# Non-linear regression DIF models estimation. Estimates parameters of non-linear regression models for DIF detection using either non-linear least squares or maximum likelihood method with various algorithms. ```r estimNLR(y, match, group, formula, method, lower, upper, start) ## S3 method for class 'estimNLR' logLik(object, ...) ## S3 method for class 'estimNLR' coef(object, ...) ## S3 method for class 'estimNLR' fitted(object, ...) ## S3 method for class 'estimNLR' residuals(object, ...) ## S3 method for class 'estimNLR' print(x, ...) ## S3 method for class 'estimNLR' vcov(object, sandwich = FALSE, ...) ``` ## Arguments - `y`: numeric: a binary vector of responses (`"1"` correct, `"0"` incorrect). - `match`: numeric: a numeric vector describing the matching criterion. - `group`: numeric: a binary vector of a group membership (`"0"` for the reference group, `"1"` for the focal group). - `formula`: formula: specification of the model. It can be obtained by the `formulaNLR()` function. - `method`: character: an estimation method to be applied. The options are `"nls"` for non-linear least squares (default), `"mle"` for the maximum likelihood method using the `"L-BFGS-B"` algorithm with constraints, `"em"` for the maximum likelihood estimation with the EM algorithm, `"plf"` for the maximum likelihood estimation with the algorithm based on parametric link function, and `"irls"` for the maximum likelihood estimation with the iteratively reweighted least squares algorithm (available for the `"2PL"` model only). See Details . - `lower`: numeric: lower bounds for item parameters of the model specified in the `formula`. - `upper`: numeric: upper bounds for item parameters of the model specified in the `formula`. - `start`: numeric: initial values of item parameters. They can be obtained by the `startNLR()` function. - `object`: an object of the `"estimNLR"` class. - `...`: other generic parameters for S3 methods. - `x`: an object of the `"estimNLR"` class. - `sandwich`: logical: should the sandwich estimator be applied for computation of the covariance matrix of item parameters when using `method = "nls"`? (the default is `FALSE`). ## Details The function offers either the non-linear least squares estimation via the `nls` function (Drabinova & Martinkova, 2017; Hladka & Martinkova, 2020), the maximum likelihood method with the `"L-BFGS-B"` algorithm with constraints via the `optim` function (Hladka & Martinkova, 2020), the maximum likelihood method with the EM algorithm (Hladka, Martinkova, & Brabec, 2025), the maximum likelihood method with the algorithm based on parametric link function (PLF; Hladka, Martinkova, & Brabec, 2025), or the maximum likelihood method with the iteratively reweighted least squares algorithm via the `glm` function. ## Examples ```r # loading data data(GMAT) y <- GMAT[, 1] # item 1 match <- scale(rowSums(GMAT[, 1:20])) # standardized total score group <- GMAT[, "group"] # group membership variable # formula for 3PL model with the same guessing for both groups, # IRT parameterization M <- formulaNLR(model = "3PLcg", type = "both", parameterization = "irt") # starting values for 3PL model with the same guessing for item 1 start <- startNLR(GMAT[, 1:20], group, model = "3PLcg", parameterization = "irt") start <- start[[1]][M$M1$parameters] # nonlinear least squares (fit_nls <- estimNLR( y = y, match = match, group = group, formula = M$M1$formula, method = "nls", lower = M$M1$lower, upper = M$M1$upper, start = start )) coef(fit_nls) logLik(fit_nls) vcov(fit_nls) vcov(fit_nls, sandwich = TRUE) fitted(fit_nls) residuals(fit_nls) # maximum likelihood method (fit_mle <- estimNLR( y = y, match = match, group = group, formula = M$M1$formula, method = "mle", lower = M$M1$lower, upper = M$M1$upper, start = start )) coef(fit_mle) logLik(fit_mle) vcov(fit_mle) fitted(fit_mle) residuals(fit_mle) # formula for 3PL model with the same guessing for both groups # intercept-slope parameterization M <- formulaNLR(model = "3PLcg", type = "both", parameterization = "is") # starting values for 3PL model with the same guessing for item 1, start <- startNLR(GMAT[, 1:20], group, model = "3PLcg", parameterization = "is") start <- start[[1]][M$M1$parameters] # EM algorithm (fit_em <- estimNLR( y = y, match = match, group = group, formula = M$M1$formula, method = "em", lower = M$M1$lower, upper = M$M1$upper, start = start )) coef(fit_em) logLik(fit_em) vcov(fit_em) fitted(fit_em) residuals(fit_em) # PLF algorithm (fit_plf <- estimNLR( y = y, match = match, group = group, formula = M$M1$formula, method = "plf", lower = M$M1$lower, upper = M$M1$upper, start = start )) coef(fit_plf) logLik(fit_plf) vcov(fit_plf) fitted(fit_plf) residuals(fit_plf) # iteratively reweighted least squares for 2PL model M <- formulaNLR(model = "2PL", parameterization = "logistic") (fit_irls <- estimNLR( y = y, match = match, group = group, formula = M$M1$formula, method = "irls" )) coef(fit_irls) logLik(fit_irls) vcov(fit_irls) fitted(fit_irls) residuals(fit_irls) ``` ## References Drabinova, A. & Martinkova, P. (2017). Detection of differential item functioning with nonlinear regression: A non-IRT approach accounting for guessing. Journal of Educational Measurement, 54(4), 498--517, tools:::Rd_expr_doi("10.1111/jedm.12158") . Hladka, A. & Martinkova, P. (2020). difNLR: Generalized logistic regression models for DIF and DDF detection. The R Journal, 12(1), 300--323, tools:::Rd_expr_doi("10.32614/RJ-2020-014") . Hladka, A. (2021). Statistical models for detection of differential item functioning. Dissertation thesis. Faculty of Mathematics and Physics, Charles University. Hladka, A., Martinkova, P., & Brabec, M. (2025). New iterative algorithms for estimation of item functioning. Journal of Educational and Behavioral Statistics. Online first, tools:::Rd_expr_doi("10.3102/10769986241312354") . ## Author(s) Adela Hladka (nee Drabinova) Institute of Computer Science of the Czech Academy of Sciences hladka@cs.cas.cz Patricia Martinkova Institute of Computer Science of the Czech Academy of Sciences martinkova@cs.cas.cz

estimNLR function