Applied Multilevel Regression Modelling

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Course Dates and Times

Monday 29 July – Friday 2 August and Monday 5 – Friday 9 August

09:00–10:30 and 11:00–12:30

Prerequisite Knowledge

To guarantee that we progress at a steady and firm pace, you'll need a thorough understanding of OLS linear regression, at a theoretical and practical level.

You should be able to interpret coefficients and model fit, work with residuals, plot marginal effects, interpret and graphically display interactions, and assess and correct regression assumption violations. You should also have basic knowledge of generalised linear models (at least binary logistic regression) and Maximum Likelihood estimation.

The course is conducted entirely in the R statistical environment, so I expect you to have practical experience with R for data management and recoding, making exploratory graphs, running OLS regressions, and plotting quantities of interest based on regression output. This experience should naturally extend to reading in data in different formats, such as Excel, SPSS, or Stata, into R.

To brush up on the above, I recommend the relevant chapters from Michael Crawley’s The R Book (2^nd edition, Wiley, 2012), or Richard Cotton’s Learning R (O’Reilly Media, 2013).

If your regression knowledge was acquired a long time ago, brush up on the essential concepts and features using John Fox’s Applied Regression Analysis and Generalized Linear Models (3^rd edition, Sage Publications, 2016).

Particularly relevant are chapters 5, 6, 7, 11, 12, and 14. A good coverage of Maximum Likelihood estimation is provided in Craig K. Enders’ Applied Missing Data Analysis (Guilford Press, 2010), chapter 3, or in Scott R. Eliason’s Maximum Likelihood Estimation: Logic and Practice (Sage Publications, QASS Series, 1993).

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Short Outline

This course will teach you the application of simple, and then increasingly complex, multilevel specifications.

The first week sets the foundations. We start from basic hierarchical linear models (HLMs), with only random intercepts, to more complex specifications that allow us to understand how an effect varies across contexts.

As part of this progression we cover estimation, 2- and 3-level configurations, what sample size considerations apply to HLMs, and how to assess models’ adequacy.

In the second week we explore alterations to the fundamental framework introduced the previous week. We cover dichotomous outcomes, applying a multilevel specification to assess change over time (growth curve modelling), as well as how to analyse non-hierarchical data configurations.

If you have no prior knowledge of multilevel models, but want in-depth coverage, this course will serve you well. Sessions will be conducted entirely in R.

ECTS Credits for this course and, below, tasks for additional credits:

4 credits Attend at least 90% of course hours, and participate fully in in-class activities. Carry out the necessary reading and/or other work prior to and after classes.
6 credits As above, plus complete two take-home assignments in the first and second weeks, comprising an interpretation task, and a multilevel modelling task:

• Task 1 Comment on published work using multilevel models, and highlight strengths and weaknesses identified in the analysis.
• Task 2 I will provide you with some data and a few model specifications that need to be estimated with this data. You will write code that generates the model estimates, in order to interpret results, determine which model fits the data best, and graphically display some of the effects uncovered in the data.

8 credits As above, plus submit a final paper:

• Final paper This will resemble a conference paper, though the literature review section can be just two or three paragraphs where you present the puzzle. Identify a few hypotheses you are interested in testing, then test them based on data of your choosing. The main parts I am interested in are the variable description (not more than a couple of pages), the analyses, and the interpretation of the results. You will be assessed based on how well you have interpreted coefficients and the model fit, whether you have explored potential problems with these models (in terms of assumption violations), and how well you were able to correct said problems. Submission deadline will be 20-25 days after the end of the Summer School.

I will provide more details about the tasks and requirements during the course itself.

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Long Course Outline

 

‘Once you know that hierarchies exist, you see them everywhere’

─ Kreft & de Leeuw, 1998, page 1

This course will introduce you to a class of statistical specifications that allows for the rigorous analysis of data that exhibit such hierarchical properties: multilevel models.

Beyond their desirable statistical properties, though, the primary sell-point of these models is that they allow us to pose, and find supportive evidence for, more complex questions about the world. They do so by treating variation at multiple levels of the nesting structure not as a nuisance but as a substantively interesting feature of the data, to be modelled rather than corrected for.

An additional desirable feature of these models is their versatility: wherever data is nested in higher-order groups, it’s a good bet that a multilevel can be adapted and applied to such data.

The two weeks are devoted, first, to covering the foundations of multilevel modelling and, second, to exploring extensions of this core framework to alternative data configurations.

In the first week we explore common linear specifications: random-intercept and random-slope models. We go over statistical notation for these models, interpretation of coefficients, presenting uncertainty, reporting results from such models, testing and displaying the effect of interactions between group-level and observation-level variables, as well as what sample size considerations should be kept in mind when analysing such data.

In the second week we allocate each day to an extension of the standard hierarchical linear framework. We introduce generalised linear mixed specifications, with an application to a dichotomous response variable. We then show how to use a multilevel specification to analyse change over time, as well as how to produce sub-national estimates of public opinion based solely on national-level data. Finally, I highlight how multilevel specifications can be applied to data structures that are not hierarchical, as well as to data that has the added complication of spatial correlations. Throughout the sessions we make intensive use of the lme4 and nlme packages, along with a variety of functions from connected packages that assist in plotting, model comparison, and data reshaping.

By the end of the course you should be able to easily identify such nested data configurations in your own field of study, e.g. voters nested in electoral districts, companies grouped in regions, parliamentarians nested in committees. You should also be able to properly specify, theoretically as well as with R/lme4 syntax, a multilevel model that fits the data structure you are faced with, and answers the substantive questions you are interested in. Finally, you will be equipped to interpret statistical output from these models, assess any misfit between model and data, and present substantive results to either a lay or a specialist audience.

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Week 1

Day 1
We start by describing problems OLS faces when applied to data that is nested, and how multilevel models (MLMs) overcome these difficulties. In addition to their statistical properties, MLMs also allow us to answer more sophisticated questions about the world. These insights will be complemented by a short practice session in R focusing on how OLS breaks down in certain situations, and how multilevel models are a compromise between two alternative strategies of analysing data in these instances.

Day 2
I introduce notation for multilevel specifications, as well as the simplest type of such models, with only a varying intercept. We then cover interpretation and inference for these models, and what the implications are of allowing for a varying intercept.

Day 3
We gradually introduce more complex specifications, allowing for the effect of a predictor to vary between groups, and trying to understand whether any group-level predictors systematically explain how this effect varies. Cross-level interactions will be presented, along with techniques to graphically present their estimates. Finally, we discuss how to do variable centering and rescaling in the case of nested data.

Day 4
We discuss how to determine the best-fitting model from a series of specifications we might have tested, as well as how to assess the quality of our model. The latter topic brings us to the issue of assumptions for multilevel models, where we cover a few diagnostic tools.

Day 5
This day is reserved for a set of smaller topics, as well as a review of the most important ideas from the previous four days. I show that the insights gained apply relatively seamlessly to data with more than two levels of nestation. I also broach the topic of sample size requirements at all levels of the nesting structure, which frequently plague empirical analyses in political science.

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Week 2

We examine advanced extensions of the standard hierarchical linear specification; these demonstrate MLM’s flexibility MLM in dealing with a variety of data configurations and outcome variables.

Day 6
We begin with a coverage of generalised linear mixed models (GLMMs), with a specific focus on dichotomous dependent variables. By working through a practical example, we cover the interpretation of estimates from these models, the presentation of marginal effects, and sample size considerations.

Day 7
I introduce multilevel models as a solution to the need to model change over time in a phenomenon, and to explain such change with time-varying and time-invariant predictors. I explain how to see such a setup as a nested data configuration, plotting and modelling trajectories of change, as well as choosing from a variety of error covariance structure options.

Day 8
We tackle cross-classified and multiple-membership models. These specifications accommodate situations where observations are simultaneously nested in two non-overlapping hierarchies, or where observations can be members of multiple higher-level units at the same time. Although this complicates our setup, we will see that multilevel models are well equipped to handle this situation. We finish the day by covering one way in which these models can help us disentangle age-period-cohort (APC) effects.

Day 9
We continue the discussion on cross-classified models by introducing multilevel regression with post-stratification (MRP). This is a specialised modelling technique that produces estimates of public opinion for sub-national units and population sub-groups from nationally representative surveys. Given the adaptability of this technique to varying types of attitudes (such as vote preferences), as well as to contexts where no census information is available (MRSP), it represents a valuable potential tool for you to master.

Day 10
We wrap up by considering the extension of multilevel models to situations where the data exhibits spatial dependence between observations. From this perspective, the distance between observations can also impact on the phenomenon we’re interested in, above and beyond the variance explained by level-1 characteristics or by group-level predictors. Following a practical example, I showcase the implementation of such models in R, as well as the interpretation of the estimates obtained.

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Note on readings Some of the days require extensive preparatory readings. Before starting the class, I strongly recommend you review regression-related considerations, so please allocate extra time before the course to familiarise yourself with the reading workload.

Day	Topic	Details
1	Multilevel models: introduction and notation	Lecture topics Where OLS breaks down; OLS-based solutions to address nested data; MLMs as a solution to these problems; Lab topics R for regression warm-up; Diagnosing and addressing heteroscedasticity; Introducing practice data for Week 1. Duration ~120 min. lecture and ~60 min. lab.
2	Random intercepts in MLM	Lecture topics MLM statistical notation. Basic setup for a multilevel specification; Group-level predictors: estimation and inference; Lab topics How MLM notation translates to R syntax; Running first model: random-intercept specification; Interpreting output from model; Gradually testing more complex specifications. Duration ~90 min. lecture and ~90 min. lab.
3	Random slopes and cross-level interactions	Lecture topics Centering predictors: grand-mean and group-mean. Adding group-level predictors for slopes; Cross-level interactions: interpretation and plotting. Lab topics More complex specifications: random slopes; Performing centering; Cross-level interactions: R code and graphical presentation; Presenting MLMs in written work. Duration ~90 min. lecture and ~90 min. lab.
4	Model fit and diagnostics	Lecture topics Model fit in MLM specifications; Model comparisons; Assessing model quality: diagnostics. Lab topics Model diagnostics: examining problems; Correcting problems in tested specifications. Duration ~90 min. lecture and ~90 min. lab.
5	Extensions of the framework: 3-level models and Week 1 recap	Lecture topics Sample size considerations in MLM: level-1 and level-2; Extending framework to 3-level models Centering and sample size in 3-level specifications; Recap of core topics from the past 4 days. Lab topics Testing a 3-level specification; Practice session for running 2-level specification with random intercepts, random slopes, and a cross-level interaction. Duration ~60 min. lecture and ~120 min. lab.
7	Modelling change over time in MLM framework	Lecture topics Growth curve modelling as a specialised application of multilevel models; Time-varying and time-invariant predictors; Cross-level interactions in growth curve modelling; Error covariance structures for growth curve modelling. Lab topics Graphical displays of change over time; Multilevel specifications with the nlme() package; Introducing time-varying and time-invariant predictors; Allowing for curvilinear change. Duration ~90 min. lecture and ~90 min. lab.
8	Cross-classified and multiple membership models	Lecture topics Data structures with no clear hierarchy; Complex error structures; Simultaneous membership in hierarchies; Extensions to modeling age-period-cohort (APC) effects. Lab topics Example of cross-classified structure; Modelling strategy in R. Duration ~90 min. lecture and ~90 min. lab.
9	Multilevel regression with post-stratification	Lecture topics Problem: studying sub-national attitudes with nationally representative data; Cross-classification as a strategy of obtaining estimates; Practical example. Lab topics Working through example of MRP. Duration ~90 min. lecture and ~90 min. lab.
6	Generalised linear mixed models: dichotomous outcomes	Lecture topics Quick review of standard logistic regression; GLMMs: the case of dichotomous outcomes; Coefficient interpretation; Sample size considerations. Lab topics GLMMs through the glmer() function; Group-level predictors, cross-level interactions, and presenting marginal effects. Duration ~90 min. lecture and ~90 min. lab.
10	Multilevel spatial modelling	Lecture topics Adapting MLM specifications to deal with spatial correlations; The W matrix; Estimation and interpretation of effects. Lab topics Working through one example of spatial analysis. Duration ~90 min. lecture and ~60 min. lab.

Day	Readings
	NB: I expect you to do the readings before the scheduled meeting The primary textbook assigned for the course is Andrew Gelman and Jennifer Hill’s Data Analysis using Regression and Multilevel/Hierarchical Models (CUP, 2007). Selected chapters have been sourced from other notable multilevel model books, such as Tom Snijders and Roel Bosker’s Multilevel Analysis (Sage, 1999), or Stephen Raudenbush and Anthony Bryk’s Hierarchical Linear Models (Sage, 2002). The primary source for the longitudinal analysis module in Week 2 is Judith Singer and John Willett’s Applied Longitudinal Data Analysis (OUP, 2003).
1	Kreft, Ita, and Jan De Leeuw. 1998 Introducing Multilevel Modeling London: Sage. Chapter 1 Gelman, Andrew, and Jennifer Hill. 2007 Data Analysis using Regression and Multilevel / Hierarchical Models New York: Cambridge University Press. Chapters 1 and 11 Optional Snijders, Tom A. B., and Roel J. Bosker. 1999 Multilevel Analysis: An introduction to basic and advanced multilevel modeling London: Sage. Chapters 2 and 3 Bickel, Robert. 2007 Multilevel Analysis for Applied Research: It’s Just Regression! New York: Guilford Press. Chapters 2 and 3 Scott, Marc A., Patrick E. Shrout, and Sharon L. Weinberg 'Multilevel Model Notation—Establishing the Commonalities' In Marc A. Scott, Jeffrey S. Simonoff, and Brian D. Marx The SAGE Handbook of Multilevel Modeling Los Angeles: Sage Publications. Chapter 2 (pp. 21–38)
2	Gelman, Andrew, and Jennifer Hill. 2007 Data Analysis using Regression and Multilevel / Hierarchical Models New York: Cambridge University Press. Chapter 12 Enders, Craig K., and Davood Tofighi. 2007 'Centering Predictor Variables in Cross-Sectional Multilevel Models: A New Look at an Old Issue' Psychological Methods 12(2): 121–138 Optional Gill, Jeff, and Andrew J. Womack. 2013 'The Multilevel Model Framework' In Marc A. Scott, Jeffrey S. Simonoff, and Brian D. Marx The SAGE Handbook of Multilevel Modeling Los Angeles: Sage Publications. Chapter 1 (pp. 3–20) Snijders, Tom A. B., and Roel J. Bosker. 1999 Multilevel Analysis: An introduction to basic and advanced multilevel modeling London: Sage. Chapter 4 Raudenbush, Stephen W., and Anthony S. Bryk. 2002 Hierarchical Linear Models: Applications and Data Analysis Methods Advanced Quantitative Techniques in the Social Sciences. Thousand Oaks, CA: Sage Publications. Chapter 2 Steenbergen, Marco R., and Bradford S. Jones. 2002 'Modeling Multilevel Data Structures' American Journal of Political Science 46(1): 218–237
3	Gelman, Andrew, and Jennifer Hill. 2007 Data Analysis using Regression and Multilevel / Hierarchical Models New York: Cambridge University Press. Chapter 13 McNeish, Daniel M., and Laura M. Stapleton. 2016 'The Effect of Small Sample Size on Two-Level Model Estimates: A Review and Illustration' Educational Psychology Review 28(2): 295–314 Optional McNeish, Daniel M. 2017 'Small Sample Methods for Multilevel Modeling: A Colloquial Elucidation of REML and the Kenward-Roger Correction' Multivariate Behavioral Research 52(5): 661–670 Snijders, Tom A. B., and Roel J. Bosker. 1999 Multilevel Analysis: An introduction to basic and advanced multilevel modeling London: Sage. Chapter 5 Brambor, T., Clark, W. R., & Golder, M. (2005) 'Understanding Interaction Models: Improving Empirical Analyses' Political Analysis, 14(1), 63–82
4	Steele, Russell. 2013 'Model Selection for Multilevel Models' In Marc A. Scott, Jeffrey S. Simonoff, and Brian D. Marx The SAGE Handbook of Multilevel Modeling Los Angeles: Sage Publications. Chapter 7 (pp. 109–126) Raudenbush, Stephen W., and Anthony S. Bryk. 2002 Hierarchical Linear Models: Applications and Data Analysis Methods (Advanced Quantitative Techniques in the Social Sciences) Thousand Oaks, CA: Sage Publications. Chapter 9 Optional Snijders, Tom A. B., and Roel J. Bosker. 1999 Multilevel Analysis: An introduction to basic and advanced multilevel modeling London: Sage. Chapter 9 Snijders, Tom A. B., and Johannes Berkhof. 2008 'Diagnostic Checks for Multilevel Models' In J. de Leeuw & E. Meijer (Eds.) Handbook of Multilevel Analysis Springer: New York (pp. 141–175)
5	Goldstein, Harvey. 2011 Multilevel Statistical Models 4th edition. London: Wiley. Chapter 3 (sections 3.1 and 3.2 only) McNeish, Daniel, and Kathryn R. Wentzel. 2017 'Accommodating Small Sample Sizes in Three-Level Models When the Third Level is Incidental' Multivariate Behavioral Research 52(2): 200–215 Brincks, Ahnalee M., Craig K. Enders, Maria M. Llabre, Rebecca J. Bulotsky-Shearer, Guillermo Prado, and Daniel J. Feaster. 2017 'Centering Predictor Variables in Three-Level Contextual Models' Multivariate Behavioral Research 52(2): 149–163 Optional Bickel, Robert. 2007 Multilevel Analysis for Applied Research: It’s Just Regression! New York: Guilford Press. Chapter 9
6	Gelman, Andrew, and Jennifer Hill. 2007 Data Analysis using Regression and Multilevel / Hierarchical Models New York: Cambridge University Press. Chapter 14 Hox, Joop J. 2010 Multilevel Analysis: Techniques and Applications 2nd edition. New York: Routledge. Chapter 6 Optional Snijders, Tom A. B., and Roel J. Bosker. 1999 Multilevel Analysis: An introduction to basic and advanced multilevel modeling London: Sage. Chapter 14 Bates, Douglas M. 2010 lme4: Mixed-effects modeling with R Chapter 6
7	Singer, Judith D., and John B. Willett. 2003 Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence New York: Oxford University Press. Chapters 3, 4, 5, and 6 Optional Singer, Judith D., and John B. Willett. 2003 Applied Longitudinal Data Analysis: Modeling Change and Event Occurrence New York: Oxford University Press. Chapter 7 Hox, Joop J. 2010 Multilevel Analysis: Techniques and Applications 2nd edition. New York: Routledge. Chapter 5 Goldstein, Harvey. 2011 Multilevel Statistical Models 4th edition. London: Wiley. Chapter 5 Laird, Nan M., and Garrett M. Fitzmaurice. 2013 'Longitudinal Data Modeling' In Marc A. Scott, Jeffrey S. Simonoff, and Brian D. Marx The SAGE Handbook of Multilevel Modeling Los Angeles: Sage Publications. Chapter 9 (pp. 141–160) Núñez-Antón, Vicente, and Dale L. Zimmerman 'Complexities in Error Structures Within Individuals' In Marc A. Scott, Jeffrey S. Simonoff, and Brian D. Marx The SAGE Handbook of Multilevel Modeling Los Angeles: Sage Publications. Chapter 10 (pp. 161–182)
8	Fielding, Antony, and Harvey Goldstein. 2006 Cross-classified and Multiple Membership Structures in Multilevel Models: An Introduction and Review Department for Education and Skills, UK Home Office: London. Research Report 791 Snijders, Tom A. B., and Roel J. Bosker. 1999 Multilevel Analysis: An introduction to basic and advanced multilevel modeling London: Sage. Chapter 11 Optional Goldstein, Harvey. 2011 Multilevel Statistical Models 4th edition. London: Wiley. Chapters 12 and 13 Yang, Yang, and Kenneth C. Land. 2008 'Age–Period–Cohort Analysis of Repeated Cross-Section Surveys: Fixed or Random Effects?' Sociological Methods & Research 36(3): 297–326 Hox, Joop J. 2010 Multilevel Analysis: Techniques and Applications 2nd edition. New York: Routledge. Chapter 9
9	Ghitza, Yair., and Andrew Gelman. 2013 'Deep Interactions with MRP: Election Turnout and Voting Patterns Among Small Electoral Subgroups' American Journal of Political Science 57(3): 762–776 Lax, Jeffrey R., Justin H. Phillips. 2009 'Gay Rights in the States: Public Opinion and Policy Responsiveness' American Political Science Review 103(3): 367–386 Optional Lax, Jeffrey R., and Justin H. Phillips. 2009 'How Should We Estimate Public Opinion in The States?' American Journal of Political Science 53(1): 107–121 Leemann, Lucas, and Fabio Wasserfallen. 2017 'Extending the Use and Prediction Precision of Subnational Public Opinion Estimation' American Journal of Political Science 61(4): 1003–1022
10	Ghitza, Y., & Gelman, A. (2013). Deep interactions with MRP: Election turnout and voting patterns among small electoral subgroups. American Journal of Political Science, 57(3), 762-776. Leemann, L., & Wasserfallen, F. (2017). Extending the use of prediction precision of subnational public opinion estimates. American Journal of Political Science, 61(4), 1003-1022. Recommended: Lax, Jeffrey, and Justin Phillips. 2009b. “How Should We Estimate Public Opinion in the States?” American Journal of Political Science 53(1): 107–21. Wang, W., Rothschild, D., Goel, S., & Gelman, A. (2015). Forecasting elections with non-representative polls. International Journal of Forecasting, 31(3), 980-991.
10	Dong, Guanpeng, Jing Ma, Richard Harris, and Gwilym Pryce. 2016 'Spatial Random Slope Multilevel Modeling Using Multivariate Conditional Autoregressive Models: A Case Study of Subjective Travel Satisfaction in Beijing' Annals of the American Association of Geographers 106(1): 19–35 Corrado, Luisa, and Bernard Fingleton. 2011 Multilevel Modeling with Spatial Effects Scottish Institute for Research in Economics: Edinburgh Optional Harris, Richard, John Moffat, and Victoria Kravtsova. 2011 'In Search of "W"' Spatial Economic Analysis 6(3): 249–270 Gelfand, Alan E., Sudipto Banerjee, C. F. Sirmans, Yong Tu, and Seow Eng Ong. 2007 'Multilevel modeling using spatial processes: Application to the Singapore housing market' Computational Statistics & Data Analysis, 51(7): 3567–3579

Software Requirements

R 3.5.2 or any newer version

RStudio 1.2.1322 or any newer version

Hardware Requirements

Please bring your own laptop to lecture and lab sessions.

Any computer or laptop bought within the last 3–4 years should be sufficient.

4 GB of RAM and 200–300 MB of free space on the hard drive are enough to run the tasks we will attempt.

Literature

The assigned readings are some of the most commonly used textbooks in the field of multilevel modelling.

If you would like to consult additional sources, particularly in terms of how to implement such models in commonly used software packages, consult the literature below:

1. Heck, R. H., & Thomas, S. L. (2015). An Introduction to Multilevel Modeling Techniques: MLM and SEM Approaches Using Mplus. New York: Routledge.
2. Finch, W. H., & Bolin, J. E. (2017). Multilevel Modeling Using Mplus. Boca Raton, FL: CRC Press.
3. Finch, W. H., Bolin, J. E., & Kelley, K. (2014). Multilevel Modeling Using R. Boca Raton, FL: Chapman & Hall/CRC.
4. Heck, R. H., Thomas, S. L., & Tabata, L. N. (2010). Multilevel and Longitudinal Modeling with IBM SPSS. New York: Routledge.
5. Rabe-Hesketh, S., & Skrondal, A. (2012). Multilevel and Longitudinal Modeling Using Stata (3rd ed.). College Station, TX: Stata Press.
6. Luke, D. A. (2004). Multilevel Modeling. Quantitative Applications in the Social Sciences. Thousand Oaks, CA: Sage Publications.
7. Beck, N., & Katz, J. N. (2011). Modeling Dynamics in Time-Series-Cross-Section Political Economy Data. Annual Review of Political Science, 14, 331–352.
8. Beretvas, S. N. (2011). Cross-Classified and Multiple-Membership Models. In J. J. Hox & K. J. Roberts (Eds.) (pp. 313–334). London: Routledge.
9. Kreft, I. G. G., de Leeuw, J., & Aiken, L. S. (1995). The Effect of Different Forms of Centering in Hierarchical Linear Models. Multivariate Behavioral Research, 30(1), 1–21.
10. Shor, B., Bafumi, J., Keele, L., & Park, D. (2007). A Bayesian Multilevel Modeling Approach to Time-Series Cross-Sectional Data. Political Analysis, 15(2), 165–181.
11. Fairbrother, M. (2014). Two Multilevel Modeling Techniques for Analyzing Comparative Longitudinal Survey Datasets. Political Science Research and Methods, 2(1), 119–140. https://doi.org/10.1017/psrm.2013.24
12. Stegmüller, D. (2013). How Many Countries for Multilevel Modeling? A Comparison of Frequentist and Bayesian Approaches. American Journal of Political Science, 57(3), 748–761.
13. Krull, J. L., & MacKinnon, D. P. (2001). Multilevel Modeling of Individual and Group Level Mediated Effects. Multivariate Behavioral Research, 36(2), 249–277.
14. Pinheiro, J. C., & Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. New York: Springer.
15. Maas, C. J. M., & Hox, J. J. (2005). Sufficient Sample Sizes for Multilevel Modeling. Methodology, 1(3), 86–92.
16. Gill, J. (2015). Bayesian Methods: A Social and Behavioral Sciences Approach (3rd ed.). Boca Raton, FL: Chapman & Hall/CRC. [chapter on hierarchical models]
17. Gelman, A., Carlin, J. B., Stern, H. S., Dunson, D. B., Vehtari, A., & Rubin, D. B. (2014). Bayesian Data Analysis. Texts in Statistical Science (3rd ed.). Boca Raton, FL: Chapman & Hall/CRC. [chapter on hierarchical models]

Recommended Courses to Cover Before this One

<p>&nbsp;</p> <p><strong>Winter School</strong></p> <p>Maximum Likelihood Estimation</p> <p>Linear Regression with R/Stata: Multiple Regression Analysis</p> <p>Logistic Regression and Generalised Linear Models</p> <p><strong>Summer School</strong></p> <p><span style="color:#00000a">Linear Regression with R/Stata:</span> Multiple Regression Analysis</p> <p>Introduction to Logistic Regression and General Linear Models: Binary, Ordered, Multinomial and Count Outcomes</p>

Recommended Courses to Cover After this One

<p>&nbsp;</p> <p><strong>Winter School</strong></p> <p>Introduction to Bayesian Inference</p> <p><strong>Summer School</strong></p> <p>Introduction to Structural Equation Modelling</p> <p>Multilevel Structural Equation Modelling</p> <p><span style="color:#00000a">Introduction to Bayesian Inference</span></p>

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Additional Information

Disclaimer

This course description may be subject to subsequent adaptations (e.g. taking into account new developments in the field, participant demands, group size, etc). Registered participants will be informed at the time of change.

By registering for this course, you confirm that you possess the knowledge required to follow it. The instructor will not teach these prerequisite items. If in doubt, please contact us before registering.