# Comparison of the ISU, NCI, MSM, and SPADE Methods for Estimating Usual Intake: A Simulation Study of Nutrients Consumed Daily

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

- The ratio of the shifted, power-transformed observed intakes is adjusted to take into account nuisance effects, such as day of the week and interview mode (telephone or in-person). Construct smoothed daily intakes by undoing the initial power transformation and shifting for the adjusted observations.
- A grafted polynomial function is fit to the normal probability plot of the smoothed intakes using least-squares. The inverse of the fitted function is used to transform the smoothed intakes to normality.
- Moment estimates of variance components are computed for the transformed intakes, and an estimate of the normal-scale usual intake distribution is obtained,
- A grafted cubic and a 9-point approximation to use to transform the normal-scale usual intake distribution to original scale.

- The observed intakes are transformed to improve normality by means of a one-parameter Box-Cox transformation, indicated by λ in this paper.
- A linear mixed effects model on the transformed intake data is fit to estimate the mean and the within- and between-person variances.
- $k$ (value to be set) pseudo-person intakes from a normal distribution is simulated with mean equal to the estimated mean and variance equal to the between-person variance.
- The simulated values by a 9-point approximation is back-transformed, which involves the estimated Box-Cox parameter and the within-person variation.

- A linear regression model is applied to the data and the residuals are used for the shrinkage part of the MSM method.
- The fitted model residuals are transformed to normality by means of a two-parameter Box-Cox transformation, with λ restricted to $1/\mathsf{\lambda}=1,2,3\dots $.
- The within- and between-person variances are estimated by means of the transformed residuals.
- The back-transformation is defined by a closed formula, involving the estimated λ and the within-person variance.
- The distribution is estimated by the inverse regression model after the back-transformation to the original scale of the residuals.

- The observed intakes are transformed by means of a one-parameter Box-Cox transformation.
- A linear mixed effects model on the transformed scale is used to estimate the mean and within-person and between-person variances.
- The mean on the transformed scale is directly back-transformed by Gaussian Quadrature, using the total variance of the model and the Box-Cox transformation parameter λ.
- The percentiles on the transformed scale correspond exactly with the percentiles on the original scale, and their back-transformation by Gaussian Quadrature involves the within-person variance and λ [19]. The distribution is calculated directly in the back-transformation step.

#### Simulations

**Box 1.**Simulation scenarios.

Scenario | n | Within-Person Variance $\left({\mathsf{\sigma}}_{\mathit{\epsilon}}^{2}\right)$ | Between-Person Variance$\left({\mathsf{\sigma}}_{\mathit{u}}^{2}\right)$ | Variance Ratio $\left({r}_{var}\right)$ |

I | 150 | 1 | 0.25 | 4 |

II | 0.11 | 9 | ||

III | 300 | 0.25 | 4 | |

IV | 0.11 | 9 | ||

V | 500 | 0.25 | 4 | |

VI | 0.11 | 9 | ||

VII | 150 | 1.2 | 0.3 | 4 |

VIII | 2.7 | 9 | ||

IX | 300 | 1.2 | 4 | |

X | 2.7 | 9 | ||

XI | 500 | 1.2 | 4 | |

XII | 2.7 | 9 |

## 3. Results

## 4. Discussion

## 5. Conclusions

## Supplementary Materials

**:**MSEs of estimates obtained with each method for each scenario $\left(\mathsf{\lambda}=0.2\right)$ and Figure S1: Boxplot of biases calculated for each method and scenario, all results for the methods that had a positive estimate for the between-person variance. Table S4: Bias and relative bias of estimates obtained with each method for each scenario $\left(\mathsf{\lambda}=0.2\right)$ and Table S5: MSEs of estimates obtained with each method for each scenario $\left(\mathsf{\lambda}=0.2\right)$, both for each method without excluded samples when other methods estimated between-person variance equal to zero. All supplementary tables and figures had more percentile results than in the article (5th, 25th, 75th, and 95th).

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Boxplot of biases calculated for each method and scenarios with n = 150 for samples with estimated between-person variance different from zero for all methods (N denotes the number of usable samples).

**Figure 2.**Boxplot of biases calculated for each method and scenarios with n = 300 for samples with estimated between-person variance different from zero for all methods (N denotes the number of usable samples).

**Figure 3.**Boxplot of biases calculated for each method and scenarios with n = 500 for samples with estimated between-person variance different from zero for all methods (N denotes the number of usable samples).

**Figure 4.**Bonferroni confidence interval for the mean bias with 95% confidence level for each scenario for samples with estimated between-person variance different from zero for all methods (n indicates the sample size and N the number of usable simulated samples).

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons by Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Laureano, G.H.C.; Torman, V.B.L.; Crispim, S.P.; Dekkers, A.L.M.; Camey, S.A.
Comparison of the ISU, NCI, MSM, and SPADE Methods for Estimating Usual Intake: A Simulation Study of Nutrients Consumed Daily. *Nutrients* **2016**, *8*, 166.
https://doi.org/10.3390/nu8030166

**AMA Style**

Laureano GHC, Torman VBL, Crispim SP, Dekkers ALM, Camey SA.
Comparison of the ISU, NCI, MSM, and SPADE Methods for Estimating Usual Intake: A Simulation Study of Nutrients Consumed Daily. *Nutrients*. 2016; 8(3):166.
https://doi.org/10.3390/nu8030166

**Chicago/Turabian Style**

Laureano, Greice H. C., Vanessa B. L. Torman, Sandra P. Crispim, Arnold L. M. Dekkers, and Suzi A. Camey.
2016. "Comparison of the ISU, NCI, MSM, and SPADE Methods for Estimating Usual Intake: A Simulation Study of Nutrients Consumed Daily" *Nutrients* 8, no. 3: 166.
https://doi.org/10.3390/nu8030166