for representing the log density function. The argument may also be a functional data object (`fd`) or a functional basis object (`Sbasis`). - `pvec`: A vector length NL containing the marker percentages. - `nfine`: The number of values in a fine grid, default as 101. ## Returns A named list containing: - **pdf_fd:**: An fd object for the probability density function values over the fine mesh. - **cdffine:**: A vector of cumulative probability values beginning with zero and ending with 1. It must not have ties. - **pdffine:**: A vector of probability values. - **logdensfd:**: A functional data object (`fd`) representing the log of the probability function for input `index`. - **C:**: The normalization constant for computing the probability density function with the command `densityfd = exp(logdensfd)/C.` - **denscdf:**: A set of unique values of the cumulative probability function defined over an equally spaced mesh of score index values of the same length as `denscdf`. - **indcdf:**: A vector of values within [0,100] corresponding to the values in `denscdf`. ## References Ramsay, J. O., Li J. and Wiberg, M. (2020) Full information optimal scoring. Journal of Educational and Behavioral Statistics, 45, 297-315. Ramsay, J. O., Li J. and Wiberg, M. (2020) Better rating scale scores with information-based psychometrics. Psych, 2, 347-360. ## Author(s) Juan Li and James Ramsay ## See Also `index_fun`, `index2info`, `mu`, `scoreDensity` ## Examples ```r # Example 1. Display the item power curves for the # short SweSAT multiple choice test with 24 items and 1000 examinees # Assemble information for estimating index density indfine <- seq(0,100,len=101) SfdList <- Quant_13B_problem_parmList$SfdList index <- Quant_13B_problem_parmList$index N <- length(index) # Define the density for only interior index values inside <- index > 0 & index < 100 indexdens <- index[inside] logdensbasis <- fda::create.bspline.basis(c(0,100), 15) index_distnList <- index_distn(index[inside], logdensbasis) denscdf <- as.numeric(index_distnList$denscdf) indcdf <- as.numeric(index_distnList$indcdf) # adjusted marker score index values are computed by interpolation markers <- c(.05, .25, .50, .75, .95) Qvec <- pracma::interp1(denscdf, indcdf, markers) result <- density_plot(indexdens, c(0,100), Qvec) ```

index_distn function