# Compute factor score estimates (a.k.a, ability estimates, latent trait estimates, etc) Computes MAP, EAP, ML (Embretson & Reise, 2000), EAP for sum-scores (Thissen et al., 1995), or WLE (Warm, 1989) factor scores with a multivariate normal prior distribution using equally spaced quadrature. EAP scores for models with more than three factors are generally not recommended since the integration grid becomes very large, resulting in slower estimation and less precision if the `quadpts` are too low. Therefore, MAP scores should be used instead of EAP scores for higher dimensional models. Multiple imputation variants are possible for each estimator if a parameter information matrix was computed, which are useful if the sample size/number of items were small. As well, if the model contained latent regression predictors this information will be used in computing MAP and EAP estimates (for these models, `full.scores=TRUE` will always be used). Finally, plausible value imputation is also available, and will also account for latent regression predictor effects. ```r fscores( object, method = "EAP", full.scores = TRUE, rotate = "oblimin", Target = NULL, response.pattern = NULL, append_response.pattern = FALSE, na.rm = FALSE, plausible.draws = 0, plausible.type = "normal", quadpts = NULL, item_weights = rep(1, extract.mirt(object, "nitems")), returnER = FALSE, T_as_X = FALSE, EAPsum.scores = FALSE, return.acov = FALSE, mean = NULL, cov = NULL, covdata = NULL, verbose = TRUE, full.scores.SE = FALSE, theta_lim = c(-6, 6), MI = 0, use_dentype_estimate = FALSE, QMC = FALSE, custom_den = NULL, custom_theta = NULL, expected.info = FALSE, min_expected = 1, max_theta = 20, start = NULL, ... ) ``` ## Arguments - `object`: a computed model object of class `SingleGroupClass`, `MultipleGroupClass`, or `DiscreteClass` - `method`: type of factor score estimation method. Can be: * `"EAP"` for the expected a-posteriori (default). For models fit using `mdirt` this will return the posterior classification probabilities * `"MAP"` for the maximum a-posteriori (i.e, Bayes modal) * `"ML"` for maximum likelihood * `"WLE"` or `"WML"` for weighted maximum-likelihood estimation * `"EAPsum"` for the expected a-posteriori for each sum score * `"plausible"` for a single plausible value imputation for each case. This is equivalent to setting `plausible.draws = 1` * `"classify"` for the posteriori classification probabilities (only applicable when the input model was of class `MixtureClass`) - `full.scores`: if `FALSE` then a summary table with factor scores for each unique pattern is displayed as a formatted `matrix` object. Otherwise, a matrix of factor scores for each response pattern in the data is returned (default) - `rotate`: prior rotation to be used when estimating the factor scores. See `summary-method` for details. If the object is not an exploratory model then this argument is ignored - `Target`: target rotation; see `summary-method` for details - `response.pattern`: an optional argument used to calculate the factor scores and standard errors for a given response vector or matrix/data.frame - `append_response.pattern`: logical; should the inputs from `response.pattern` also be appended to the factor score output? - `na.rm`: logical; remove rows with any missing values? This is generally not required due to the nature of computing factors scores, however for the "EAPsum" method this may be necessary to ensure that the sum-scores correspond to the same composite score - `plausible.draws`: number of plausible values to draw for future researchers to perform secondary analyses of the latent trait scores. Typically used in conjunction with latent regression predictors (see `mirt` for details), but can also be generated when no predictor variables were modelled. If `plausible.draws` is greater than 0 a list of plausible values will be returned - `plausible.type`: type of plausible values to obtain. Can be either `'normal'` (default) to use a normal approximation based on the ACOV matrix, or `'MH'` to obtain Metropolis-Hastings samples from the posterior (silently passes object to `mirt`, therefore arguments like `technical` can be supplied to increase the number of burn-in draws and discarded samples) - `quadpts`: number of quadrature to use per dimension. If not specified, a suitable one will be created which decreases as the number of dimensions increases (and therefore for estimates such as EAP, will be less accurate). This is determined from the switch statement `quadpts <- switch(as.character(nfact), '1'=121, '2'=61, '3'=31, '4'=19,'5'=11, '6'=7, 5)` - `item_weights`: a user-defined weight vector used in the likelihood expressions to add more/less weight for a given observed response. Default is a vector of 1's, indicating that all the items receive the same weight - `returnER`: logical; return empirical reliability (also known as marginal reliability) estimates as a numeric values? - `T_as_X`: logical; should the observed variance be equal to `var(X) = var(T) + E(E^2)` or `var(X) = var(T)` when computing empirical reliability estimates? Default (`FALSE`) uses the former - `EAPsum.scores`: logical; include the model-implied expected values and variance for the item and total scores when using `method = 'EAPsum'` with `full.scores=FALSE`? This information is included in the hidden `'fit'` attribute which can be extracted via `attr(., 'fit')` for later use - `return.acov`: logical; return a list containing covariance matrices instead of factors scores? `impute = TRUE` not supported with this option - `mean`: a vector for custom latent variable means. If NULL, the default for 'group' values from the computed mirt object will be used - `cov`: a custom matrix of the latent variable covariance matrix. If NULL, the default for 'group' values from the computed mirt object will be used - `covdata`: when latent regression model has been fitted, and the `response.pattern` input is used to score individuals, then this argument is used to include the latent regression covariate terms for each row vector supplied to `response.pattern` - `verbose`: logical; print verbose output messages? - `full.scores.SE`: logical; when `full.scores == TRUE`, also return the standard errors associated with each respondent? Default is `FALSE` - `theta_lim`: lower and upper range to evaluate latent trait integral for each dimension. If omitted, a range will be generated automatically based on the number of dimensions - `MI`: a number indicating how many multiple imputation draws to perform. Default is 0, indicating that no MI draws will be performed - `use_dentype_estimate`: logical; if the density of the latent trait was estimated in the model (e.g., via Davidian curves or empirical histograms), should this information be used to compute the latent trait estimates? Only applicable for EAP-based estimates (EAP, EAPsum, and plausible) - `QMC`: logical; use quasi-Monte Carlo integration? If `quadpts` is omitted the default number of nodes is 5000 - `custom_den`: a function used to define the integration density (if required). The NULL default assumes that the multivariate normal distribution with the 'GroupPars' hyper-parameters are used. At the minimum must be of the form: `function(Theta, ...)` where Theta is a matrix of latent trait values (will be a grid of values if `method == 'EAPsum'` or `method == 'EAP'`, otherwise Theta will have only 1 row). Additional arguments may included and are caught through the `fscores(...)` input. The function **must** return a numeric vector of density weights (one for each row in Theta) - `custom_theta`: a matrix of custom integration nodes to use instead of the default, where each column corresponds to the respective dimension in the model - `expected.info`: logical; instead of using the observed information when `method = 'ML'` or `method = 'WLE'` use the expected information computed via `testinfo`? Only currently supported for unidimensional models - `min_expected`: when computing goodness of fit tests when `method = 'EAPsum'`, this value is used to collapse across the conditioned total scores until the expected values are greater than this value. Note that this only affect the goodness of fit tests and not the returned EAP for sum scores table - `max_theta`: the maximum/minimum value any given factor score estimate will achieve using any modal estimator method (e.g., MAP, WLE, ML) - `start`: a matrix of starting values to use for iterative estimation methods. Default will start at a vector of 0's for each response pattern, or will start at the EAP estimates (unidimensional models only). Must be in the form that matches `full.scores = FALSE` (mostly used in the `mirtCAT` package) - `...`: additional arguments to be passed to `nlm` ## Details The function will return either a table with the computed scores and standard errors, the original data matrix with scores appended to the rightmost column, or the scores only. By default the latent means and covariances are determined from the estimated object, though these can be overwritten. Iterative estimation methods can be estimated in parallel to decrease estimation times if a `mirtCluster` object is available. If the input object is a discrete latent class object estimated from `mdirt` then the returned results will be with respect to the posterior classification for each individual. The method inputs for `'DiscreteClass'` objects may only be `'EAP'`, for posterior classification of each response pattern, or `'EAPsum'` for posterior classification based on the raw sum-score. For more information on these algorithms refer to the `mirtCAT` package and the associated JSS paper (Chalmers, 2016). ## Examples ```r mod <- mirt(Science) tabscores <- fscores(mod, full.scores = FALSE) head(tabscores) # convert scores into expected total score information with 95% CIs E.total <- expected.test(mod, Theta=tabscores[,'F1']) E.total_2.5 <- expected.test(mod, Theta=tabscores[,'F1'] + tabscores[,'SE_F1'] * qnorm(.05/2)) E.total_97.5 <- expected.test(mod, Theta=tabscores[,'F1'] + tabscores[,'SE_F1'] * qnorm(1-.05/2)) data.frame(Total_score=rowSums(tabscores[,1:4]), E.total, E.total_2.5, E.total_97.5) |> head() fullscores <- fscores(mod) fullscores_with_SE <- fscores(mod, full.scores.SE=TRUE) head(fullscores) head(fullscores_with_SE) # convert scores into expected total score information with 95% CIs E.total <- expected.test(mod, Theta=fullscores[,'F1']) E.total_2.5 <- expected.test(mod, Theta=fullscores_with_SE[,'F1'] + fullscores_with_SE[,'SE_F1'] * qnorm(.05/2)) E.total_97.5 <- expected.test(mod, Theta=fullscores_with_SE[,'F1'] + fullscores_with_SE[,'SE_F1'] * qnorm(1-.05/2)) data.frame(Total_score=rowSums(Science), E.total, E.total_2.5, E.total_97.5) |> head() # change method argument to use MAP estimates fullscores <- fscores(mod, method='MAP') head(fullscores) # calculate MAP for a given response vector fscores(mod, method='MAP', response.pattern = c(1,2,3,4)) # or matrix fscores(mod, method='MAP', response.pattern = rbind(c(1,2,3,4), c(2,2,1,3))) # return only the scores and their SEs fscores(mod, method='MAP', response.pattern = c(1,2,3,4)) # use custom latent variable properties (diffuse prior for MAP is very close to ML) fscores(mod, method='MAP', cov = matrix(1000), full.scores = FALSE) fscores(mod, method='ML', full.scores = FALSE) # EAPsum table of values based on total scores (fs <- fscores(mod, method = 'EAPsum', full.scores = FALSE)) # convert expected counts back into marginal probability distribution within(fs, `P(y)` <- expected / sum(observed)) # list of error VCOV matrices for EAPsum (works for other estimators as well) acovs <- fscores(mod, method = 'EAPsum', full.scores = FALSE, return.acov = TRUE) acovs # WLE estimation, run in parallel using available cores if(interactive()) mirtCluster() head(fscores(mod, method='WLE', full.scores = FALSE)) # multiple imputation using 30 draws for EAP scores. Requires information matrix mod <- mirt(Science, 1, SE=TRUE) fs <- fscores(mod, MI = 30) head(fs) # plausible values for future work pv <- fscores(mod, plausible.draws = 5) lapply(pv, function(x) c(mean=mean(x), var=var(x), min=min(x), max=max(x))) ## define a custom_den function (*must* return a numeric vector). # EAP with a uniform prior between -3 and 3 fun <- function(Theta, ...) as.numeric(dunif(Theta, min = -3, max = 3)) head(fscores(mod, custom_den = fun)) # compare EAP estimators with same modified prior fun <- function(Theta, ...) as.numeric(dnorm(Theta, mean=.5)) head(fscores(mod, custom_den = fun)) head(fscores(mod, method = 'EAP', mean=.5)) ## Not run: # custom MAP prior: standard truncated normal between 5 and -2 library(msm) # need the :: scope for parallel to see the function (not require if no mirtCluster() defined) fun <- function(Theta, ...) msm::dtnorm(Theta, mean = 0, sd = 1, lower = -2, upper = 5) head(fscores(mod, custom_den = fun, method = 'MAP', full.scores = FALSE)) ## End(Not run) #################### # scoring via response.pattern input (with latent regression structure) # simulate data set.seed(1234) N <- 1000 # covariates X1 <- rnorm(N); X2 <- rnorm(N) covdata <- data.frame(X1, X2) Theta <- matrix(0.5 * X1 + -1 * X2 + rnorm(N, sd = 0.5)) # items and response data a <- matrix(1, 20); d <- matrix(rnorm(20)) dat <- simdata(a, d, 1000, itemtype = '2PL', Theta=Theta) # conditional model using X1 and X2 as predictors of Theta mod <- mirt(dat, 1, 'Rasch', covdata=covdata, formula = ~ X1 + X2) coef(mod, simplify=TRUE) # all EAP estimates that include latent regression information fs <- fscores(mod, full.scores.SE=TRUE) head(fs) # score only two response patterns rp <- dat[1:2, ] cd <- covdata[1:2, ] fscores(mod, response.pattern=rp, covdata=cd) fscores(mod, response.pattern=rp[2,], covdata=cd[2,]) # just one pattern ``` ## References Chalmers, R., P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment. **Journal of Statistical Software, 48**(6), 1-29. tools:::Rd_expr_doi("10.18637/jss.v048.i06") Chalmers, R. P. (2016). Generating Adaptive and Non-Adaptive Test Interfaces for Multidimensional Item Response Theory Applications. **Journal of Statistical Software, 71**(5), 1-39. tools:::Rd_expr_doi("10.18637/jss.v071.i05") Embretson, S. E. & Reise, S. P. (2000). Item Response Theory for Psychologists. Erlbaum. Thissen, D., Pommerich, M., Billeaud, K., & Williams, V. S. L. (1995). Item Response Theory for Scores on Tests Including Polytomous Items with Ordered Responses. **Applied Psychological Measurement, 19**, 39-49. Warm, T. A. (1989). Weighted likelihood estimation of ability in item response theory. **Psychometrika, 54**, 427-450. ## See Also `averageMI` ## Author(s) Phil Chalmers rphilip.chalmers@gmail.com

fscores function