Two Taylor-series approximation methods for nonlinear mixed models

Bayesian joint models for multi-regional clinical trials

In recent years, multi-regional clinical trials (MRCTs) have increased in popularity in the pharmaceutical industry due to their ability to accelerate the global drug development process. To address potential challenges with MRCTs, the International Council for Harmonisation released the E17 guidance document which suggests the use of statistical methods that utilize information borrowing across regions if regional sample sizes are small. We develop an approach that allows for information borrowing via Bayesian model averaging in the context of a joint analysis of survival and longitudinal data from MRCTs. In this novel application of joint models to MRCTs, we use Laplace's method to integrate over subject-specific random effects and to approximate posterior distributions for region-specific treatment effects on the time-to-event outcome. Through simulation studies, we demonstrate that the joint modeling approach can result in an increased rejection rate when testing the global treatment effect compared with methods that analyze survival data alone. We then apply the proposed approach to data from a cardiovascular outcomes MRCT.

Flexible modeling of multiple nonlinear longitudinal trajectories with censored and non-ignorable missing outcomes

Multivariate nonlinear mixed-effects models (MNLMMs) have become a promising tool for analyzing multi-outcome longitudinal data following nonlinear trajectory patterns. However, such a classical analysis can be challenging due to censorship induced by detection limits of the quantification assay or non-response occurring when participants missed scheduled visits intermittently or discontinued participation. This article proposes an extension of the MNLMM approach, called the MNLMM-CM, by taking the censored and non-ignorable missing responses into account simultaneously. The non-ignorable missingness is described by the selection-modeling factorization to tackle the missing not at random mechanism. A Monte Carlo expectation conditional maximization algorithm coupled with the first-order Taylor approximation is developed for parameter estimation. The techniques for the calculation of standard errors of fixed effects, estimation of unobservable random effects, imputation of censored and missing responses and prediction of future values are also provided. The proposed methodology is motivated and illustrated by the analysis of a clinical HIV/AIDS dataset with censored RNA viral loads and the presence of missing CD4 and CD8 cell counts. The superiority of our method on the provision of more adequate estimation is validated by a simulation study.

Efficient inference and identifiability analysis for differential equation models with random parameters

Heterogeneity is a dominant factor in the behaviour of many biological processes. Despite this, it is common for mathematical and statistical analyses to ignore biological heterogeneity as a source of variability in experimental data. Therefore, methods for exploring the identifiability of models that explicitly incorporate heterogeneity through variability in model parameters are relatively underdeveloped. We develop a new likelihood-based framework, based on moment matching, for inference and identifiability analysis of differential equation models that capture biological heterogeneity through parameters that vary according to probability distributions. As our novel method is based on an approximate likelihood function, it is highly flexible; we demonstrate identifiability analysis using both a frequentist approach based on profile likelihood, and a Bayesian approach based on Markov-chain Monte Carlo. Through three case studies, we demonstrate our method by providing a didactic guide to inference and identifiability analysis of hyperparameters that relate to the statistical moments of model parameters from independent observed data. Our approach has a computational cost comparable to analysis of models that neglect heterogeneity, a significant improvement over many existing alternatives. We demonstrate how analysis of random parameter models can aid better understanding of the sources of heterogeneity from biological data.

Constrained Nonlinear and Mixed Effects Integral Differential Equation Models for Dynamic Cell Polarity Signaling

Polar cell growth is a process that couples the establishment of cell polarity with growth and is extremely important in the growth, development, and reproduction of eukaryotic organisms, such as pollen tube growth during plant fertilization and neuronal axon growth in animals. Pollen tube growth requires dynamic but polarized distribution and activation of a signaling protein named ROP1 to the plasma membrane via three processes: positive feedback and negative feedback regulation of ROP1 activation and its lateral diffusion along the plasma membrane. In this paper, we introduce a mechanistic integro-differential equation (IDE) along with constrained semiparametric regression to quantitatively describe the interplay among these three processes that lead to the polar distribution of active ROP1 at a steady state. Moreover, we introduce a population variability by a constrained nonlinear mixed model. Our analysis of ROP1 activity distributions from multiple pollen tubes revealed that the equilibrium between the positive and negative feedbacks for pollen tubes with similar shapes are remarkably stable, permitting us to infer an inherent quantitative relationship between the positive and negative feedback loops that defines the tip growth of pollen tubes and the polarity of tip growth.

The optimization of combinatorial drug therapies: Strategies and laboratorial platforms

Designing optimal combinatorial drug therapies is challenging, because the drug interactions depend not only on the drugs involved, but also on their doses. With recent advances, combinatorial drug therapy is closer than ever to clinical application. Herein, we summarize approaches and advances over the past decade for identifying and optimizing drug combination therapies, with innovations across research fields, covering physical laboratory platforms for combination screening to computational models and algorithms designed for synergism prediction and optimization. By comparing different types of approach, we detail a three-step workflow that could maximize the overall optimization efficiency, thus enabling the application of personalized optimization of combinatorial drug therapy.

Examination of Nonlinear and Functional Mixed-Effects Models with Nonparametrically Generated Data

Previous research has shown functional mixed-effects models and traditional mixed-effects models perform similarly when recovering individual trajectories when data were generated following a parametric structure. We extend this previous work and compare nonlinear mixed-effects (NMEM) and functional mixed-effects models' (FMEM) ability to recover underlying trajectories when generated from an inherently nonparametric process. Nonlinear trajectories were generated using B-splines, NMEMs and FMEMs were estimated, and the accuracy of the estimated curves was examined. Sample size, number of time points per curve, and measurement design were varied across simulation conditions. Results showed the FMEMs recovered the underlying mean curve more accurately than the NMEMs, and that, the FMEMs tended to recover the underlying individual curves more accurately than the NMEMs. Progesterone cycle data were then analyzed to demonstrate the utility of both approaches, and models performed similarly when analyzing these data.

Therapeutic drug monitoring of anti-tumour necrosis factor-α agents in inflammatory bowel disease: Limits and improvements

AIMS

Since the publication of the American Gastroenterological Association's recommendations in 2017, there have been no significant changes in the biological monitoring recommendations in inflammatory bowel disease. Possible limitations are the lack of evidence to recommend proactive therapeutic drug monitoring (pTDM) over reactive TDM (rTDM), and the limited information about individualized dosing methods. This article aims to review the TDM strategy updates and the use of individualized dosing methods.

METHODS

For the analysis of the TDM strategies and individualized dosing method, a search was carried out in PubMed and Cochrane Central. In the TDM case, since August 2017.

RESULTS

A total of 263 publications were found, but only 7 related to proactive TDM. Five of these publications directly compared pTDM vs rTDM and 2 were randomized clinical trials. Six studies found benefits of pTDM and 1 found no differences. Regarding the individualized dosing method, 229 distinct results were found. Population pharmacokinetics was the most widely used method to develop individual dosage models and to analyse the influence of factors on drug concentrations (albumin concentration, weight, presence of anti-drug antibodies etc).

CONCLUSION

We have found no major changes in TDM strategies. There is a growing trend towards the use of pTDM because it has shown a longer duration of treatment response, lower rates of discontinuation and relapses. However, the available evidence is limited and of low quality. Despite the common use of population pharmacokinetic methods to analyse pharmacokinetic factors, they are not commonly used for personalized dosing.

Comparison of empirical and dynamic models for HIV viral load rebound after treatment interruption

OBJECTIVE

To compare empirical and mechanistic modeling approaches for describing HIV-1 RNA viral load trajectories after antiretroviral treatment interruption and for identifying factors that predict features of viral rebound process.

METHODS

We apply and compare two modeling approaches in analysis of data from 346 participants in six AIDS Clinical Trial Group studies. From each separate analysis, we identify predictors for viral set points and delay in rebound. Our empirical model postulates a parametric functional form whose parameters represent different features of the viral rebound process, such as rate of rise and viral load set point. The viral dynamics model augments standard HIV dynamics models-a class of mathematical models based on differential equations describing biological mechanisms-by including reactivation of latently infected cells and adaptive immune response. We use Monolix, which makes use of a Stochastic Approximation of the Expectation-Maximization algorithm, to fit non-linear mixed effects models incorporating observations that were below the assay limit of quantification.

RESULTS

Among the 346 participants, the median age at treatment interruption was 42. Ninety-three percent of participants were male and sixty-five percent, white non-Hispanic. Both models provided a reasonable fit to the data and can accommodate atypical viral load trajectories. The median set points obtained from two approaches were similar: 4.44 log10 copies/mL from the empirical model and 4.59 log10 copies/mL from the viral dynamics model. Both models revealed that higher nadir CD4 cell counts and ART initiation during acute/recent phase were associated with lower viral set points and identified receiving a non-nucleoside reverse transcriptase inhibitor (NNRTI)-based pre-ATI regimen as a predictor for a delay in rebound.

CONCLUSION

Although based on different sets of assumptions, both models lead to similar conclusions regarding features of viral rebound process.

Maximum likelihood estimation of nonlinear mixed-effects models with crossed random effects by combining first-order conditional linearization and sequential quadratic programming

Nonlinear mixed-effects (NLME) models have become popular in various disciplines over the past several decades. However, the existing methods for parameter estimation implemented in standard statistical packages such as SAS and R/S-Plus are generally limited to single- or multi-level NLME models that only allow nested random effects and are unable to cope with crossed random effects within the framework of NLME modeling. In this study, we propose a general formulation of NLME models that can accommodate both nested and crossed random effects, and then develop a computational algorithm for parameter estimation based on normal assumptions. The maximum likelihood estimation is carried out using the first-order conditional expansion (FOCE) for NLME model linearization and sequential quadratic programming (SQP) for computational optimization while ensuring positive-definiteness of the estimated variance-covariance matrices of both random effects and error terms. The FOCE-SQP algorithm is evaluated using the height and diameter data measured on trees from Korean larch (L. olgensis var. Changpaiensis) experimental plots as well as simulation studies. We show that the FOCE-SQP method converges fast with high accuracy. Applications of the general formulation of NLME models are illustrated with an analysis of the Korean larch data.

A Note on Recurring Misconceptions When Fitting Nonlinear Mixed Models

Nonlinear mixed-effects (NLME) models are used when analyzing continuous repeated measures data taken on each of a number of individuals where the focus is on characteristics of complex, nonlinear individual change. Challenges with fitting NLME models and interpreting analytic results have been well documented in the statistical literature. However, parameter estimates as well as fitted functions from NLME analyses in recent articles have been misinterpreted, suggesting the need for clarification of these issues before these misconceptions become fact. These misconceptions arise from the choice of popular estimation algorithms, namely, the first-order linearization method (FO) and Gaussian-Hermite quadrature (GHQ) methods, and how these choices necessarily lead to population-average (PA) or subject-specific (SS) interpretations of model parameters, respectively. These estimation approaches also affect the fitted function for the typical individual, the lack-of-fit of individuals' predicted trajectories, and vice versa.

Estimation of genetic (co)variances of Gompertz growth function parameters in pigs

The objective of this study was to estimate genetic (co)variances for the Gompertz growth function parameters, asymptotic mature weight (A), the ratio of mature weight to birthweight (B) and rate of maturation (k), using alternative modelling approaches. The data set consisted of 51 893 live weight records from 10 201 growing pigs. The growth of each pig was modelled using the Gompertz model employing either a two-step fixed effect or mixed model approach or a one-step mixed model approach using restricted maximum likelihood for the estimation of genetic (co)variance. Heritability estimates for the Gompertz growth function parameters, A (0.40), B (0.69) and k (0.45), were greatest for the one-step approach, compared with the two-step fixed effects approach, A (0.10), B (0.33) and k (0.13), and the two-step mixed model approach, A (0.17), B (0.32) and k (0.18). Inferred genetic correlations (i.e. correlations of estimated breeding values) between growth function parameters within models ranged from -0.78 to 0.76, and across models ranged from 0.28 to 0.73 for parameter A, 0.75 to 0.88 for parameter B and 0.09 to 0.37 for parameter k. Correlations between predicted daily sire live weights based on the Gompertz growth curve parameters' estimated breeding values from 60 to 200 days of age between all three modelled approaches were moderately to strongly correlated (0.75 to 0.95). Results from this study provide heritability estimates for biologically interpretable parameters of pig growth through the quantification of genetic (co)variances, thereby facilitating the estimation of breeding values for inclusion in breeding objectives to aid in breeding and selection decisions.

Fitting and Calibrating a Multilevel Mixed-Effects Stem Taper Model for Maritime Pine in NW Spain

Stem taper data are usually hierarchical (several measurements per tree, and several trees per plot), making application of a multilevel mixed-effects modelling approach essential. However, correlation between trees in the same plot/stand has often been ignored in previous studies. Fitting and calibration of a variable-exponent stem taper function were conducted using data from 420 trees felled in even-aged maritime pine (Pinus pinaster Ait.) stands in NW Spain. In the fitting step, the tree level explained much more variability than the plot level, and therefore calibration at plot level was omitted. Several stem heights were evaluated for measurement of the additional diameter needed for calibration at tree level. Calibration with an additional diameter measured at between 40 and 60% of total tree height showed the greatest improvement in volume and diameter predictions. If additional diameter measurement is not available, the fixed-effects model fitted by the ordinary least squares technique should be used. Finally, we also evaluated how the expansion of parameters with random effects affects the stem taper prediction, as we consider this a key question when applying the mixed-effects modelling approach to taper equations. The results showed that correlation between random effects should be taken into account when assessing the influence of random effects in stem taper prediction.

Segmented mixed models with random changepoints: a maximum likelihood approach with application to treatment for depression study

We present a simple and effective iterative procedure to estimate segmented mixed models in a likelihood based framework. Random effects and covariates are allowed for each model parameter, including the changepoint. The method is practical and avoids the computational burdens related to estimation of nonlinear mixed effects models. A conventional linear mixed model with proper covariates that account for the changepoints is the key to our estimating algorithm. We illustrate the method via simulations and using data from a randomized clinical trial focused on change in depressive symptoms over time which characteristically show two separate phases of change.

Shrinkage in nonlinear mixed-effects population models: quantification, influencing factors, and impact

Shrinkage of empirical Bayes estimates (EBEs) of posterior individual parameters in mixed-effects models has been shown to obscure the apparent correlations among random effects and relationships between random effects and covariates. Empirical quantification equations have been widely used for population pharmacokinetic/pharmacodynamic models. The objectives of this manuscript were (1) to compare the empirical equations with theoretically derived equations, (2) to investigate and confirm the influencing factor on shrinkage, and (3) to evaluate the impact of shrinkage on estimation errors of EBEs using Monte Carlo simulations. A mathematical derivation was first provided for the shrinkage in nonlinear mixed effects model. Using a linear mixed model, the simulation results demonstrated that the shrinkage estimated from the empirical equations matched those based on the theoretically derived equations. Simulations with a two-compartment pharmacokinetic model verified that shrinkage has a reversed relationship with the relative ratio of interindividual variability to residual variability. Fewer numbers of observations per subject were associated with higher amount of shrinkage, consistent with findings from previous research. The influence of sampling times appeared to be larger when fewer PK samples were collected for each individual. As expected, sample size has very limited impact on shrinkage of the PK parameters of the two-compartment model. Assessment of estimation error suggested an average 1:1 relationship between shrinkage and median estimation error of EBEs.

Non-compartment model to compartment model pharmacokinetics transformation meta-analysis--a multivariate nonlinear mixed model

BACKGROUND

To fulfill the model based drug development, the very first step is usually a model establishment from published literatures. Pharmacokinetics model is the central piece of model based drug development. This paper proposed an important approach to transform published non-compartment model pharmacokinetics (PK) parameters into compartment model PK parameters. This meta-analysis was performed with a multivariate nonlinear mixed model. A conditional first-order linearization approach was developed for statistical estimation and inference.

RESULTS

Using MDZ as an example, we showed that this approach successfully transformed 6 non-compartment model PK parameters from 10 publications into 5 compartment model PK parameters. In simulation studies, we showed that this multivariate nonlinear mixed model had little relative bias (<1%) in estimating compartment model PK parameters if all non-compartment PK parameters were reported in every study. If there missing non-compartment PK parameters existed in some published literatures, the relative bias of compartment model PK parameter was still small (<3%). The 95% coverage probabilities of these PK parameter estimates were above 85%.

CONCLUSIONS

This non-compartment model PK parameter transformation into compartment model meta-analysis approach possesses valid statistical inference. It can be routinely used for model based drug development.

When are summary ROC curves appropriate for diagnostic meta-analyses?

Diagnostic tests are increasingly evaluated with systematic reviews and this has lead to the recent developments of statistical methods to analyse such data. The most commonly used method is the summary receiver operating characteristic (SROC) curve, which can be fitted with a non-linear bivariate random-effects model. This paper focuses on the practical problems of interpreting and presenting data from such analyses. First, many meta-analyses may be underpowered to obtain reliable estimates of the SROC parameters. Second, the SROC model may be inappropriate. In these situations, a summary with two univariate meta-analyses of the true and false positive rates (TPRs and FPRs) may be more appropriate. We characterize the type of problems that can occur in fitting these models and present an algorithm to guide the analyst of such studies, with illustrations from analyses of published data. A set of R functions, freely available to perform these analyses, can be downloaded from (www.diagmeta.info).

The Bayesian Bias Correction Method of the First-Order Approximation of Nonlinear Mixed-Effects Models for Population Pharmacokinetics

Population pharmacokinetic analysis usually employs nonlinear mixed-effects models. To estimate the parameters, Beal and Sheiner (1982) proposed the first-order method that employs a first-order Taylor series expansion around the means of random individual parameters. Because of the small computational burden and the high convergence proportion of maximization of the log likelihood function, this method is often used in practice. However, it is known that the estimates are biased. This paper proposes a simple procedure to reduce the bias. The proposed method maximizes the nonapproximated log likelihood functions of each individual given estimates of the population parameters derived from the first-order method, and the derived Bayes estimates of the random individual parameters are utilized to improve the estimates of the population mean parameters. We confirmed that the proposed method reduced the bias using simulated data and actual erythropoietin concentration data.

Ordinary Differential Equation PK/PD Models Using the SAS Macro NLINMIX

We describe some theory and recent enhancements for the SAS macro NLINMIX (Wolfinger, R. D. (1993). Laplace's approximation for nonlinear mixed effects models. Biometrika 80:791-795) that enable model calculation to take place within the interaction matrix language SAS/IML (SAS Institute Inc. (1999a). SAS/IML User's Guide Version 7. Cary, NC: SAS Institute Inc.). They provide greater flexibility and scope for the specification and analysis of complex nonlinear mixed models. For example, using data from a frequently sampled intravenous glucose test, we fit a two-compartment kinetics model that has no closed-form representation. It is derived as the solution of a system of ordinary differential equations and specified as such in SAS/IML. Additional details and example NLINMIX code are available in Appendix A.

Cytotoxic assays for screening anticancer agents

In the process of identifying potential anticancer agents, the ability of a new agent is tested for cytotoxic activity against a panel of standard cancer cell lines. The National Cancer Institute (NCI) present the cytotoxic profile for each agent as a set of estimates of the dose required to inhibit the growth of each cell line. The NCI estimates are obtained from a linear interpolation method applied to the dose-response curves. In this paper non-linear fits are proposed as an alternative to interpolation. This is illustrated with data from two agents recently submitted to NCI for potential anticancer activity. Fitting of individual non-linear curves proved difficult, but a non-linear mixed model applied to the full set of cell lines overcame most of the problems. Two non-linear functional forms were fitted using random effect models by both maximum likelihood and a full Bayesian approach. Model-based toxicity estimates have some advantages over those obtained from interpolation. They provide standard errors for toxicity estimates and other derived quantities, allow model comparisons. Examples of each are illustrated.

Statistical inference in generalized linear mixed models: a review

We present a review of statistical inference in generalized linear mixed models (GLMMs). GLMMs are an extension of generalized linear models and are suitable for the analysis of non-normal data with a clustered structure. A GLMM contains parameters common to all clusters (fixed regression effects and variance components) and cluster-specific parameters. The latter parameters are assumed to be randomly drawn from a population distribution. The parameters of this population distribution (the variance components) have to be estimated together with the fixed effects. We focus on the case in which the cluster-specific parameters are normally distributed. The cluster-specific effects are integrated out of the likelihood so that the fixed effects and variance components can be estimated. Unfortunately, the integral over the cluster-specific effects is intractable for most GLMMs with a normal mixing distribution. Within a classical statistical framework, we distinguish between two broad classes of methods to handle this intractable integral: methods that rely on a numerical approximation to the integral and methods that use an analytical approximation to the integrand. Finally, we present an overview of available methods for testing hypotheses about the parameters of GLMMs.

Prediction discrepancies for the evaluation of nonlinear mixed-effects models

Reliable estimation methods for non-linear mixed-effects models are now available and, although these models are increasingly used, only a limited number of statistical developments for their evaluation have been reported. We develop a criterion and a test to evaluate nonlinear mixed-effects models based on the whole predictive distribution. For each observation, we define the prediction discrepancy (pd) as the percentile of the observation in the whole marginal predictive distribution under H(0). We propose to compute prediction discrepancies using Monte Carlo integration which does not require model approximation. If the model is valid, these pd should be uniformly distributed over (0, 1) which can be tested by a Kolmogorov-Smirnov test. In a simulation study based on a standard population pharmacokinetic model, we compare and show the interest of this criterion with respect to the one most frequently used to evaluate nonlinear mixed-effects models: standardized prediction errors (spe) which are evaluated using a first order approximation of the model. Trends in pd can also be evaluated via several plots to check for specific departures from the model.

A Population Pharmacokinetic Model with Time-Dependent Covariates Measured with Errors

We propose a population pharmacokinetic (PK) model with time-dependent covariates measured with errors. This model is used to model S-oxybutynin's kinetics following an oral administration of Ditropan, and allows the distribution rate to depend on time-dependent covariates blood pressure and heart rate, which are measured with errors. We propose two two-step estimation methods: the second-order two-step method with numerical solutions of differential equations (2orderND), and the second-order two-step method with closed form approximate solutions of differential equations (2orderAD). The proposed methods are computationally easy and require fitting a linear mixed model at the first step and a nonlinear mixed model at the second step. We apply the proposed methods to the analysis of the Ditropan data, and evaluate their performance using a simulation study. Our results show that the 2orderND method performs well, while the 2orderAD method can yield PK parameter estimators that are subject to considerable biases.

Multivariate Multilevel Nonlinear Mixed Effects Models for Timber Yield Predictions

Nonlinear mixed effects models have become important tools for growth and yield modeling in forestry. To date, applications have concentrated on modeling single growth variables such as tree height or bole volume. Here, we propose multivariate multilevel nonlinear mixed effects models for describing several plot-level timber quantity characteristics simultaneously. We describe how such models can be used to produce future predictions of timber volume (yield). The class of models and methods of estimation and prediction are developed and then illustrated on data from a University of Georgia study of the effects of various site preparation methods on the growth of slash pine (Pinus elliottii Engelm.).

A nonlinear mixed model framework for item response theory

Mixed models take the dependency between observations based on the same cluster into account by introducing 1 or more random effects. Common item response theory (IRT) models introduce latent person variables to model the dependence between responses of the same participant. Assuming a distribution for the latent variables, these IRT models are formally equivalent with nonlinear mixed models. It is shown how a variety of IRT models can be formulated as particular instances of nonlinear mixed models. The unifying framework offers the advantage that relations between different IRT models become explicit and that it is rather straightforward to see how existing IRT models can be adapted and extended. The approach is illustrated with a self-report study on anger.

Detection of quantitative trait loci influencing dairy traits using a model for longitudinal data

A longitudinal-linkage analysis approach was developed and applied to an outbred population. Nonlinear mixed-effects models were used to describe the lactation patterns and were extended to include marker information following single-marker and interval mapping models. Quantitative trait loci (QTL) affecting the shape and scale of lactation curves for production and health traits in dairy cattle were mapped in three U.S. Holstein families (Dairy Bull DNA Repository families one, four, and five) using the granddaughter design. Information on 81 informative markers on six Bos taurus autosomes (BTA) was combined with milk yield, fat, and protein percentage and somatic cell score (SCS) test-day records. Six percent of the single-marker tests surpassed the experiment-wise significance threshold. Marker BL41 on BTA3 was associated with decrease in milk yield during mid-lactation in family one. The scale and shape of the protein percentage lactation curve in family four varied with BMC4203 (BTA6) allele that the son received from the grandsire. Some map locations were associated with variation in the lactation pattern of multiple traits. In family four, the marker HUJI177 (BTA3) was associated with changes in the milk yield and protein percentage curves suggesting a QTL with pleiotropic effects or multiple QTL in the region. The interval mapping model uncovered a QTL on BTA7 associated with variation in milk-yield pattern in family four and a QTL on BTA21 affecting SCS in family five. The developed approach can be extended to random regressions, covariance functions, spline, gametic and variance component models. The results from the longitudinal-QTL approach will help to understand the genetic factors acting at different stages of lactation and will assist in positional candidate gene research. Identified positions can be incorporated into marker-assisted selection decisions to alter the persistency and peak production or the fluctuation of SCS during a lactation.