# Comparisons of statistical models for growth curves from 90-day rat feeding studies

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## Abstract

The objective of this work was to compare several models of body weight data from 90-day rodent feeding trials. Polynomial and nonlinear functions relating time and weight were examined as were the use of Toeplitz error covariance structures and random coefficients. The models were evaluated by fitting them to five publicly available datasets from rat feeding studies. Model performance was assessed in terms of their ability to capture the complexity of the growth patterns, validity of necessary assumptions, and information criteria scores. The results demonstrated the importance of selecting a curve function that effectively reflects the mean response. Toeplitz error covariance structures resulted in superior model fit, while failing to address deviations from model assumptions. Models using the Richards function and random coefficients were generally superior to the other models evaluated and dramatically improved upon linear models with complex error structures.

## Keywords

Error covariance structures Gompertz function Linear mixed models Nonlinear mixed models Random coefficients Richards function## Introduction

Animal feeding studies are used to evaluate the safety of agricultural, pharmaceutical, industrial, and food chemicals. Additionally, if a compositional analysis of food or feed derived from a genetically modified (GM) crop suggests compositional differences from conventional crop varieties that could be potential health hazards, a 90-day feeding study in rodents has been recommended to provide additional information used in the comparative risk assessment (EFSA 2011a; ILSI Task Force 2004).

Internationally harmonized guidance on the design and conduct of 90-day toxicity studies has been developed by the Organisation for Economic Co-operation and Development ((OECD) 408 chemical testing guideline (2003)). The European Food Safety Authority (EFSA) requires certain adaptations of these methods for testing whole foods from GM crops (EFSA 2011b). This paper focuses on the statistical evaluation of data from studies following the EFSA guidance.

Traditionally, fixed effect linear models have been applied to compare treatment groups separately week-by-week, as described by Schmidt et al. (2016). As this approach considers each week separately, the chance of at least one false-positive result is inflated relative to comparing final weights among treatments. This approach also has reduced power to detect treatment effects that are small but consistent across time. Schmidt et al. (2016) proposed the use of linear mixed models with weight increasing linearly with time and a complex error correlation structure. This method improved interpretability by providing a single parameter to describe the response of weight over time. However, the linear relationship between weight and time prevents this model from estimating the curvature apparent in visual inspections of raw data curves from feeding trials. Polynomial or nonlinear curves of moderate complexity might be sufficient to fully capture the relationships. Kuhi et al. (2010) provide an extensive review of mathematical functions for modeling growth in poultry. Similar considerations should be made when evaluating options for rodent data.

The use of parametric structures to model patterns of correlation/covariance among error terms within an individual is well-established for models of repeated measurements (Littell et al. 2006, ch. 5). Such structures can capture the correlations between multiple measurements on the same rodent or cage, reflecting consistent effects of each individual rodent or cage. Additionally, error covariance structures can capture the tendency of measurements made close together in time to be more similar than those with greater separation.

An alternative to modeling covariance among repeated measures is the modeling of random coefficients in linear mixed models. Polynomial models with randomly distributed coefficients that follow parametric covariance structures are described by Vonesh and Chinchilli (1997) and Sjoblad et al. (1992). These models can also be adapted to non-polynomial relationships between the response and explanatory variables.

The objective of this work was to compare models for body weight data from 90-day rodent feeding trials. Polynomial and nonlinear functions were examined as were types of covariance structures. Performance of the models was compared in terms of empirical assessments of suitability for multiple trials.

## Materials and methods

### Motivating data

The motivating data we considered are from five rat feeding trials conducted as part of the European Commission funded GMO Risk Assessment and Communication of Evidence (GRACE) project. Detailed methods for the trials have been published previously (Schmidt et al. 2017; Zeljenková et al. 2014, 2016). Raw data and initial statistical analysis reports of all five trials can be found at the CADIMA website (https://www.cadima.info/index.php/area/publicAnimalFeedingTrials; search Project column for GRACE).

All five trials (denoted A, B, C, D, and E) included diets incorporating material from GM maize containing the same event at two different incorporation rates: 11% and 33%. We abbreviate these diets as GMO11 and GMO33. Diets made with material from a near-isogenic control of the GM maize were also included at the same incorporation rates in all five trials. Trials A, B, and C also included diets made with different varieties of conventional maize at the same incorporation rates.

In all trials, two rats of the same sex were housed in each cage and fed together, making cages the experimental units. Equal numbers of male and female rats were assigned to each diet treatment group. Each diet was fed to 8, 8, 10, 5, and 5 cages of male and female rats in trials A, B, C, D, and E, respectively. Rats used in the trials were selected to be relatively uniform in initial body weight, and were allocated to treatment groups such that the average weight among treatment groups was similar. Initiation of feeding on diet treatments and measurement of body weights was considered week 0 in modeling and graphing. Each rat was weighed weekly for 13 weeks (approximating 90 days) after feeding was begun. The duration of the 13th week was 5 days for trials A and B, and 6 days for trials D and E. All rats survived until the end of the studies, and there were no missing data. Although trial C was conducted for a full year, we only considered the first 90 days of weight data to match the other trials.

#### Linear mixed models (LMMs)

**X**and

**Z**are the design matrices for fixed effects and random effects, and \({\varvec{e}}\) is the error vector. We assume \({\varvec{u}}\) and \({\varvec{e}}\) are independent and \({\varvec{u}} \sim N\left( {0,{\varvec{G}}} \right), {\varvec{e}} \sim N\left( {0,{\varvec{R}}} \right)\). Following the common convention, we refer to these as “G-side” and “R-side” random effects and the associated covariance matrices \({\varvec{G}}\) and \({\varvec{R}}\) as “G-side” and “R-side” covariance structures. For additional theoretical discussion of LMMs, interested readers can refer to Vonesh and Chinchilli (1997), Sjoblad et al. (1992), and Littell et al. (2006, pp. 734–756).

#### Polynomial models with R-side covariance structures

In a Toeplitz covariance structure, all observations have the same variance, and covariance parameters depend on the time lag between observations. Alternative structures for \({\varvec{\Sigma}}\) could also be fit as discussed later.

#### Polynomial models with G-side random effects

All \(a_{ij}\), \(b_{ij}\), and \(g_{ij}\) were modeled as independent from each other, and from \(\epsilon_{ijt}\). Models 2L and 2C were constructed in a similar manner.

#### Nonlinear mixed models

Nonlinear functions have been used extensively in growth curve modeling (Hammill et al. 1995; Kuhi et al. 2010; Richards 1959) and might be expected to provide better model fit than polynomial models. Here we describe our application of the Gompertz and Richards nonlinear functions evaluated with Toeplitz R-side structures and separately with G-side random coefficients using the same approaches as the polynomial models.

\(a_{ij}\), \(o_{ij}\), and \(k_{ij}\) were modeled as independent from each other and from \(\varepsilon_{ijt}\). Model 1G was fit using the Toeplitz R-side covariance structure for \({\varvec{\Sigma}}\), and a constant variance structure for \({\varvec{\Sigma}}\) was fit for model 2G. Note that \(\alpha\), the initial weight, was assumed to be constant for all treatment groups since the diets could not cause effects before the rats were fed.

\(a_{ij}\), \(o_{ij}\), and \(m_{ij}\) were modeled as independent from each other and from \(\epsilon_{ijt}\). Model 1R was fit using the Toeplitz R-side covariance structure for \({\varvec{\Sigma}}\) and a constant variance structure for \({\varvec{\Sigma}}\) was fit for model 2R. Again, \(\alpha\) was assumed to be constant for all treatment groups. In order to simplify the models, we treated \(\varsigma\) as a study-sex-specific parameter with a constant value for all cages in all treatment groups.

#### Model comparisons

The models were evaluated for each study by visual inspection of simulation and residual plots and by comparison of information criteria. The MIXED and NLMIXED procedures in SAS (Version 9.4) were used to fit models to the data and generate studentized residual plots. Once a model was fit, we assumed the fitted model with the estimated parameters was the data generating mechanism for the raw data and simulated a data set based on this mechanism. Observations at each time point were simulated and plotted for a number of cages for each treatment equal to the quantities tested in the study. Features of the simulated dataset that substantially differed from the raw data could indicate flaws in the model’s approximation of the true data generating mechanism. Note that each simulation plot is just one possible realization of the fitted model, and may include random effects or error values from the extremes of the distributions simulated. The same can be said of the raw data. Thus, limited emphasis should be placed on interpretation of a single curve or observation that does not match between the raw and simulated data. Despite these limitations, the plotted simulation is a useful diagnostic tool that allows for intuitive assessment and comparisons among models. We used the maximum likelihood method to estimate model parameters, since use of the restricted maximum likelihood (REML) method would have made the information criteria non-comparable among the methods (Littell et al. 2006, pp. 752-754).

Comparisons among the models were made after being fit separately to the ten data sets consisting of male and female data from the five trials. Results from all five trials can be found in the supplemental material. Since results were generally similar among the ten data sets, the results from the female trial A data set are presented and discussed, as this trial was of intermediate size among the five and the data do not include any obvious outliers. Any differences in results among the sexes or trials are discussed.

## Results and discussion

### Polynomial models with R-side covariance structures

Information criteria from three R-side models for female trial A data

Model 1L | Model 1Q | Model 1C | |
---|---|---|---|

AIC | 3356.8 | 3127.2 | 3066.6 |

AICC | 3358.4 | 3129.7 | 3070.1 |

BIC | 3390.6 | 3169.4 | 3117.2 |

The failure of these models to capture heteroscedasticity in the data might hamper their use to accurately estimate the relationship between time and weight for a particular treatment or to identify differences among diets (Littell et al. 2006, p. 794; Scheffé 1959). Simulations by Ferron et al. (2002) demonstrated that misspecification of the error covariance structure in growth curves led to biased estimates of variance parameters. One may consider fitting a heteroscedastic covariance structure, by allowing variance heterogeneity in the Toeplitz structure or another structure. Cubic models with compound symmetric and AR(1) heterogeneous variance structures were applied to all ten data sets and residuals were plotted against predicted values (Supplemental Fig. 31 to 40). Unresolvable convergence issues occurred for most of the data sets when a heteroscedastic Toeplitz structure was attempted. In our evaluations of these heterogeneous variance structures, we observed moderate to extreme biases in the fixed effect estimates appearing as uneven spread of residuals about the zero line.

### Polynomial models with G-side random effects

Information criteria from three G-side models for female trial A data

Model 2L | Model 2Q | Model 2C | |
---|---|---|---|

AIC | 4410.9 | 3475.2 | 3142.2 |

AICC | 4411.2 | 3476.1 | 3143.8 |

BIC | 4426.1 | 3500.5 | 3176.0 |

As shown in Fig. 4, the random coefficient models resulted in increasing separation among simulated growth curves of each cage over time. This better reflects the raw data pattern as compared to the models with Toeplitz error covariance structures. As observed with the models including R-side structures, model 2L was overly simplistic, whereas the quadratic and cubic models better captured the curvature observed in the raw data. Results for the other data sets were similar (Supplemental Fig. 5, 8, 11, 14, 17, 20, 23, 26, and 29).

The studentized residuals from the linear and quadratic models applied to female trial A data showed asymmetric patterns (Fig. 5). These patterns indicate that these models did not adequately fit the data, whereas the cubic model performed better. When applied to the other data sets, the cubic model resulted in modest to substantial improvements in model fit (Supplemental Fig. 6, 9, 12, 15, 18, 21, 24, 27, and 30). However, the residuals based on the cubic model still showed asymmetry for the female data from trial C and the male data from trials A, C, D, and E.

The information criteria (Table 2) indicate that the higher order polynomials led to better performance for the female trial A data. Comparing Tables 1 and 2, the information criteria for the three random coefficient models were uniformly greater (indicating worse fit) than those of the corresponding Toeplitz structure models with smaller differences for higher order polynomials. The same pattern of results was observed for both sexes in trials A, B, and C, and male data in trial D (Supplemental Tables 1 to 6 and 8). For the remaining three data sets from trials D and E, the information criteria for random coefficient cubic models were slightly better than those for model 1C. Trials D and E included fewer treatments and cages per diet, suggesting that the Toeplitz R-side structure may require more data to provide better model fit than G-side random coefficients when used with fairly complex (cubic) polynomial models.

The model comparisons above demonstrated that while easily interpreted, LMMs with only a single slope parameter per treatment were insufficient to capture the growth curve pattern in the raw data, regardless of whether an R-side or G-side covariance structure was used. When random coefficients were used in the model, substantial improvements were observed with the higher order polynomials. However, the asymmetric spread of residuals for model 2C suggested that the mean response function could be improved beyond the cubic. This was not apparent for model 1C, because the R-side covariance structure captured the deviations from the cubic curve as correlated random error.

Neither random coefficient models nor Toeplitz R-side structure models were indisputably superior with any of the polynomial parameterizations evaluated. Likely due to the limited number of observations in these experiments, convergence could not be achieved for models attempted with both Toeplitz error covariance structures and random coefficients. Models combining random coefficients and simpler error covariance structures are a potential option for further optimization which we did not pursue. Besides the graphs and information criteria, some additional considerations favor the random coefficient models. First, the random coefficient models required fewer parameters than the LMM with Toeplitz structures. This difference would increase with more time points, since the Toeplitz structure requires the same number of parameters as time points modeled. Additionally, random coefficients allow each cage to be modeled with its own growth curve. Such models may better reflect biological reality than using the Toeplitz structure which models curvature differences among experimental units as correlated errors.

### Nonlinear mixed models

Information criteria from models for female trial A data

Toeplitz models | Model 1L | Model 1Q | Model 1C | Model 1G | Model 1R |
---|---|---|---|---|---|

AIC | 3356.8 | 3127.2 | 3066.6 | 3067.1 | 3055.0 |

AICC | 3358.4 | 3129.7 | 3070.1 | 3069.5 | 3060.1 |

BIC | 3390.6 | 3169.4 | 3117.2 | 3109.3 | 3115.8 |

Random coefficients models | Model 2L | Model 2Q | Model 2C | Model 2G | Model 2R |
---|---|---|---|---|---|

AIC | 4410.9 | 3475.2 | 3142.2 | 3209.3 | 3057.7 |

AICC | 4411.2 | 3476.1 | 3143.8 | 3210.2 | 3058.7 |

BIC | 4426.1 | 3500.5 | 3176.0 | 3234.6 | 3084.7 |

As with the simulation plots, the residual plots show evidence that the Toeplitz models failed to account for heteroscedasticity. The residuals of model 2G form an obvious curved pattern in the plot, indicating that the Gompertz function failed to capture the full complexity of the growth curve (Fig. 7). The residuals show little to no curvature in models 2R, 1G, and 1R, with decreasing curvature in that order, indicating that these models are capturing most or all of the relationships between time and weight. Both random effects models fit to male data from trials D and E resulted in curvatures in residual plots due to relatively small growth rates in the beginning weeks for these two data sets (Supplemental Fig. 24 and 30). Furthermore, for the other eight data sets, models 2R, 1G, and 1R displayed little to no curvature for each data set successfully fit, whereas model 2G had curvature for all but the female data of trials D and E.

The Richards function resulted in similar or better information criteria values as compared to the other functions evaluated. For the successfully fit Toeplitz models, the Gompertz and Richards functions resulted in similar or slightly better model fit scores compared to the cubic function. When random coefficients were used, the Richards function resulted in substantially lower information criteria values than the Gompertz and cubic models. The Toeplitz models resulted in better model fit based on information criteria values as compared to the corresponding random coefficient models for most data sets. The exception was Richards function which resulted in lower information criteria with random coefficients for two of the three data sets where the Toeplitz version was successfully fit. It is important to recall that Toeplitz models failed to account for heteroscedasticity as demonstrated in the residual diagnostic plots. This suggests that when the underlying mean response function poorly captures the overall growth pattern, the correlations in the Toeplitz error structure can capture the deviations from the mean growth pattern at each time point. However, when an adequate function (Richards) is chosen to model the mean growth pattern, random coefficients can efficiently model variability among individual cages and effectively account for heteroscedasticity over time.

From a practical standpoint, we note that the Toeplitz structure was difficult to implement with the nonlinear functions. For the majority of the data sets, no starting values could be found to allow model convergence and properly conditioned final Hessian matrices for models 1G and 1R, even with repeated attempts. Additionally, it took much longer to fit the Toeplitz models compared to the corresponding random coefficients models using the NLMIXED procedure. The complexity of the Toeplitz structure could be even more problematic for data sets with more time points, since the number of parameters in the Toeplitz structure equals the number of time points.

The preceding discussion evaluated various aspects of modeling 90-day rat studies including mean response function (polynomials and nonlinear growth curves), cage to cage variability structure (G-side random effects), and within cage error (R-side random effects). The options evaluated are not exhaustive. Some additional options include G-side structures that do not assume independence of the random factors and simpler R-side structures combined with G-side structures. Although Model 2R appeared to be the best model choice for most of the data sets we evaluated, it would not be appropriate to conclude that it should always be chosen for the analysis of 90-day rat studies. Instead, it would be advisable for researchers to attempt multiple models of greater and lesser complexity, choosing the one with superior information criteria scores and residual plots that do not show major flaws.

### Treatment comparisons

When modeling body weight data from 90-day rodent feeding studies, the goal is to estimate and compare the effects of diet treatments on rodent growth. Once a model is chosen and fit to the data, the parameters for that model can be estimated specifically for each diet. Confidence intervals for the parameter estimates can be provided to establish the accuracy of those estimates. Approximate *F*-tests or *t*-tests can be used to evaluate the statistical significance of hypotheses of interest. Conducting hypothesis tests on parameters that apply to the whole growth curve reduces the risk of multiplicity as compared to separately comparing treatment effects at each time point.

The datasets from the GRACE project would not be expected to show any true effects of the GM diets on rat growth, since extensive evaluations beyond weight measurements were previously conducted and showed no biologically relevant effects of the GM diets (Schmidt et al. 2017; Zeljenková et al. 2016, 2014). Therefore, any observed treatment effects for these datasets would be false positives. In analyses of other datasets, interpretation of statistically significant treatment comparisons would require expert evaluation within the context of all the results of the study, as statistical significance does not necessarily or automatically imply biological relevance (EFSA 2011c).

The scope of this investigation prevents us from evaluating the ability of these models to detect true adverse effects in terms of growth rates or final 13th week weights. Such an evaluation would require analysis of feeding trials with positive controls that have known biologically relevant effects on growth rate and/or final weight. Once biologically relevant effect sizes are determined, simulations could be conducted to compare these models in terms of statistical power and false-positive error rates. Since deviations from assumptions generally lead to loss of statistical power or increased false-positive rates, it is likely that similar models would be preferred.

## Conclusions

For body weight data from the five 90-day rat feeding trials conducted as part of the GRACE project, models using the Richards function and random coefficients were generally superior to the other models evaluated and dramatically improved upon one-parameter linear models with complex error structures. In terms of model fit, the complex Toeplitz error variance structure compensated for overly simple models of the relationship between time and weight. However, these models may be incorrectly modeling true complexity in the relationship between time and weight as error correlation. Additionally, the Toeplitz models did not effectively model the observed heteroscedasticity over time. Random coefficient models captured heteroscedasticity over time while partitioning out variation among individual cages with modest increases in parameter numbers. Inferior model fit was observed when random coefficients were used with simple functions. Only when a sufficiently complex function was used could superior model fit be achieved along with the benefits of random coefficients. The Richards function with random coefficients was well suited for most of the data sets analyzed, but may not be optimal for other growth curve data sets. Other investigators should take care to observe the residual plots from applying this model to ensure that it adequately captures the growth pattern. Application of these models to make comparisons among diet treatments will also require expert evaluation of biological relevance within the context of all study results.

This paper focused on evaluations of methods for statistical analysis of body weight data from studies that follow the EFSA guidance. However, similar model evaluations could be made for other animal feeding studies, and some of the same approaches or conclusions may apply.

## Notes

### Acknowledgements

We would like to acknowledge Bonnie Hong for initiating this project and George Milliken for his important early work on this project. We also thank them for providing valuable reviews and suggestions.

### Author contributions

KH was responsible for statistical analysis. CW was responsible for manuscript preparation. Both authors contributed ideas and research.

### Compliance with ethical standards

### Conflict of interest

During the conduct of the study, KH was employed by Iowa State University and contracted by DuPont Pioneer, and CW was employed by DuPont Pioneer. The work was funded by DuPont Pioneer (now Corteva Agriscience).

## Supplementary material

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