1921:
6907:
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the wells. The overall recent commercial success rate of horizontal wells in the United States is known to be 65%, which implies that only 2 out of 3 drilled wells will be commercially successful. For this reason, one of the crucial tasks of petroleum engineers is to quantify the uncertainty associated with oil or gas production from shale reservoirs, and further, to predict an approximated production behavior of a new well at a new location given specific completion data before actual drilling takes place to save a large degree of well construction costs.
391:
6168:
1082:
2538:
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6902:{\displaystyle =\underbrace {\pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}}|\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2})} _{Stage1:Individual-LevelModel}\times \underbrace {\pi (\{\theta _{li}\}_{i=1,l=1}^{N,K}|\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage2:PopulationModel}\times \underbrace {p(\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})} _{Stage3:Prior}}
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review, defining a problem and specifying the research question and hypothesis. Bayesian-specific workflow comprises three sub-steps: (b)–(i) formalizing prior distributions based on background knowledge and prior elicitation; (b)–(ii) determining the likelihood function based on a nonlinear function
2511:
can be formulated as nonlinear mixed-effects models. The mixed-model approach allows modeling of both population level and individual differences in effects that have a nonlinear effect on the observed outcomes, for example the rate at which a compound is being metabolized or distributed in the body.
1948:
may both directly affect the measured outcome (e.g. organisms with lack of nutrients end up smaller), but possibly also timing (e.g. organisms with lack of nutrients grow at a slower pace). If a model fails to account for the differences in timing, the estimated population-level curves may smooth out
2528:
The platform of the nonlinear mixed effect models can be used to describe infection trajectories of subjects and understand some common features shared across the subjects. In epidemiological problems, subjects can be countries, states, or counties, etc. This can be particularly useful in estimating
4412:
The framework of
Bayesian hierarchical modeling is frequently used in diverse applications. Particularly, Bayesian nonlinear mixed-effects models have recently received significant attention. A basic version of the Bayesian nonlinear mixed-effects models is represented as the following three-stage:
2550:
and gas reservoirs, because of very low permeability, and a flow mechanism very different from that of conventional reservoirs, estimates for the well construction cost often contain high levels of uncertainty, and oil companies need to make heavy investment in the drilling and completion phase of
1047:
in certain classes of nonlinear mixed-effects models – typically under the assumption of normally distributed random variables. A popular approach is the
Lindstrom-Bates algorithm which relies on iteratively optimizing a nonlinear problem, locally linearizing the model around this optimum and then
6912:
The panel on the right displays
Bayesian research cycle using Bayesian nonlinear mixed-effects model. A research cycle using the Bayesian nonlinear mixed-effects model comprises two steps: (a) standard research cycle and (b) Bayesian-specific workflow. Standard research cycle involves literature
2541:
Prediction of oil production rate decline curve obtained by latent kriging. 324 training wells and two test wells in the Eagle Ford Shale
Reservoir of South Texas (top left); A schematic example of a hydraulically fractured horizontal well (bottom left); Predicted curves at test wells via latent
5192:{\displaystyle \sigma ^{2}\sim \pi (\sigma ^{2}),\quad \alpha _{l}\sim \pi (\alpha _{l}),\quad (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP})\sim \pi (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP}),\quad \omega _{l}^{2}\sim \pi (\omega _{l}^{2}),\quad l=1,\ldots ,K.}
3539:
2749:
4425:
2564:
2219:
3010:
1069:, the temporal patterns of progression on outcome variables may follow a nonlinear temporal shape that is similar between patients. However, the stage of disease of an individual may not be known or only partially known from what can be measured. Therefore, a
5873:
5579:
3530:{\displaystyle \beta _{lj}|\lambda _{lj},\tau _{l},\sigma _{l}\sim N(0,\sigma _{l}^{2}\tau _{l}^{2}\lambda _{lj}^{2}),\quad \sigma ,\lambda _{lj},\tau _{l},\sigma _{l}\sim C^{+}(0,1),\quad \quad \quad \quad \quad \quad \quad l=1,2,3,\,j=1,\cdots ,p,}
4395:
techniques have been employed in the latent level, this technique is called latent kriging. The right panels show the prediction results of the latent kriging method applied to the two test wells in the Eagle Ford Shale
Reservoir of South Texas.
634:
907:
4687:
2495:
Basic pharmacokinetic processes affecting the fate of ingested substances. Nonlinear mixed-effects modeling can be used to estimate the population-level effects of these processes while also modeling the individual variation between
1291:
4321:
1885:(ADAS-Cog) is shown in the box. As shown, the inclusion of fixed effects of baseline categorization (MCI or dementia relative to normal cognition) and the random effect of individual continuous disease stage
3679:{\displaystyle \alpha _{l}\sim \pi (\alpha )\propto 1,\quad \sigma _{l}^{2}\sim \pi (\sigma ^{2})\propto 1/\sigma ^{2},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad l=1,2,3,}
3001:{\displaystyle \theta _{li}=\theta _{l}(s_{i})=\alpha _{l}+\sum _{j=1}^{p}\beta _{lj}x_{j}+\epsilon _{l}(s_{i})+\eta _{l}(s_{i}),\quad \epsilon _{l}(\cdot )\sim GWN(\sigma _{l}^{2}),\quad \quad l=1,2,3,}
4389:
3757:
4669:{\displaystyle {y}_{ij}=f(t_{ij};\theta _{1i},\theta _{2i},\ldots ,\theta _{li},\ldots ,\theta _{Ki})+\epsilon _{ij},\quad \epsilon _{ij}\sim N(0,\sigma ^{2}),\quad i=1,\ldots ,N,\,j=1,\ldots ,M_{i}.}
1977:
may underestimate the magnitude of the pubertal height spurt because age is not synchronized with biological development. The differences in biological development can be modeled using random effects
2737:{\displaystyle {y}_{it}=\mu (t;\theta _{1i},\theta _{2i},\theta _{3i})+\epsilon _{it},\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad i=1,\ldots ,N,\,t=1,\ldots ,T_{i},}
3900:
3816:
5410:
2063:
5482:
3255:{\displaystyle \eta _{l}(\cdot )\sim GP(0,K_{\gamma _{l}}(\cdot ,\cdot )),\quad K_{\gamma _{l}}(s_{i},s_{j})=\gamma _{l}^{2}\exp(-e^{\rho _{l}}\|s_{i}-s_{j}\|^{2}),\quad \quad \quad l=1,2,3,}
2400:
2055:
6157:{\displaystyle \propto \pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}},\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})}
4186:
4002:
1171:
5862:{\displaystyle \pi (\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K}|\{y_{ij}\}_{i=1,j=1}^{N,M_{i}})}
4060:
2429:
2004:
956:
1563:
4130:
1924:
Estimation of a mean height curve for boys from the
Berkeley Growth Study with and without warping. Warping model is fitted as a nonlinear mixed-effects model using the pavpop
1592:
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1871:
1008:
1209:
491:
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1702:
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823:
761:
1253:
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2354:
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1283:
927:
791:
1910:
1819:
689:
4889:{\displaystyle \theta _{li}=\alpha _{l}+\sum _{b=1}^{P}\beta _{lb}x_{ib}+\eta _{li},\quad \eta _{li}\sim N(0,\omega _{l}^{2}),\quad i=1,\ldots ,N,\,l=1,\ldots ,K.}
1285:). These longitudinal trajectories can be modeled using a nonlinear mixed effects model that allows differences in disease state based on baseline categorization:
1518:{\displaystyle {y}_{ij}=f_{\tilde {\beta }}(t_{ij}+A_{i}^{MCI}\beta ^{MCI}+A_{i}^{DEM}\beta ^{DEM}+b_{i})+\epsilon _{ij},\quad i=1,\ldots ,M,\,j=1,\ldots ,n_{i}}
485:
is an example of a nonlinear mixed-effects model, the most commonly used models are members of the class of nonlinear mixed-effects models for repeated measures
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976:
731:
709:
660:
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The platform of the nonlinear mixed effect models can be extended to consider the spatial association by incorporating the geostatistical processes such as
449:
or when there are dependencies between measurements on related statistical units. Nonlinear mixed-effects models are applied in many fields including
7404:
Lee, Se Yoon; Mallick, Bani (2021). "Bayesian
Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas".
1920:
1961:
Models for estimating the mean curves of human height and weight as a function of age and the natural variation around the mean are used to create
4225:
1073:
time variable that describe individual disease stage (i.e. where the patient is along the nonlinear mean curve) can be included in the model.
7208:
Raket LL, Sommer S, Markussen B (2014). "A nonlinear mixed-effects model for simultaneous smoothing and registration of functional data".
2254:
is a function that represents the height development of a typical child as a function of age. Its shape is determined by the parameters
1953:
between organisms. Nonlinear mixed-effects models enable simultaneous modeling of individual differences in growth outcomes and timing.
4408:
Bayesian research cycle using
Bayesian nonlinear mixed effects model: (a) standard research cycle and (b) Bayesian-specific workflow.
1048:
employing conventional methods from linear mixed-effects models to do maximum likelihood estimation. Stochastic approximation of the
445:. Like linear mixed-effects models, they are particularly useful in settings where there are multiple measurements within the same
421:
2463:
is a random variable describing additive variation (e.g. consistent differences in height between children and measurement noise).
1792:
are parameters that model the difference in disease progression of the MCI and dementia groups relative to the cognitively normal,
4326:
3694:
2524:
Extrapolated infection trajectories of 40 countries severely affected by COVID-19 and grand (population) average through May 14th
331:
3759:
is a function that models the mean time-profile of log-scaled oil production rate whose shape is determined by the parameters
2529:
a future trend of the epidemic in an early stage of pendemic where nearly little information is known regarding the disease.
2402:
is a warping function that maps age to biological development to synchronize. Its shape is determined by the random effects
6933:; and (b)–(iii) making a posterior inference. The resulting posterior inference can be used to start a new research cycle.
3827:
3762:
2214:{\displaystyle {y}_{ij}=f_{\beta }(v(t_{ij},{\boldsymbol {w}}_{i}))+\epsilon _{ij},\quad i=1,\ldots ,M,\,j=1,\ldots ,n_{i}}
321:
5354:
1096:
is characterized by a progressive cognitive deterioration. However, patients may differ widely in cognitive ability and
5435:
2532:
1965:. The growth of children can however become desynchronized due to both genetic and environmental factors. For example,
1049:
5574:
A central task in the application of the
Bayesian nonlinear mixed-effect models is to evaluate the posterior density:
7020:
6962:
2361:
2016:
4142:
3935:
2546:
The eventual success of petroleum development projects relies on a large degree of well construction costs. As for
285:
1105:
1565:
is a function that models the mean time-profile of cognitive decline whose shape is determined by the parameters
1111:
1044:
336:
274:
94:
69:
6952:
2504:
1028:
196:
1040:
1020:
2475:
of time (i.e. additive shifts in biological age and differences in rate of maturation), while the so-called
4029:
2405:
1980:
932:
414:
4215:
1534:
357:
7476:"Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications"
7433:"Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications"
7523:
4099:
1568:
326:
295:
222:
629:{\displaystyle {y}_{ij}=f(\phi _{ij},{v}_{ij})+\epsilon _{ij},\quad i=1,\ldots ,M,\,j=1,\ldots ,n_{i}}
4065:
1212:
316:
305:
269:
176:
7059:
5507:
2533:
Example: Prediction of oil production curve of shale oil wells at a new location with latent kriging
2436:
1912:
aligns the trajectories of cognitive deterioration to reveal a common pattern of cognitive decline.
1846:
983:
902:{\displaystyle \phi _{ij}={\boldsymbol {A}}_{ij}\beta +{\boldsymbol {B}}_{ij}{\boldsymbol {b}}_{i},}
6977:
1925:
1176:
1086:
377:
248:
171:
64:
43:
1762:
1729:
1974:
1669:
1631:
1024:
407:
300:
7116:
Kuhn, E; Lavielle, M (2005). "Maximum likelihood estimation in nonlinear mixed effects models".
5537:
2230:
798:
736:
3819:
1093:
264:
259:
201:
7073:
Lindstrom, MJ; Bates, DM (1990). "Nonlinear mixed effects models for repeated measures data".
5504:
is a `nonlinear' function and describes the temporal trajectory of individuals. In the model,
1222:
2472:
352:
48:
5564:
describe within-individual variability and between-individual variability, respectively. If
5284:
5254:
5204:
4193:
2329:
2279:
2257:
1599:
1258:
912:
766:
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3907:
3903:
1888:
1797:
1036:
667:
372:
362:
243:
211:
166:
145:
53:
5570:
is not considered, then the model reduces to a frequentist nonlinear mixed-effect model.
2467:
There exists several methods and software packages for fitting such models. The so-called
1085:
Example of disease progression modeling of longitudinal ADAS-Cog scores using the progmod
8:
6947:
4137:
1104:
at a single time point can often only be used to coarsely group individuals in different
1066:
290:
191:
186:
140:
89:
79:
24:
7345:"Estimation of COVID-19 spread curves integrating global data and borrowing information"
1081:
7487:
7444:
7381:
7356:
7344:
7320:
7295:
7276:
7263:
7236:
7187:
7174:
7147:
7090:
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5487:
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5334:
5314:
5234:
4007:
3913:
2547:
2508:
2309:
1937:
1878:
1824:
1707:
1019:
When the model is only nonlinear in fixed effects and the random effects are
Gaussian,
961:
716:
694:
645:
395:
124:
109:
1970:
1065:
Nonlinear mixed-effects models have been used for modeling progression of disease. In
7386:
7325:
7268:
7191:
7179:
7098:
7016:
6957:
1941:
1933:
1101:
1097:
474:
438:
390:
181:
84:
38:
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7307:
7258:
7248:
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7159:
7125:
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7008:
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4133:
3818:. The function is obtained from taking logarithm to the rate decline curve used in
2555:
446:
206:
135:
7371:
7221:
2007:
1950:
1070:
367:
74:
7417:
7129:
2486:
1032:
119:
1966:
1211:
individuals that are each categorized as having either normal cognition (CN),
733:
is a real-valued differentiable function of a group-specific parameter vector
7517:
7164:
7007:. Statistics and Computing. New York: Springer Science & Business Media.
1076:
482:
454:
238:
114:
5351:-th subject. Parameters involved in the model are written in Greek letters.
2537:
7390:
7329:
7272:
7183:
2500:
1962:
1945:
478:
458:
104:
7102:
7037:
4404:
7502:
7475:
7459:
7432:
7253:
6942:
4316:{\displaystyle (\theta _{1i},\theta _{2i},\theta _{3i}),(i=1,\cdots ,N),}
2057:. A simple nonlinear mixed-effects model with this structure is given by
442:
150:
99:
7296:"Nonlinearity detection: advantages of nonlinear mixed-effects modeling"
7094:
2480:
1052:
gives an alternative approach for doing maximum-likelihood estimation.
7311:
1039:. In the more general setting, there exist several methods for doing
7086:
2520:
1626:
represents observation time (e.g. time since baseline in the study),
7492:
7449:
7361:
7012:
2491:
1882:
1216:
450:
3902:
represents covariates obtained from the completion process of the
7148:"Statistical disease progression modeling in Alzheimer's disease"
4392:
4219:
2483:
warping functions. An example of the latter is shown in the box.
462:
4218:
used on the latent level (the second stage) eventually produce
4004:
represents the spatial location (longitude, latitude) of the
2515:
2487:
Example: Population Pharmacokinetic/pharmacodynamic modeling
4384:{\displaystyle \mu (t;\theta _{1},\theta _{2},\theta _{3})}
3752:{\displaystyle \mu (t;\theta _{1},\theta _{2},\theta _{3})}
2471:
model can fit such models using warping functions that are
1108:. Now suppose we have a set of longitudinal cognitive data
1077:
Example: Modeling cognitive decline in Alzheimer's disease
4399:
1881:
mean function fitted to longitudinal measurements of the
4062:
represents the Gaussian white noise with error variance
1060:
7234:
1973:
can vary several years between adolescents. Therefore,
1883:
Alzheimer's Disease Assessment Scale-Cognitive Subscale
1932:
Growth phenomena often follow nonlinear patters (e.g.
7237:"SITAR—a useful instrument for growth curve analysis"
6919:
6171:
5876:
5582:
5540:
5510:
5490:
5438:
5418:
5357:
5337:
5317:
5287:
5257:
5237:
5207:
4910:
4690:
4428:
4329:
4228:
4196:
4145:
4102:
4068:
4032:
4010:
3938:
3916:
3895:{\displaystyle x_{i}=(x_{i1},\cdots ,x_{ip})^{\top }}
3830:
3811:{\displaystyle (\theta _{1},\theta _{2},\theta _{3})}
3765:
3697:
3542:
3267:
3013:
2752:
2567:
2439:
2408:
2364:
2332:
2312:
2282:
2260:
2233:
2066:
2019:
1983:
1891:
1849:
1827:
1800:
1765:
1732:
1710:
1672:
1634:
1602:
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1537:
1294:
1261:
1225:
1179:
1114:
986:
964:
935:
915:
831:
801:
769:
739:
719:
697:
670:
648:
494:
7207:
6998:
6996:
5405:{\displaystyle f(t;\theta _{1},\ldots ,\theta _{K})}
958:
is a vector of random effects associated with group
7293:
7203:
7201:
6925:
6901:
6156:
5861:
5556:
5526:
5496:
5476:
5424:
5404:
5343:
5323:
5303:
5273:
5243:
5223:
5191:
4888:
4668:
4383:
4315:
4202:
4180:
4124:
4086:
4054:
4016:
3996:
3922:
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3810:
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3678:
3529:
3254:
3000:
2736:
2455:
2423:
2394:
2348:
2318:
2298:
2266:
2246:
2213:
2049:
1998:
1904:
1865:
1833:
1813:
1784:
1751:
1716:
1696:
1658:
1618:
1586:
1557:
1517:
1277:
1247:
1203:
1165:
1002:
970:
950:
921:
901:
817:
785:
755:
725:
703:
683:
654:
628:
6993:
5477:{\displaystyle (\theta _{1},\ldots ,\theta _{K})}
1956:
1821:is the difference in disease stage of individual
7515:
7343:Lee, Se Yoon; Lei, Bowen; Mallick, Bani (2020).
1724:has MCI or dementia at baseline and 0 otherwise,
7198:
2395:{\displaystyle v(\cdot ,{\boldsymbol {w}}_{i})}
2050:{\displaystyle v(\cdot ,{\boldsymbol {w}}_{i})}
1873:is a random variable describing additive noise.
1010:is a random variable describing additive noise.
7072:
4181:{\displaystyle K_{\gamma _{l}}(\cdot ,\cdot )}
7342:
7002:
3997:{\displaystyle s_{i}=(s_{i1},s_{i2})^{\top }}
2558:on the second stage of the model as follows:
2006:that describe a mapping of observed age to a
1704:are dummy variables that are 1 if individual
415:
7294:Jonsson, EN; Karlsson, MO; Wade, JR (2000).
7235:Cole TJ, Donaldson MD, Ben-Shlomo Y (2010).
7118:Computational Statistics & Data Analysis
7115:
6831:
6817:
6776:
6759:
6736:
6722:
6598:
6584:
6543:
6526:
6503:
6489:
6446:
6429:
6269:
6252:
6202:
6185:
6131:
6117:
6076:
6059:
6036:
6022:
5968:
5951:
5903:
5886:
5811:
5794:
5769:
5755:
5714:
5697:
5674:
5660:
5606:
5589:
4391:on the date level (the first level). As the
3210:
3183:
7141:
7139:
1915:
1166:{\displaystyle (y_{i1},\ldots ,y_{in_{i}})}
825:is modeled as a linear mixed-effects model
7403:
2516:Example: COVID-19 epidemiological modeling
1841:relative to his/her baseline category, and
422:
408:
7501:
7491:
7458:
7448:
7380:
7370:
7360:
7319:
7262:
7252:
7173:
7163:
5412:is a known function parameterized by the
4861:
4634:
4323:that dictate the shape of the mean curve
4210:represents the horseshoe shrinkage prior.
3502:
2702:
2182:
1486:
597:
7136:
4403:
2536:
2519:
2490:
2326:corresponding to the height measurement
1919:
1080:
5231:denotes the continuous response of the
2411:
2379:
2122:
2034:
1986:
938:
886:
871:
850:
7516:
7035:
4400:Bayesian nonlinear mixed-effects model
691:is the number of observations for the
7241:International Journal of Epidemiology
7145:
4055:{\displaystyle \epsilon _{l}(\cdot )}
2424:{\displaystyle {\boldsymbol {w}}_{i}}
1999:{\displaystyle {\boldsymbol {w}}_{i}}
1061:Example: Disease progression modeling
951:{\displaystyle {\boldsymbol {b}}_{i}}
7005:Mixed-effects models in S and S-PLUS
4222:predictors for the curve parameters
7473:
7430:
1877:An example of such a model with an
1558:{\displaystyle f_{\tilde {\beta }}}
13:
3989:
3887:
1219:(DEM) at the baseline visit (time
1050:expectation-maximization algorithm
14:
7535:
6963:Mixed-design analysis of variance
4125:{\displaystyle \eta _{l}(\cdot )}
2010:biological age using a so-called
1587:{\displaystyle {\tilde {\beta }}}
1035:may differ from the conventional
929:is a vector of fixed effects and
662:is the number of groups/subjects,
4094:(also called the nugget effect),
389:
7467:
7424:
7397:
7039:Ecological models and data in R
7003:Pinheiro, J; Bates, DM (2006).
5164:
5118:
4982:
4946:
4836:
4786:
4609:
4564:
4418:Stage 1: Individual-Level Model
4087:{\displaystyle \sigma _{l}^{2}}
3651:
3650:
3649:
3648:
3647:
3646:
3645:
3644:
3643:
3642:
3641:
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3471:
3394:
3227:
3226:
3225:
3089:
2973:
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2916:
2677:
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2675:
2674:
2673:
2672:
2671:
2670:
2669:
2668:
2667:
2666:
2665:
2505:exposure-response relationships
2157:
1461:
1055:
1045:maximum a posteriori estimation
572:
337:Least-squares spectral analysis
275:Generalized estimating equation
95:Multinomial logistic regression
70:Vector generalized linear model
7336:
7287:
7228:
7109:
7066:
7029:
6953:Generalized linear mixed model
6851:
6706:
6618:
6485:
6426:
6320:
6248:
6182:
6151:
5883:
5856:
5790:
5586:
5527:{\displaystyle \epsilon _{ij}}
5471:
5439:
5399:
5361:
5251:-th subject at the time point
5158:
5140:
5112:
5052:
5043:
4983:
4976:
4963:
4940:
4927:
4830:
4806:
4603:
4584:
4542:
4450:
4378:
4333:
4307:
4283:
4277:
4229:
4175:
4163:
4119:
4113:
4049:
4043:
3985:
3952:
3883:
3844:
3805:
3766:
3746:
3701:
3612:
3599:
3565:
3559:
3465:
3453:
3388:
3331:
3282:
3219:
3160:
3133:
3107:
3083:
3080:
3068:
3042:
3030:
3024:
2966:
2948:
2933:
2927:
2910:
2897:
2881:
2868:
2792:
2779:
2643:
2589:
2456:{\displaystyle \epsilon _{ij}}
2389:
2368:
2135:
2132:
2101:
2095:
2044:
2023:
1957:Example: Modeling human height
1866:{\displaystyle \epsilon _{ij}}
1578:
1548:
1439:
1330:
1323:
1160:
1115:
1003:{\displaystyle \epsilon _{ij}}
550:
516:
435:Nonlinear mixed-effects models
1:
7042:. Princeton University Press.
6986:
1949:finer details due to lack of
1255:corresponding to measurement
1204:{\displaystyle i=1,\ldots ,M}
1041:maximum-likelihood estimation
1021:maximum-likelihood estimation
1014:
468:
156:Nonlinear mixed-effects model
7372:10.1371/journal.pone.0236860
7222:10.1016/j.patrec.2013.10.018
4216:Gaussian process regressions
1785:{\displaystyle \beta ^{DEM}}
1752:{\displaystyle \beta ^{MCI}}
7:
7210:Pattern Recognition Letters
6936:
1697:{\displaystyle A_{i}^{DEM}}
1659:{\displaystyle A_{i}^{MCI}}
443:linear mixed-effects models
358:Mean and predicted response
10:
7540:
7418:10.1007/s13571-020-00245-8
7130:10.1016/j.csda.2004.07.002
5557:{\displaystyle \eta _{li}}
2479:model can fit models with
2247:{\displaystyle f_{\beta }}
818:{\displaystyle \phi _{ij}}
756:{\displaystyle \phi _{ij}}
151:Linear mixed-effects model
4680:Stage 2: Population Model
1213:mild cognitive impairment
317:Least absolute deviations
7165:10.3389/fdata.2020.00024
6978:Repeated measures design
1916:Example: Growth analysis
1248:{\displaystyle t_{i1}=0}
65:Generalized linear model
1975:cross-sectional studies
1967:age at onset of puberty
1025:nonlinear least squares
763:and a covariate vector
6927:
6903:
6158:
5863:
5558:
5528:
5498:
5478:
5426:
5406:
5345:
5325:
5305:
5304:{\displaystyle x_{ib}}
5275:
5274:{\displaystyle t_{ij}}
5245:
5225:
5224:{\displaystyle y_{ij}}
5193:
4890:
4740:
4670:
4409:
4385:
4317:
4204:
4203:{\displaystyle \beta }
4182:
4126:
4088:
4056:
4018:
3998:
3924:
3896:
3820:decline curve analysis
3812:
3753:
3680:
3531:
3256:
3002:
2831:
2738:
2543:
2542:kriging method (right)
2525:
2497:
2473:affine transformations
2457:
2425:
2396:
2350:
2349:{\displaystyle y_{ij}}
2320:
2300:
2299:{\displaystyle t_{ij}}
2268:
2267:{\displaystyle \beta }
2248:
2215:
2051:
2000:
1929:
1906:
1867:
1835:
1815:
1786:
1753:
1718:
1698:
1660:
1620:
1619:{\displaystyle t_{ij}}
1588:
1559:
1519:
1279:
1278:{\displaystyle y_{i1}}
1249:
1205:
1167:
1090:
1004:
972:
952:
923:
922:{\displaystyle \beta }
903:
819:
787:
786:{\displaystyle v_{ij}}
757:
727:
705:
685:
656:
630:
437:constitute a class of
396:Mathematics portal
322:Iteratively reweighted
7474:Lee, Se Yoon (2022).
7431:Lee, Se Yoon (2022).
7152:Frontiers in Big Data
6928:
6904:
6159:
5864:
5559:
5529:
5499:
5479:
5427:
5407:
5346:
5331:-th covariate of the
5326:
5306:
5276:
5246:
5226:
5194:
4891:
4720:
4671:
4407:
4386:
4318:
4205:
4183:
4127:
4089:
4057:
4019:
3999:
3925:
3897:
3813:
3754:
3681:
3532:
3257:
3003:
2811:
2739:
2540:
2523:
2494:
2458:
2426:
2397:
2351:
2321:
2301:
2269:
2249:
2216:
2052:
2001:
1923:
1907:
1905:{\displaystyle b_{i}}
1868:
1836:
1816:
1814:{\displaystyle b_{i}}
1787:
1754:
1719:
1699:
1661:
1621:
1589:
1560:
1520:
1280:
1250:
1206:
1168:
1084:
1029:asymptotic properties
1005:
973:
953:
924:
904:
820:
788:
758:
728:
706:
686:
684:{\displaystyle n_{i}}
657:
631:
353:Regression validation
332:Bayesian multivariate
49:Polynomial regression
7503:10.3390/math10060898
7460:10.3390/math10060898
6973:Random effects model
6917:
6169:
5874:
5580:
5538:
5508:
5488:
5436:
5432:-dimensional vector
5416:
5355:
5335:
5315:
5285:
5255:
5235:
5205:
4908:
4688:
4426:
4327:
4226:
4194:
4143:
4100:
4066:
4030:
4008:
3936:
3914:
3908:directional drilling
3904:hydraulic fracturing
3828:
3763:
3695:
3540:
3265:
3011:
2750:
2565:
2437:
2406:
2362:
2330:
2310:
2306:is the age of child
2280:
2258:
2231:
2064:
2017:
1981:
1889:
1847:
1825:
1798:
1763:
1730:
1708:
1670:
1632:
1600:
1569:
1535:
1292:
1259:
1223:
1177:
1112:
1037:general linear model
984:
962:
933:
913:
829:
799:
767:
737:
717:
695:
668:
646:
492:
378:Gauss–Markov theorem
373:Studentized residual
363:Errors and residuals
197:Principal components
167:Nonlinear regression
54:General linear model
7036:Bolker, BM (2008).
6948:Fixed effects model
6850:
6813:
6755:
6617:
6580:
6522:
6483:
6306:
6246:
6150:
6113:
6055:
6005:
5947:
5855:
5788:
5751:
5693:
5643:
5157:
5133:
4829:
4138:covariance function
4083:
3592:
3387:
3369:
3354:
3153:
2965:
1969:and its associated
1946:nutrient deficiency
1944:). Factors such as
1693:
1655:
1409:
1369:
1094:Alzheimer's disease
1067:progressive disease
223:Errors-in-variables
90:Logistic regression
80:Binomial regression
25:Regression analysis
19:Part of a series on
7254:10.1093/ije/dyq115
7146:Raket, LL (2020).
6923:
6899:
6898:
6858:
6830:
6775:
6735:
6695:
6625:
6597:
6542:
6502:
6445:
6415:
6327:
6268:
6201:
6154:
6130:
6075:
6035:
5967:
5902:
5859:
5810:
5768:
5713:
5673:
5605:
5554:
5524:
5494:
5474:
5422:
5402:
5341:
5321:
5301:
5271:
5241:
5221:
5189:
5143:
5119:
4886:
4815:
4666:
4410:
4381:
4313:
4200:
4178:
4122:
4084:
4069:
4052:
4014:
3994:
3920:
3892:
3808:
3749:
3676:
3578:
3527:
3370:
3355:
3340:
3252:
3139:
2998:
2951:
2734:
2548:unconventional oil
2544:
2526:
2498:
2453:
2421:
2392:
2346:
2316:
2296:
2264:
2244:
2211:
2047:
1996:
1938:exponential growth
1930:
1902:
1863:
1831:
1811:
1782:
1749:
1714:
1694:
1673:
1656:
1635:
1616:
1584:
1555:
1515:
1389:
1349:
1275:
1245:
1201:
1163:
1091:
1031:of estimators and
1027:methods, although
1023:can be done using
1000:
968:
948:
919:
899:
815:
783:
753:
723:
701:
681:
652:
626:
439:statistical models
110:Multinomial probit
7524:Regression models
6958:Linear regression
6926:{\displaystyle f}
6701:
6699:
6421:
6419:
6177:
6175:
5497:{\displaystyle f}
5425:{\displaystyle K}
5344:{\displaystyle i}
5324:{\displaystyle b}
5244:{\displaystyle i}
4017:{\displaystyle i}
3923:{\displaystyle i}
2319:{\displaystyle i}
1942:hyperbolic growth
1834:{\displaystyle i}
1717:{\displaystyle i}
1581:
1551:
1326:
1106:stages of disease
1102:cognitive testing
971:{\displaystyle i}
726:{\displaystyle f}
711:th group/subject,
704:{\displaystyle i}
655:{\displaystyle M}
475:statistical model
447:statistical units
432:
431:
85:Binary regression
44:Simple regression
39:Linear regression
7531:
7508:
7507:
7505:
7495:
7471:
7465:
7464:
7462:
7452:
7428:
7422:
7421:
7401:
7395:
7394:
7384:
7374:
7364:
7340:
7334:
7333:
7323:
7312:10.1208/ps020332
7291:
7285:
7284:
7266:
7256:
7232:
7226:
7225:
7205:
7196:
7195:
7177:
7167:
7143:
7134:
7133:
7124:(4): 1020–1038.
7113:
7107:
7106:
7070:
7064:
7063:
7057:
7053:
7051:
7043:
7033:
7027:
7026:
7000:
6968:Multilevel model
6932:
6930:
6929:
6924:
6908:
6906:
6905:
6900:
6897:
6859:
6854:
6849:
6844:
6829:
6828:
6812:
6801:
6774:
6773:
6754:
6749:
6734:
6733:
6718:
6717:
6694:
6626:
6621:
6616:
6611:
6596:
6595:
6579:
6568:
6541:
6540:
6521:
6516:
6501:
6500:
6488:
6482:
6471:
6444:
6443:
6414:
6328:
6323:
6319:
6318:
6305:
6294:
6267:
6266:
6251:
6245:
6244:
6243:
6227:
6200:
6199:
6163:
6161:
6160:
6155:
6149:
6144:
6129:
6128:
6112:
6101:
6074:
6073:
6054:
6049:
6034:
6033:
6018:
6017:
6004:
5993:
5966:
5965:
5946:
5945:
5944:
5928:
5901:
5900:
5868:
5866:
5865:
5860:
5854:
5853:
5852:
5836:
5809:
5808:
5793:
5787:
5782:
5767:
5766:
5750:
5739:
5712:
5711:
5692:
5687:
5672:
5671:
5656:
5655:
5642:
5631:
5604:
5603:
5563:
5561:
5560:
5555:
5553:
5552:
5533:
5531:
5530:
5525:
5523:
5522:
5503:
5501:
5500:
5495:
5483:
5481:
5480:
5475:
5470:
5469:
5451:
5450:
5431:
5429:
5428:
5423:
5411:
5409:
5408:
5403:
5398:
5397:
5379:
5378:
5350:
5348:
5347:
5342:
5330:
5328:
5327:
5322:
5310:
5308:
5307:
5302:
5300:
5299:
5280:
5278:
5277:
5272:
5270:
5269:
5250:
5248:
5247:
5242:
5230:
5228:
5227:
5222:
5220:
5219:
5198:
5196:
5195:
5190:
5156:
5151:
5132:
5127:
5111:
5110:
5089:
5088:
5067:
5066:
5042:
5041:
5020:
5019:
4998:
4997:
4975:
4974:
4956:
4955:
4939:
4938:
4920:
4919:
4895:
4893:
4892:
4887:
4828:
4823:
4799:
4798:
4782:
4781:
4766:
4765:
4753:
4752:
4739:
4734:
4716:
4715:
4703:
4702:
4675:
4673:
4672:
4667:
4662:
4661:
4602:
4601:
4577:
4576:
4560:
4559:
4541:
4540:
4519:
4518:
4497:
4496:
4481:
4480:
4465:
4464:
4443:
4442:
4434:
4390:
4388:
4387:
4382:
4377:
4376:
4364:
4363:
4351:
4350:
4322:
4320:
4319:
4314:
4276:
4275:
4260:
4259:
4244:
4243:
4209:
4207:
4206:
4201:
4187:
4185:
4184:
4179:
4162:
4161:
4160:
4159:
4134:Gaussian process
4131:
4129:
4128:
4123:
4112:
4111:
4093:
4091:
4090:
4085:
4082:
4077:
4061:
4059:
4058:
4053:
4042:
4041:
4023:
4021:
4020:
4015:
4003:
4001:
4000:
3995:
3993:
3992:
3983:
3982:
3967:
3966:
3948:
3947:
3929:
3927:
3926:
3921:
3901:
3899:
3898:
3893:
3891:
3890:
3881:
3880:
3859:
3858:
3840:
3839:
3817:
3815:
3814:
3809:
3804:
3803:
3791:
3790:
3778:
3777:
3758:
3756:
3755:
3750:
3745:
3744:
3732:
3731:
3719:
3718:
3685:
3683:
3682:
3677:
3635:
3634:
3625:
3611:
3610:
3591:
3586:
3552:
3551:
3536:
3534:
3533:
3528:
3452:
3451:
3439:
3438:
3426:
3425:
3413:
3412:
3386:
3381:
3368:
3363:
3353:
3348:
3324:
3323:
3311:
3310:
3298:
3297:
3285:
3280:
3279:
3261:
3259:
3258:
3253:
3218:
3217:
3208:
3207:
3195:
3194:
3182:
3181:
3180:
3179:
3152:
3147:
3132:
3131:
3119:
3118:
3106:
3105:
3104:
3103:
3067:
3066:
3065:
3064:
3023:
3022:
3007:
3005:
3004:
2999:
2964:
2959:
2926:
2925:
2909:
2908:
2896:
2895:
2880:
2879:
2867:
2866:
2854:
2853:
2844:
2843:
2830:
2825:
2807:
2806:
2791:
2790:
2778:
2777:
2765:
2764:
2743:
2741:
2740:
2735:
2730:
2729:
2661:
2660:
2642:
2641:
2626:
2625:
2610:
2609:
2582:
2581:
2573:
2556:Gaussian process
2481:smoothly-varying
2462:
2460:
2459:
2454:
2452:
2451:
2430:
2428:
2427:
2422:
2420:
2419:
2414:
2401:
2399:
2398:
2393:
2388:
2387:
2382:
2355:
2353:
2352:
2347:
2345:
2344:
2325:
2323:
2322:
2317:
2305:
2303:
2302:
2297:
2295:
2294:
2273:
2271:
2270:
2265:
2253:
2251:
2250:
2245:
2243:
2242:
2220:
2218:
2217:
2212:
2210:
2209:
2153:
2152:
2131:
2130:
2125:
2116:
2115:
2094:
2093:
2081:
2080:
2072:
2056:
2054:
2053:
2048:
2043:
2042:
2037:
2012:warping function
2005:
2003:
2002:
1997:
1995:
1994:
1989:
1911:
1909:
1908:
1903:
1901:
1900:
1872:
1870:
1869:
1864:
1862:
1861:
1840:
1838:
1837:
1832:
1820:
1818:
1817:
1812:
1810:
1809:
1791:
1789:
1788:
1783:
1781:
1780:
1758:
1756:
1755:
1750:
1748:
1747:
1723:
1721:
1720:
1715:
1703:
1701:
1700:
1695:
1692:
1681:
1665:
1663:
1662:
1657:
1654:
1643:
1625:
1623:
1622:
1617:
1615:
1614:
1593:
1591:
1590:
1585:
1583:
1582:
1574:
1564:
1562:
1561:
1556:
1554:
1553:
1552:
1544:
1524:
1522:
1521:
1516:
1514:
1513:
1457:
1456:
1438:
1437:
1425:
1424:
1408:
1397:
1385:
1384:
1368:
1357:
1345:
1344:
1329:
1328:
1327:
1319:
1309:
1308:
1300:
1284:
1282:
1281:
1276:
1274:
1273:
1254:
1252:
1251:
1246:
1238:
1237:
1210:
1208:
1207:
1202:
1172:
1170:
1169:
1164:
1159:
1158:
1157:
1156:
1130:
1129:
1009:
1007:
1006:
1001:
999:
998:
977:
975:
974:
969:
957:
955:
954:
949:
947:
946:
941:
928:
926:
925:
920:
908:
906:
905:
900:
895:
894:
889:
883:
882:
874:
862:
861:
853:
844:
843:
824:
822:
821:
816:
814:
813:
792:
790:
789:
784:
782:
781:
762:
760:
759:
754:
752:
751:
732:
730:
729:
724:
710:
708:
707:
702:
690:
688:
687:
682:
680:
679:
661:
659:
658:
653:
635:
633:
632:
627:
625:
624:
568:
567:
549:
548:
540:
531:
530:
509:
508:
500:
477:containing both
424:
417:
410:
394:
393:
301:Ridge regression
136:Multilevel model
16:
15:
7539:
7538:
7534:
7533:
7532:
7530:
7529:
7528:
7514:
7513:
7512:
7511:
7472:
7468:
7429:
7425:
7402:
7398:
7355:(7): e0236860.
7341:
7337:
7292:
7288:
7233:
7229:
7206:
7199:
7144:
7137:
7114:
7110:
7087:10.2307/2532087
7071:
7067:
7055:
7054:
7045:
7044:
7034:
7030:
7023:
7001:
6994:
6989:
6983:
6939:
6918:
6915:
6914:
6911:
6860:
6845:
6834:
6824:
6820:
6802:
6779:
6766:
6762:
6750:
6739:
6729:
6725:
6713:
6709:
6702:
6700:
6627:
6612:
6601:
6591:
6587:
6569:
6546:
6533:
6529:
6517:
6506:
6496:
6492:
6484:
6472:
6449:
6436:
6432:
6422:
6420:
6329:
6314:
6310:
6295:
6272:
6259:
6255:
6247:
6239:
6235:
6228:
6205:
6192:
6188:
6178:
6176:
6170:
6167:
6166:
6145:
6134:
6124:
6120:
6102:
6079:
6066:
6062:
6050:
6039:
6029:
6025:
6013:
6009:
5994:
5971:
5958:
5954:
5940:
5936:
5929:
5906:
5893:
5889:
5875:
5872:
5871:
5848:
5844:
5837:
5814:
5801:
5797:
5789:
5783:
5772:
5762:
5758:
5740:
5717:
5704:
5700:
5688:
5677:
5667:
5663:
5651:
5647:
5632:
5609:
5596:
5592:
5581:
5578:
5577:
5573:
5545:
5541:
5539:
5536:
5535:
5515:
5511:
5509:
5506:
5505:
5489:
5486:
5485:
5465:
5461:
5446:
5442:
5437:
5434:
5433:
5417:
5414:
5413:
5393:
5389:
5374:
5370:
5356:
5353:
5352:
5336:
5333:
5332:
5316:
5313:
5312:
5292:
5288:
5286:
5283:
5282:
5262:
5258:
5256:
5253:
5252:
5236:
5233:
5232:
5212:
5208:
5206:
5203:
5202:
5152:
5147:
5128:
5123:
5103:
5099:
5081:
5077:
5059:
5055:
5034:
5030:
5012:
5008:
4990:
4986:
4970:
4966:
4951:
4947:
4934:
4930:
4915:
4911:
4909:
4906:
4905:
4824:
4819:
4791:
4787:
4774:
4770:
4758:
4754:
4745:
4741:
4735:
4724:
4711:
4707:
4695:
4691:
4689:
4686:
4685:
4657:
4653:
4597:
4593:
4569:
4565:
4552:
4548:
4533:
4529:
4511:
4507:
4489:
4485:
4473:
4469:
4457:
4453:
4435:
4430:
4429:
4427:
4424:
4423:
4402:
4372:
4368:
4359:
4355:
4346:
4342:
4328:
4325:
4324:
4268:
4264:
4252:
4248:
4236:
4232:
4227:
4224:
4223:
4195:
4192:
4191:
4155:
4151:
4150:
4146:
4144:
4141:
4140:
4132:represents the
4107:
4103:
4101:
4098:
4097:
4078:
4073:
4067:
4064:
4063:
4037:
4033:
4031:
4028:
4027:
4009:
4006:
4005:
3988:
3984:
3975:
3971:
3959:
3955:
3943:
3939:
3937:
3934:
3933:
3915:
3912:
3911:
3906:and horizontal
3886:
3882:
3873:
3869:
3851:
3847:
3835:
3831:
3829:
3826:
3825:
3799:
3795:
3786:
3782:
3773:
3769:
3764:
3761:
3760:
3740:
3736:
3727:
3723:
3714:
3710:
3696:
3693:
3692:
3630:
3626:
3621:
3606:
3602:
3587:
3582:
3547:
3543:
3541:
3538:
3537:
3447:
3443:
3434:
3430:
3421:
3417:
3405:
3401:
3382:
3374:
3364:
3359:
3349:
3344:
3319:
3315:
3306:
3302:
3290:
3286:
3281:
3272:
3268:
3266:
3263:
3262:
3213:
3209:
3203:
3199:
3190:
3186:
3175:
3171:
3170:
3166:
3148:
3143:
3127:
3123:
3114:
3110:
3099:
3095:
3094:
3090:
3060:
3056:
3055:
3051:
3018:
3014:
3012:
3009:
3008:
2960:
2955:
2921:
2917:
2904:
2900:
2891:
2887:
2875:
2871:
2862:
2858:
2849:
2845:
2836:
2832:
2826:
2815:
2802:
2798:
2786:
2782:
2773:
2769:
2757:
2753:
2751:
2748:
2747:
2725:
2721:
2653:
2649:
2634:
2630:
2618:
2614:
2602:
2598:
2574:
2569:
2568:
2566:
2563:
2562:
2535:
2518:
2503:for describing
2489:
2444:
2440:
2438:
2435:
2434:
2415:
2410:
2409:
2407:
2404:
2403:
2383:
2378:
2377:
2363:
2360:
2359:
2337:
2333:
2331:
2328:
2327:
2311:
2308:
2307:
2287:
2283:
2281:
2278:
2277:
2259:
2256:
2255:
2238:
2234:
2232:
2229:
2228:
2205:
2201:
2145:
2141:
2126:
2121:
2120:
2108:
2104:
2089:
2085:
2073:
2068:
2067:
2065:
2062:
2061:
2038:
2033:
2032:
2018:
2015:
2014:
1990:
1985:
1984:
1982:
1979:
1978:
1959:
1951:synchronization
1934:logistic growth
1918:
1896:
1892:
1890:
1887:
1886:
1854:
1850:
1848:
1845:
1844:
1826:
1823:
1822:
1805:
1801:
1799:
1796:
1795:
1770:
1766:
1764:
1761:
1760:
1737:
1733:
1731:
1728:
1727:
1709:
1706:
1705:
1682:
1677:
1671:
1668:
1667:
1644:
1639:
1633:
1630:
1629:
1607:
1603:
1601:
1598:
1597:
1573:
1572:
1570:
1567:
1566:
1543:
1542:
1538:
1536:
1533:
1532:
1509:
1505:
1449:
1445:
1433:
1429:
1414:
1410:
1398:
1393:
1374:
1370:
1358:
1353:
1337:
1333:
1318:
1317:
1313:
1301:
1296:
1295:
1293:
1290:
1289:
1266:
1262:
1260:
1257:
1256:
1230:
1226:
1224:
1221:
1220:
1178:
1175:
1174:
1152:
1148:
1144:
1140:
1122:
1118:
1113:
1110:
1109:
1079:
1063:
1058:
1033:test statistics
1017:
991:
987:
985:
982:
981:
963:
960:
959:
942:
937:
936:
934:
931:
930:
914:
911:
910:
890:
885:
884:
875:
870:
869:
854:
849:
848:
836:
832:
830:
827:
826:
806:
802:
800:
797:
796:
774:
770:
768:
765:
764:
744:
740:
738:
735:
734:
718:
715:
714:
696:
693:
692:
675:
671:
669:
666:
665:
647:
644:
643:
620:
616:
560:
556:
541:
536:
535:
523:
519:
501:
496:
495:
493:
490:
489:
471:
428:
388:
368:Goodness of fit
75:Discrete choice
12:
11:
5:
7537:
7527:
7526:
7510:
7509:
7466:
7423:
7396:
7335:
7286:
7247:(6): 1558–66.
7227:
7197:
7135:
7108:
7081:(3): 673–687.
7065:
7056:|website=
7028:
7021:
7013:10.1007/b98882
6991:
6990:
6988:
6985:
6981:
6980:
6975:
6970:
6965:
6960:
6955:
6950:
6945:
6938:
6935:
6922:
6896:
6893:
6890:
6887:
6884:
6881:
6878:
6875:
6872:
6869:
6866:
6863:
6857:
6853:
6848:
6843:
6840:
6837:
6833:
6827:
6823:
6819:
6816:
6811:
6808:
6805:
6800:
6797:
6794:
6791:
6788:
6785:
6782:
6778:
6772:
6769:
6765:
6761:
6758:
6753:
6748:
6745:
6742:
6738:
6732:
6728:
6724:
6721:
6716:
6712:
6708:
6705:
6698:
6693:
6690:
6687:
6684:
6681:
6678:
6675:
6672:
6669:
6666:
6663:
6660:
6657:
6654:
6651:
6648:
6645:
6642:
6639:
6636:
6633:
6630:
6624:
6620:
6615:
6610:
6607:
6604:
6600:
6594:
6590:
6586:
6583:
6578:
6575:
6572:
6567:
6564:
6561:
6558:
6555:
6552:
6549:
6545:
6539:
6536:
6532:
6528:
6525:
6520:
6515:
6512:
6509:
6505:
6499:
6495:
6491:
6487:
6481:
6478:
6475:
6470:
6467:
6464:
6461:
6458:
6455:
6452:
6448:
6442:
6439:
6435:
6431:
6428:
6425:
6418:
6413:
6410:
6407:
6404:
6401:
6398:
6395:
6392:
6389:
6386:
6383:
6380:
6377:
6374:
6371:
6368:
6365:
6362:
6359:
6356:
6353:
6350:
6347:
6344:
6341:
6338:
6335:
6332:
6326:
6322:
6317:
6313:
6309:
6304:
6301:
6298:
6293:
6290:
6287:
6284:
6281:
6278:
6275:
6271:
6265:
6262:
6258:
6254:
6250:
6242:
6238:
6234:
6231:
6226:
6223:
6220:
6217:
6214:
6211:
6208:
6204:
6198:
6195:
6191:
6187:
6184:
6181:
6174:
6153:
6148:
6143:
6140:
6137:
6133:
6127:
6123:
6119:
6116:
6111:
6108:
6105:
6100:
6097:
6094:
6091:
6088:
6085:
6082:
6078:
6072:
6069:
6065:
6061:
6058:
6053:
6048:
6045:
6042:
6038:
6032:
6028:
6024:
6021:
6016:
6012:
6008:
6003:
6000:
5997:
5992:
5989:
5986:
5983:
5980:
5977:
5974:
5970:
5964:
5961:
5957:
5953:
5950:
5943:
5939:
5935:
5932:
5927:
5924:
5921:
5918:
5915:
5912:
5909:
5905:
5899:
5896:
5892:
5888:
5885:
5882:
5879:
5858:
5851:
5847:
5843:
5840:
5835:
5832:
5829:
5826:
5823:
5820:
5817:
5813:
5807:
5804:
5800:
5796:
5792:
5786:
5781:
5778:
5775:
5771:
5765:
5761:
5757:
5754:
5749:
5746:
5743:
5738:
5735:
5732:
5729:
5726:
5723:
5720:
5716:
5710:
5707:
5703:
5699:
5696:
5691:
5686:
5683:
5680:
5676:
5670:
5666:
5662:
5659:
5654:
5650:
5646:
5641:
5638:
5635:
5630:
5627:
5624:
5621:
5618:
5615:
5612:
5608:
5602:
5599:
5595:
5591:
5588:
5585:
5567:Stage 3: Prior
5551:
5548:
5544:
5521:
5518:
5514:
5493:
5473:
5468:
5464:
5460:
5457:
5454:
5449:
5445:
5441:
5421:
5401:
5396:
5392:
5388:
5385:
5382:
5377:
5373:
5369:
5366:
5363:
5360:
5340:
5320:
5298:
5295:
5291:
5268:
5265:
5261:
5240:
5218:
5215:
5211:
5188:
5185:
5182:
5179:
5176:
5173:
5170:
5167:
5163:
5160:
5155:
5150:
5146:
5142:
5139:
5136:
5131:
5126:
5122:
5117:
5114:
5109:
5106:
5102:
5098:
5095:
5092:
5087:
5084:
5080:
5076:
5073:
5070:
5065:
5062:
5058:
5054:
5051:
5048:
5045:
5040:
5037:
5033:
5029:
5026:
5023:
5018:
5015:
5011:
5007:
5004:
5001:
4996:
4993:
4989:
4985:
4981:
4978:
4973:
4969:
4965:
4962:
4959:
4954:
4950:
4945:
4942:
4937:
4933:
4929:
4926:
4923:
4918:
4914:
4900:Stage 3: Prior
4885:
4882:
4879:
4876:
4873:
4870:
4867:
4864:
4860:
4857:
4854:
4851:
4848:
4845:
4842:
4839:
4835:
4832:
4827:
4822:
4818:
4814:
4811:
4808:
4805:
4802:
4797:
4794:
4790:
4785:
4780:
4777:
4773:
4769:
4764:
4761:
4757:
4751:
4748:
4744:
4738:
4733:
4730:
4727:
4723:
4719:
4714:
4710:
4706:
4701:
4698:
4694:
4665:
4660:
4656:
4652:
4649:
4646:
4643:
4640:
4637:
4633:
4630:
4627:
4624:
4621:
4618:
4615:
4612:
4608:
4605:
4600:
4596:
4592:
4589:
4586:
4583:
4580:
4575:
4572:
4568:
4563:
4558:
4555:
4551:
4547:
4544:
4539:
4536:
4532:
4528:
4525:
4522:
4517:
4514:
4510:
4506:
4503:
4500:
4495:
4492:
4488:
4484:
4479:
4476:
4472:
4468:
4463:
4460:
4456:
4452:
4449:
4446:
4441:
4438:
4433:
4401:
4398:
4380:
4375:
4371:
4367:
4362:
4358:
4354:
4349:
4345:
4341:
4338:
4335:
4332:
4312:
4309:
4306:
4303:
4300:
4297:
4294:
4291:
4288:
4285:
4282:
4279:
4274:
4271:
4267:
4263:
4258:
4255:
4251:
4247:
4242:
4239:
4235:
4231:
4212:
4211:
4199:
4189:
4177:
4174:
4171:
4168:
4165:
4158:
4154:
4149:
4136:with Gaussian
4121:
4118:
4115:
4110:
4106:
4095:
4081:
4076:
4072:
4051:
4048:
4045:
4040:
4036:
4025:
4013:
3991:
3987:
3981:
3978:
3974:
3970:
3965:
3962:
3958:
3954:
3951:
3946:
3942:
3931:
3919:
3889:
3885:
3879:
3876:
3872:
3868:
3865:
3862:
3857:
3854:
3850:
3846:
3843:
3838:
3834:
3823:
3807:
3802:
3798:
3794:
3789:
3785:
3781:
3776:
3772:
3768:
3748:
3743:
3739:
3735:
3730:
3726:
3722:
3717:
3713:
3709:
3706:
3703:
3700:
3675:
3672:
3669:
3666:
3663:
3660:
3657:
3654:
3638:
3633:
3629:
3624:
3620:
3617:
3614:
3609:
3605:
3601:
3598:
3595:
3590:
3585:
3581:
3576:
3573:
3570:
3567:
3564:
3561:
3558:
3555:
3550:
3546:
3526:
3523:
3520:
3517:
3514:
3511:
3508:
3505:
3501:
3498:
3495:
3492:
3489:
3486:
3483:
3480:
3470:
3467:
3464:
3461:
3458:
3455:
3450:
3446:
3442:
3437:
3433:
3429:
3424:
3420:
3416:
3411:
3408:
3404:
3400:
3397:
3393:
3390:
3385:
3380:
3377:
3373:
3367:
3362:
3358:
3352:
3347:
3343:
3339:
3336:
3333:
3330:
3327:
3322:
3318:
3314:
3309:
3305:
3301:
3296:
3293:
3289:
3284:
3278:
3275:
3271:
3251:
3248:
3245:
3242:
3239:
3236:
3233:
3230:
3224:
3221:
3216:
3212:
3206:
3202:
3198:
3193:
3189:
3185:
3178:
3174:
3169:
3165:
3162:
3159:
3156:
3151:
3146:
3142:
3138:
3135:
3130:
3126:
3122:
3117:
3113:
3109:
3102:
3098:
3093:
3088:
3085:
3082:
3079:
3076:
3073:
3070:
3063:
3059:
3054:
3050:
3047:
3044:
3041:
3038:
3035:
3032:
3029:
3026:
3021:
3017:
2997:
2994:
2991:
2988:
2985:
2982:
2979:
2976:
2971:
2968:
2963:
2958:
2954:
2950:
2947:
2944:
2941:
2938:
2935:
2932:
2929:
2924:
2920:
2915:
2912:
2907:
2903:
2899:
2894:
2890:
2886:
2883:
2878:
2874:
2870:
2865:
2861:
2857:
2852:
2848:
2842:
2839:
2835:
2829:
2824:
2821:
2818:
2814:
2810:
2805:
2801:
2797:
2794:
2789:
2785:
2781:
2776:
2772:
2768:
2763:
2760:
2756:
2745:
2744:
2733:
2728:
2724:
2720:
2717:
2714:
2711:
2708:
2705:
2701:
2698:
2695:
2692:
2689:
2686:
2683:
2680:
2664:
2659:
2656:
2652:
2648:
2645:
2640:
2637:
2633:
2629:
2624:
2621:
2617:
2613:
2608:
2605:
2601:
2597:
2594:
2591:
2588:
2585:
2580:
2577:
2572:
2534:
2531:
2517:
2514:
2488:
2485:
2465:
2464:
2450:
2447:
2443:
2432:
2418:
2413:
2391:
2386:
2381:
2376:
2373:
2370:
2367:
2357:
2343:
2340:
2336:
2315:
2293:
2290:
2286:
2275:
2263:
2241:
2237:
2222:
2221:
2208:
2204:
2200:
2197:
2194:
2191:
2188:
2185:
2181:
2178:
2175:
2172:
2169:
2166:
2163:
2160:
2156:
2151:
2148:
2144:
2140:
2137:
2134:
2129:
2124:
2119:
2114:
2111:
2107:
2103:
2100:
2097:
2092:
2088:
2084:
2079:
2076:
2071:
2046:
2041:
2036:
2031:
2028:
2025:
2022:
1993:
1988:
1958:
1955:
1917:
1914:
1899:
1895:
1875:
1874:
1860:
1857:
1853:
1842:
1830:
1808:
1804:
1793:
1779:
1776:
1773:
1769:
1746:
1743:
1740:
1736:
1725:
1713:
1691:
1688:
1685:
1680:
1676:
1653:
1650:
1647:
1642:
1638:
1627:
1613:
1610:
1606:
1595:
1580:
1577:
1550:
1547:
1541:
1526:
1525:
1512:
1508:
1504:
1501:
1498:
1495:
1492:
1489:
1485:
1482:
1479:
1476:
1473:
1470:
1467:
1464:
1460:
1455:
1452:
1448:
1444:
1441:
1436:
1432:
1428:
1423:
1420:
1417:
1413:
1407:
1404:
1401:
1396:
1392:
1388:
1383:
1380:
1377:
1373:
1367:
1364:
1361:
1356:
1352:
1348:
1343:
1340:
1336:
1332:
1325:
1322:
1316:
1312:
1307:
1304:
1299:
1272:
1269:
1265:
1244:
1241:
1236:
1233:
1229:
1200:
1197:
1194:
1191:
1188:
1185:
1182:
1162:
1155:
1151:
1147:
1143:
1139:
1136:
1133:
1128:
1125:
1121:
1117:
1078:
1075:
1062:
1059:
1057:
1054:
1016:
1013:
1012:
1011:
997:
994:
990:
979:
967:
945:
940:
918:
898:
893:
888:
881:
878:
873:
868:
865:
860:
857:
852:
847:
842:
839:
835:
812:
809:
805:
794:
780:
777:
773:
750:
747:
743:
722:
712:
700:
678:
674:
663:
651:
637:
636:
623:
619:
615:
612:
609:
606:
603:
600:
596:
593:
590:
587:
584:
581:
578:
575:
571:
566:
563:
559:
555:
552:
547:
544:
539:
534:
529:
526:
522:
518:
515:
512:
507:
504:
499:
483:random effects
470:
467:
430:
429:
427:
426:
419:
412:
404:
401:
400:
399:
398:
383:
382:
381:
380:
375:
370:
365:
360:
355:
347:
346:
342:
341:
340:
339:
334:
329:
324:
319:
311:
310:
309:
308:
303:
298:
293:
288:
280:
279:
278:
277:
272:
267:
262:
254:
253:
252:
251:
246:
241:
233:
232:
228:
227:
226:
225:
217:
216:
215:
214:
209:
204:
199:
194:
189:
184:
179:
177:Semiparametric
174:
169:
161:
160:
159:
158:
153:
148:
146:Random effects
143:
138:
130:
129:
128:
127:
122:
120:Ordered probit
117:
112:
107:
102:
97:
92:
87:
82:
77:
72:
67:
59:
58:
57:
56:
51:
46:
41:
33:
32:
28:
27:
21:
20:
9:
6:
4:
3:
2:
7536:
7525:
7522:
7521:
7519:
7504:
7499:
7494:
7489:
7485:
7481:
7477:
7470:
7461:
7456:
7451:
7446:
7442:
7438:
7434:
7427:
7419:
7415:
7411:
7407:
7400:
7392:
7388:
7383:
7378:
7373:
7368:
7363:
7358:
7354:
7350:
7346:
7339:
7331:
7327:
7322:
7317:
7313:
7309:
7305:
7301:
7300:AAPS PharmSci
7297:
7290:
7282:
7278:
7274:
7270:
7265:
7260:
7255:
7250:
7246:
7242:
7238:
7231:
7223:
7219:
7215:
7211:
7204:
7202:
7193:
7189:
7185:
7181:
7176:
7171:
7166:
7161:
7157:
7153:
7149:
7142:
7140:
7131:
7127:
7123:
7119:
7112:
7104:
7100:
7096:
7092:
7088:
7084:
7080:
7076:
7069:
7061:
7049:
7041:
7040:
7032:
7024:
7022:0-387-98957-9
7018:
7014:
7010:
7006:
6999:
6997:
6992:
6984:
6979:
6976:
6974:
6971:
6969:
6966:
6964:
6961:
6959:
6956:
6954:
6951:
6949:
6946:
6944:
6941:
6940:
6934:
6920:
6909:
6894:
6891:
6888:
6885:
6882:
6879:
6876:
6873:
6870:
6867:
6864:
6861:
6855:
6846:
6841:
6838:
6835:
6825:
6821:
6814:
6809:
6806:
6803:
6798:
6795:
6792:
6789:
6786:
6783:
6780:
6770:
6767:
6763:
6756:
6751:
6746:
6743:
6740:
6730:
6726:
6719:
6714:
6710:
6703:
6696:
6691:
6688:
6685:
6682:
6679:
6676:
6673:
6670:
6667:
6664:
6661:
6658:
6655:
6652:
6649:
6646:
6643:
6640:
6637:
6634:
6631:
6628:
6622:
6613:
6608:
6605:
6602:
6592:
6588:
6581:
6576:
6573:
6570:
6565:
6562:
6559:
6556:
6553:
6550:
6547:
6537:
6534:
6530:
6523:
6518:
6513:
6510:
6507:
6497:
6493:
6479:
6476:
6473:
6468:
6465:
6462:
6459:
6456:
6453:
6450:
6440:
6437:
6433:
6423:
6416:
6411:
6408:
6405:
6402:
6399:
6396:
6393:
6390:
6387:
6384:
6381:
6378:
6375:
6372:
6369:
6366:
6363:
6360:
6357:
6354:
6351:
6348:
6345:
6342:
6339:
6336:
6333:
6330:
6324:
6315:
6311:
6307:
6302:
6299:
6296:
6291:
6288:
6285:
6282:
6279:
6276:
6273:
6263:
6260:
6256:
6240:
6236:
6232:
6229:
6224:
6221:
6218:
6215:
6212:
6209:
6206:
6196:
6193:
6189:
6179:
6172:
6164:
6146:
6141:
6138:
6135:
6125:
6121:
6114:
6109:
6106:
6103:
6098:
6095:
6092:
6089:
6086:
6083:
6080:
6070:
6067:
6063:
6056:
6051:
6046:
6043:
6040:
6030:
6026:
6019:
6014:
6010:
6006:
6001:
5998:
5995:
5990:
5987:
5984:
5981:
5978:
5975:
5972:
5962:
5959:
5955:
5948:
5941:
5937:
5933:
5930:
5925:
5922:
5919:
5916:
5913:
5910:
5907:
5897:
5894:
5890:
5880:
5877:
5869:
5849:
5845:
5841:
5838:
5833:
5830:
5827:
5824:
5821:
5818:
5815:
5805:
5802:
5798:
5784:
5779:
5776:
5773:
5763:
5759:
5752:
5747:
5744:
5741:
5736:
5733:
5730:
5727:
5724:
5721:
5718:
5708:
5705:
5701:
5694:
5689:
5684:
5681:
5678:
5668:
5664:
5657:
5652:
5648:
5644:
5639:
5636:
5633:
5628:
5625:
5622:
5619:
5616:
5613:
5610:
5600:
5597:
5593:
5583:
5575:
5571:
5569:
5568:
5549:
5546:
5542:
5519:
5516:
5512:
5491:
5484:. Typically,
5466:
5462:
5458:
5455:
5452:
5447:
5443:
5419:
5394:
5390:
5386:
5383:
5380:
5375:
5371:
5367:
5364:
5358:
5338:
5318:
5296:
5293:
5289:
5266:
5263:
5259:
5238:
5216:
5213:
5209:
5199:
5186:
5183:
5180:
5177:
5174:
5171:
5168:
5165:
5161:
5153:
5148:
5144:
5137:
5134:
5129:
5124:
5120:
5115:
5107:
5104:
5100:
5096:
5093:
5090:
5085:
5082:
5078:
5074:
5071:
5068:
5063:
5060:
5056:
5049:
5046:
5038:
5035:
5031:
5027:
5024:
5021:
5016:
5013:
5009:
5005:
5002:
4999:
4994:
4991:
4987:
4979:
4971:
4967:
4960:
4957:
4952:
4948:
4943:
4935:
4931:
4924:
4921:
4916:
4912:
4903:
4902:
4901:
4896:
4883:
4880:
4877:
4874:
4871:
4868:
4865:
4862:
4858:
4855:
4852:
4849:
4846:
4843:
4840:
4837:
4833:
4825:
4820:
4816:
4812:
4809:
4803:
4800:
4795:
4792:
4788:
4783:
4778:
4775:
4771:
4767:
4762:
4759:
4755:
4749:
4746:
4742:
4736:
4731:
4728:
4725:
4721:
4717:
4712:
4708:
4704:
4699:
4696:
4692:
4683:
4682:
4681:
4676:
4663:
4658:
4654:
4650:
4647:
4644:
4641:
4638:
4635:
4631:
4628:
4625:
4622:
4619:
4616:
4613:
4610:
4606:
4598:
4594:
4590:
4587:
4581:
4578:
4573:
4570:
4566:
4561:
4556:
4553:
4549:
4545:
4537:
4534:
4530:
4526:
4523:
4520:
4515:
4512:
4508:
4504:
4501:
4498:
4493:
4490:
4486:
4482:
4477:
4474:
4470:
4466:
4461:
4458:
4454:
4447:
4444:
4439:
4436:
4431:
4421:
4420:
4419:
4414:
4406:
4397:
4394:
4373:
4369:
4365:
4360:
4356:
4352:
4347:
4343:
4339:
4336:
4330:
4310:
4304:
4301:
4298:
4295:
4292:
4289:
4286:
4280:
4272:
4269:
4265:
4261:
4256:
4253:
4249:
4245:
4240:
4237:
4233:
4221:
4217:
4197:
4190:
4172:
4169:
4166:
4156:
4152:
4147:
4139:
4135:
4116:
4108:
4104:
4096:
4079:
4074:
4070:
4046:
4038:
4034:
4026:
4011:
3979:
3976:
3972:
3968:
3963:
3960:
3956:
3949:
3944:
3940:
3932:
3917:
3909:
3905:
3877:
3874:
3870:
3866:
3863:
3860:
3855:
3852:
3848:
3841:
3836:
3832:
3824:
3821:
3800:
3796:
3792:
3787:
3783:
3779:
3774:
3770:
3741:
3737:
3733:
3728:
3724:
3720:
3715:
3711:
3707:
3704:
3698:
3691:
3690:
3689:
3686:
3673:
3670:
3667:
3664:
3661:
3658:
3655:
3652:
3636:
3631:
3627:
3622:
3618:
3615:
3607:
3603:
3596:
3593:
3588:
3583:
3579:
3574:
3571:
3568:
3562:
3556:
3553:
3548:
3544:
3524:
3521:
3518:
3515:
3512:
3509:
3506:
3503:
3499:
3496:
3493:
3490:
3487:
3484:
3481:
3478:
3468:
3462:
3459:
3456:
3448:
3444:
3440:
3435:
3431:
3427:
3422:
3418:
3414:
3409:
3406:
3402:
3398:
3395:
3391:
3383:
3378:
3375:
3371:
3365:
3360:
3356:
3350:
3345:
3341:
3337:
3334:
3328:
3325:
3320:
3316:
3312:
3307:
3303:
3299:
3294:
3291:
3287:
3276:
3273:
3269:
3249:
3246:
3243:
3240:
3237:
3234:
3231:
3228:
3222:
3214:
3204:
3200:
3196:
3191:
3187:
3176:
3172:
3167:
3163:
3157:
3154:
3149:
3144:
3140:
3136:
3128:
3124:
3120:
3115:
3111:
3100:
3096:
3091:
3086:
3077:
3074:
3071:
3061:
3057:
3052:
3048:
3045:
3039:
3036:
3033:
3027:
3019:
3015:
2995:
2992:
2989:
2986:
2983:
2980:
2977:
2974:
2969:
2961:
2956:
2952:
2945:
2942:
2939:
2936:
2930:
2922:
2918:
2913:
2905:
2901:
2892:
2888:
2884:
2876:
2872:
2863:
2859:
2855:
2850:
2846:
2840:
2837:
2833:
2827:
2822:
2819:
2816:
2812:
2808:
2803:
2799:
2795:
2787:
2783:
2774:
2770:
2766:
2761:
2758:
2754:
2731:
2726:
2722:
2718:
2715:
2712:
2709:
2706:
2703:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2662:
2657:
2654:
2650:
2646:
2638:
2635:
2631:
2627:
2622:
2619:
2615:
2611:
2606:
2603:
2599:
2595:
2592:
2586:
2583:
2578:
2575:
2570:
2561:
2560:
2559:
2557:
2552:
2549:
2539:
2530:
2522:
2513:
2510:
2506:
2502:
2493:
2484:
2482:
2478:
2474:
2470:
2448:
2445:
2441:
2433:
2416:
2384:
2374:
2371:
2365:
2358:
2341:
2338:
2334:
2313:
2291:
2288:
2284:
2276:
2261:
2239:
2235:
2227:
2226:
2225:
2206:
2202:
2198:
2195:
2192:
2189:
2186:
2183:
2179:
2176:
2173:
2170:
2167:
2164:
2161:
2158:
2154:
2149:
2146:
2142:
2138:
2127:
2117:
2112:
2109:
2105:
2098:
2090:
2086:
2082:
2077:
2074:
2069:
2060:
2059:
2058:
2039:
2029:
2026:
2020:
2013:
2009:
1991:
1976:
1972:
1968:
1964:
1963:growth charts
1954:
1952:
1947:
1943:
1939:
1935:
1927:
1922:
1913:
1897:
1893:
1884:
1880:
1858:
1855:
1851:
1843:
1828:
1806:
1802:
1794:
1777:
1774:
1771:
1767:
1744:
1741:
1738:
1734:
1726:
1711:
1689:
1686:
1683:
1678:
1674:
1651:
1648:
1645:
1640:
1636:
1628:
1611:
1608:
1604:
1596:
1575:
1545:
1539:
1531:
1530:
1529:
1510:
1506:
1502:
1499:
1496:
1493:
1490:
1487:
1483:
1480:
1477:
1474:
1471:
1468:
1465:
1462:
1458:
1453:
1450:
1446:
1442:
1434:
1430:
1426:
1421:
1418:
1415:
1411:
1405:
1402:
1399:
1394:
1390:
1386:
1381:
1378:
1375:
1371:
1365:
1362:
1359:
1354:
1350:
1346:
1341:
1338:
1334:
1320:
1314:
1310:
1305:
1302:
1297:
1288:
1287:
1286:
1270:
1267:
1263:
1242:
1239:
1234:
1231:
1227:
1218:
1214:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1153:
1149:
1145:
1141:
1137:
1134:
1131:
1126:
1123:
1119:
1107:
1103:
1099:
1095:
1088:
1083:
1074:
1072:
1068:
1053:
1051:
1046:
1042:
1038:
1034:
1030:
1026:
1022:
995:
992:
988:
980:
965:
943:
916:
896:
891:
879:
876:
866:
863:
858:
855:
845:
840:
837:
833:
810:
807:
803:
795:
778:
775:
771:
748:
745:
741:
720:
713:
698:
676:
672:
664:
649:
642:
641:
640:
621:
617:
613:
610:
607:
604:
601:
598:
594:
591:
588:
585:
582:
579:
576:
573:
569:
564:
561:
557:
553:
545:
542:
537:
532:
527:
524:
520:
513:
510:
505:
502:
497:
488:
487:
486:
484:
480:
479:fixed effects
476:
466:
464:
460:
456:
455:public health
452:
448:
444:
441:generalizing
440:
436:
425:
420:
418:
413:
411:
406:
405:
403:
402:
397:
392:
387:
386:
385:
384:
379:
376:
374:
371:
369:
366:
364:
361:
359:
356:
354:
351:
350:
349:
348:
344:
343:
338:
335:
333:
330:
328:
325:
323:
320:
318:
315:
314:
313:
312:
307:
304:
302:
299:
297:
294:
292:
289:
287:
284:
283:
282:
281:
276:
273:
271:
268:
266:
263:
261:
258:
257:
256:
255:
250:
247:
245:
242:
240:
239:Least squares
237:
236:
235:
234:
230:
229:
224:
221:
220:
219:
218:
213:
210:
208:
205:
203:
200:
198:
195:
193:
190:
188:
185:
183:
180:
178:
175:
173:
172:Nonparametric
170:
168:
165:
164:
163:
162:
157:
154:
152:
149:
147:
144:
142:
141:Fixed effects
139:
137:
134:
133:
132:
131:
126:
123:
121:
118:
116:
115:Ordered logit
113:
111:
108:
106:
103:
101:
98:
96:
93:
91:
88:
86:
83:
81:
78:
76:
73:
71:
68:
66:
63:
62:
61:
60:
55:
52:
50:
47:
45:
42:
40:
37:
36:
35:
34:
30:
29:
26:
23:
22:
18:
17:
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7479:
7469:
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7436:
7426:
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7299:
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7230:
7213:
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7151:
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7117:
7111:
7078:
7074:
7068:
7038:
7031:
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6982:
6910:
6165:
5870:
5576:
5572:
5566:
5565:
5200:
4904:
4899:
4898:
4897:
4684:
4679:
4678:
4677:
4422:
4417:
4416:
4415:
4411:
4213:
3687:
2746:
2553:
2545:
2527:
2507:such as the
2501:PK/PD models
2499:
2476:
2468:
2466:
2223:
2011:
1971:height spurt
1960:
1931:
1876:
1527:
1092:
1064:
1056:Applications
1018:
638:
472:
459:pharmacology
434:
433:
296:Non-negative
155:
7480:Mathematics
7437:Mathematics
6943:Mixed model
1879:exponential
306:Regularized
270:Generalized
202:Least angle
100:Mixed logit
7493:2201.12430
7486:(6): 898.
7450:2201.12430
7443:(6): 898.
7362:2005.00662
7306:(3): E32.
7075:Biometrics
6987:References
2509:Emax model
1015:Estimation
473:While any
469:Definition
345:Background
249:Non-linear
231:Estimation
7406:Sankhya B
7192:221105601
7058:ignored (
7048:cite book
6856:⏟
6822:ω
6764:β
6727:α
6711:σ
6697:×
6623:⏟
6589:ω
6531:β
6494:α
6434:θ
6424:π
6417:×
6382:−
6325:⏟
6312:σ
6257:θ
6180:π
6122:ω
6064:β
6027:α
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5956:θ
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5760:ω
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5513:ϵ
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4509:θ
4502:…
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4071:σ
4047:⋅
4035:ϵ
4024:-th well,
3990:⊤
3930:-th well,
3888:⊤
3864:⋯
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3784:θ
3771:θ
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3725:θ
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3628:σ
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3604:σ
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3580:σ
3569:∝
3563:α
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3545:α
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3441:∼
3432:σ
3419:τ
3403:λ
3396:σ
3372:λ
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3342:σ
3326:∼
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3304:τ
3288:λ
3270:β
3211:‖
3197:−
3184:‖
3173:ρ
3164:−
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3141:γ
3097:γ
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3072:⋅
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2889:η
2860:ϵ
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2813:∑
2800:α
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2755:θ
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2691:…
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2496:subjects.
2442:ϵ
2372:⋅
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2240:β
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2143:ϵ
2091:β
2027:⋅
1852:ϵ
1768:β
1735:β
1579:~
1576:β
1549:~
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1500:…
1475:…
1447:ϵ
1412:β
1372:β
1324:~
1321:β
1215:(MCI) or
1193:…
1135:…
989:ϵ
917:β
864:β
834:ϕ
804:ϕ
742:ϕ
611:…
586:…
558:ϵ
521:ϕ
212:Segmented
7518:Category
7412:: 1–43.
7391:32726361
7349:PLOS ONE
7330:11741248
7281:17816715
7273:20647267
7184:33693397
6937:See also
3910:for the
1928:package.
1217:dementia
1089:package.
451:medicine
327:Bayesian
265:Weighted
260:Ordinary
192:Isotonic
187:Quantile
7382:7390340
7321:2761142
7264:2992626
7216:: 1–7.
7175:7931952
7103:2242409
7095:2532087
5311:is the
4393:kriging
4220:kriging
3688:where
2224:where
1528:where
1098:reserve
639:where
463:ecology
286:Partial
125:Poisson
7389:
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7318:
7279:
7271:
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7182:
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7158:: 24.
7101:
7093:
7019:
5281:, and
5201:Here,
2477:pavpop
2008:latent
1940:, and
1173:from
1071:latent
909:where
461:, and
244:Linear
182:Robust
105:Probit
31:Models
7488:arXiv
7445:arXiv
7357:arXiv
7277:S2CID
7188:S2CID
7091:JSTOR
5534:and
2469:SITAR
1100:, so
978:, and
291:Total
207:Local
7387:PMID
7326:PMID
7269:PMID
7180:PMID
7099:PMID
7060:help
7017:ISBN
4214:The
1759:and
1666:and
481:and
7498:doi
7455:doi
7414:doi
7377:PMC
7367:doi
7316:PMC
7308:doi
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7218:doi
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7160:doi
7126:doi
7083:doi
7009:doi
3155:exp
1043:or
7520::
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3188:s
3177:l
3168:e
3161:(
3150:2
3145:l
3137:=
3134:)
3129:j
3125:s
3121:,
3116:i
3112:s
3108:(
3101:l
3092:K
3087:,
3084:)
3081:)
3075:,
3069:(
3062:l
3053:K
3049:,
3046:0
3043:(
3040:P
3037:G
3031:)
3025:(
3020:l
2996:,
2993:3
2990:,
2987:2
2984:,
2981:1
2978:=
2975:l
2970:,
2967:)
2962:2
2957:l
2949:(
2946:N
2943:W
2940:G
2934:)
2928:(
2923:l
2914:,
2911:)
2906:i
2902:s
2898:(
2893:l
2885:+
2882:)
2877:i
2873:s
2869:(
2864:l
2856:+
2851:j
2847:x
2841:j
2838:l
2828:p
2823:1
2820:=
2817:j
2809:+
2804:l
2796:=
2793:)
2788:i
2784:s
2780:(
2775:l
2767:=
2762:i
2759:l
2732:,
2727:i
2723:T
2719:,
2713:,
2710:1
2707:=
2704:t
2700:,
2697:N
2694:,
2688:,
2685:1
2682:=
2679:i
2663:,
2658:t
2655:i
2647:+
2644:)
2639:i
2636:3
2628:,
2623:i
2620:2
2612:,
2607:i
2604:1
2596:;
2593:t
2590:(
2584:=
2579:t
2576:i
2571:y
2449:j
2446:i
2431:,
2417:i
2412:w
2390:)
2385:i
2380:w
2375:,
2369:(
2366:v
2356:,
2342:j
2339:i
2335:y
2314:i
2292:j
2289:i
2285:t
2274:,
2236:f
2207:i
2203:n
2199:,
2193:,
2190:1
2187:=
2184:j
2180:,
2177:M
2174:,
2168:,
2165:1
2162:=
2159:i
2155:,
2150:j
2147:i
2139:+
2136:)
2133:)
2128:i
2123:w
2118:,
2113:j
2110:i
2106:t
2102:(
2099:v
2096:(
2087:f
2083:=
2078:j
2075:i
2070:y
2045:)
2040:i
2035:w
2030:,
2024:(
2021:v
1992:i
1987:w
1926:R
1898:i
1894:b
1859:j
1856:i
1829:i
1807:i
1803:b
1778:M
1775:E
1772:D
1745:I
1742:C
1739:M
1712:i
1690:M
1687:E
1684:D
1679:i
1675:A
1652:I
1649:C
1646:M
1641:i
1637:A
1612:j
1609:i
1605:t
1594:,
1540:f
1511:i
1507:n
1503:,
1497:,
1494:1
1491:=
1488:j
1484:,
1481:M
1478:,
1472:,
1469:1
1466:=
1463:i
1459:,
1454:j
1451:i
1443:+
1440:)
1435:i
1431:b
1427:+
1422:M
1419:E
1416:D
1406:M
1403:E
1400:D
1395:i
1391:A
1387:+
1382:I
1379:C
1376:M
1366:I
1363:C
1360:M
1355:i
1351:A
1347:+
1342:j
1339:i
1335:t
1331:(
1315:f
1311:=
1306:j
1303:i
1298:y
1271:1
1268:i
1264:y
1243:0
1240:=
1235:1
1232:i
1228:t
1199:M
1196:,
1190:,
1187:1
1184:=
1181:i
1161:)
1154:i
1150:n
1146:i
1142:y
1138:,
1132:,
1127:1
1124:i
1120:y
1116:(
1087:R
996:j
993:i
966:i
944:i
939:b
897:,
892:i
887:b
880:j
877:i
872:B
867:+
859:j
856:i
851:A
846:=
841:j
838:i
811:j
808:i
793:,
779:j
776:i
772:v
749:j
746:i
721:f
699:i
677:i
673:n
650:M
622:i
618:n
614:,
608:,
605:1
602:=
599:j
595:,
592:M
589:,
583:,
580:1
577:=
574:i
570:,
565:j
562:i
554:+
551:)
546:j
543:i
538:v
533:,
528:j
525:i
517:(
514:f
511:=
506:j
503:i
498:y
423:e
416:t
409:v
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