Smoothing techniques are designed to improve the accuracy of equating functions. The main purpose of this dissertation was to propose a new statistic (CS) and compare it to existing model selection strategies in selecting smoothing parameters for polynomial loglinear presmoothing (C) and cubic spline postsmoothing (S) for mixed-format tests under a random groups design. For polynomial loglinear presmoothing, CS was compared to seven existing model selection strategies in selecting the C parameters: likelihood ratio chi-square test (G2), Pearson chi-square test (PC), likelihood ratio chi-square difference test (G2diff), Pearson chi-square difference test (PCdiff), Akaike Information Criterion (AIC), Bayesian Information Criterion (BIC), and Consistent Akaike Information Criterion (CAIC). For cubic spline postsmoothing, CS was compared to the ± 1 standard error of equating (± 1 SEE) rule. In this dissertation, both the pseudo-test data, Biology long and short, and Environmental Science long and short, and the simulated data were used to evaluate the performance of the CS statistic and the existing model selection strategies. For both types of data, sample sizes of 500, 1000, 2000, and 3000 were investigated. In addition, No Equating Needed conditions and Equating Needed conditions were investigated for the simulated data. For polynomial loglinear presmoothing, mean absolute difference (MAD), average squared bias (ASB), average squared error (ASE), and mean squared errors (MSE) were computed to evaluate the performance of all model selection strategies based on three sets of criteria: cumulative relative frequency distribution (CRFD), relative frequency distribution (RFD), and the equipercentile equating relationship. For cubic spline postsmoothing, the evaluation of different model selection procedures was only based on the MAD, ASB, ASE, and MSE of equipercentile equating. The main findings based on the pseudo-test data and simulated data were as follows: (1) As sample sizes increased, the average C values increased and the average S values decreased for all model selection strategies. (2) For polynomial loglinear presmoothing, compared to the results without smoothing, all model selection strategies always introduced bias of RFD and significantly reduced the standard errors and mean squared errors of RFD; only AIC reduced the MSE of CRFD and MSE of equipercentile equating across all sample sizes and all test forms; the best CS procedure tended to yield an equivalent or smaller MSE of equipercentile equating than the AIC and G2diff statistics. (3) For cubic spline postsmoothing, both the ± 1 SEE rule and the CS procedure tended to perform reasonably well in reducing the ASE and MSE of equipercentile equating. (4) Among all existing model selection strategies, the ±1 SEE rule in postsmoothing tended to perform better than any of the seven existing model selection strategies in presmoothing in terms of the reduction of random error and total error; (5) pseudo-test data and the simulated data tended to yield similar results. The limitations of the study and possible future research are discussed in the dissertation.