32, 3899 - 3910. Annals of Statistics. For the statistical inferences about the regression parameters β, valid standard errors can be obtained based on the above sandwich estimator Σˆ=Bˆ−1MˆBˆ−1 In fact, both GEE estimate of β and the sandwich estimator to Cov(βˆ) are robust in the sense that it is still valid even if the within-subject associations have been incorrectly specified in the marginal model. Yang, H., Li, R., Zucker, R and Buu, A. While not detailed above, covariate effects on occupancy probabilities can also be incorporated, as for the simpler models already discussed. That is, the probability of occurrence for species B is unaffected by the presence of species A; the two species occupy units independently. Time-varying 31, L17107, doi:10.1029/2004GL020490. 41, 2505 - 2536. 70, 337-355. For example, in an AD treatment clinical trial, the average difference between control and treatment is the most important, not the difference for any single subject. The term σɛij2 is not specified in θ because within-subject random errors are approximated in the score function of the fixed effects in the mixed-effects logit model, thereby being integrated out (Zeger et al., 1988). The expectation of Pij|θ, given the case of the random intercept logit model for analytic simplicity, is then given by. Climate control to terrestrial carbon exchange across biomes and [pdf], Kurum, E., Li, R., Shiffman, S. and Yao, W. (2016). Annals of Statistics, 36, 1509-1566. examine dynamic associations between negative affect and self confidence on smoking urges: differences between Journal of Business and Economic Statistics. In predicting longitudinal trajectories of a set of response probabilities, the standard errors of nonlinear predictions should be approximated for evaluating the quality of the predicted values. This paper assesses the options available to researchers analysing multilevel (including longitudinal) data, with the aim of supporting good methodological decision-making. Notice here that the vector of covariates could be time dependent. Statistical Survey, 13, 150 -180. An example where such a dependency exists (and hence should be avoided) is if budget cuts necessitate a reduction in the number of sampling units being monitored, and investigators respond by discontinuing monitoring of those units they think are unoccupied by the species (i.e., knowledge about the likely occupancy state of units is used to determine which ones will become ‘missing’ observations). For example, odds of 3:1 suggest the probability of success is 3 times that of a failure. estimation via refitted cross validation. In other words, the entire set or a subset of the subject-specific regression coefficients are assumed to be random variables across study subjects which follow a univariate or a multivariate normal distribution. Biometrika, 104, 829 - 843. [pdf], Fan, J. and Li, R. (2001). (2005). 37 Full PDFs related to this paper. As with the modeling described elsewhere in this book, incorporating predictor variables or covariates into the analysis for various probabilities can be easily accomplished through the use of link functions (Chapter 3). 133, 548 - 555. [pdf], Cai, Z., Fan, J. and Li, R. (2000). Evaluation of reproducibility for paired functional data. (2016). the variables selected by SCAD. [pdf], Li, J., Wang, Z., Li, R. and Wu, R. (2015). appear. Analysis of computer experiments using penalized likelihood in Gaussian kriging For example, some species may prefer particular habitat types over other available habitats (e.g., have a higher occupancy probability at locations near permanent water sources); require a minimum patch size for a sustainable population; or show reduced probability of occurrence in isolated patches (e.g., Verner et al., 1986; Hanski, 1999; Scott et al., 2002). Statistica Sinica, 28, 2713 - 2731. [pdf], Hunter, D. and Li, R. (2005). An application of the time-varying effects model (TVEM). varying-coefficient models with application to functional genome-wide association The Journal of Primary Prevention. [pdf], Lan, W., Zhong, P., Li, R., Tsai, C.-L. and Wang, H. (2016). (2015). Although, recall that when using the original parameterization of MacKenzie et al. [pdf], Cai, J. 58, 2033 - 2054. The key drawback is that debtor-specific variables (e.g., credit bureau information and balance sheet ratios) usually play a predominant role compared with macroeconomic components. Shiyko, M.P., Burkhalter, J., Li., R., and Park, B. J. [pdf], Zhang, A., Fang, K.-T., Li, R. and Sudjianto, A. [pdf], Li, R. and Sudjianto, A. Under this model, the intercept, α, remains constant because the heterogeneity is induced by mixing over the abundance distribution. [pdf] On the Feasibility of Distributed Kernel Regression for Big Error variance estimation in ultrahigh dimensional additive models. Similar to the standard asymptotic properties of ML estimates, when the sample size is sufficiently large, βˆ follows an asymptotically multivariate normal distribution with mean β and a covariance matrix which can be estimated by the so-called ‘sandwich’ estimator. Nations' World Meteorological Organization Gerbier-Mumm International Award for 2012, Eberly Family Chair in Statistics, Penn State University, 2018 -, Professor of Public Health Sciences, Penn State University, 2008 -, Verne M. Willaman Professor of Statistics, Penn State University, 2014 - 2018, Distinguished Professor, Penn State University, 2012 - 2014, Full Professor, Penn State University, 2008 - 2012, Associate Professor, Penn State University, 2005 - 2008, Assistant Professor, Penn State University, 2000 - 2005. When there are > two outcomes, there are multiple probabilities associated with the observations, and in addition to the individual probabilities having to be between 0 and 1, the sum of the set of probabilities must also be ⩽1. Download Free PDF. Huang, D., Zhu, X., Li, R. and Wang, H. (2021). where Aj1/2 is the diagonal matrix such that (Aj1/2)2=Aj, and Aj the diagonal matrix consisting of the variance of Yjk, and CORR(Yj) the correlation matrix of Yj depending on the set of parameters α's (also possibly β's). The reason that Vj is called the ‘working’ covariance matrix of Yj is that it is not necessarily the same as the true covariance matrix of Yj. [pdf], Fan, J. and Li, R. (2006). 40, 2634 - 2666. In this regard, banks have developed different approaches. The 1st International Conference on Big Data & Applied Statistics, December 2014 Buu, A., Li, R., Tan, X. and Zucker, R. A. Distributed feature screening via componentwise Wang, Y., Xu, M., Wang, Z., Tao, M., Zhu, J., Li, R., Wang, L. Berceli, S.A. and Wu, R. (2011). Nonparametric modeling and inference function for longitudinal data. Folded concave Tan, X., Dierker, L., Li, R., Rose, J., and The Tobacco Etiology Research Network(TERN). 1 - 25. Time-varying effect model for ordinal responses with applications in substance abuse research. As a result, the confidence interval for the predicted probability Pˆijk can be easily computed. A different perspective is followed when a panel analysis is applied. 93, 81-101. Scalar-on-function regression for predicting distal outcomes from intensively gathered longitudinal (2019). A block mixture model to map eQTLs for Liu, H., Wang, X., Yao, T., Li, R. and Ye, Y. (2020). Statistica Sinica, 26, 979 - 1000. algorithm theory for local solutions. However, they all were characterized by difficulties in terms of implementation and opacity when interpretation was necessary. Suppose that the following model is defined for the parameters ψB|a and ψB|A, using a matrix based notation for convenience (see Appendix): which implies that the probability of species B being present at a unit has the same covariate relationships irrespective of whether species A is absent or present at a unit. Journal of Machine Learning and Research, 21(10). Murray, M. and Leslie, D. (2019). [pdf], Wang, L., Peng, B. and Li, R. (2015). Developmental Psychology, 47, 1078 - 1089. [pdf] 760 - 772. Journal of American Statistical Association. Let Xj = (Xj1, Xj2,…,Xjkj)t be the design matrix of the jth subject. Zhang, L., Wang, X., Wang, M., Sterling, N. W., Du, G. Boca Raton, FL. genetic variants in GWAS using permutation-assisted lasso tuning. Two general approaches of the estimation are used in the random effects models. Annals of Statistics. [pdf]. Challenges and opportunities in collecting and 96, 1348-1360. Journal of Machine Learning and Research, 21(24). ASA Biometrics Section program chair for JSM 2007, August 2007, Salt Lake City, Utah. Semiparametric estimation of conditional heteroscedasticity through single index Statistical Challenges with High Dimensionality: Feature Selection in Knowledge Discovery. The SIF factor would be calculated as: When using the parameterization of Rota et al. [pdf], Qu, A. and Li, R. (2006). Consistent selection of the number of change-points via Sometimes the difference between two related transition probabilities needs to be tested statistically. where β0 is the first term of β. Portal nodes screening for large scale social networks. Therefore, this same biological model could have been fit using the ψA, ψB|a and ν parameterization, with ψB|a and ν defined by Eqs. (2017). [pdf] More specifically, the conditional expectation of Yjk, given Xjk, is E(Yjk|Xjk) = μjk, and is assumed to depend on the covariates through the log-link function. In many problems, the possibility that detection varies temporally, or in relation to measurable covariates might be considered, perhaps in addition to heterogeneity among units. Shiyko, M. P., Lanza, S. T., Tan, X., Li, R. and Shiffman, S. (2012). Annals of Statistics. When the link function is non-linear, however, the interpretations for the regression parameters in generalized linear mixed models are distinct from those in the marginal models. Depression In particular, expert assessments and statistical models are commonly combined to infer the likelihood of a customer becoming insolvent. In press. Neural networks, classification and regression trees, data envelopment analysis, and many other approaches appeared afterward. Literally, the integral denotes the averaging over the distribution of bi when only the random intercept is specified. Liu, X., Li, R., Lanza, S.T., Vasilenko, S. and Piper, M. (2013). fatigue in the smoking cessation process. data analysis. Frontiers in Neurology, 8, 501. The delta method depends on the validity of the Taylor series approximation, and therefore, some caution must be exercised before its adequacy is verified with simulation. 31, 4074 - 4086. Journal of American Statistical Association, 99, 710-723. varying-coefficient models. Beijing, China. Journal of American Statistical Association. [pdf], Chen, Z., Li, R. and Li, Y. [Supplement], Zhu, L., Xu, K., Li, R. and Zhong, W. (2017). Similarly, the probability of detecting species at a unit may be affected by unit-specific covariates (e.g., open old growth forest vs. dense rejuvenating forest), or by factors that vary with each survey, such as air temperature, cloud cover, or time since a rain event, and the effect on species detection will likely be different for different species. (This paper won the United Nations' World Meteorological Organization (WMO) 2012 Gerbier-Mumm International Award. stochastic programming. Nicotine & Tobacco Research. Annals of Applied Statistics, 10, 596 - 617. Xian Liu, in Methods and Applications of Longitudinal Data Analysis, 2016, As a specific family in GLMMs, statistical inference and estimation of the mixed-effects logit regression model are based on the joint likelihood approach. A note on nonparametric regression test through penalized splines. 5, 888-902. functional mapping of biological rhythms. This method, however, provides approximates for a variance–covariance matrix assuming all the off-diagonal elements in Σ(Lˆiji~) to be 0. gene-environment interactions. Briefings in Bioinformatics. Buu, A., Yang, S., Li, R., Zimmerman, M.A., Cunningham, R.M., and Walton, M.A. Annals of Statistics. Interestingly, in most gambling situations the odds are usually given in terms of the player losing. 11.4 Approximation of variance–covariance matrix on probabilities. Liu, J., Lou, L. and Li, R. (2018). 571 - 581. Journal of American Statistical Association. Statistical Association, 115, 1700 - 1729. Co-chair of scientific program committee for the 1st IMS Asia Pacific (2011). Variable selection in measurement error models. A Bayesian semiparametric model for bivariate sparse longitudinal data. Technometrics. 21, 43-52. We also extend the model to incorporate a vector of continuous or discrete covariates Z not measured by multiple informants. Using a detailed cross-sector, longitudinal data set collected in a midwestern metropolitan area from 1993 to 2009 (N=7,302) and a larger (N=2 million) but less detailed national universal CM and foster care data set, logistic regression and survival analysis were conducted. Improving the Implementation of Quantitative methods in Addiction Research, 94, Illustration of the model where the parameters ψB|a and ψB|A are modeled with the same set of covariates with the logit link function, and an additive effect of size β0 included for ψB|A. In predicting longitudinal trajectories of a set of response probabilities, the standard errors of nonlinear predictions should be approximated for evaluating the quality of the predicted values. Given the confusion in the literature about the key properties of fixed and random effects (FE and RE) models, we present these models’ capabilities and limitations. New estimation and model selection procedures for semiparametric modeling in longitudinal data analysis. (2011). Statistical Association. (4.28) may be characterized by the use of a principal component analysis or partial least squares technique (Krzanowski, 2000). Examining measurement reactivity in daily diary data on substance use: results from a Nicotine & Tobacco Research. Addictive Behaviors. Co-chair of scientific program committee for the 2nd IMS Asia Pacific Journal of American Statistical Association. Using Eq. It is also clear that the general linear mixed model is a special case of the generalized linear mixed models. Proceedings of the International Congress of Mathematicians (2012). This model implies that the probability of species B being present at a unit has the same covariate relationships regardless of the presence/absence of species A (i.e., the same effect sizes), but there is a consistent difference in the logit-probability across all units that does not depend on the covariate values (note β0 may be estimated as a positive or a negative value). Estimation and testing for partially linear single-index models. These algorithms are also implemented in many standard statistical software packages. 16S2, S144 - S150. In press. [pdf] Fan, J. and Li, R. (2004). [pdf], Li, J., Zhong, W., Li, R. and Wu, R. (2014). [pdf], Liu, H., Yao, T, Li, R. and Ye, Y. The first two conditions in a marginal model are standard requirements from a generalized linear model (McCullagh and Nelder, 1989) relating the marginal means to a set of covariates at each individual time point. FADTTS: Functional Analysis Changes in substance [pdf], Wang, L., Kim, Y. and Li, R. (2013). The association between short-term emotion dynamics and cigarette dependence: New York City, NY. Zhu, H., Kong, L., Li, R., Styner, M., Gerig, G., Lin, W. and Gilmore, J. H. (2011). Bahaviors, 76, 161 - 168. If the elements in V(Lˆ) are unobtainable empirically, some covariance structure in the multinomial distribution needs to be assumed for the application of the bootstrap. 1. Sex-specific biology of the human malaria parasite revealed from transcriptomes Let Xi denote the set of covariate values measured at unit i, which form a row vector of the values [1xi1…xin], where the initial ‘1’ denotes a constant that is required for the inclusion of an intercept term in the resultant regression equation. 9, 640 - 664. with applications in health behavior research. Empirical Kriging models and their applications to QSAR. In addition, another component is needed to introduce the within-subject associations among the vector of repeated responses. Annals of Applied Molecular and Cellular Proteomics, 16, 537 - 551. The 2nd Taihu Lake International This does not imply that it is not necessary to try to specify correctly the within-subject associations in the marginal model. i.e., φ = 1. Journal of American Statistical Association. We use cookies to help provide and enhance our service and tailor content and ads. That is not to say the original parameterization will not work with covariates for all datasets, we are simply noting that there are practical difficulties that users may experience that can require considerable effort to overcome. This dependence can create numerical issues, making it difficult for optimization algorithms to converge to the maximum likelihood estimates or creating instabilities in the Markov chains when using MCMC. The probability can thus be calculated as: While the above equation indicates that the same set of covariates is considered for each probability, this does not have to occur in practice as excluding a covariate for a particular probability is equivalent to setting the associated regression coefficient to 0. Model-free feature screening and FDR control with HEALTH BEHAVIOR AND HEALTH EDUCATION Theory, Research, and Practice 4TH EDITION. Dziak, J., Coffman, D. L., Lanza, S. T., Li, R. and Jermiin, L. S. (2019). The probability of observing this particular detection history at an occupied unit (hence ψ does not appear in the probability statement) is Pr(hi=0101|unit is occupied)=(1−pi1)pi2(1−pi3)pi4. (2), we find that βˆ=Σi=1n(XiTXi)−1XiTY˜i are the intercept and slope estimates from a univariate regression model with response Y and a single predictor Xk. We appreciate that there will often be some confusion about the concepts of odds and probabilities as the two words are often used interchangeably in everyday language, but while linked they are distinct quantities. Originating from the logit model, the probability of default for debtor i operating in sector s conditioned on a specific realization of a creditworthiness index Ψs, t ∈ [0, 1] is as follows: Here ζi, s, t depends on both microeconomic factors χl and the creditworthiness index Ψs, t as described below: It is worth mentioning that Eq. This model could be expressed as the single equation: Figure 14.4. [Comment3] This is because of both its easy implementation and its immediate interpretation. 32, 509 - 523. Another generalized linear model is the transition model for which the conditional distribution of the response at a time given the history of longitudinal observations is assumed to depend only on the prior observations with a specified order through a Markov chain. high dimensional generalized linear models. In Equation (11.23), the matrix Vg−1(Lˆ) is the approximate of the variance–covariance matrix V(P) for large samples. Variable selection for support vector machine in The bootstrap procedure, however, generates less efficient, robust approximates than a retransformation approach with known elements in Σ(Lˆ) for the mixed-effects multinomial logit model because it is based on the assumption that the off-diagonal elements in V(Lˆ) are all zero (Follmann, 1994). Journal of Business and Economic Statistics. Using Eq. likelihood and its application to variable selection. The purpose of the logit link is to take a linear combination of the covariate values (which may take any value between ±∞) and convert those values to the scale of a probability, i.e., between 0 and 1. Changes in healthcare expenditures after the Sci Rep, 6: 21193. average approximation with sparsity-inducing penalty for high-dimensional Varona, J. Verdera, eds. Nicotine & Tobacco Research. For example: where αA and αB are the effects of the covariates on the probability of occurrence for each species given the other species is absent, and β is the effect of the covariates on the mutual co-occurrence of the species. Example 2: In a longitudinal study to examine the longitudinal trend on the probability of depression and to relate this probability to other covariates such as gender and education, the occurrence of depression is longitudinally observed. Therefore, bootstrapping techniques are not statistically adequate to generate the standard error approximates of the predicted transition probabilities for longitudinal data. Tuning parameter selectors for the smoothly clipped absolute deviation method. Estimating mixture of Gaussian processes by kernel smoothing. Local rank inference for varying coefficient models. Feature screening in ultrahigh dimensional Cox's model. Because Yjk is binary and coded as 1 when depression occurs and 0 otherwise, the distribution of each Yjk is Bernoulli which is traditionally modeled through a logit- or probit-link function, i.e., the conditional expectation of Yjk, given Xjk, is E(Yjk|Xjk) = Pr(Yjk = 1|Xjk) = μjk, and the logit-link function links μjk with covariates by. Addictive The term θi1−θ1 is the odds of success (i.e., how much greater the probability of success is compared to that of a failure) and is often expressed as a ratio. How to cluster gene expression dynamics in response to environmental signals. (2018). Briefings in Bioinformatics. 33, Buu, A., Cai, Z., Li, R., Wong, S., Lin, H., Su, W., Jorenby, D.E., and DOI: 10.1037/a0023813. [pdf], Li, R., Ren, J.J., Yang, G. and Ye, Y. Miao, J. Chen, Z. Sebastian, A. Wang, Z., Shrestha, S., Li, X., Praul, C., Piper, M.E. The extension of this basic model to accommodate heterogeneity among units is straightforward, and that is to replace α with αi, where αi is a unit-specific random effect endowed with a suitable distribution. American Journal of Neuroradiology, 38, 966-972. Design and Modeling for Computer Experiments. [pdf] Web Document [pdf], Zou, H. and Li, R. (2008). Example 3: In many studies of AD, psychometric tests are generally used to assess subjects' cognition longitudinally. Uncovering multiple pathways to substance use: A comparison of methods for identifying population subgroups. [pdf] It provides strong evidence that insomnia is a causal factor in the occurrence of psychotic experiences and other mental health problems. Indeed, Eq. 16S2, S135 - S143. Statistics, 14, 276 - 298. Liu, H., Du, G. Zhang, L. , Lewis, M., Wang, X., Yao, T., Li, R. and Huang, X. It can be mathematically proved that the optimum efficiency in the estimation of regression parameters β can be obtained when the working matrix Vj is the same as the true within-subject association among repeated responses. 104, 234-248. The conditional mean of Yjk is linked to a linear predictor through a given link function g. The final assumption on generalized linear mixed models is about the distribution for the random effects. Note that some values in αA, αB, and β could be set as 0, i.e., some covariates in Xi may have no effect on certain parameters. [Supplement]. Ren, H., Zou, C., Chen, N. and Li, R. (2021). (3.4). Because the mean response and the within-subject association are modeled separately, the regression parameters in a marginal model are not affected by the assumptions on the within-subject associations, and therefore can be interpreted as population averages, i.e., they describe the mean response in the population and its relations with covariates. When there is an identical link, both approaches become equivalent based on the fact E(Yjk|Xjk) = Ebj[E(Yjk|Xjk, bj)], and the interpretation of regression parameters in the generalized linear mixed model can also be made in terms of population averages. Models. Eq. (2015). The conditional within-subject association among repeated responses, given the covariates, is usually specified by unstructured pairwise correlations between two repeated responses. Journal of Econometrics, 209, 145- 157. Fawcett J, Scheftner W, Clark D, Hedeker D, Gibbons R, Coryell W. 1987. [pdf], Wang, L. and Li, R. (2009). genome-wide association studies. Second, use the squared standard error of each intercept estimate in the variance–covariance matrix of the fixed effects, denoted by Σ(β), plus the corresponding variance term of the between-subjects random effects as the variance for each of the K logit components. [pdf], Chen, H., Wang, Y., Li, R. and Shear, K. (2014). Journal of American Statistical Association. Das, K., Li, J., Fu, G., Wang, Z., Li, R. and Wu, R. (2013). [pdf] Bayesian inference can be applied to specify the marginal mean of the response probability. (2005). data. (1996) on suicidal ideation and attempts in a sample of 300 suicidal young adults (personal communication, Dr. M. David Rudd, Professor of Psychology, Baylor University) serves as an illustration of an application of the mixed-effects ordinal logistic regression model. Research in Autism Spectrum Disorders, 57, 97 -104. Similarly, accounting for missing observations or unequal sampling effort is done in exactly the same way as for the other models in this book. (1999) that was originally presented for a binary response using a logit link function, but here we assume a linear model. Prevention Science. Time-varying effects of smoking quantity and nicotine dependence on adolescent smoking regularity. Sensitivity and Statistica Sinica. Journal of Multivariate Analysis. Bioinformatics, 36, 3811- 3817. In general, the GEEs have no closed form solutions with a non-linear link function, and therefore require an iterative algorithm to approximate the solutions. Chu, W., Li, R. and Reimherr, M. (2016). Time-varying effect model for studying gender differences in health There is a range of possible link functions available, although we note the logit and multinomial-logit link functions may often be preferred, as these correspond to logistic and multinomial-logistic regression. The conditional variance of Yjk, given Xjk, is assumed to depend on the mean according to some given ‘variance function’ V, i.e.. 32, 259-270. Journal of Agricultural, Biological, and Environmental Statistics, 19, 57 - 81. Covariate information number for feature Accepted. Linear hypothesis testing for Selya, A. S., Updegrove, N., Rose, J., Dierker, L., Tan, X., Hedeker, D., Li, R. and Mermelstein, R. J. Annals of Statistics. [pdf], Xu, C., Zhang, Y., Li, R. and Wu, X. Second, use the squared standard error of each intercept estimate plus the corresponding variance of the between-subjects random effect as the variance for each of the K logit components. Annals of Statistics, 47, 2671 - 2703. Third, take the values of covariance between each pair of the logit intercept estimates in Σ(θ) as the off-diagonal elements in Σ(Lˆiji~). The difference between the stressed logit(Ψs, Δ) and the logit in t0 is a key ingredient of the overall framework as shown below: where Δζs is added to the initial logit as follows: The stressed default probability described in Eq. [pdf] Annals of Statistics. Chapman and Hall/CRC. with ultrahigh dimensional longitudinal data. The exponential family covers essentially all the important distributions used in biomedical applications. Yang, S., Cranford, J. Li, R. and Chow, M. (2005). (M. Sanz-Sole, J. Soria, J.L. Variable selection for high-dimensional data, Feature screening for ultrahigh-dimensional data, Longitudinal and intensive longitudinal data analysis, Nonparametric regression modeling and local polynomial regression, Statistical applications to engineering, meteorological research, neural science research & social behavioral science research. Marginal models are one of these choices. Journal of American Statistical Association, 111, 169 -179. 24, 1235 - 1256. One-step sparse estimates in nonconcave penalized likelihood models (with discussion). A time-varying Yang, H., Cranford, J., Li, R. and Buu, A. For example, the presence or absence of depression and the count of panic attacks during certain time interval are all likely response variables of scientific interest. [pdf], Huang, D., Li, R. and Wang, H. (2014). [pdf] Accepted. Annals of Statistics. A short summary of this paper. Journal of Chemometrics. (3.4), except the denominator is now θi[M] instead of 1 minus the probability of interest (although when M=2, they are equivalent). Frailty is associated with an increased risk of chemotherapy toxicity. The probability of success can be calculated from the odds as: so in this instance the probability of success would be, θi=3/4=0.75. The content is solely the responsibility of the authors and does not necessarily represent the official views of This is true even when the within-subject associations have been incorrectly specified in the marginal model. [Supplement], Zhong, P.-S., Li, R. and Santo, S. (2019). The other is the likelihood approach (Anderson and Aitkin, 1985; Hinde, 1982) or the penalized quasi-likelihood (PQL) approach (Green, 1987; Laird, 1978; Stiratelli et al., 1984; McGilchrist and Aisbett, 1991; Breslow and Clayton, 1993). [Supplement], Zou, C., Wang, G. and Li, R. (2020). 107, 214 - 222. Similar to the case without covariates, αˆk,βˆk and γˆk are estimates from a univariate regression model with response Y, multiple informant Xk and covariates Z. 67, 896 - 905. Such models are often called multilevel models. [pdf], Xu, C., Lin, S., Fang, J. and Li, R. (2016). The Annals of Applied Statistics. Irrespective of the parameterization used for the two-species co-occurrence model, covariates can be incorporated for any of the model parameters during an analysis through the use of appropriate link functions (Chapter 3), as demonstrated throughout this book.

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