Reparameterized Model Research Articles

Experimental design and re-parameterization of the Neue-Kuss model for accurate and precise prediction of isocratic retention factors from gradient measurements in reversed phase liquid chromatography

The present work describes a re-parameterization of the Neue Kuss (NK) model for describing retention in liquid chromatography, and this re-parameterized model is used to fit a large set of isocratic retention measurements with improved convergence properties relative to the original parameterization of the model. Next, an experimental design for retention measurements using mobile phase gradient elution conditions is proposed for the purpose of obtaining accurate and precise NK parameters. Simulated retention data for mobile phase gradient elution conditions with two different levels of noise, as well as an essentially zero noise level were fit with the re-parameterized model. The results showed that the re-parameterized fits yielded average (absolute value) prediction errors for the parameters at the highest noise level of 7.2 % for S1,ref, 18 % for S2,ref and 6.2 % for kref (the re-parameterized NK model parameters). These errors were significantly smaller than those for the original parameterization of the NK model, where the errors were 23 % for S1, 25 % for S2 and 160 % for kw (the original NK model parameters). Furthermore, isocratic retention factors predicted using these model parameters were found to have an average magnitude of error of 0.51 % for the re-parameterized model, as opposed to 6800 % for the model with the original parameterization. A further test of this approach was carried out for independent experimental measurements for five solutes on a C18 column. The average magnitude of error of the isocratic retention factors predicted from parameters obtained from fits of gradient data was 1.6 %, provided that the range of organic solvent compositions that the solute sampled in the mobile phase gradient experiments was consistent with the isocratic experiments. These results indicate that the re-parameterization of the NK model allows for significant improvements in the fitting process, and that the proposed experimental design allows for NK parameters to be extracted from mobile phase gradient experiments, with prediction accuracies of isocratic retention factors on the order of 1-2 %.

Combining field data and modeling to better understand maize growth response to phosphorus (P) fertilizer application and soil P dynamics in calcareous soils

We used field experimental data to evaluate the ability of the agricultural production system model (APSIM) to simulate soil P availability, maize biomass and grain yield in response to P fertilizer applications on a fluvo-aquic soil in the North China Plain. Crop and soil data from a 2-year experiment with three P fertilizer application rates (0, 75 and 300 kg P2O5 ha−1) were used to calibrate the model. Sensitivity analysis was carried out to investigate the influence of APSIM SoilP parameters on the simulated P availability in soil and maize growth. Crop and soil P parameters were then derived by matching or relating the simulation results to observed crop biomass, yield, P uptake and Olsen-P in soil. The re-parameterized model was further validated against 2 years of independent data at the same sites. The re-parameterized model enabled good simulation of the maize leaf area index (LAI), biomass, grain yield, P uptake, and grain P content in response to different levels of P additions against both the calibration and validation datasets. Our results showed that APSIM needs to be re-parameterized for simulation of maize LAI dynamics through modification of leaf size curve and a reduction in the rate of leaf senescence for modern stay-green maize cultivars in China. The P concentration limits (maximum and minimum P concentrations in organs) at different stages also need to be adjusted. Our results further showed a curvilinear relationship between the measured Olsen-P concentration and simulated labile P content, which could facilitate the initialization of APSIM P pools in the NCP with Olsen-P measurements in future studies. It remains difficult to parameterize the APSIM SoilP module due to the conceptual nature of the pools and simplified conceptualization of key P transformation processes. A fundamental understanding still needs to be developed for modelling and predicting the fate of applied P fertilizers in soils with contrasting physical and chemical characteristics.

AMDNet: A Modern UAV RGB Remote-Sensing Tree Species Image Segmentation Model Based on Dual-Attention Residual and Structure Re-Parameterization

Tree species classification is an important and challenging task in image recognition and the management of forest resources. Moreover, the task of tree species classification based on remote sensing images can significantly improve the efficiency of the tree species survey and save costs. In recent years, many large models have achieved high accuracy in the task of tree species classification in an airborne remote-sensing manner, but due to their fixed geometric structure, traditional convolutional neural networks are inherently limited to the local receptive field and can only provide segmental context information. The limitation of insufficient context information greatly affects the segmentation accuracy. In this paper, a dual-attention residual network (AMDNet) and a re-parameterized model approach are proposed to capture the global context information, fuse the weight, reduce the model volume, and maintain the computational efficiency. Firstly, we propose MobileNetV2 as the backbone network for feature extraction to further improve the feature identification by modeling semantic dependencies in the spatial dimension and channel dimension and adding the output of the two attention modules. Then, the attention perception features are generated by stacking the attention modules, and the in-depth residual attention network is trained using attention residual learning, through which more accurate segmentation results can be obtained. Secondly, we adopt the approach of structure re-parameterization, use a multi-branch topology for training, carry out weighted averaging on multiple trained models, and fuse multiple branch modules into a completely equivalent module in inference. The proposed approach results in a reduction in the number of parameters and an accelerated inference speed while also achieving improved classification accuracy. In addition, the model training strategy is optimized based on Transformer to enhance the accuracy of segmentation. The model was used to conduct classification experiments on aerial orthophotos of Hongya Forest Farm in Sichuan, China, and the MIOU of tree species recognition using the test equipment reached 93.8%. Compared with current models such as UNet, our model exhibits a better performance in terms of both speed and accuracy, in addition to its enhanced deployment capacity, and its speed advantage is more conducive to real-time segmentation, thereby representing a novel approach for the classification of tree species in remote sensing imagery with significant potential for practical applications.

Reparameterization of the Weibull model for practical uses in food science.

The reparameterization of the Weibull cumulative distribution function and its survival function was performed to obtain meaningful parameters in food and biological sciences such as the lag phase (λ), the maximum rate ( ), and the maximum increase/decrease of the function (A). The application of the Lambert function was crucial in order to achieve an explicit mathematical solution. Since the reparameterized model is applicable only when the shape parameter (α) is greater than one, the Weibull model was modified with the introduction of a new parameter ( ) that represents the model rate at time β (scale parameter). All models were applied to literature data on food technology and microbiology topics: Microbial growth, thermal microbial inactivation, thermal degradation kinetics, and particle size distributions. The Weibull model and the reparameterized versions showed identical fitting performance in terms of coefficient of determination, residual mean standard error, values of residuals, and estimated values of the parameters. Some faults in the datasets used in this study permitted to re-mark the criticality of a good experimental plan when data modeling is approached. The parameter resulted in an interesting new rate parameter that is not correlated with the scale parameter ( =0.64 ± 0.37)and highly correlated with the shape parameter ( =0.90 ± 0.11). Also, the reparameterization of the Weibull probability density function was performed by using both the standard and new parameters and applied to experimental data and gave useful information from the distribution curve, such as the value of the mode ( ) and a measure of the curve skewness (λ).

The re-parameterized inverse Gaussian regression to model length of stay of COVID-19 patients in the public health care system of Piracicaba, Brazil

Among the models applied to analyze survival data, a standout is the inverse Gaussian distribution, which belongs to the class of models to analyze positive asymmetric data. However, the variance of this distribution depends on two parameters, which prevents establishing a functional relation with a linear predictor when the assumption of constant variance does not hold. In this context, the aim of this paper is to re-parameterize the inverse Gaussian distribution to enable establishing an association between a linear predictor and the variance. We propose deviance residuals to verify the model assumptions. Some simulations indicate that the distribution of these residuals approaches the standard normal distribution and the mean squared errors of the estimators are small for large samples. Further, we fit the new model to hospitalization times of COVID-19 patients in Piracicaba (Brazil) which indicates that men spend more time hospitalized than women, and this pattern is more pronounced for individuals older than 60 years. The re-parameterized inverse Gaussian model proved to be a good alternative to analyze censored data with non-constant variance.

Bridging atomistic simulations and thermodynamic hydration models of aqueous electrolyte solutions.

Chemical thermodynamic models of solvent and solute activities predict the equilibrium behavior of aqueous solutions. However, these models are semi-empirical. They represent micro-scale ion and solvent behaviors controlling the macroscopic properties using small numbers of parameters whose values are obtained by fitting to activities and other partial derivatives of the Gibbs energy measured for the bulk solutions. We have conducted atomistic simulations of aqueous electrolyte solutions (MgCl2 and CaCl2) to determine the parameters of thermodynamic hydration models. We have implemented a cooperative hydration model to categorize the water molecules in electrolyte solutions into different subpopulations. The value of the electrolyte-specific parameter, k, was determined from the ion-affected subpopulation with the lowest absolute value of the free energy of removing the water molecule. The other equilibrium constant parameter, K1, associated with the first degree of hydration, was computed from the free energy of hydration of hydrated clusters. The hydration number, h, was determined from a reorientation dynamic analysis of the water subpopulations compared to bulk-like behavior. The reparameterized models [R. H. Stokes and R. H. Robinson, J. Solution Chem. 2, 173 (1973) and Balomenos et al., Fluid Phase Equilib. 243, 29 (2006)] using the computed values of the parameters lead to the osmotic coefficients of MgCl2 solutions that are consistent with measurements. Such an approach removes the dependence on the availability of experimental data and could lead to aqueous thermodynamic models capable of estimating the values of solute and solvent activities as well as thermal and volumetric properties for a wide range of compositions and concentrations.

The Interaction Between Genetic Variant ZNF804A rs1344706 and Alcohol Withdrawal on Impulsivity: Evidence for the Diathesis-Stress Model.

Objective: Alcohol use disorder (AUD) is the most common substance use disorder, which may relate to increased impulsivity. A more detailed understanding of the potential moderating factor on association between AUD and impulsivity is likely to have far-reaching effects. This study aims to examine whether the interaction between a genetic variant ZNF804A rs1344706 and alcohol use is related to impulsivity in Chinese Han adult males diagnosed with AUD.Methods: A total of 455 Chinese Han adult males diagnosed with AUD were included in this study. Impulsivity was assessed using Barratt Impulsiveness Scale. Alcohol dependence was measured by Michigan Alcoholism Screening Test. Genomic DNA was extracted from peripheral blood of participants and genotyped.Results: Hierarchical multiple regression yielded a significant interaction between ZNF804A rs1344706 and alcohol use (β = 0.20, p = 0.0237). Then, A region of significance (RoS) test was performed to interpret the interaction effect. Re-parameterized regression models revealed that the interaction between ZNF804A rs1344706 and alcohol problem severity fit to the weak diathesis-stress model (R2 = 0.15, p < 0.0010), indicating that the T allele carriers are more susceptible to alcohol problem severity, jointly contributing to impulsivity.Conclusions: This study, which analyzed a specific gene-environment interaction, demonstrated that carriers of the T allele of ZNF804A rs1344706 may be more susceptible to alcohol problem severity, correlated with higher levels of impulsivity during withdrawal.

English

Hormesis is a biphasic dose (concentration)-response phenomenon characterized by low dose stimulatory and high dose inhibitory effects exerted by stressors on living organisms. Recently, there has been increased interest in the phenomenon and statistical models for its exploration in toxicological studies. The bilogistic model of Beckon et al. is one of such models for modeling biphasic dose-response relationships in toxicological studies. However, there is no explicit formula for the estimation of effective doses (EDK) with the model. In this study, a simple general approach was suggested to reparameterize the model, leading to a range of mathematical models for determination of effective doses at both stimulatory and inhibitory ranges in inverted U-shaped hormetic dose-response relationships. The reparameterized models were tested on experimental data from three different in vitro experimental systems obtained from literature and our experiment. They were successfully applied to test for significance of hormesis and estimate effective doses and their statistical properties. In addition, reparameterization of the model for a particular effective dose (EDK) did not affect estimation of other parameters (such as x1, x2, &beta;1, &beta;2 and M). The reparameterized models provided useful tools for adequate exploration of the tested hormetic dose-response relationships. The extended models could hopefully be versatile in characterization of variable hormetic dose response relationships in many toxicological disciplines. &nbsp; Key words: Toxicity, hormesis, hormesis quantities, model extension, concentration-responses.

AutoRepar: A method to obtain identifiable and observable reparameterizations of dynamic models with mechanistic insights

AbstractMechanistic dynamic models of biological systems allow for a quantitative and systematic interpretation of data and the generation of testable hypotheses. However, these models are often over‐parameterized, leading to nonidentifiability and nonobservability, that is, the impossibility of inferring their parameters and state variables. The lack of structural identifiability and observability (SIO) compromises a model's ability to make predictions and provide insight. Here we present a methodology, AutoRepar, that corrects SIO deficiencies of nonlinear ODE models automatically, yielding reparameterized models that are structurally identifiable and observable. The reparameterization preserves the mechanistic meaning of selected variables, and has the exact same dynamics and input‐output mapping as the original model. We implement AutoRepar as an extension of the STRIKE‐GOLDD software toolbox for SIO analysis, applying it to several models from the literature to demonstrate its ability to repair their structural deficiencies. AutoRepar increases the applicability of mechanistic models, enabling them to provide reliable information about their parameters and dynamics.

Improving the representation of forests in hydrological models

Forests play a critical role in the hydrologic cycle, impacting the surface and groundwater dynamics of watersheds through transpiration, interception, shading, and modification of the atmospheric boundary layer. It is therefore critical that forest dynamics are adequately represented in watershed models, such as the widely applied Soil and Water Assessment Tool (SWAT). SWAT’s default parameterization generally produces unrealistic forest growth predictions, which we address here through an improved representation of forest dynamics using species-specific re-parameterizations. We applied this methodology to the two dominant pine species in the southeastern U.S., loblolly pine (Pinus taeda L.) and slash pine (Pinus elliotti). Specifically, we replaced unrealistic parameter values related to tree growth with physically meaningful parameters derived from publicly available remote-sensing products, field measurements, published literature, and expert knowledge. Outputs of the default and re-parameterized models were compared at four pine plantation sites across a range of management, soil, and climate conditions. Results were validated against MODIS-derived leaf area index (LAI) and evapotranspiration (ET), as well as field observations of total biomass. The re-parameterized model outperformed the default model in simulating LAI, biomass accumulation, and ET at all sites. The two parametrizations also resulted in substantially different mean annual water budgets for all sites, with reductions in water yield ranging from 13 to 45% under the new parameterization, highlighting the importance of properly parameterizing forest dynamics in watershed models. Importantly, our re-parameterization methodology does not require alteration to the SWAT code, allowing it to be readily adapted and applied in ongoing and future watershed modeling studies.

Spatiotemporal modeling of first and second wave outbreak dynamics of COVID-19 in Germany

The COVID-19 pandemic has kept the world in suspense for the past year. In most federal countries such as Germany, locally varying conditions demand for state- or county-level decisions to adapt to the disease dynamics. However, this requires a deep understanding of the mesoscale outbreak dynamics between microscale agent models and macroscale global models. Here, we use a reparameterized SIQRD network model that accounts for local political decisions to predict the spatiotemporal evolution of the pandemic in Germany at county resolution. Our optimized model reproduces state-wise cumulative infections and deaths as reported by the Robert Koch Institute and predicts the development for individual counties at convincing accuracy during both waves in spring and fall of 2020. We demonstrate the dominating effect of local infection seeds and identify effective measures to attenuate the rapid spread. Our model has great potential to support decision makers on a state and community politics level to individually strategize their best way forward during the months to come.

Latent Growth Modeling with Categorical Response Data: A Methodological Investigation of Model Parameterization, Estimation, and Missing Data

ABSTRACT Measuring change in an educational or psychological construct over time is often achieved by repeatedly administering the same items to the same examinees over time and fitting a second-order latent growth curve model. However, latent growth modeling with full information maximum likelihood (FIML) estimation becomes computationally challenging when the observed response data are categorical. This study first discusses some possible options that researchers can take regarding model specification and estimation (e.g., limited-information and various FIML estimators) to circumvent the challenge. To explore the utility of a stochastic Newton-Raphson type of algorithm (i.e., Metropolis Hastings-Robbins Monro; MH-RM) implemented primarily for multidimensional item response model, a re-parameterized latent growth model is also introduced. The viability of each option is examined via Monte-Carlo simulations. Insights on the pros and cons of these options and the conditions under which they are applicable are provided for researchers.

Bayesian differential analysis of gene regulatory networks exploiting genetic perturbations

BackgroundGene regulatory networks (GRNs) can be inferred from both gene expression data and genetic perturbations. Under different conditions, the gene data of the same gene set may be different from each other, which results in different GRNs. Detecting structural difference between GRNs under different conditions is of great significance for understanding gene functions and biological mechanisms.ResultsIn this paper, we propose a Bayesian Fused algorithm to jointly infer differential structures of GRNs under two different conditions. The algorithm is developed for GRNs modeled with structural equation models (SEMs), which makes it possible to incorporate genetic perturbations into models to improve the inference accuracy, so we name it BFDSEM. Different from the naive approaches that separately infer pair-wise GRNs and identify the difference from the inferred GRNs, we first re-parameterize the two SEMs to form an integrated model that takes full advantage of the two groups of gene data, and then solve the re-parameterized model by developing a novel Bayesian fused prior following the criterion that separate GRNs and differential GRN are both sparse.ConclusionsComputer simulations are run on synthetic data to compare BFDSEM to two state-of-the-art joint inference algorithms: FSSEM and ReDNet. The results demonstrate that the performance of BFDSEM is comparable to FSSEM, and is generally better than ReDNet. The BFDSEM algorithm is also applied to a real data set of lung cancer and adjacent normal tissues, the yielded normal GRN and differential GRN are consistent with the reported results in previous literatures. An open-source program implementing BFDSEM is freely available in Additional file 1.

Modeling portal-drained viscera and liver fluxes of essential amino acids in dairy cows

The objective of this work was to predict essential amino acid (EAA) use and release by the portal-drained viscera (PDV) and liver of dairy cows. Previously derived equations were tested using data assembled from the literature, refit to the data, and modifications were undertaken to determine the best model for each EAA. The refitted model has the same structure as the original equations but is parameterized using a database of group means, as the original equations were derived using a single study with individual cow data and found to be biased. The PDV clearance model predicted portal vein concentrations given inputs of absorbed and arterial fluxes of EAA with root mean squared errors (RMSE) ranging from 3.3 to 12.1% of the observed means, and concordance correlation coefficients (CCC) ranging from 0.86 to 0.99 when using previously reported parameters. The reparameterized model generated from the assembled data set resulted in predictions of EAA portal vein concentrations with RMSE ranging from 3.2 to 8.6% and CCC ranging from 0.93 to 1.00. Slope bias ranged from 12.4 to 55.3% of mean squared errors and was correlated with arterial EAA concentrations. Modifying the model to allow rate constants to vary as a function of arterial EAA concentrations reduced slope bias, resulting in RMSE ranging from 1.9 to 6.5% and CCC from 0.97 to 1.00. Alternatively, splitting the model to account for use of EAA from absorption separately from arterial use resulted in poorer predictions and biologically infeasible parameter estimates. The liver clearance model predicted hepatic vein concentrations from arterial and portal vein input fluxes with RMSE across EAA ranging from 1.9 to 6.8% and CCC ranging from 0.97 to 1.00 when using reported parameters. The reparameterized model generated from the assembled data set resulted in predictions of EAA hepatic vein concentrations with RMSE ranging from 1.9 to 6.7% and CCC ranging from 0.97 to 1.00. Significant slope bias was present for Arg, His, Lys, Phe, Thr, and Val. Altering the model to represent the clearance rate constant as a function of arterial concentrations resulted in RMSE ranging from 1.8 to 6.5% and CCC ranging from 0.97 to 1.00. The combination of PDV and liver clearance models provided predictions of total splanchnic use similar to those of an empirical model representing splanchnic use as a fractional proportion of absorption that had RMSE ranging from 3.0 to 8.6% and CCC ranging from 0.95 to 0.99, with significant slope bias for the majority of EAA.

Latent Growth Models with Floors, Ceilings, and Random Knots

In longitudinal/developmental studies, individual growth trajectories are sometimes bounded by a floor at the beginning of the observation period and/or a ceiling toward the end of the observation period (or vice versa), resulting in inherently nonlinear growth patterns. If the trajectories between the floor and ceiling are approximately linear, such longitudinal growth patterns can be described with a linear piecewise (spline) model in which segments join at knots. In these scenarios, it may be of specific interest for researchers to examine the timing when transition occurs, and in some occasions also to examine the levels of the floors and/or ceilings if they are not known and fixed. In the current study, we propose a reparameterized piecewise latent growth curve model so that a direct estimation of the random knots (and, if needed, a direct estimation of random floors and ceilings) is possible. We derive the model reparameterization using a 4-step structured latent curve modeling approach. We provide two illustrative examples to demonstrate how the proposed reparameterized models can be fitted to longitudinal growth data using the popular SEM software Mplus and we supply the full coding for applied researchers’ reference.

Careful Prior Specification Avoids Incautious Inference for Log-Gaussian Cox Point Processes

SummaryHyperprior specifications for random fields in spatial point process modelling can have a major influence on the results. In fitting log-Gaussian Cox processes to rainforest tree species, we consider a reparameterized model combining a spatially structured and an unstructured random field into a single component. This component has one hyperparameter accounting for marginal variance, whereas an additional hyperparameter governs the fraction of the variance that is explained by the spatially structured effect. This facilitates interpretation of the hyperparameters, and significance of covariates is studied for a range of hyperprior specifications. Appropriate scaling makes the analysis invariant to grid resolution.

Comparison of Cognitive Diagnosis Models under Changing Conditions: DINA, RDINA, HODINA and HORDINA

The application of CDMs to fraction subtraction data revealed problems on the classification of examinees, latent class sizes, and the use of higher-order models. Additionally, selecting the most appropriate model assumes critical importance if there are several appropriate models available for the data. In the present study, DINA–RDINA and HODINA–HORDINA models were compared under changing conditions (i.e., number of attributes, g and s item parameter values, and number of items) with simulated and real data. The results show that for conditions where the g–s parameter values and the number of attributes were low (0.1 and 3, respectively), the reparameterized models generated values that were virtually identical to those obtained using DINA models. However, when the g–s parameter values and the number of attributes were increased (0.5 and 5, respectively), the parameter estimations obtained from the models, latent class estimates, AIC, and BIC show differences through the values from the models.

Update of model to predict sensible heat loss in broilers

The present study was conducted to adjust and adapt some parameters of the model of production and heat loss by convection and conduction, so as to predict the actual feed intake (aFI) of broilers reared in sheds. The re-parameterised models were the sensible heat loss by convection from surface (HS) and by conduction (HC) in birds. The HS model was re-parameterised to calculate the heat loss of poultry reared in sheds and the parameters of thermal resistance of feathers (RF) and skin (RS) of poultry were inserted. The HC model was re-parameterised for birds in sheds and the RF, RS and the thermal resistance of the litter (R) were inserted. The re-parameterised HS model was HS = [A × QV × (TB – TA)]/[(TB – 17) × (RF + RS)], where TA is the air temperature, QV is the volume factor, TB is the surface temperature of the bird (°C) and A was estimated to be 11.94 watts (W). The values found in the model ranged from 0.75 W for birds with 100 g BW subjected to 33°C TA, 50% HU, 0.1 m/s wind speed (V) and 12.53 W for birds with 4100 g subjected to 33°C TA, 80% HU and 0.1 m/s V. The values found in the re-parameterised HC model (HC = [(TB – TC) × k × AR × QA]/[L × (RF + RS + R)], where K is the thermal conductivity of the litter, AR is the contact area of bird with the litter and QA is the area factor, and L is the litter height) ranging from 0.017 W to chickens with 100 g BW in comfortable conditions and 0.17 W for birds with 4100 g in thermal discomfort condition. The present study showed that the re-parameterisation of heat-loss equations is more accurate to predict the heat flux in broilers under different environmental conditions.

Estimation and Inference with a (Nearly) Singular Jacobian

This paper develops extremum estimation and inference results for nonlinear models with very general forms of potential identification failure when the source of this identification failure is known. We examine models that may have a general deficient rank Jacobian in certain parts of the parameter space. When identification fails in one of these models, it becomes under-identified and the identification status of individual parameters is not generally straightforward to characterize. We provide a systematic reparameterization procedure that leads to a reparameterized model with straightforward identification status. Using this reparameterization, we determine the asymptotic behavior of standard extremum estimators and Wald statistics under a comprehensive class of parameter sequences characterizing the strength of identification of the model parameters, ranging from non-identification to strong identification. Using the asymptotic results, we propose hypothesis testing methods that make use of a standard Wald statistic and data-dependent critical values, leading to tests with correct asymptotic size regardless of identification strength and good power properties. Importantly, this allows one to directly conduct uniform inference on low-dimensional functions of the model parameters, including one-dimensional subvectors. The paper illustrates these results in three examples: a sample selection model, a triangular threshold crossing model and a collective model for household expenditures.

Adjusted priors for Bayes factors involving reparameterized order constraints

Many psychological theories that are instantiated as statistical models imply order constraints on the model parameters. To fit and test such restrictions, order constraints of the form θi≤θj can be reparameterized with auxiliary parameters η∈[0,1] to replace the original parameters by θi=η⋅θj. This approach is especially common in multinomial processing tree (MPT) modeling because the reparameterized, less complex model also belongs to the MPT class. Here, we discuss the importance of adjusting the prior distributions for the auxiliary parameters of a reparameterized model. This adjustment is important for computing the Bayes factor, a model selection criterion that measures the evidence in favor of an order constraint by trading off model fit and complexity. We show that uniform priors for the auxiliary parameters result in a Bayes factor that differs from the one that is obtained using a multivariate uniform prior on the order-constrained original parameters. As a remedy, we derive the adjusted priors for the auxiliary parameters of the reparameterized model. The practical relevance of the problem is underscored with a concrete example using the multi-trial pair-clustering model.