- Research article
- Open Access
Novel metrics for growth model selection
- Matthew R. Grigsby^{1},
- Junrui Di^{2},
- Andrew Leroux^{2},
- Vadim Zipunnikov^{2},
- Luo Xiao^{3},
- Ciprian Crainiceanu^{2} and
- William Checkley^{1, 2}Email author
https://doi.org/10.1186/s12982-018-0072-z
© The Author(s) 2018
- Received: 16 May 2017
- Accepted: 14 February 2018
- Published: 23 February 2018
Abstract
Background
Literature surrounding the statistical modeling of childhood growth data involves a diverse set of potential models from which investigators can choose. However, the lack of a comprehensive framework for comparing non-nested models leads to difficulty in assessing model performance. This paper proposes a framework for comparing non-nested growth models using novel metrics of predictive accuracy based on modifications of the mean squared error criteria.
Methods
Three metrics were created: normalized, age-adjusted, and weighted mean squared error (MSE). Predictive performance metrics were used to compare linear mixed effects models and functional regression models. Prediction accuracy was assessed by partitioning the observed data into training and test datasets. This partitioning was constructed to assess prediction accuracy for backward (i.e., early growth), forward (i.e., late growth), in-range, and on new-individuals. Analyses were done with height measurements from 215 Peruvian children with data spanning from near birth to 2 years of age.
Results
Functional models outperformed linear mixed effects models in all scenarios tested. In particular, prediction errors for functional concurrent regression (FCR) and functional principal component analysis models were approximately 6% lower when compared to linear mixed effects models. When we weighted subject-specific MSEs according to subject-specific growth rates during infancy, we found that FCR was the best performer in all scenarios.
Conclusion
With this novel approach, we can quantitatively compare non-nested models and weight subgroups of interest to select the best performing growth model for a particular application or problem at hand.
Background
Childhood growth modeling plays an important role in understanding and surveilling health outcomes at both individual and population levels. Specific uses include predicting health outcomes based on current trajectories (e.g. failure to thrive, obesity, stunting, wasting) and understanding associations between growth outcomes and childhood exposures (e.g. environmental, gestational, disease) [1, 2]. Many types of statistical approaches have been proposed to model growth measurements as functions of age and related baseline covariates [3–11]. Frequently used statistical models such as linear mixed effects, quantile regression, and functional principal components methods provide great modeling flexibility and are often able to address key features of growth data such as sparsity of sampling, cross-sectional skewness, and smoothness of growth trajectories [12–14].
Comparing models requires an objective criterion that can be uniformly applied to all of them. Nested models can be compared via metrics such as the likelihood ratio test (LRT) or F-test, and penalization for parametrization with the Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC). However, comparing non-nested models is complicated because not all models optimize the same objective functions. Therefore, a comprehensive model selection strategy among competing, often non-nested, models necessitates development of a universal selection criterion.
We propose a novel approach based on modifications of the mean squared error, including normalization, age-stratification, and weighting for subject-specific growth rates. These methods differ from those mentioned above in that they measure model predictive performance rather than model fit. Quantifying predictive accuracy at the subpopulation level is critically important in auxology applications. For example, subpopulations representing lower quantiles of growth often contain children who are either stunted or faltering and may require special attention. In such scenarios, model choice may necessarily be driven disproportionately by accuracy of predicting outcomes among said subpopulations. These proposed modifications are centered on an idea of using out-of-sample prediction accuracy as universal measures of model performance.
Methods
Study setting
This analysis used data collected in the CONTENT study, located in the two peri-urban communities of Pampas de San Juan Miraflores and Nuevo Paraíso. Both were high density populations located approximately 25 km south of Lima [15]. The original purpose of this study was to examine the impact of Helicobacter pylori on child growth using World Health Organization Multicentre Growth Reference Study standards for calculating height and weight Z scores [15]. Further characterization of these regions can be found in previous publications [15, 16].
Study design
Data was collected longitudinally between May 2007 and February 2011 [15, 16]. Children were not included if they had severe disease requiring hospitalization, were part of a multiple pregnancy, had a birth weight less than 1500 grams, and/or their parents had intentions of moving during the period of the study [15]. Data was collected at birth with follow up lasting until the age of 24 months. Additional information on study design, including more specific details on information collected, can be found in the original publication [15].
Biostatistical models
When studying growth-related health outcomes and exposures, height and weight are usually collected at multiple time points to assess individual growth trajectories [1, 4, 17–20]. Notable features of longitudinal data include within-subject correlation, heterogeneity of individual baseline, and dynamic growth [21]. In this study, we employ traditional growth models such as linear mixed effects (LME), as well as less well known techniques such as functional concurrent regression (FCR) and functional principal component analysis (fPCA) [13, 21–24]. For simplicity, we used height as our growth outcome in this study. Let \(Y_{ij}\) denote the height of child \(i\) at time point \(j\), and \(t_{ij}\) is the corresponding age for child \(i\) at time point \(j\), where \(i = 1, 2, \ldots , 215\), and \(j = 1, 2, \ldots , m_{i}\). Sex effect was included in LME and FCR models, and we denote \(X_{i}\) to be the sex for subject \(i\). Even though linear regression with truncated cubic splines is well known and simple to implement, Grajeda et al. showed they were inaccurate when modeling longitudinal growth because they did not account for the nature of repeated measurements clustered within subjects and because the assumption on independence between measurements was violated [21].
Parametric, linear mixed effects model
Inclusion of subject-specific random effects is a convenient way to account for subject level clustering and is easy to implement in most statistical software packages [3, 13, 21, 25].
Since growth exhibits a pronounced non-linear association with age, population mean growth is modeled using truncated cubic splines with knots at 3, 6, 12, and 18 months. Random slopes and intercepts were used to capture the heterogeneity in growth curves. Specifically, random intercepts depict shifts (up or down) of subject-level growth from the population-level intercept, while random slopes represent subject-level growth velocity around the population prediction.
Nonparametric, functional models
It has been noted that some parametric models may not be sufficiently flexible to fully capture the non-linearity in individual growth trajectories [24]. Therefore, nonparametric approaches have gained popularity in recent years to deal with longitudinal data. One reason to think of repeated measurements as functions at different time points is because the derivatives could be of interest as well (e.g. growth rates of children). Two functional approaches are discussed next.
Definition of comparison criteria metrics
In this section, we introduce three metrics to perform growth model comparison. Let \(\hat{Y}_{ij}^{k}\) be the fitted value obtained from model \(k\), and \(i = 1, \ldots ,215\), \(j = 1, \ldots ,m_{i}\), and \(k = 1, \ldots ,3\).
Mean squared error (MSE)
Normalized mean squared error (nMSE)
Age-stratified mean squared error (aMSE)
Weighted mean squared error (wMSE)
Model comparisons
Error was measured by holding out a portion of the data (out-of-sample), fitting models to in-sample data, and then measuring predictive accuracy on the observations held out. With forward, backward, and in-range, analysis was performed by randomly selecting 50% of the children and subsequently holding out 10, 20, and 50% of their data. For new individual prediction, we randomly selected 10, 20, and 50% of the children to hold out. Primary analysis was performed using the 20% method, with the 10 and 50% used for comparison in sensitivity analysis.
Model performance will be presented as median and interquartile range (IQR) of MSE, nMSE, wMSE, or aMSE for each of the three model types (i.e. LME, fPCA, and FCR).
Results
Population characteristics
The final sample included 215 with complete data out of 304 eligible children. Eleven (3.6%) had incomplete anthropometric data and 78 (25.7%) did not have follow up past the age of 1 year. There were 39 observations per individual on average, with males representing 49% (n = 106) of the sample. Median lag between observations was 14 days (interquartile range 11–27).
In-range
Median and interquartile range for MSE stratified by location of prediction and model type
Backward | Forward | In-range | New individuals | |
---|---|---|---|---|
LME | 0.58 (0.29, 1.40) | 0.79 (0.39, 1.74) | 0.41 (0.27, 0.73) | 0.39 (0.29, 0.47) |
FCR | 0.49 (0.26, 0.92) | 0.71 (0.29, 1.29) | 0.23 (0.17, 0.36) | 0.17 (0.13, 0.22) |
fPCA | 0.48 (0.24, 0.90) | 0.62 (0.32, 1.42) | 0.24 (0.17, 0.35) | 0.18 (0.14, 0.22) |
Median and interquartile range for nMSE, stratified by location of prediction and model type
Backward | Forward | In-range | New individuals | |
---|---|---|---|---|
LME | 18.19 (9.77, 43.90) | 11.67 (6.25, 24.59) | 10.14 (5.59, 18.08) | 8.55 (6.75, 12.21) |
FCR | 17.44 (8.11, 31.11) | 10.84 (4.47, 18.78) | 5.57 (3.73, 7.67) | 3.85 (3.02, 4.78) |
fPCA | 16.17 (8.05, 32.35) | 9.78 (4.72, 19.86) | 5.72 (3.81, 7.98) | 4.10 (3.17, 4.76) |
Median and interquartile range for wMSE, stratified by location of prediction and model type
Backward | Forward | In-range | New individuals | |
---|---|---|---|---|
LME | 0.23 (0.12, 0.68) | 0.39 (0.17, 0.76) | 0.18 (0.11, 0.39) | 0.17 (0.12, 0.27) |
FCR | 0.19 (0.10, 0.28) | 0.24 (0.12, 0.45) | 0.10 (0.06, 0.16) | 0.07 (0.05, 0.12) |
fPCA | 0.21 (0.12, 0.45) | 0.25 (0.10, 0.62) | 0.12 (0.07, 0.19) | 0.08 (0.06, 0.13) |
Forward
Model error with forward prediction, using median nMSE, ranged from 0.79 (IQR 0.39–1.74) for LME to 0.62 (0.32–1.42) for fPCA. Model error was lowest for fPCA when utilizing MSE and nMSE, but FCR slightly outperformed fPCA with wMSE (Tables 1, 2, and 3). There were similar results using aMSE with fPCA performing best, followed by FCR and LME (Fig. 4). Intra-strata comparison showed a trend, with FCR, fPCA, and LME performing better in ages 12–18 months compared to ages 18–24 months.
Backward
Backward prediction revealed similar results. fPCA performed best using MSE and nMSE, but FCR performed best with wMSE (Tables 1, 2, and 3). Median nMSE ranged from 18E−5 (IQR 9.8E−5 to 43.9E−5) for LME to 16.2E−5 (8.1E−5 to 32.3E−5) for fPCA. All predicted points in backward prediction fell between ages 0–6 months. Therefore, the aMSE did not stratify the data and was interpreted as the standard MSE.
New individuals
When predicting in-range on new individuals, FCR slightly outperformed fPCA for all metrics and sampling methods. Median nMSE ranged from 8.6E−5 (IQR 6.6E−5 to 12.2E−5) for LME to 3.9E−5 (3.0E−5 to 4.8E−5) for FCR. Error distributions using aMSE were consistent with the above findings, with FCR and fPCA performing best followed by LME (Fig. 4). Between-strata differences were more apparent for LME, with LME showing less error at higher age ranges.
Sensitivity analyses
Discussion
This analysis demonstrates how to compare growth models (both nested and non-nested) by measuring prediction error via nMSE, wMSE, and aMSE. Each metric is subject-specific and can be used in a variety of real world situations. Sampling techniques can be adjusted to replicate exact scenarios of interest. Utilizing the nMSE and aMSE addresses the issue of the MSE favoring larger measurements. Furthermore, the aMSE can illuminate intra-age group performance differences and the wMSE demonstrates the ability to weight specific subgroups of interest, potentially helping to further detect performance gaps between growth models.
Based on the results of this study, functional models outperformed traditional linear models in all scenarios. Even when utilizing proven techniques with LME (i.e. truncated cubic splines and autoregressive correlation correction), FCR and fPCA performed better in all scenarios tested [21, 50]. The difference in prediction error between FCR and fPCA in most situations was relatively small. Employing the wMSE revealed a shift in the best performing model when predicting backward, forward, and in-range. In these situations, the MSE and nMSE preferred fPCA as the best performing model while the wMSE showed FCR outperforming fPCA (Tables 1, 2, and 3). While functional models consistently outperformed LME, it seems they were more sensitive to the proportion of data removed when predicting backward and in-range (Fig. 5). One possible explanation is that shapes of curves are well defined for LME with cubic splines; however, for functional approaches, it is more difficult to predict trajectories of growth curves with limited amount of data.
A limitation of the MSE is its tendency to be inflated by outliers. Using subject-specific estimates partially addresses this, but there is still the possibility of having outliers within subjects. Sensitivity analysis should be performed to assess whether more robust (outlier-insensitive) approaches are necessary. There are a few other limitations to this study. First, even though we used a variety of sampling strategies, they do not comprehensively represent real world situations. There are more scenarios that could not be included in this analysis, such as predicting backward and forward on new individuals as well as choosing different hold out percentages. Second, aMSE can be less useful in certain situations. For example, age-stratification may not be needed when predicting over a relatively short age range or if data is sparse with fewer observations in each age group.
Our study also has some potential strengths. First, the proposed method is a novel approach of transforming the subject-specific MSE (i.e. nMSE, aMSE, and wMSE) to assess prediction error differences between both nested and non-nested growth models. Alternative methods such as AIC, BIC, F-test, and the LRT only work for nested models. Second, our approach is flexible, allowing adaptation to specific real-world situations. The ability to weight subgroups of interest and adapt the age ranges used with aMSE contributes to this. Third, the CONTENT dataset is of high quality and high resolution. There were very few outliers regarding growth trends and the average number of observations per child was approximately 40 within a 2-year span. Finally, this analysis employed modern growth modeling techniques. FCR, fPCA, and LME are proven effective techniques for longitudinal growth modeling [13, 22, 24, 26–29, 32–39, 51–53].
Conclusion
Subject-specific normalized mean squared error, age-stratified mean squared error, and weighted mean squared error are useful metrics for comparing both nested and non-nested growth models. We applied these metrics to three competing modeling methods and demonstrated the ability to weight subgroups of interest and evaluate performance gaps.
Declarations
Authors’ contributions
MG, JD, and AL were responsible for conducting analysis. MG, JD, AL, VZ, and WC were responsible for data interpretation and creating the first draft of the manuscript. LX provided mentoring and assistance with coding statistical models in R. WC and CC provided mentoring. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Availability of data and materials
The datasets used and/or analyzed during the current study are available from the corresponding author on reasonable request.
Consent for publication
Not applicable.
Ethics approval and consent to participate
This study was approved by the European Union Ethics Committee, A.B. PRISMA and Universidad Peruana Cayetano Heredia, Lima, Peru, and the Johns Hopkins Bloomberg School of Public Health, Baltimore, USA.
Funding
Our work was supported by the Bill and Melinda Gates Foundation [OPP1114097].
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Authors’ Affiliations
References
- Fairley L, Petherick ES, Howe LD, Tilling K, Cameron N, Lawlor DA, et al. Describing differences in weight and length growth trajectories between white and Pakistani infants in the UK: analysis of the Born in Bradford birth cohort study using multilevel linear spline models. Arch Dis Child. 2013;98:274.View ArticlePubMedPubMed CentralGoogle Scholar
- Regnault N, Gillman MW. Importance of characterizing growth trajectories. Ann Nutr Metab. 2014;65(2–3):110–3.View ArticlePubMedPubMed CentralGoogle Scholar
- Borghi E, De Onis M, Garza C, Van den Broeck J, Frongillo E, Grummer-Strawn L, et al. Construction of the world health organization child growth standards: selection of methods for attained growth curves. Stat Med. 2006;25(2):247–65.View ArticlePubMedGoogle Scholar
- Howe LD, Tilling K, Matijasevich A, Petherick ES, Santos AC, Fairley L, et al. Linear spline multilevel models for summarising childhood growth trajectories: a guide to their application using examples from five birth cohorts. Stat Methods Med Res. 2016;25(5):1854–74.View ArticlePubMedGoogle Scholar
- Richard SA, McCormick BJ, Miller MA, Caulfield LE, Checkley W. Modeling environmental influences on child growth in the MAL-ED cohort study: opportunities and challenges. Clin Infect Dis. 2014;59(suppl 4):S255–60.View ArticlePubMedPubMed CentralGoogle Scholar
- Tilling K, Macdonald-Wallis C, Lawlor DA, Hughes RA, Howe LD. Modelling childhood growth using fractional polynomials and linear splines. Ann Nutr Metab. 2014;65(2–3):129–38.View ArticlePubMedPubMed CentralGoogle Scholar
- Berkey CS, Reed RB. A model for describing normal and abnormal growth in early childhood. Hum Biol. 1987;56:973–87.Google Scholar
- Count EW. Growth patterns of the human physique: an approach to kinetic anthropometry: Part I. Hum Biol. 1943;15(1):1–32.Google Scholar
- Guo S, Roche AF, Fomon SJ, Nelson SE, Chumlea WC, Rogers RR, et al. Reference data on gains in weight and length during the first two years of life. J Pediatr. 1991;119(3):355–62.View ArticlePubMedGoogle Scholar
- Karlberg J, Jalil F, Lam B, Low L, Yeung C. Linear growth retardation in relation to the three phases of growth. Eur J Clin Nutr. 1994;48:S25–43 (Discussion S-4).PubMedGoogle Scholar
- Wingerd J. The relation of growth from birth to 2 years to sex, parental size and other factors, using Rao’s method of the transformed time scale. Hum Biol. 1970;42:105–31.PubMedGoogle Scholar
- Koenker R. Quantile regression. Cambridge: Cambridge University Press; 2005.View ArticleGoogle Scholar
- Laird NM, Ware JH. Random-effects models for longitudinal data. Biometrics. 1982;38:963–74.View ArticlePubMedGoogle Scholar
- Yao F, Müller H-G, Wang J-L. Functional data analysis for sparse longitudinal data. J Am Stat Assoc. 2005;100(470):577–90.View ArticleGoogle Scholar
- Jaganath D, Saito M, Gilman RH, Queiroz DM, Rocha GA, Cama V, et al. First detected Helicobacter pylori infection in infancy modifies the association between diarrheal disease and childhood growth in Peru. Helicobacter. 2014;19(4):272–9.View ArticlePubMedGoogle Scholar
- Checkley W, Gilman RH, Black RE, Lescano AG, Cabrera L, Taylor DN, et al. Effects of nutritional status on diarrhea in Peruvian children. J Pediatr. 2002;140(2):210–8.View ArticlePubMedGoogle Scholar
- Checkley W, Epstein LD, Gilman RH, Black RE, Cabrera L, Sterling CR. Effects of Cryptosporidium parvum infection in Peruvian children: growth faltering and subsequent catch-up growth. Am J Epidemiol. 1998;148(5):497–506.View ArticlePubMedGoogle Scholar
- Checkley W, Epstein LD, Gilman RH, Cabrera L, Black RE. Effects of acute diarrhea on linear growth in Peruvian children. Am J Epidemiol. 2003;157(2):166–75.View ArticlePubMedGoogle Scholar
- Roth DE, Perumal N, Al Mahmud A, Baqui AH. Maternal vitamin D3 supplementation during the third trimester of pregnancy: effects on infant growth in a longitudinal follow-up study in Bangladesh. J Pediatr. 2013;163(6):1605–11.View ArticlePubMedGoogle Scholar
- Tilling K, Davies N, Windmeijer F, Kramer MS, Bogdanovich N, Matush L, et al. Is infant weight associated with childhood blood pressure? Analysis of the promotion of breastfeeding intervention trial (PROBIT) cohort. Int J Epidemiol. 2011;40(5):1227–37.View ArticlePubMedPubMed CentralGoogle Scholar
- Grajeda LM, Ivanescu A, Saito M, Crainiceanu C, Jaganath D, Gilman RH, et al. Modelling subject-specific childhood growth using linear mixed-effect models with cubic regression splines. Emerg Themes Epidemiol. 2016;13(1):1.View ArticlePubMedPubMed CentralGoogle Scholar
- Besse P, Ramsay JO. Principal components analysis of sampled functions. Psychometrika. 1986;51(2):285–311.View ArticleGoogle Scholar
- Crainiceanu CM, Ruppert D, Wand MP. Bayesian analysis for penalized spline regression using Win BUGS. J Stat Softw. 2005;14:1–24.View ArticleGoogle Scholar
- Guo W. Functional mixed effects models. Biometrics. 2002;58(1):121–8.View ArticlePubMedGoogle Scholar
- Goldstein H. Efficient statistical modelling of longitudinal data. Ann Hum Biol. 1986;13(2):129–41.View ArticlePubMedGoogle Scholar
- Besse PC, Cardot H, Ferraty F. Simultaneous non-parametric regressions of unbalanced longitudinal data. Comput Stat Data Anal. 1997;24(3):255–70.View ArticleGoogle Scholar
- Kneip A. Nonparametric estimation of common regressors for similar curve data. Ann Stat. 1994;22:1386–427.View ArticleGoogle Scholar
- Ramsay JO, Dalzell C. Some tools for functional data analysis. J R Stat Soc Ser B (Methodol). 1991;53:539–72.Google Scholar
- Staniswalis JG, Lee JJ. Nonparametric regression analysis of longitudinal data. J Am Stat Assoc. 1998;93(444):1403–18.View ArticleGoogle Scholar
- Xiao L, Zipunnikov V, Ruppert D, Crainiceanu C. Fast covariance estimation for high-dimensional functional data. Stat Comput. 2016;26(1–2):409–21.View ArticlePubMedGoogle Scholar
- Xiao L, Li C, Checkley W, Crainiceanu CM. Fast covariance estimation for sparse functional data. arXiv preprint arXiv:1603.05758; 2016.
- Cederbaum J, Pouplier M, Hoole P, Greven S. Functional linear mixed models for irregularly or sparsely sampled data. Stat Model. 2016;16(1):67–88.View ArticleGoogle Scholar
- Chen H, Wang Y. A penalized spline approach to functional mixed effects model analysis. Biometrics. 2011;67(3):861–70.View ArticlePubMedGoogle Scholar
- Di C-Z, Crainiceanu CM, Caffo BS, Punjabi NM. Multilevel functional principal component analysis. Ann Appl Stat. 2009;3(1):458.View ArticlePubMedPubMed CentralGoogle Scholar
- Hadjipantelis PZ, Aston JA, Müller H-G, Evans JP. Unifying amplitude and phase analysis: a compositional data approach to functional multivariate mixed-effects modeling of Mandarin Chinese. J Am Stat Assoc. 2015;110(510):545–59.View ArticlePubMedPubMed CentralGoogle Scholar
- Reimherr M, Nicolae D. Estimating variance components in functional linear models with applications to genetic heritability. J Am Stat Assoc. 2016;111(513):407–22.View ArticleGoogle Scholar
- Yuan Y, Gilmore JH, Geng X, Martin S, Chen K, J-l Wang, et al. FMEM: functional mixed effects modeling for the analysis of longitudinal white matter tract data. NeuroImage. 2014;84:753–64.View ArticlePubMedGoogle Scholar
- Zhou L, Huang JZ, Martinez JG, Maity A, Baladandayuthapani V, Carroll RJ. Reduced rank mixed effects models for spatially correlated hierarchical functional data. J Am Stat Assoc. 2010;105(489):390–400.View ArticlePubMedPubMed CentralGoogle Scholar
- Zhu H, Brown PJ, Morris JS. Robust, adaptive functional regression in functional mixed model framework. J Am Stat Assoc. 2012;106:1167.View ArticleGoogle Scholar
- Wood SN. Generalized additive models: an introduction with R. Boston: CRC Press; 2017.Google Scholar
- Wood SN, Li Z, Shaddick G, Augustin NH. Generalized additive models for gigadata: modeling the UK black smoke network daily data. J Am Stat Assoc. 2017;112:1–12.View ArticleGoogle Scholar
- Leroux A, Xiao L, Crainiceanu C, Checkley W. Dynamic prediction in functional concurrent regression with an application to child growth. Stat Med. 2017. https://doi.org/10.1002/sim.7582 PubMedGoogle Scholar
- R Core Team. R: A language and environment for statistical computing. Vienna: R Foundation for Statistical Computing; 2017.Google Scholar
- Carbone R, Armstrong JS. Note. Evaluation of extrapolative forecasting methods: Results of a survey of academicians and practitioners. J Forecast. 1982;1(2):215–7.View ArticleGoogle Scholar
- Armstrong JS, Collopy F. Error measures for generalizing about forecasting methods: empirical comparisons. Int J Forecast. 1992;8(1):69–80.View ArticleGoogle Scholar
- Chatfield C. Apples, oranges and mean square error. Int J Forecast. 1988;4(4):515–8.View ArticleGoogle Scholar
- Flores BE. A pragmatic view of accuracy measurement in forecasting. Omega. 1986;14(2):93–8.View ArticleGoogle Scholar
- Makridakis S. Accuracy measures: theoretical and practical concerns. Int J Forecast. 1993;9(4):527–9.View ArticleGoogle Scholar
- Sanders NR. Measuring forecast accuracy: some practical suggestions. Prod Inventory Manag J. 1997;38(1):43.Google Scholar
- Funatogawa I, Funatogawa T, Ohashi Y. An autoregressive linear mixed effects model for the analysis of longitudinal data which show profiles approaching asymptotes. Stat Med. 2007;26(9):2113–30.View ArticlePubMedGoogle Scholar
- Davidian M, Giltinan DM. Nonlinear models for repeated measurement data. Boston: CRC Press; 1995.Google Scholar
- Lindstrom MJ, Bates DM. Nonlinear mixed effects models for repeated measures data. Biometrics. 1990;46:673–87.View ArticlePubMedGoogle Scholar
- Ramsay JO. Functional data analysis. New York: Wiley; 2006.View ArticleGoogle Scholar