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BMC Medical Research Methodology
Published in: BMC Medical Research Methodology 1/2020

Open Access 01-12-2020 | Imipramine | Research article

A graphical approach to assess the goodness-of-fit of random-effects linear models when the goal is to measure individual benefits of medical treatments in severely ill patients

Authors: Zhiwen Wang, Francisco J. Diaz

Published in: BMC Medical Research Methodology | Issue 1/2020

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Abstract

Background

Two-dimensional personalized medicine (2-PM) models are tools for measuring individual benefits of medical treatments for chronic diseases which have potential applications in personalized medicine. These models assume normality for the distribution of random effects. It is necessary to examine the appropriateness of this assumption. Here, we propose a graphical approach to assessing the goodness-of-fit of 2-PM models with continuous responses.

Methods

We propose benefit quantile-quantile (BQQ) plots which compare the empirical quantiles of individual benefits from a patient sample predicted through an empirical Bayes (EB) approach versus the quantiles of the theoretical distribution of individual benefits derived from the assumption of normality for the random effects. We examine the performance of the approach by conducting a simulation study that compared 2-PM models with non-normal distributions for the random effects versus models with comparable normal distributions. Cramer-von Mises discrepancies were used to quantify the performance of the approach. The approach was illustrated with data from a clinical trial of imipramine for patients with depression.

Results

Simulations showed that BQQ plots were able to capture deviations from the normality assumption for the random effects and did not show any asymmetric deviations from the y = x line when the random effects were normally distributed. For the depression data, the points of the BQQ plot were scattered around closely to the y = x line, without presenting any asymmetric deviations. This implied the adequacy of the normality assumption for the random effects and the goodness-of-fit of the 2-PM model for the imipramine data.

Conclusion

BQQ plots are sensitive to violations of the normality assumption for the random effects, suggesting that the approach is a useful tool for examining the goodness-of-fit of random-effects linear models when the goal is to measure individual treatment benefits.
Appendix
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Literature
1.
Diaz FJ. Estimating individual benefits of medical or behavioral treatments in severely ill patients. Stat Methods Med Res. 2019;28:911–27.PubMed
2.
Diaz FJ. Measuring the individual benefit of a medical or behavioral treatment using generalized linear mixed-effects models. Stat Med. 2016;35:4077–92.PubMedPubMedCentral
3.
Laird NM, Ware JH. Random-effects models for longitudinal data. Biometrics. 1982;38:963–74.PubMed
4.
Fitzmaurice GM, Davidian M, Verbeke G, Molenberghs G. Longitudinal data analysis. Boca Raton: CRC Press; 2009.
5.
Fitzmaurice GM, Laird NM, Ware JH. Applied longitudinal analysis. 2nd ed. Hoboken: Wiley; 2011.
6.
Hedeker D, Gibbons RD. Longitudinal data analysis. Hoboken: Wiley-Interscience; 2006.
7.
Diaz FJ, Rivera TE, Josiassen RC, de Leon J. Individualizing drug dosage by using a random intercept linear model. Stat Med. 2007;26:2052–73.PubMed
8.
Diaz FJ, Cogollo MR, Spina E, Santoro V, Rendon DM, de Leon J. Drug dosage individualization based on a random-effects linear model. J Biopharm Stat. 2012;22:463–84.PubMed
9.
Diaz FJ, Yeh HW, de Leon J. Role of statistical random-effects linear models in personalized medicine. Curr Pharmacogenomics Person Med. 2012;10:22–32.PubMedPubMedCentral
10.
Diaz FJ, de Leon J. The mathematics of drug dose individualization should be built with random-effects linear models. Ther Drug Monit. 2013;35:276–7.PubMed
11.
Diaz FJ. Construction of the design matrix for generalized linear mixed-effects models in the context of clinical trials of treatment sequences. Rev Colomb Estadística. 2018;41:191–233.
12.
Verbeke G, Molenberghs G. Linear mixed models for longitudinal data. New York: Springer; 2000.
13.
Frees EW. Longitudinal and panel data : analysis and applications in the social sciences. Cambridge; New York: Cambridge University Press; 2004.
14.
Nasserinejad K, De Kort W, Baart M, Komárek A, Van Rosmalen J, Lesaffre E. Predicting hemoglobin levels in whole blood donors using transition models and mixed effects models. BMC Med Res Methodol. 2013;13:62.PubMedPubMedCentral
15.
McCulloch CE, Neuhaus JM. Misspecifying the shape of a random effects distribution: why getting it wrong may not matter. Stat Sci. 2011;26:388–402.
16.
Mcculloch CE, Neuhaus JM. Prediction of random effects in linear and generalized linear models under model misspecification. Biometrics. 2011;67:270–9.PubMedPubMedCentral
17.
Wang Z. Prediction of random effects in mixed effects models under violations of the normality assumption for the random effects and a graphical approach to detect violations. Doctoral dissertation. Kansas: University of Kansas; 2019.
18.
Verbeke G, Lesaffre E. The effect of misspecifying the random-effects distribution in linear mixed models for longitudinal data. Comput Stat Data Anal. 1997;23:541–56.
19.
Verbeke G, Lesaffre E. A linear mixed-effects model with heterogeneity in the random-effects population. J Am Stat Assoc. 1996;91:217–21.
20.
Agresti A, Caffo B, Ohman-Strickland P. Examples in which misspecification of a random effects distribution reduces efficiency, and possible remedies. Comput Stat Data Anal. 2004;47:639–53.
21.
Litière S, Alonso A, Molenberghs G. Type I and type II error unde random-effects misspecification in generalized linear mixed models. Biometrics. 2007;63:1038–44.PubMed
22.
Litière S, Alonso A, Molenberghs G. The impact of a misspecified random-effects distribution on the estimation and the performance of inferential procedures in generalized linear mixed models. Stat Med. 2008;27:3125–44.PubMed
23.
Nobre JS, Da Motta Singer J. Residual analysis for linear mixed models. Biom J. 2007;49:863–75.PubMed
24.
Bates DM, Pinheiro JC. Mixed-effects models in S and S-PLUS. 1st ed. New York: Springer-Verlag; 2000.
25.
Gregoire TG, Schabenberger O, Barrett JP. Linear modelling of irregularly spaced, unbalanced, longitudinal data from permanent-plot measurements. Can J For Res. 1995;25:137–56.
26.
Waternaux C, Laird NM, Ware JH. Methods for analysis of longitudinal data: blood-lead concentrations and cognitive development. J Am Stat Assoc. 1989;84:33–41.
27.
Wilk MB, Gnanadesikan R. Probability plotting methods for the analysis of data. Biometrika. 1968;55:1–17.PubMed
28.
Verbeke G, Molenberghs G. The gradient function as an exploratory goodness-of-fit assessment of the random-effects distribution in mixed models. Biostatistics. 2013;14:477–90.PubMed
29.
Pan ZY, Lin DY. Goodness-of-fit methods for generalized linear mixed models. Biometrics. 2005;61:1000–9.PubMed
30.
Grady JJ, Helms RW. Model selection techniques for the covariance matrix for incomplete longitudinal data. Stat Med. 1995;14:1397–416.PubMed
31.
Diaz FJ, Santoro V, Spina E, Cogollo M, Rivera TE, Botts S, et al. Estimating the size of the effects of co-medications on plasma clozapine concentrations using a model that controls for clozapine doses and confounding variables. Pharmacopsychiatry. 2008;41:81–91.PubMed
32.
Efendi A, Drikvandi R, Verbeke G, Molenberghs G. A goodness-of-fit test for the random-effects distribution in mixed models. Stat Methods Med Res. 2017;26:970–83.PubMed
33.
Drikvandi R, Verbeke G, Molenberghs G. Diagnosing misspecification of the random-effects distribution in mixed models. Biometrics. 2017;73:63–71.PubMed
34.
Alonso A, Litière S, Molenberghs G. A family of tests to detect misspecifications in the random-effects structure of generalized linear mixed models. Comput Stat Data Anal. 2008;52:4474–86.
35.
Abad AA, Litière S, Molenberghs G. Testing for misspecification in generalized linear mixed models. Biostatistics. 2010;11:771–86.PubMed
36.
McCulloch C, Searle S, Neuhaus J. Generalized, linear, and mixed models. 2nd ed. Hoboken: Wiley Series in Probability and Statistics; 2008.
37.
Ten Have TR, Localio AR. Empirical Bayes estimation of random effects parameters in mixed effects logistic regression models. Biometrics. 1999;55:1022–9.PubMed
38.
Robinson GK. That BLUP is a good thing: the estimation of random effects. Stat Sci. 1991;6:15–32.
39.
Reisby N, Gram LF, Bech P, Nagy A, Petersen GO, Ortmann J, et al. Imipramine: clinical effects and pharmacokinetic variability. Psychopharmacology. 1977;54:263–72.PubMed
40.
Buja A, Cook D, Hofmann H, Lawrence M, Lee EK, Swayne DF, et al. Statistical inference for exploratory data analysis and model diagnostics. Philos Trans R Soc A Math Phys Eng Sci. 2009;367:4361–83.
41.
Loy A, Follett L, Hofmann H. Variations of Q–Q plots: the power of our eyes! Am Stat. 2016;70:202–14.
42.
Anderson TW. An introduction to multivariate statistical analysis. 3rd ed. New York: Wiley-Interscience; 2003.
43.
Anderson TW. On the distribution of the two-sample Cramer-von Mises criterion. Ann Math Stat. 1962;33:1148–59.
44.
CSöRgő S, Faraway JJ. The exact and asymptotic distributions of Cramér-Von Mises statistics. J R Stat Soc Ser B. 1996;58:221–34.
45.
Darling DA. The Kolmogorov-Smirnov, Cramer-von Mises Tests. Ann Math Stat. 1957;28:823–38.
46.
Lange N, Ryan L. Assessing normality in random effects models. Ann Stat. 1989;17:624–42.
47.
Shapiro SS, Wilk MB. An analysis of variance test for normality (complete samples). Biometrika. 1965;52:591–611.
48.
Razali NM, Wah YB. Power comparisons of Shapiro-Wilk , Kolmogorov-Smirnov, Lilliefors and Anderson-Darling tests. J Stat Model Anal. 2011;2:21–33.
49.
Anderson TW, Darling DA. Asymptotic theory of certain “goodness of fit” criteria based on stochastic processes. Ann Math Stat. 1952;23:193–212.
50.
Pinheiro JC, Liu C, Nianwu Y. Efficient algorithms for robust estimation in linear mixed-effects models using the multivariatetdistribution. J Comput Graph Stat. 2001;10:249–76.
51.
Yavuz FG, Arslan O. Linear mixed model with Laplace distribution (LLMM). Stat Pap. 2018;59:271–89.
52.
Lin TI, Lee JC. Estimation and prediction in linear mixed models with skew-normal random effects for longitudinal data. Stat Med. 2008;27:1490–507.PubMed
53.
García Ben M, Yohai VJ. Quantile-quantile plot for deviance residuals in the generalized linear model. J Comput Graph Stat. 2004;13:36–47.
54.
Aldor-Noiman S, Brown LD, Buja A, Rolke W, Stine RA. The power to see: a new graphical test of normality. Am Stat. 2013;67:249–60.
55.
Stine RA. Explaining normal quantile-quantile plots through animation: the water-filling analogy. Am Stat. 2017;71:145–7.
56.
Loy A, Hofmann H, Cook D. Model choice and diagnostics for linear mixed-effects models using statistics on street corners. J Comput Graph Stat. 2017;26:478–92.
57.
Johnson RA, Wichern DW. Applied multivariate statistical analysis. 6th ed. New Jersey: Pearson; 2008.
58.
Waagepetersen R. A simulation-based goodness-of-fit test for random effects in generalized linear mixed models. Scand J Stat. 2006;33:721–31.
59.
Tchetgen EJ, Coull BA. A diagnostic test for the mixing distribution in a generalised linear mixed model. Biometrika. 2006;93:1003–10.
Metadata
Title
A graphical approach to assess the goodness-of-fit of random-effects linear models when the goal is to measure individual benefits of medical treatments in severely ill patients
Authors
Zhiwen Wang
Francisco J. Diaz
Publication date
01-12-2020
Publisher
BioMed Central
Keyword
Imipramine
Published in
BMC Medical Research Methodology / Issue 1/2020
Electronic ISSN: 1471-2288
DOI
https://doi.org/10.1186/s12874-020-01054-3

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