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research with sufficient power, large sample sizes are required in multilevel models. However, the number of individual observations in groups is not as important as the number of groups in a study. In order to detect cross-level interactions, given that the group sizes are not too small, recommendations have been made that at least 20 groups are needed, although many fewer can be used if one is only interested in inference on the fixed effects and the random effects are control, or "nuisance", variables. The issue of statistical power in multilevel models is complicated by the fact that power varies as a function of effect size and intraclass correlations, it differs for fixed effects versus random effects, and it changes depending on the number of groups and the number of individual observations per group.
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example, assign class variables to the individual level). The problem with this approach is that it would violate the assumption of independence, and thus could bias our results. This is known as atomistic fallacy. Another way to analyze the data using traditional statistical approaches is to aggregate individual level variables to higher-order variables and then to conduct an analysis on this higher level. The problem with this approach is that it discards all within-group information (because it takes the average of the individual level variables). As much as 80–90% of the variance could be wasted, and the relationship between aggregated variables is inflated, and thus distorted. This is known as
1956:. However, it would also predict, for example, that a white person might have an average income $ 7,000 above a black person, and a 65-year-old might have an income $ 3,000 below a 45-year-old, in both cases regardless of location. A multilevel model, however, would allow for different regression coefficients for each predictor in each location. Essentially, it would assume that people in a given location have correlated incomes generated by a single set of regression coefficients, whereas people in another location have incomes generated by a different set of coefficients. Meanwhile, the coefficients themselves are assumed to be correlated and generated from a single set of
2032:
population of group intercepts and slopes. This allows for an analysis in which one can assume that slopes are fixed but intercepts are allowed to vary. However this presents a problem, as individual components are independent but group components are independent between groups, but dependent within groups. This also allows for an analysis in which the slopes are random; however, the correlations of the error terms (disturbances) are dependent on the values of the individual-level variables. Thus, the problem with using a random-coefficients model in order to analyze hierarchical data is that it is still not possible to incorporate higher order variables.
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4735:{\displaystyle {\begin{aligned}=&~\left.{\pi (\{y_{ij}\}_{i=1,j=1}^{N,M_{i}}|\{\theta _{li}\}_{i=1,l=1}^{N,K},\sigma ^{2})}\right\}{\text{Stage 1: Individual-Level Model}}\\{\phantom {spacer}}\\\times &~\left.{\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})}\right\}{\text{Stage 2: Population Model}}\\{\phantom {spacer}}\\\times &~\left.{p(\sigma ^{2},\{\alpha _{l}\}_{l=1}^{K},\{\beta _{lb}\}_{l=1,b=1}^{K,P},\{\omega _{l}\}_{l=1}^{K})}\right\}{\text{Stage 3: Prior}}\end{aligned}}}
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3122:{\displaystyle {\begin{aligned}&\sigma ^{2}\sim \pi (\sigma ^{2}),\\{\phantom {spacer}}\\&\alpha _{l}\sim \pi (\alpha _{l}),\\{\phantom {spacer}}\\&(\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP})\sim \pi (\beta _{l1},\ldots ,\beta _{lb},\ldots ,\beta _{lP}),\\{\phantom {spacer}}\\&\omega _{l}^{2}\sim \pi (\omega _{l}^{2}),\\{\phantom {spacer}}\\&l=1,\ldots ,K.\end{aligned}}}
2685:
509:). The units of analysis are usually individuals (at a lower level) who are nested within contextual/aggregate units (at a higher level). While the lowest level of data in multilevel models is usually an individual, repeated measurements of individuals may also be examined. As such, multilevel models provide an alternative type of analysis for univariate or
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1695:, which assesses the difference between models. The likelihood-ratio test can be employed for model building in general, for examining what happens when effects in a model are allowed to vary, and when testing a dummy-coded categorical variable as a single effect. However, the test can only be used when models are
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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
1656:
A random intercepts model is a model in which intercepts are allowed to vary, and therefore, the scores on the dependent variable for each individual observation are predicted by the intercept that varies across groups. This model assumes that slopes are fixed (the same across different contexts). In
2003:
Cross-level interactions may also be of substantive interest; for example, when a slope is allowed to vary randomly, a level-2 predictor may be included in the slope formula for the level-1 covariate. For example, one may estimate the interaction of race and neighborhood to obtain an estimate of the
1776:
Independence is an assumption of general linear models, which states that cases are random samples from the population and that scores on the dependent variable are independent of each other. One of the main purposes of multilevel models is to deal with cases where the assumption of independence is
2057:
Multilevel modeling is frequently used in diverse applications and it can be formulated by the
Bayesian framework. 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
1987:
Multilevel models have been used in education research or geographical research, to estimate separately the variance between pupils within the same school, and the variance between schools. In psychological applications, the multiple levels are items in an instrument, individuals, and families. In
1936:
As a simple example, consider a basic linear regression model that predicts income as a function of age, class, gender and race. It might then be observed that income levels also vary depending on the city and state of residence. A simple way to incorporate this into the regression model would be
1755:
The assumption of normality states that the error terms at every level of the model are normally distributed. However, most statistical software allows one to specify different distributions for the variance terms, such as a
Poisson, binomial, logistic. The multilevel modelling approach can be used
2031:
Another way to analyze hierarchical data would be through a random-coefficients model. This model assumes that each group has a different regression model—with its own intercept and slope. Because groups are sampled, the model assumes that the intercepts and slopes are also randomly sampled from a
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Statistical power for multilevel models differs depending on whether it is level 1 or level 2 effects that are being examined. Power for level 1 effects is dependent upon the number of individual observations, whereas the power for level 2 effects is dependent upon the number of groups. To conduct
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The assumption of linearity states that there is a rectilinear (straight-line, as opposed to non-linear or U-shaped) relationship between variables. However, the model can be extended to nonlinear relationships. Particularly, when the mean part of the level 1 regression equation is replaced with a
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In order to conduct a multilevel model analysis, one would start with fixed coefficients (slopes and intercepts). One aspect would be allowed to vary at a time (that is, would be changed), and compared with the previous model in order to assess better model fit. There are three different questions
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can also be computed. When computing a t-test, it is important to keep in mind the degrees of freedom, which will depend on the level of the predictor (e.g., level 1 predictor or level 2 predictor). For a level 1 predictor, the degrees of freedom are based on the number of level 1 predictors, the
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Before conducting a multilevel model analysis, a researcher must decide on several aspects, including which predictors are to be included in the analysis, if any. Second, the researcher must decide whether parameter values (i.e., the elements that will be estimated) will be fixed or random. Fixed
4744:
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
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At Level 1, both the intercepts and slopes in the groups can be either fixed (meaning that all groups have the same values, although in the real world this would be a rare occurrence), non-randomly varying (meaning that the intercepts and/or slopes are predictable from an independent variable at
1944:
to account for the location (i.e. a set of additional binary predictors and associated regression coefficients, one per location). This would have the effect of shifting the mean income up or down—but it would still assume, for example, that the effect of race and gender on income is the same
2040:
Multilevel models have two error terms, which are also known as disturbances. The individual components are all independent, but there are also group components, which are independent between groups but correlated within groups. However, variance components can differ, as some groups are more
2023:
There are several alternative ways of analyzing hierarchical data, although most of them have some problems. First, traditional statistical techniques can be used. One could disaggregate higher-order variables to the individual level, and thus conduct an analysis on this individual level (for
1669:
A random slopes model is a model in which slopes are allowed to vary according to a correlation matrix, and therefore, the slopes are different across grouping variable such as time or individuals. This model assumes that intercepts are fixed (the same across different contexts).
525:, where scores on the dependent variable are adjusted for covariates (e.g. individual differences) before testing treatment differences. Multilevel models are able to analyze these experiments without the assumptions of homogeneity-of-regression slopes that is required by ANCOVA.
2387:{\displaystyle {\begin{aligned}&{y}_{ij}=f(t_{ij};\theta _{1i},\theta _{2i},\ldots ,\theta _{li},\ldots ,\theta _{Ki})+\epsilon _{ij},\\{\phantom {spacer}}\\&\epsilon _{ij}\sim N(0,\sigma ^{2}),\\{\phantom {spacer}}\\&i=1,\ldots ,N,\,j=1,\ldots ,M_{i}.\end{aligned}}}
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A model that includes both random intercepts and random slopes is likely the most realistic type of model, although it is also the most complex. In this model, both intercepts and slopes are allowed to vary across groups, meaning that they are different in different contexts.
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parameters are composed of a constant over all the groups, whereas a random parameter has a different value for each of the groups. Additionally, the researcher must decide whether to employ a maximum likelihood estimation or a restricted maximum likelihood estimation type.
1815:. This assumption is testable but often ignored, rendering the estimator inconsistent. If this assumption is violated, the random-effect must be modeled explicitly in the fixed part of the model, either by using dummy variables or including cluster means of all
3802:
1853:
The type of statistical tests that are employed in multilevel models depend on whether one is examining fixed effects or variance components. When examining fixed effects, the tests are compared with the standard error of the fixed effect, which results in a
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everywhere. In reality, this is unlikely to be the case—different local laws, different retirement policies, differences in level of racial prejudice, etc. are likely to cause all of the predictors to have different sorts of effects in different locales.
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that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of
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2680:{\displaystyle {\begin{aligned}&\theta _{li}=\alpha _{l}+\sum _{b=1}^{P}\beta _{lb}x_{ib}+\eta _{li},\\{\phantom {spacer}}\\&\eta _{li}\sim N(0,\omega _{l}^{2}),\\{\phantom {spacer}}\\&i=1,\ldots ,N,\,l=1,\ldots ,K.\end{aligned}}}
1768:, also known as homogeneity of variance, assumes equality of population variances. However, different variance-correlation matrix can be specified to account for this, and the heterogeneity of variance can itself be modeled.
528:
Multilevel models can be used on data with many levels, although 2-level models are the most common and the rest of this article deals only with these. The dependent variable must be examined at the lowest level of analysis.
1960:. Additional levels are possible: For example, people might be grouped by cities, and the city-level regression coefficients grouped by state, and the state-level coefficients generated from a single hyper-hyperparameter.
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Different covariables may be relevant on different levels. They can be used for longitudinal studies, as with growth studies, to separate changes within one individual and differences between individuals.
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the level at which it was measured. In this example "test score" might be measured at pupil level, "teacher experience" at class level, "school funding" at school level, and "urban" at district level.
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violated; multilevel models do, however, assume that 1) the level 1 and level 2 residuals are uncorrelated and 2) The errors (as measured by the residuals) at the highest level are uncorrelated.
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4086:{\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})}
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that a researcher would ask in assessing a model. First, is it a good model? Second, is a more complex model better? Third, what contribution do individual predictors make to the model?
3791:{\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}})}
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number of groups and the number of individual observations. For a level 2 predictor, the degrees of freedom are based on the number of level 2 predictors and the number of groups.
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regressors. This assumption is probably the most important assumption the estimator makes, but one that is misunderstood by most applied researchers using these types of models.
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research, data from individuals must often be nested within teams or other functional units. They are often used in ecological research as well under the more general term
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Level 2), or randomly varying (meaning that the intercepts and/or slopes are different in the different groups, and that each have their own overall mean and variance).
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can not be described by the linear relationship, then one can find some non linear functional relationship between the response and predictor, and extend the model to
3239:
3209:
3159:
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1813:
1699:(meaning that a more complex model includes all of the effects of a simpler model). When testing non-nested models, comparisons between models can be made using the
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992:
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refers to the score on the dependent variable for an individual observation at Level 1 (subscript i refers to individual case, subscript j refers to the group).
502:), although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.
5589:
Bryk, Anthony S.; Raudenbush, Stephen W. (1 January 1988). "Heterogeneity of variance in experimental studies: A challenge to conventional interpretations".
1460:
refers to the overall intercept. This is the grand mean of the scores on the dependent variable across all the groups when all the predictors are equal to 0.
5386:"Fixed effects models versus mixed effects models for clustered data: Reviewing the approaches, disentangling the differences, and making recommendations"
5138:
Lee, Se Yoon; Mallick, Bani (2021). "Bayesian
Hierarchical Modeling: Application Towards Production Results in the Eagle Ford Shale of South Texas".
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Multilevel models are particularly appropriate for research designs where data for participants are organized at more than one level (i.e.,
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When there are multiple level 1 independent variables, the model can be expanded by substituting vectors and matrices in the equation.
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Bayesian research cycle using
Bayesian nonlinear mixed effects model: (a) standard research cycle and (b) Bayesian-specific workflow.
2013:
1989:
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1201:
1133:
827:
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347:
5823:"Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model?"
4969:"Should I use fixed effects or random effects when I have fewer than five levels of a grouping factor in a mixed-effects model?"
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The dependent variables are the intercepts and the slopes for the independent variables at Level 1 in the groups of Level 2.
4765:; and (b)–(iii) making a posterior inference. The resulting posterior inference can be used to start a new research cycle.
337:
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1971:
and arbitrary relationships among the different variables. Multilevel analysis has been extended to include multilevel
3365:
759:
refers to the slope for the relationship in group j (Level 2) between the Level 1 predictor and the dependent variable.
3503:
A central task in the application of the
Bayesian nonlinear mixed-effect models is to evaluate the posterior density:
1988:
sociological applications, multilevel models are used to examine individuals embedded within regions or countries. In
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4779:
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In other words, a simple linear regression model might, for example, predict that a given randomly sampled person in
1638:
refers to the deviation in group j from the average slope between the dependent variable and the Level 1 predictor.
301:
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In order to assess models, different model fit statistics would be examined. One such statistic is the chi-square
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290:
110:
85:
1704:
212:
2028:, and statistically, this type of analysis results in decreased power in addition to the loss of information.
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171:
1907:
However, if one were studying multiple schools and multiple school districts, a 4-level model could include
1972:
1727:), but some of the assumptions are modified for the hierarchical nature of the design (i.e., nested data).
1700:
430:
5442:"Bridging Methodological Divides Between Macro- and Microresearch: Endogeneity and Methods for Panel Data"
5311:
1964:
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5621:"Bayesian Nonlinear Models for Repeated Measurement Data: An Overview, Implementation, and Applications"
5268:
Goldstein, Harvey (1991). "Nonlinear
Multilevel Models, with an Application to Discrete Response Data".
791:
refers to the random errors of prediction for the Level 1 equation (it is also sometimes referred to as
5892:
5335:"On Ignoring the Random Effects Assumption in Multilevel Models: Review, Critique, and Recommendations"
342:
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is a `nonlinear' function and describes the temporal trajectory of individuals. In the model,
1661:, which are helpful in determining whether multilevel models are required in the first place.
1692:
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refer to the effect of the Level 2 predictor on the Level 1 intercept and slope respectively.
510:
368:
64:
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describe within-individual variability and between-individual variability, respectively. If
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is not considered, then the model reduces to a frequentist nonlinear mixed-effect model.
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105:
95:
40:
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5068:"Estimation of COVID-19 spread curves integrating global data and borrowing information"
1719:
Multilevel models have the same assumptions as other major general linear models (e.g.,
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1976:
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Bliese, Paul D.; Schepker, Donald J.; Essman, Spenser M.; Ployhart, Robert E. (2020).
5854:
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refers to the average slope between the dependent variable and the Level 1 predictor.
1117:
514:
499:
486:
406:
197:
100:
54:
5788:; Tavlas, George S. (2001). "Random Coefficient Models". In Baltagi, Badi H. (ed.).
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1968:
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non-linear parametric function, then such a model framework is widely called the
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may be examined. Furthermore, multilevel models can be used as an alternative to
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5767:
Multilevel
Analysis: an Introduction to Basic and Advanced Multilevel Modeling
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would have an average yearly income $ 10,000 higher than a similar person in
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Using SPSS for
Windows and Macintosh : analyzing and understanding data
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Applied multiple regression/correlation analysis for the behavioral sciences
3281:-th subject. Parameters involved in the model are written in Greek letters.
2004:
interaction between an individual's characteristics and the social context.
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When there is a single level 1 independent variable, the level 1 model is
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Hierarchical linear models : applications and data analysis methods
2007:
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refers to the intercept of the dependent variable for individual case i.
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5490:
Econometric Analysis of Cross Section and Panel Data, second edition
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5756:
Hierarchical Linear Models: Applications and Data Analysis Methods
2018:
5672:
Data Analysis Using Regression and Multilevel/Hierarchical Models
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1949:
1424:{\displaystyle \beta _{1j}=\gamma _{10}+\gamma _{11}w_{j}+u_{1j}}
1341:{\displaystyle \beta _{0j}=\gamma _{00}+\gamma _{01}w_{j}+u_{0j}}
27:
Statistical models of parameters that vary at more than one level
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1859:
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522:
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refers to the deviation in group j from the overall intercept.
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The concept of level is the keystone of this approach. In an
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5302:
4832:(2. ed., ed.). Thousand Oaks, CA : Sage Publications.
1258:{\displaystyle u_{1j}\sim {\mathcal {N}}(0,\sigma _{3}^{2})}
1190:{\displaystyle u_{0j}\sim {\mathcal {N}}(0,\sigma _{2}^{2})}
884:{\displaystyle e_{ij}\sim {\mathcal {N}}(0,\sigma _{1}^{2})}
5333:
Antonakis, John; Bastardoz, Nicolas; Rönkkö, Mikko (2021).
5233:(4th ed.). Upper Saddle River, NJ: Pearson Education.
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The regressors must not correlate with the random effects,
623:{\displaystyle Y_{ij}=\beta _{0j}+\beta _{1j}X_{ij}+e_{ij}}
5675:. New York: Cambridge University Press. pp. 235–299.
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5175:
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Applications to longitudinal (repeated measures) data
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3335:{\displaystyle f(t;\theta _{1},\ldots ,\theta _{K})}
1967:, which are general models with multiple levels of
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5523:(Repr. ed.). London: Sage Publications Ltd.
5305:"Introduction to Multilevel Modeling Using HLM 6"
3407:{\displaystyle (\theta _{1},\ldots ,\theta _{K})}
1896:example, the levels for a 2-level model might be
5879:
5736:Multilevel Analysis: Techniques and Applications
5066:Lee, Se Yoon; Lei, Bowen; Mallick, Bani (2020).
4890:Fidell, Barbara G. Tabachnick, Linda S. (2007).
4828:Bryk, Stephen W. Raudenbush, Anthony S. (2002).
1682:
1657:addition, this model provides information about
2019:Alternative ways of analyzing hierarchical data
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4933:(3. repr. ed.). Thousand Oaks, CA: Sage.
1123:
994:is the cumulative infection trajectory of the
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1781:Orthogonality of regressors to random effects
1116:for each country may show a shape similar to
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5177:. Thousand Oaks, Calif.: Sage Publications.
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1756:for all forms of Generalized Linear models.
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5565:(Reprint. ed.). Mahwah, NJ : Erlbaum.
4962:
4960:
4958:
4956:
4954:
4952:
4950:
900:When the relationship between the response
5764:Snijders, T. A. B.; Bosker, R. J. (2011).
5665:
5486:
5225:Salkind, Samuel B. Green, Neil J. (2004).
5137:
1651:
438:
424:
5848:
5838:
5809:Linear Mixed Models for Longitudinal Data
5689:
5646:
5636:
5487:Wooldridge, Jeffrey M. (1 October 2010).
5267:
5173:editor, G. David Garson (10 April 2012).
5128:
5111:
5101:
5083:
4994:
4984:
4885:
4883:
3342:is a known function parameterized by the
2648:
2348:
2014:Multilevel Modeling for Repeated Measures
5758:(2nd ed.). Thousand Oaks, CA: Sage.
5036:
5034:
4947:
4881:
4879:
4877:
4875:
4873:
4871:
4869:
4867:
4865:
4863:
2048:
1865:
1734:
5792:. Oxford: Blackwell. pp. 410–429.
5790:A Companion to Theoretical Econometrics
5754:Raudenbush, S. W.; Bryk, A. S. (2002).
5584:
5582:
5224:
5220:
5218:
5216:
5214:
5032:
5030:
5028:
5026:
5024:
5022:
5020:
5018:
5016:
5014:
3161:denotes the continuous response of the
1924:The researcher must establish for each
1064:-th time points, then the ordered pair
14:
5880:
5514:
5512:
5510:
5172:
4889:
4823:
4821:
4819:
4817:
4815:
2045:Bayesian nonlinear mixed-effects model
1664:
5821:Gomes, Dylan G.E. (20 January 2022).
5820:
5807:Verbeke, G.; Molenberghs, G. (2013).
5739:(2nd ed.). New York: Routledge.
5518:
5383:McNeish, Daniel; Kelley, Ken (2019).
5328:
5326:
5324:
5166:
5045:(3. ed.). Mahwah, NJ : Erlbaum.
5040:
4967:Gomes, Dylan G.E. (20 January 2022).
4966:
4860:
5711:Hedeker, D.; Gibbons, R. D. (2012).
5579:
5211:
5011:
4928:
4827:
1963:Multilevel models are a subclass of
1873:
1848:
5729:
5618:
5557:
5551:
5507:
4922:
4812:
24:
5659:
5321:
5303:ATS Statistical Consulting Group.
1707:(BIC), among others. See further
1674:Random intercepts and slopes model
1642:
1223:
1155:
849:
25:
5904:
5866:
5715:(2nd ed.). New York: Wiley.
5519:Leeuw, Ita Kreft, Jan de (1998).
4780:Mixed-design analysis of variance
1979:, and other more general models.
964:. For example, when the response
1518:refers to the Level 2 predictor.
695:refers to the Level 1 predictor.
405:
5873:Centre for Multilevel Modelling
5760:This concentrates on education.
5696:(4th ed.). London: Wiley.
5521:Introducing multilevel modeling
5480:
5433:
5376:
5342:Organizational Research Methods
5296:
5261:
5041:Cohen, Jacob (3 October 2003).
4268:Stage 1: Individual-Level Model
2063:Stage 1: Individual-Level Model
1882:
1109:{\displaystyle (X_{ij},Y_{ij})}
353:Least-squares spectral analysis
291:Generalized estimating equation
111:Multinomial logistic regression
86:Vector generalized linear model
5770:(2nd ed.). London: Sage.
5059:
4715:
4570:
4510:
4377:
4318:
4258:
4186:
4120:
4080:
3812:
3785:
3719:
3515:
3457:{\displaystyle \epsilon _{ij}}
3401:
3369:
3329:
3291:
3181:-th subject at the time point
3052:
3034:
2974:
2914:
2905:
2845:
2804:
2791:
2736:
2723:
2585:
2561:
2285:
2266:
2192:
2100:
2035:
1714:
1705:Bayesian information criterion
1252:
1228:
1184:
1160:
1103:
1071:
878:
854:
13:
1:
5693:Multilevel Statistical Models
4892:Using multivariate statistics
4805:
4795:Nonlinear mixed-effects model
1745:nonlinear mixed-effects model
1683:Developing a multilevel model
962:nonlinear mixed-effects model
172:Nonlinear mixed-effects model
5103:10.1371/journal.pone.0236860
1973:structural equation modeling
1965:hierarchical Bayesian models
1701:Akaike information criterion
1567:{\displaystyle \gamma _{11}}
1540:{\displaystyle \gamma _{01}}
1482:{\displaystyle \gamma _{10}}
1453:{\displaystyle \gamma _{00}}
517:. Individual differences in
18:Hierarchical linear modeling
7:
5603:10.1037/0033-2909.104.3.396
4768:
1124:Level 2 regression equation
752:{\displaystyle \beta _{1j}}
720:{\displaystyle \beta _{0j}}
533:Level 1 regression equation
374:Mean and predicted response
10:
5909:
5713:Longitudinal Data Analysis
5152:10.1007/s13571-020-00245-8
3487:{\displaystyle \eta _{li}}
2011:
1931:
455:hierarchical linear models
167:Linear mixed-effects model
4929:Luke, Douglas A. (2004).
4520:Stage 2: Population Model
2398:Stage 2: Population Model
2041:homogeneous than others.
1990:organizational psychology
459:linear mixed-effect model
333:Least absolute deviations
5458:10.1177/0149206319868016
5354:10.1177/1094428119877457
4800:Restricted randomization
1887:
81:Generalized linear model
2058:following three-stage:
1982:
1659:intraclass correlations
1652:Random intercepts model
479:random parameter models
5690:Goldstein, H. (2011).
5591:Psychological Bulletin
5282:10.1093/biomet/78.1.45
4759:
4736:
4087:
3792:
3488:
3458:
3428:
3408:
3356:
3336:
3275:
3255:
3235:
3234:{\displaystyle x_{ib}}
3205:
3204:{\displaystyle t_{ij}}
3175:
3155:
3154:{\displaystyle y_{ij}}
3123:
2681:
2463:
2388:
2054:
1870:
1839:
1838:{\displaystyle X_{ij}}
1809:
1808:{\displaystyle u_{0j}}
1739:
1632:
1631:{\displaystyle u_{1j}}
1600:
1599:{\displaystyle u_{0j}}
1568:
1541:
1512:
1483:
1454:
1425:
1342:
1259:
1191:
1110:
1058:
1038:
1037:{\displaystyle X_{ij}}
1008:
988:
987:{\displaystyle Y_{ij}}
954:
953:{\displaystyle X_{ij}}
924:
923:{\displaystyle Y_{ij}}
885:
815:
814:{\displaystyle r_{ij}}
785:
784:{\displaystyle e_{ij}}
753:
721:
689:
688:{\displaystyle X_{ij}}
657:
656:{\displaystyle Y_{ij}}
624:
412:Mathematics portal
338:Iteratively reweighted
5619:Lee, Se Yoon (2022).
5446:Journal of Management
5391:Psychological Methods
4760:
4737:
4088:
3793:
3489:
3459:
3429:
3409:
3357:
3337:
3276:
3261:-th covariate of the
3256:
3236:
3206:
3176:
3156:
3124:
2682:
2443:
2389:
2052:
2012:Further information:
1977:latent class modeling
1937:to add an additional
1869:
1840:
1810:
1738:
1693:likelihood-ratio test
1633:
1601:
1569:
1542:
1513:
1511:{\displaystyle w_{j}}
1484:
1455:
1426:
1343:
1260:
1192:
1111:
1059:
1039:
1009:
989:
955:
925:
886:
816:
786:
754:
722:
690:
658:
625:
511:multivariate analysis
475:random-effects models
369:Regression validation
348:Bayesian multivariate
65:Polynomial regression
5888:Analysis of variance
5648:10.3390/math10060898
5317:on 31 December 2010.
4790:Random effects model
4749:
4548:
4296:
4098:
3803:
3509:
3468:
3438:
3418:
3366:
3362:-dimensional vector
3346:
3285:
3265:
3245:
3215:
3185:
3165:
3135:
3083:
3005:
2835:
2767:
2699:
2616:
2534:
2406:
2316:
2239:
2071:
1942:categorical variable
1894:educational research
1819:
1789:
1612:
1580:
1551:
1524:
1495:
1466:
1437:
1353:
1270:
1202:
1134:
1068:
1048:
1018:
998:
968:
934:
904:
828:
795:
765:
733:
701:
669:
637:
543:
394:Gauss–Markov theorem
389:Studentized residual
379:Errors and residuals
213:Principal components
183:Nonlinear regression
70:General linear model
5840:10.7717/peerj.12794
5669:; Hill, J. (2007).
5094:2020PLoSO..1536860L
4986:10.7717/peerj.12794
4931:Multilevel modeling
4785:Multiscale modeling
4714:
4677:
4619:
4529:
4509:
4472:
4414:
4375:
4277:
4244:
4184:
4079:
4042:
3984:
3934:
3876:
3784:
3717:
3680:
3622:
3572:
3064:
3051:
3027:
2986:
2816:
2748:
2597:
2584:
2515:
2297:
2220:
1665:Random slopes model
1251:
1183:
877:
239:Errors-in-variables
106:Logistic regression
96:Binomial regression
41:Regression analysis
35:Part of a series on
5786:Swamy, P. A. V. B.
5403:10.1037/met0000182
5203:has generic name (
4755:
4732:
4730:
4694:
4639:
4599:
4489:
4434:
4394:
4337:
4206:
4139:
4083:
4059:
4004:
3964:
3896:
3831:
3788:
3739:
3697:
3642:
3602:
3534:
3484:
3454:
3424:
3404:
3352:
3332:
3271:
3251:
3231:
3201:
3171:
3151:
3119:
3117:
3037:
3013:
2677:
2675:
2570:
2384:
2382:
2055:
2026:ecological fallacy
1871:
1835:
1805:
1764:The assumption of
1740:
1628:
1596:
1564:
1537:
1508:
1479:
1450:
1421:
1338:
1255:
1237:
1187:
1169:
1106:
1054:
1034:
1004:
984:
950:
920:
881:
863:
811:
781:
749:
717:
685:
653:
620:
487:statistical models
483:split-plot designs
471:random coefficient
467:nested data models
126:Multinomial probit
5893:Regression models
5799:978-0-631-21254-6
5746:978-1-84872-845-5
5722:978-0-470-88918-3
5703:978-0-470-74865-7
5682:978-0-521-68689-1
5572:978-0-8058-3219-8
5530:978-0-7619-5141-4
5500:978-0-262-29679-3
5240:978-0-13-146597-8
5184:978-1-4129-9885-7
5052:978-0-8058-2223-6
4940:978-0-7619-2879-9
4901:978-0-205-45938-4
4839:978-0-7619-1904-9
4758:{\displaystyle f}
4726:
4562:
4521:
4310:
4269:
4112:
3427:{\displaystyle f}
3355:{\displaystyle K}
3274:{\displaystyle i}
3254:{\displaystyle b}
3174:{\displaystyle i}
1874:Statistical power
1849:Statistical tests
1118:logistic function
1057:{\displaystyle j}
1014:-th country, and
1007:{\displaystyle i}
515:repeated measures
500:linear regression
451:Multilevel models
448:
447:
101:Binary regression
60:Simple regression
55:Linear regression
16:(Redirected from
5900:
5862:
5852:
5842:
5812:
5803:
5781:
5759:
5750:
5726:
5707:
5686:
5653:
5652:
5650:
5640:
5616:
5607:
5606:
5586:
5577:
5576:
5555:
5549:
5548:
5542:
5534:
5516:
5505:
5504:
5484:
5478:
5477:
5437:
5431:
5430:
5388:
5380:
5374:
5373:
5339:
5330:
5319:
5318:
5316:
5310:. Archived from
5309:
5300:
5294:
5293:
5265:
5259:
5258:
5252:
5244:
5232:
5222:
5209:
5208:
5202:
5198:
5196:
5188:
5170:
5164:
5163:
5135:
5126:
5125:
5115:
5105:
5087:
5063:
5057:
5056:
5038:
5009:
5008:
4998:
4988:
4964:
4945:
4944:
4926:
4920:
4919:
4913:
4905:
4887:
4858:
4857:
4851:
4843:
4825:
4764:
4762:
4761:
4756:
4741:
4739:
4738:
4733:
4731:
4727:
4724:
4722:
4718:
4713:
4708:
4693:
4692:
4676:
4665:
4638:
4637:
4618:
4613:
4598:
4597:
4582:
4581:
4560:
4550:
4549:
4522:
4519:
4517:
4513:
4508:
4503:
4488:
4487:
4471:
4460:
4433:
4432:
4413:
4408:
4393:
4392:
4380:
4374:
4363:
4336:
4335:
4308:
4298:
4297:
4270:
4267:
4265:
4261:
4257:
4256:
4243:
4232:
4205:
4204:
4189:
4183:
4182:
4181:
4165:
4138:
4137:
4110:
4092:
4090:
4089:
4084:
4078:
4073:
4058:
4057:
4041:
4030:
4003:
4002:
3983:
3978:
3963:
3962:
3947:
3946:
3933:
3922:
3895:
3894:
3875:
3874:
3873:
3857:
3830:
3829:
3797:
3795:
3794:
3789:
3783:
3782:
3781:
3765:
3738:
3737:
3722:
3716:
3711:
3696:
3695:
3679:
3668:
3641:
3640:
3621:
3616:
3601:
3600:
3585:
3584:
3571:
3560:
3533:
3532:
3493:
3491:
3490:
3485:
3483:
3482:
3463:
3461:
3460:
3455:
3453:
3452:
3433:
3431:
3430:
3425:
3413:
3411:
3410:
3405:
3400:
3399:
3381:
3380:
3361:
3359:
3358:
3353:
3341:
3339:
3338:
3333:
3328:
3327:
3309:
3308:
3280:
3278:
3277:
3272:
3260:
3258:
3257:
3252:
3240:
3238:
3237:
3232:
3230:
3229:
3210:
3208:
3207:
3202:
3200:
3199:
3180:
3178:
3177:
3172:
3160:
3158:
3157:
3152:
3150:
3149:
3128:
3126:
3125:
3120:
3118:
3089:
3085:
3084:
3050:
3045:
3026:
3021:
3011:
3007:
3006:
2973:
2972:
2951:
2950:
2929:
2928:
2904:
2903:
2882:
2881:
2860:
2859:
2841:
2837:
2836:
2803:
2802:
2784:
2783:
2773:
2769:
2768:
2735:
2734:
2716:
2715:
2705:
2686:
2684:
2683:
2678:
2676:
2622:
2618:
2617:
2583:
2578:
2554:
2553:
2540:
2536:
2535:
2505:
2504:
2489:
2488:
2476:
2475:
2462:
2457:
2439:
2438:
2426:
2425:
2412:
2393:
2391:
2390:
2385:
2383:
2376:
2375:
2322:
2318:
2317:
2284:
2283:
2259:
2258:
2245:
2241:
2240:
2210:
2209:
2191:
2190:
2169:
2168:
2147:
2146:
2131:
2130:
2115:
2114:
2093:
2092:
2084:
2077:
1969:random variables
1844:
1842:
1841:
1836:
1834:
1833:
1814:
1812:
1811:
1806:
1804:
1803:
1766:homoscedasticity
1760:Homoscedasticity
1637:
1635:
1634:
1629:
1627:
1626:
1605:
1603:
1602:
1597:
1595:
1594:
1573:
1571:
1570:
1565:
1563:
1562:
1546:
1544:
1543:
1538:
1536:
1535:
1517:
1515:
1514:
1509:
1507:
1506:
1488:
1486:
1485:
1480:
1478:
1477:
1459:
1457:
1456:
1451:
1449:
1448:
1430:
1428:
1427:
1422:
1420:
1419:
1404:
1403:
1394:
1393:
1381:
1380:
1368:
1367:
1347:
1345:
1344:
1339:
1337:
1336:
1321:
1320:
1311:
1310:
1298:
1297:
1285:
1284:
1264:
1262:
1261:
1256:
1250:
1245:
1227:
1226:
1217:
1216:
1196:
1194:
1193:
1188:
1182:
1177:
1159:
1158:
1149:
1148:
1115:
1113:
1112:
1107:
1102:
1101:
1086:
1085:
1063:
1061:
1060:
1055:
1043:
1041:
1040:
1035:
1033:
1032:
1013:
1011:
1010:
1005:
993:
991:
990:
985:
983:
982:
959:
957:
956:
951:
949:
948:
929:
927:
926:
921:
919:
918:
890:
888:
887:
882:
876:
871:
853:
852:
843:
842:
820:
818:
817:
812:
810:
809:
790:
788:
787:
782:
780:
779:
758:
756:
755:
750:
748:
747:
726:
724:
723:
718:
716:
715:
694:
692:
691:
686:
684:
683:
662:
660:
659:
654:
652:
651:
629:
627:
626:
621:
619:
618:
603:
602:
590:
589:
574:
573:
558:
557:
498:(in particular,
440:
433:
426:
410:
409:
317:Ridge regression
152:Multilevel model
32:
31:
21:
5908:
5907:
5903:
5902:
5901:
5899:
5898:
5897:
5878:
5877:
5869:
5800:
5778:
5747:
5723:
5704:
5683:
5662:
5660:Further reading
5657:
5656:
5617:
5610:
5587:
5580:
5573:
5556:
5552:
5536:
5535:
5531:
5517:
5508:
5501:
5485:
5481:
5438:
5434:
5381:
5377:
5337:
5331:
5322:
5314:
5307:
5301:
5297:
5266:
5262:
5246:
5245:
5241:
5223:
5212:
5200:
5199:
5190:
5189:
5185:
5171:
5167:
5136:
5129:
5078:(7): e0236860.
5064:
5060:
5053:
5039:
5012:
4965:
4948:
4941:
4927:
4923:
4907:
4906:
4902:
4888:
4861:
4845:
4844:
4840:
4826:
4813:
4808:
4771:
4750:
4747:
4746:
4729:
4728:
4723:
4709:
4698:
4688:
4684:
4666:
4643:
4630:
4626:
4614:
4603:
4593:
4589:
4577:
4573:
4566:
4563:
4558:
4552:
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4528:
4527:
4524:
4523:
4518:
4504:
4493:
4483:
4479:
4461:
4438:
4425:
4421:
4409:
4398:
4388:
4384:
4376:
4364:
4341:
4328:
4324:
4314:
4311:
4306:
4300:
4299:
4276:
4275:
4272:
4271:
4266:
4252:
4248:
4233:
4210:
4197:
4193:
4185:
4177:
4173:
4166:
4143:
4130:
4126:
4116:
4113:
4108:
4101:
4099:
4096:
4095:
4074:
4063:
4053:
4049:
4031:
4008:
3995:
3991:
3979:
3968:
3958:
3954:
3942:
3938:
3923:
3900:
3887:
3883:
3869:
3865:
3858:
3835:
3822:
3818:
3804:
3801:
3800:
3777:
3773:
3766:
3743:
3730:
3726:
3718:
3712:
3701:
3691:
3687:
3669:
3646:
3633:
3629:
3617:
3606:
3596:
3592:
3580:
3576:
3561:
3538:
3525:
3521:
3510:
3507:
3506:
3475:
3471:
3469:
3466:
3465:
3445:
3441:
3439:
3436:
3435:
3419:
3416:
3415:
3395:
3391:
3376:
3372:
3367:
3364:
3363:
3347:
3344:
3343:
3323:
3319:
3304:
3300:
3286:
3283:
3282:
3266:
3263:
3262:
3246:
3243:
3242:
3222:
3218:
3216:
3213:
3212:
3192:
3188:
3186:
3183:
3182:
3166:
3163:
3162:
3142:
3138:
3136:
3133:
3132:
3116:
3115:
3087:
3086:
3063:
3062:
3059:
3058:
3046:
3041:
3022:
3017:
3009:
3008:
2985:
2984:
2981:
2980:
2965:
2961:
2943:
2939:
2921:
2917:
2896:
2892:
2874:
2870:
2852:
2848:
2839:
2838:
2815:
2814:
2811:
2810:
2798:
2794:
2779:
2775:
2771:
2770:
2747:
2746:
2743:
2742:
2730:
2726:
2711:
2707:
2702:
2700:
2697:
2696:
2674:
2673:
2620:
2619:
2596:
2595:
2592:
2591:
2579:
2574:
2546:
2542:
2538:
2537:
2514:
2513:
2510:
2509:
2497:
2493:
2481:
2477:
2468:
2464:
2458:
2447:
2434:
2430:
2418:
2414:
2409:
2407:
2404:
2403:
2381:
2380:
2371:
2367:
2320:
2319:
2296:
2295:
2292:
2291:
2279:
2275:
2251:
2247:
2243:
2242:
2219:
2218:
2215:
2214:
2202:
2198:
2183:
2179:
2161:
2157:
2139:
2135:
2123:
2119:
2107:
2103:
2085:
2080:
2079:
2074:
2072:
2069:
2068:
2047:
2038:
2021:
2016:
2010:
1985:
1958:hyperparameters
1954:Mobile, Alabama
1934:
1890:
1885:
1876:
1851:
1826:
1822:
1820:
1817:
1816:
1796:
1792:
1790:
1787:
1786:
1717:
1709:Model selection
1685:
1676:
1667:
1654:
1645:
1643:Types of models
1619:
1615:
1613:
1610:
1609:
1587:
1583:
1581:
1578:
1577:
1558:
1554:
1552:
1549:
1548:
1531:
1527:
1525:
1522:
1521:
1502:
1498:
1496:
1493:
1492:
1473:
1469:
1467:
1464:
1463:
1444:
1440:
1438:
1435:
1434:
1412:
1408:
1399:
1395:
1389:
1385:
1376:
1372:
1360:
1356:
1354:
1351:
1350:
1329:
1325:
1316:
1312:
1306:
1302:
1293:
1289:
1277:
1273:
1271:
1268:
1267:
1246:
1241:
1222:
1221:
1209:
1205:
1203:
1200:
1199:
1178:
1173:
1154:
1153:
1141:
1137:
1135:
1132:
1131:
1126:
1094:
1090:
1078:
1074:
1069:
1066:
1065:
1049:
1046:
1045:
1044:represents the
1025:
1021:
1019:
1016:
1015:
999:
996:
995:
975:
971:
969:
966:
965:
941:
937:
935:
932:
931:
911:
907:
905:
902:
901:
872:
867:
848:
847:
835:
831:
829:
826:
825:
802:
798:
796:
793:
792:
772:
768:
766:
763:
762:
740:
736:
734:
731:
730:
708:
704:
702:
699:
698:
676:
672:
670:
667:
666:
644:
640:
638:
635:
634:
611:
607:
595:
591:
582:
578:
566:
562:
550:
546:
544:
541:
540:
535:
453:(also known as
444:
404:
384:Goodness of fit
91:Discrete choice
28:
23:
22:
15:
12:
11:
5:
5906:
5896:
5895:
5890:
5876:
5875:
5868:
5867:External links
5865:
5864:
5863:
5818:
5804:
5798:
5782:
5776:
5761:
5751:
5745:
5727:
5721:
5708:
5702:
5687:
5681:
5661:
5658:
5655:
5654:
5608:
5597:(3): 396–404.
5578:
5571:
5550:
5529:
5506:
5499:
5479:
5432:
5375:
5348:(2): 443–483.
5320:
5295:
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5239:
5210:
5183:
5165:
5127:
5058:
5051:
5010:
4946:
4939:
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4838:
4810:
4809:
4807:
4804:
4803:
4802:
4797:
4792:
4787:
4782:
4777:
4775:Hyperparameter
4770:
4767:
4754:
4725:Stage 3: Prior
4721:
4717:
4712:
4707:
4704:
4701:
4697:
4691:
4687:
4683:
4680:
4675:
4672:
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4407:
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4391:
4387:
4383:
4379:
4373:
4370:
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4362:
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4347:
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4280:
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4255:
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4242:
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4236:
4231:
4228:
4225:
4222:
4219:
4216:
4213:
4209:
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4200:
4196:
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4180:
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4115:
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4020:
4017:
4014:
4011:
4007:
4001:
3998:
3994:
3990:
3987:
3982:
3977:
3974:
3971:
3967:
3961:
3957:
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3950:
3945:
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3937:
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3899:
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3890:
3886:
3882:
3879:
3872:
3868:
3864:
3861:
3856:
3853:
3850:
3847:
3844:
3841:
3838:
3834:
3828:
3825:
3821:
3817:
3814:
3811:
3808:
3787:
3780:
3776:
3772:
3769:
3764:
3761:
3758:
3755:
3752:
3749:
3746:
3742:
3736:
3733:
3729:
3725:
3721:
3715:
3710:
3707:
3704:
3700:
3694:
3690:
3686:
3683:
3678:
3675:
3672:
3667:
3664:
3661:
3658:
3655:
3652:
3649:
3645:
3639:
3636:
3632:
3628:
3625:
3620:
3615:
3612:
3609:
3605:
3599:
3595:
3591:
3588:
3583:
3579:
3575:
3570:
3567:
3564:
3559:
3556:
3553:
3550:
3547:
3544:
3541:
3537:
3531:
3528:
3524:
3520:
3517:
3514:
3497:Stage 3: Prior
3481:
3478:
3474:
3451:
3448:
3444:
3423:
3403:
3398:
3394:
3390:
3387:
3384:
3379:
3375:
3371:
3351:
3331:
3326:
3322:
3318:
3315:
3312:
3307:
3303:
3299:
3296:
3293:
3290:
3270:
3250:
3228:
3225:
3221:
3198:
3195:
3191:
3170:
3148:
3145:
3141:
3114:
3111:
3108:
3105:
3102:
3099:
3096:
3093:
3090:
3088:
3082:
3079:
3076:
3073:
3070:
3067:
3061:
3060:
3057:
3054:
3049:
3044:
3040:
3036:
3033:
3030:
3025:
3020:
3016:
3012:
3010:
3004:
3001:
2998:
2995:
2992:
2989:
2983:
2982:
2979:
2976:
2971:
2968:
2964:
2960:
2957:
2954:
2949:
2946:
2942:
2938:
2935:
2932:
2927:
2924:
2920:
2916:
2913:
2910:
2907:
2902:
2899:
2895:
2891:
2888:
2885:
2880:
2877:
2873:
2869:
2866:
2863:
2858:
2855:
2851:
2847:
2844:
2842:
2840:
2834:
2831:
2828:
2825:
2822:
2819:
2813:
2812:
2809:
2806:
2801:
2797:
2793:
2790:
2787:
2782:
2778:
2774:
2772:
2766:
2763:
2760:
2757:
2754:
2751:
2745:
2744:
2741:
2738:
2733:
2729:
2725:
2722:
2719:
2714:
2710:
2706:
2704:
2691:Stage 3: Prior
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2651:
2647:
2644:
2641:
2638:
2635:
2632:
2629:
2626:
2623:
2621:
2615:
2612:
2609:
2606:
2603:
2600:
2594:
2593:
2590:
2587:
2582:
2577:
2573:
2569:
2566:
2563:
2560:
2557:
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2549:
2545:
2541:
2539:
2533:
2530:
2527:
2524:
2521:
2518:
2512:
2511:
2508:
2503:
2500:
2496:
2492:
2487:
2484:
2480:
2474:
2471:
2467:
2461:
2456:
2453:
2450:
2446:
2442:
2437:
2433:
2429:
2424:
2421:
2417:
2413:
2411:
2379:
2374:
2370:
2366:
2363:
2360:
2357:
2354:
2351:
2347:
2344:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2321:
2315:
2312:
2309:
2306:
2303:
2300:
2294:
2293:
2290:
2287:
2282:
2278:
2274:
2271:
2268:
2265:
2262:
2257:
2254:
2250:
2246:
2244:
2238:
2235:
2232:
2229:
2226:
2223:
2217:
2216:
2213:
2208:
2205:
2201:
2197:
2194:
2189:
2186:
2182:
2178:
2175:
2172:
2167:
2164:
2160:
2156:
2153:
2150:
2145:
2142:
2138:
2134:
2129:
2126:
2122:
2118:
2113:
2110:
2106:
2102:
2099:
2096:
2091:
2088:
2083:
2078:
2076:
2046:
2043:
2037:
2034:
2020:
2017:
2009:
2006:
1984:
1981:
1933:
1930:
1922:
1921:
1918:
1915:
1912:
1905:
1904:
1901:
1889:
1886:
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1622:
1618:
1607:
1593:
1590:
1586:
1575:
1561:
1557:
1534:
1530:
1519:
1505:
1501:
1490:
1476:
1472:
1461:
1447:
1443:
1418:
1415:
1411:
1407:
1402:
1398:
1392:
1388:
1384:
1379:
1375:
1371:
1366:
1363:
1359:
1335:
1332:
1328:
1324:
1319:
1315:
1309:
1305:
1301:
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1288:
1283:
1280:
1276:
1254:
1249:
1244:
1240:
1236:
1233:
1230:
1225:
1220:
1215:
1212:
1208:
1186:
1181:
1176:
1172:
1168:
1165:
1162:
1157:
1152:
1147:
1144:
1140:
1125:
1122:
1105:
1100:
1097:
1093:
1089:
1084:
1081:
1077:
1073:
1053:
1031:
1028:
1024:
1003:
981:
978:
974:
947:
944:
940:
930:and predictor
917:
914:
910:
880:
875:
870:
866:
862:
859:
856:
851:
846:
841:
838:
834:
823:
822:
808:
805:
801:
778:
775:
771:
760:
746:
743:
739:
728:
714:
711:
707:
696:
682:
679:
675:
664:
650:
647:
643:
617:
614:
610:
606:
601:
598:
594:
588:
585:
581:
577:
572:
569:
565:
561:
556:
553:
549:
534:
531:
446:
445:
443:
442:
435:
428:
420:
417:
416:
415:
414:
399:
398:
397:
396:
391:
386:
381:
376:
371:
363:
362:
358:
357:
356:
355:
350:
345:
340:
335:
327:
326:
325:
324:
319:
314:
309:
304:
296:
295:
294:
293:
288:
283:
278:
270:
269:
268:
267:
262:
257:
249:
248:
244:
243:
242:
241:
233:
232:
231:
230:
225:
220:
215:
210:
205:
200:
195:
193:Semiparametric
190:
185:
177:
176:
175:
174:
169:
164:
162:Random effects
159:
154:
146:
145:
144:
143:
138:
136:Ordered probit
133:
128:
123:
118:
113:
108:
103:
98:
93:
88:
83:
75:
74:
73:
72:
67:
62:
57:
49:
48:
44:
43:
37:
36:
26:
9:
6:
4:
3:
2:
5905:
5894:
5891:
5889:
5886:
5885:
5883:
5874:
5871:
5870:
5860:
5856:
5851:
5846:
5841:
5836:
5832:
5828:
5824:
5819:
5816:
5810:
5805:
5801:
5795:
5791:
5787:
5783:
5779:
5777:9781446254332
5773:
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5757:
5752:
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5742:
5738:
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5728:
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5714:
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5554:
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5515:
5513:
5511:
5502:
5496:
5493:. MIT Press.
5492:
5491:
5483:
5475:
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4997:
4992:
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3414:. Typically,
3396:
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2996:
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2364:
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2339:
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2330:
2327:
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2252:
2248:
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2233:
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2227:
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2206:
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2199:
2195:
2187:
2184:
2180:
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2173:
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2162:
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2124:
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2111:
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2104:
2097:
2094:
2089:
2086:
2081:
2066:
2065:
2064:
2059:
2051:
2042:
2033:
2029:
2027:
2015:
2005:
2001:
1997:
1995:
1991:
1980:
1978:
1975:, multilevel
1974:
1970:
1966:
1961:
1959:
1955:
1951:
1946:
1943:
1940:
1929:
1927:
1919:
1916:
1913:
1910:
1909:
1908:
1902:
1899:
1898:
1897:
1895:
1880:
1868:
1864:
1861:
1857:
1846:
1830:
1827:
1823:
1800:
1797:
1793:
1780:
1779:
1778:
1771:
1770:
1769:
1767:
1759:
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1757:
1750:
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1748:
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1737:
1730:
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1728:
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1722:
1712:
1710:
1706:
1703:(AIC) or the
1702:
1698:
1694:
1689:
1680:
1671:
1662:
1660:
1649:
1623:
1620:
1616:
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1588:
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1532:
1528:
1520:
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1470:
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1179:
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912:
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891:
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806:
803:
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680:
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645:
641:
633:
632:
631:
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612:
608:
604:
599:
596:
592:
586:
583:
579:
575:
570:
567:
563:
559:
554:
551:
547:
538:
530:
526:
524:
520:
519:growth curves
516:
512:
508:
503:
501:
497:
496:linear models
492:
488:
484:
480:
476:
472:
468:
464:
460:
456:
452:
441:
436:
434:
429:
427:
422:
421:
419:
418:
413:
408:
403:
402:
401:
400:
395:
392:
390:
387:
385:
382:
380:
377:
375:
372:
370:
367:
366:
365:
364:
360:
359:
354:
351:
349:
346:
344:
341:
339:
336:
334:
331:
330:
329:
328:
323:
320:
318:
315:
313:
310:
308:
305:
303:
300:
299:
298:
297:
292:
289:
287:
284:
282:
279:
277:
274:
273:
272:
271:
266:
263:
261:
258:
256:
255:Least squares
253:
252:
251:
250:
246:
245:
240:
237:
236:
235:
234:
229:
226:
224:
221:
219:
216:
214:
211:
209:
206:
204:
201:
199:
196:
194:
191:
189:
188:Nonparametric
186:
184:
181:
180:
179:
178:
173:
170:
168:
165:
163:
160:
158:
157:Fixed effects
155:
153:
150:
149:
148:
147:
142:
139:
137:
134:
132:
131:Ordered logit
129:
127:
124:
122:
119:
117:
114:
112:
109:
107:
104:
102:
99:
97:
94:
92:
89:
87:
84:
82:
79:
78:
77:
76:
71:
68:
66:
63:
61:
58:
56:
53:
52:
51:
50:
46:
45:
42:
39:
38:
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33:
30:
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5692:
5671:
5628:
5624:
5594:
5590:
5562:
5553:
5520:
5489:
5482:
5452:(1): 70–99.
5449:
5445:
5435:
5397:(1): 20–35.
5394:
5390:
5378:
5345:
5341:
5312:the original
5298:
5276:(1): 45–51.
5273:
5269:
5263:
5228:
5174:
5168:
5143:
5139:
5075:
5071:
5061:
5042:
4976:
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2397:
2396:
2395:
2067:
2062:
2061:
2060:
2056:
2039:
2030:
2022:
2002:
1998:
1994:mixed models
1986:
1962:
1947:
1935:
1923:
1906:
1891:
1883:Applications
1877:
1852:
1784:
1775:
1763:
1754:
1741:
1718:
1690:
1686:
1677:
1668:
1655:
1646:
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1266:
1198:
1130:
1127:
899:
896:
892:
824:
539:
536:
527:
504:
482:
478:
474:
470:
466:
463:mixed models
462:
458:
454:
450:
449:
312:Non-negative
151:
29:
5811:. Springer.
5625:Mathematics
5201:|last=
2036:Error terms
1939:independent
1715:Assumptions
507:nested data
322:Regularized
286:Generalized
218:Least angle
116:Mixed logit
5882:Categories
5833:: e12794.
5731:Hox, J. J.
5667:Gelman, A.
5638:2201.12430
5631:(6): 898.
5270:Biometrika
5085:2005.00662
4979:: e12794.
4806:References
1725:regression
491:parameters
361:Background
265:Non-linear
247:Estimation
5813:Includes
5559:Hox, Joop
5539:cite book
5474:202288849
5466:0149-2063
5411:1939-1463
5370:210355362
5362:1094-4281
5249:cite book
5193:cite book
5160:234027590
5140:Sankhya B
4910:cite book
4848:cite book
4686:ω
4628:β
4591:α
4575:σ
4556:×
4481:ω
4423:β
4386:α
4326:θ
4316:π
4304:×
4250:σ
4195:θ
4118:π
4051:ω
3993:β
3956:α
3940:σ
3885:θ
3810:π
3807:∝
3689:ω
3631:β
3594:α
3578:σ
3523:θ
3513:π
3473:η
3443:ϵ
3393:θ
3386:…
3374:θ
3321:θ
3314:…
3302:θ
3104:…
3039:ω
3032:π
3029:∼
3015:ω
2963:β
2956:…
2941:β
2934:…
2919:β
2912:π
2909:∼
2894:β
2887:…
2872:β
2865:…
2850:β
2796:α
2789:π
2786:∼
2777:α
2728:σ
2721:π
2718:∼
2709:σ
2662:…
2637:…
2572:ω
2556:∼
2544:η
2495:η
2466:β
2445:∑
2432:α
2416:θ
2362:…
2337:…
2277:σ
2261:∼
2249:ϵ
2200:ϵ
2181:θ
2174:…
2159:θ
2152:…
2137:θ
2121:θ
1751:Normality
1731:Linearity
1556:γ
1529:γ
1471:γ
1442:γ
1387:γ
1374:γ
1358:β
1304:γ
1291:γ
1275:β
1239:σ
1219:∼
1171:σ
1151:∼
865:σ
845:∼
738:β
706:β
580:β
564:β
228:Segmented
5859:35116198
5733:(2010).
5561:(2002).
5427:44145669
5419:29863377
5146:: 1–43.
5122:32726361
5072:PLOS ONE
5005:35116198
4769:See also
1926:variable
1920:district
343:Bayesian
281:Weighted
276:Ordinary
208:Isotonic
203:Quantile
5850:8784019
5290:2336894
5113:7390340
5090:Bibcode
4996:8784019
3241:is the
1950:Seattle
1932:Example
302:Partial
141:Poisson
5857:
5847:
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3211:, and
3131:Here,
1917:school
1860:t-test
1856:Z-test
1697:nested
523:ANCOVA
485:) are
260:Linear
198:Robust
121:Probit
47:Models
5827:PeerJ
5633:arXiv
5470:S2CID
5423:S2CID
5366:S2CID
5338:(PDF)
5315:(PDF)
5308:(PDF)
5286:JSTOR
5156:S2CID
5080:arXiv
4973:PeerJ
3464:and
1914:class
1911:pupil
1903:class
1900:pupil
1888:Level
1721:ANOVA
481:, or
307:Total
223:Local
5855:PMID
5817:code
5794:ISBN
5772:ISBN
5741:ISBN
5717:ISBN
5698:ISBN
5677:ISBN
5567:ISBN
5545:link
5525:ISBN
5495:ISBN
5462:ISSN
5415:PMID
5407:ISSN
5358:ISSN
5255:link
5235:ISBN
5205:help
5179:ISBN
5118:PMID
5047:ISBN
5001:PMID
4935:ISBN
4916:link
4896:ISBN
4854:link
4834:ISBN
1983:Uses
1858:. A
1547:and
5845:PMC
5835:doi
5815:SAS
5643:doi
5599:doi
5595:104
5454:doi
5399:doi
5350:doi
5278:doi
5148:doi
5108:PMC
5098:doi
4991:PMC
4981:doi
513:of
489:of
5884::
5853:.
5843:.
5831:10
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4584:,
4579:2
4571:(
4568:p
4546:r
4543:e
4540:c
4537:a
4534:p
4531:s
4515:}
4511:)
4506:K
4501:1
4498:=
4495:l
4491:}
4485:l
4477:{
4474:,
4469:P
4466:,
4463:K
4458:1
4455:=
4452:b
4449:,
4446:1
4443:=
4440:l
4436:}
4430:b
4427:l
4419:{
4416:,
4411:K
4406:1
4403:=
4400:l
4396:}
4390:l
4382:{
4378:|
4372:K
4369:,
4366:N
4361:1
4358:=
4355:l
4352:,
4349:1
4346:=
4343:i
4339:}
4333:i
4330:l
4322:{
4319:(
4294:r
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4288:c
4285:a
4282:p
4279:s
4263:}
4259:)
4254:2
4246:,
4241:K
4238:,
4235:N
4230:1
4227:=
4224:l
4221:,
4218:1
4215:=
4212:i
4208:}
4202:i
4199:l
4191:{
4187:|
4179:i
4175:M
4171:,
4168:N
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4154:,
4151:1
4148:=
4145:i
4141:}
4135:j
4132:i
4128:y
4124:{
4121:(
4106:=
4081:)
4076:K
4071:1
4068:=
4065:l
4061:}
4055:l
4047:{
4044:,
4039:P
4036:,
4033:K
4028:1
4025:=
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4019:,
4016:1
4013:=
4010:l
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4000:b
3997:l
3989:{
3986:,
3981:K
3976:1
3973:=
3970:l
3966:}
3960:l
3952:{
3949:,
3944:2
3936:,
3931:K
3928:,
3925:N
3920:1
3917:=
3914:l
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3908:1
3905:=
3902:i
3898:}
3892:i
3889:l
3881:{
3878:,
3871:i
3867:M
3863:,
3860:N
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3843:1
3840:=
3837:i
3833:}
3827:j
3824:i
3820:y
3816:{
3813:(
3786:)
3779:i
3775:M
3771:,
3768:N
3763:1
3760:=
3757:j
3754:,
3751:1
3748:=
3745:i
3741:}
3735:j
3732:i
3728:y
3724:{
3720:|
3714:K
3709:1
3706:=
3703:l
3699:}
3693:l
3685:{
3682:,
3677:P
3674:,
3671:K
3666:1
3663:=
3660:b
3657:,
3654:1
3651:=
3648:l
3644:}
3638:b
3635:l
3627:{
3624:,
3619:K
3614:1
3611:=
3608:l
3604:}
3598:l
3590:{
3587:,
3582:2
3574:,
3569:K
3566:,
3563:N
3558:1
3555:=
3552:l
3549:,
3546:1
3543:=
3540:i
3536:}
3530:i
3527:l
3519:{
3516:(
3480:i
3477:l
3450:j
3447:i
3422:f
3402:)
3397:K
3389:,
3383:,
3378:1
3370:(
3350:K
3330:)
3325:K
3317:,
3311:,
3306:1
3298:;
3295:t
3292:(
3289:f
3269:i
3249:b
3227:b
3224:i
3220:x
3197:j
3194:i
3190:t
3169:i
3147:j
3144:i
3140:y
3113:.
3110:K
3107:,
3101:,
3098:1
3095:=
3092:l
3081:r
3078:e
3075:c
3072:a
3069:p
3066:s
3056:,
3053:)
3048:2
3043:l
3035:(
3024:2
3019:l
3003:r
3000:e
2997:c
2994:a
2991:p
2988:s
2978:,
2975:)
2970:P
2967:l
2959:,
2953:,
2948:b
2945:l
2937:,
2931:,
2926:1
2923:l
2915:(
2906:)
2901:P
2898:l
2890:,
2884:,
2879:b
2876:l
2868:,
2862:,
2857:1
2854:l
2846:(
2833:r
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2824:a
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2818:s
2808:,
2805:)
2800:l
2792:(
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2759:c
2756:a
2753:p
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2740:,
2737:)
2732:2
2724:(
2713:2
2671:.
2668:K
2665:,
2659:,
2656:1
2653:=
2650:l
2646:,
2643:N
2640:,
2634:,
2631:1
2628:=
2625:i
2614:r
2611:e
2608:c
2605:a
2602:p
2599:s
2589:,
2586:)
2581:2
2576:l
2568:,
2565:0
2562:(
2559:N
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2532:r
2529:e
2526:c
2523:a
2520:p
2517:s
2507:,
2502:i
2499:l
2491:+
2486:b
2483:i
2479:x
2473:b
2470:l
2460:P
2455:1
2452:=
2449:b
2441:+
2436:l
2428:=
2423:i
2420:l
2378:.
2373:i
2369:M
2365:,
2359:,
2356:1
2353:=
2350:j
2346:,
2343:N
2340:,
2334:,
2331:1
2328:=
2325:i
2314:r
2311:e
2308:c
2305:a
2302:p
2299:s
2289:,
2286:)
2281:2
2273:,
2270:0
2267:(
2264:N
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2253:i
2237:r
2234:e
2231:c
2228:a
2225:p
2222:s
2212:,
2207:j
2204:i
2196:+
2193:)
2188:i
2185:K
2177:,
2171:,
2166:i
2163:l
2155:,
2149:,
2144:i
2141:2
2133:,
2128:i
2125:1
2117:;
2112:j
2109:i
2105:t
2101:(
2098:f
2095:=
2090:j
2087:i
2082:y
1831:j
1828:i
1824:X
1801:j
1798:0
1794:u
1624:j
1621:1
1617:u
1592:j
1589:0
1585:u
1504:j
1500:w
1417:j
1414:1
1410:u
1406:+
1401:j
1397:w
1383:+
1370:=
1365:j
1362:1
1334:j
1331:0
1327:u
1323:+
1318:j
1314:w
1300:+
1287:=
1282:j
1279:0
1253:)
1248:2
1243:3
1235:,
1232:0
1229:(
1224:N
1214:j
1211:1
1207:u
1185:)
1180:2
1175:2
1167:,
1164:0
1161:(
1156:N
1146:j
1143:0
1139:u
1104:)
1099:j
1096:i
1092:Y
1088:,
1083:j
1080:i
1076:X
1072:(
1052:j
1030:j
1027:i
1023:X
1002:i
980:j
977:i
973:Y
946:j
943:i
939:X
916:j
913:i
909:Y
879:)
874:2
869:1
861:,
858:0
855:(
850:N
840:j
837:i
833:e
807:j
804:i
800:r
777:j
774:i
770:e
745:j
742:1
713:j
710:0
681:j
678:i
674:X
649:j
646:i
642:Y
616:j
613:i
609:e
605:+
600:j
597:i
593:X
587:j
584:1
576:+
571:j
568:0
560:=
555:j
552:i
548:Y
439:e
432:t
425:v
20:)
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