About

Stephen Raudenbush is the Lewis-Sebring Distinguished Service Professor in the Department of Sociology, the College, and the Harris School of Public Policy Studies at the University of Chicago. He holds a B.A., Ed.M., and Ed.D. from Harvard University. His research interests include the sociology of education and quantitative methods, with a focus on statistical models for child and youth development within social settings such as classrooms, schools, and neighborhoods. Raudenbush is best known for developing hierarchical linear models, which have broad applications in the design and analysis of longitudinal and multilevel research. His current work involves studying the development of literacy and math skills in early childhood with implications for instruction, as well as methods for assessing school and classroom quality. He is a member of the National Academy of Sciences and the American Academy of Arts and Sciences, and has received the American Educational Research Association award for Distinguished Contributions to Educational Research.

Research topics

• Computer Science
• Mathematics education
• Psychology
• Artificial Intelligence
• Developmental psychology
• Machine Learning
• Sociology
• Social Science
• Econometrics
• Management science
• Cognitive psychology
• Data science
• Medicine
• Engineering
• Linguistics
• Mathematics
• Statistics

Selected publications

• Multiple Imputation to Estimate Hierarchical Models From Data Missing at Random: Latent Covariates, Random Coefficients, and Statistical Interactions

  Journal of Educational and Behavioral Statistics · 2025-12-10

  articleSenior author

  Consider the conventional multilevel model <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Y</mml:mi> <mml:mo>=</mml:mo> <mml:mi mathvariant="normal">C</mml:mi> <mml:mi mathvariant="normal">γ</mml:mi> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">Zu</mml:mi> <mml:mo>+</mml:mo> <mml:mi mathvariant="normal">e</mml:mi> </mml:mrow> </mml:math> where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">γ</mml:mi> </mml:mrow> </mml:math> represents fixed effects and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">u</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> are multivariate normal random effects. The continuous outcomes <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Y</mml:mi> </mml:mrow> </mml:math> and covariates <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:math> are fully observed with a subset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Z</mml:mi> </mml:mrow> </mml:math> of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:math> . The parameters are <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">θ</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">γ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>var</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">u</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">var</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">e</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> . Dempster, Rubin and Tsutakawa framed the estimation as a missing data problem, where <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">u</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> are the complete data and the random effects <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> are conceived as missing data. Viewed in this way, the Expectation-Maximization (EM) algorithm has proven to be a natural and popular approach to estimation. However, when <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:math> is partially observed or subject to measurement error, it is natural to formulate a multilevel model for <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">C</mml:mi> </mml:mrow> </mml:math> that includes random effects, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi>ν</mml:mi> </mml:mrow> </mml:math> . In this article, we extend this thinking to allow estimation of the joint distribution of data <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">Y</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">Y</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">C</mml:mi> <mml:mo stretchy="false">)</mml:mo> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Y</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">o</mml:mi> </mml:mrow> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mrow> <mml:mi mathvariant="normal">Y</mml:mi> </mml:mrow> <mml:mrow> <mml:mi mathvariant="normal">m</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> and random effects <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" overflow="scroll"> <mml:mrow> <mml:mi mathvariant="normal">b</mml:mi> <mml:mo>=</mml:mo> <mml:mo stretchy="false">(</mml:mo> <mml:mi mathvariant="normal">u</mml:mi> <mml:mo>,</mml:mo> <mml:mi mathvariant="normal">v</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:math> from observed data

  PublisherDOI

• Variability in Causal Effects and Noncompliance in a Multisite Trial: A Bivariate Hierarchical Generalized Random Coefficients Model for a Binary Outcome

  Statistics in Medicine · 2024-10-15 · 1 citations

  articleOpen accessSenior author

  of the experimental patients accessed e-assist while no controls were provided the access. Of interest are the average causal effect of assignment to treatment and the complier average causal effect as well as the variation of these causal effects across physicians. Each physician generates probabilities of screening for experimental compliers (experimental patients who accessed e-assist), control compliers (controls who would have accessed e-assist had they been assigned to e-assist), and never takers (patients who would have avoided e-assist no matter what). Estimating physician-specific probabilities jointly over physicians poses novel challenges. We address these challenges by maximum likelihood, factoring a "complete-data likelihood" uniquely into the conditional distribution of screening and partially observed compliance given random effects and the distribution of random effects. We marginalize this likelihood using adaptive Gauss-Hermite quadrature. The approach is doubly iterative in that the conditional distribution defies analytic evaluation. Because the small sample size per physician constrains estimability of multiple random effects, we reduce their dimensionality using a shared random effects model having a factor analytic structure. We assess estimators and recommend sample sizes to produce reasonably accurate and precise estimates by simulation, and analyze data from a trial of a CRCS intervention.

  PublisherOA PDFDOI

• Hierarchical Linear Models (HLM) and Multilevel Causal Inference

  2023-01-01

  report1st authorCorresponding

  This seminar introduces you to the theory and practice of multilevel modeling, and the logic of causal inference at multiple levels of analysis. The first day introduces you to two-level and three-level analysis, with an emphasis on how to build and interpret HLMs with a theoretically-rigorous foundation to guide estimation and inference. The second day considers how modern methods of causal inference can be applied to multilevel experimental and quasi-experimental designs. The new updated version of the HLM software will be used to illustrate model building and inference in a hands-on way. An official Instats certificate of completion with Professor Raudenbush is provided at the conclusion of the seminar. The seminar offers 2 ECTS Equivalent points for European PhD students.

  PublisherDOI

• Maximum Likelihood Estimation of Hierarchical Linear Models from Incomplete Data: Random Coefficients, Statistical Interactions, and Measurement Error

  Figshare · 2023-01-01

  datasetOpen accessSenior author

  We consider two-level models where a continuous response <i>R</i> and continuous covariates <i>C</i> are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data Robs and Cobs. However, if the model for <i>R</i> given <i>C</i> includes random coefficients, interactions, or polynomial terms, their joint distribution will be nonstandard. We propose a family of unique factorizations involving selected “provisionally known random effects” <i>u</i> such that h(Robs,Cobs|u) is normally distributed and <i>u</i> is a low-dimensional normal random vector; we approximate h(Robs,Cobs)=∫h(Robs,Cobs|u)g(u)du via adaptive Gauss-Hermite quadrature. For polynomial models, the approximation is exact but, in any case, can be made as accurate as required given sufficient computation time. The model incorporates random effects as explanatory variables, reducing bias due to measurement error. By construction, our factorizations solve problems of compatibility among fully conditional distributions that have arisen in Bayesian imputation based on the Gibbs Sampler. We spell out general rules for selecting <i>u</i>, and show that our factorizations can support fully compatible Bayesian methods of imputation using the Gibbs Sampler. Supplementary materials for this article are available online.

  PublisherDOI

• Maximum Likelihood Estimation of Hierarchical Linear Models from Incomplete Data: Random Coefficients, Statistical Interactions, and Measurement Error

  Figshare · 2023-01-01

  datasetOpen accessSenior author

  Abstract–We consider two-level models where a continuous response <i>R</i> and continuous covariates <i>C</i> are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data <i>R_obs </i> and <i>C_obs </i>. However, if the model for <i>R</i> given <i>C</i> includes random coefficients, interactions, or polynomial terms, their joint distribution will be nonstandard. We propose a family of unique factorizations involving selected “provisionally known random effects” <i>u</i> such that h(Robs,Cobs|u) is normally distributed and <i>u</i> is a low-dimensional normal random vector; we approximate h(Robs,Cobs)=∫h(Robs,Cobs|u)g(u)du via adaptive Gauss-Hermite quadrature. For polynomial models, the approximation is exact but, in any case, can be made as accurate as required given sufficient computation time. The model incorporates random effects as explanatory variables, reducing bias due to measurement error. By construction, our factorizations solve problems of compatibility among fully conditional distributions that have arisen in Bayesian imputation based on the Gibbs Sampler. We spell out general rules for selecting <i>u</i>, and show that our factorizations can support fully compatible Bayesian methods of imputation using the Gibbs Sampler.

  PublisherDOI

• Maximum Likelihood Estimation of Hierarchical Linear Models from Incomplete Data: Random Coefficients, Statistical Interactions, and Measurement Error

  Journal of Computational and Graphical Statistics · 2023-07-13 · 7 citations

  articleSenior author

  –We consider two-level models where a continuous response R and continuous covariates C are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data Robs and Cobs . However, if the model for R given C includes random coefficients, interactions, or polynomial terms, their joint distribution will be nonstandard. We propose a family of unique factorizations involving selected “provisionally known random effects” u such that h(Robs,Cobs|u) is normally distributed and u is a low-dimensional normal random vector; we approximate h(Robs,Cobs)=∫h(Robs,Cobs|u)g(u)du via adaptive Gauss-Hermite quadrature. For polynomial models, the approximation is exact but, in any case, can be made as accurate as required given sufficient computation time. The model incorporates random effects as explanatory variables, reducing bias due to measurement error. By construction, our factorizations solve problems of compatibility among fully conditional distributions that have arisen in Bayesian imputation based on the Gibbs Sampler. We spell out general rules for selecting u, and show that our factorizations can support fully compatible Bayesian methods of imputation using the Gibbs Sampler.

  PublisherDOI

• Hierarchical Linear Models (HLM) and Multilevel Causal Inference

  2022-01-01

  report1st authorCorresponding

  This seminar, taught by Professor Raudenbush, will introduce you to the theory and practice of multilevel modeling, and the logic of causal inference at multiple levels of analysis. The first day introduces you to two-level and three-level analysis, with an emphasis on how to build and interpret HLMs with a theoretically-rigorous foundation that will guide estimation and inference. The second day considers how modern methods of causal inference can be applied to multilevel experimental and quasi-experimental designs. The new and substantially updated version of the HLM software will be used to illustrate model building and inference in a hands-on way. All participants will receive a free 60-day trial license to this exciting update to the classic HLM software from Scientific Software International, and a 20% discount on a future purchase of HLM. An official Instats certificate of completion with Professor Raudenbush is provided at the conclusion of the seminar. The seminar offers 2 ECTS Equivalent points for European PhD students.

  PublisherDOI

• Effects of Time-Varying Parent Input on Children’s Language Outcomes Differ for Vocabulary and Syntax

  Psychological Science · 2021 · 30 citations

  Senior authorCorresponding
  • Psychology
  • Linguistics
  • Developmental psychology

  Early linguistic input is a powerful predictor of children's language outcomes. We investigated two novel questions about this relationship: Does the impact of language input vary over time, and does the impact of time-varying language input on child outcomes differ for vocabulary and for syntax? Using methods from epidemiology to account for baseline and time-varying confounding, we predicted 64 children's outcomes on standardized tests of vocabulary and syntax in kindergarten from their parents' vocabulary and syntax input when the children were 14 and 30 months old. For vocabulary, children whose parents provided diverse input earlier as well as later in development were predicted to have the highest outcomes. For syntax, children whose parents' input substantially increased in syntactic complexity over time were predicted to have the highest outcomes. The optimal sequence of parents' linguistic input for supporting children's language acquisition thus varies for vocabulary and for syntax.

  PublisherDOI

• Effects of double-dose algebra on college persistence and degree attainment

  Proceedings of the National Academy of Sciences · 2021-06-28 · 13 citations

  articleOpen accessCorresponding

  Significance To obtain high-paying jobs, students from low-income backgrounds need to complete college at higher rates. Could remedial math coursework in high school help them do so? Some have argued that adolescence is too late to intervene, or that schools cannot overcome systemic social problems that hinder student success. Yet, evidence from Chicago suggests otherwise. We find that ninth graders assigned to “double-dose algebra”—an algebra class plus a second class designed to bolster algebra skills—were more likely to stay in, and complete, college compared to similar students who did not take double-dose algebra. However, when students were placed in double-dose classes with much-lower-skilled peers the program had no effect.

  PublisherOA PDFDOI

• Front Matter

  American Journal of Sociology · 2020-07-01

  paratextOpen access
  PublisherOA PDFDOI

Recent grants

• NIH Grant R01MH056600

  NIH · $399k · 2001

• Enviromental & Biological Variation and Language Growth

  NIH · $34.1M · 2002–2020

Frequent coauthors

• Robert J. Sampson

  Harvard University

  387 shared

• Felton J. Earls

  374 shared

• Jeanne Brooks‐Gunn

  Columbia University

  347 shared

• J. Douglas Willms

  Learning Partnership

  35 shared

• Anthony S. Bryk

  21 shared

• Sean F. Reardon

  Stanford University

  17 shared

• Jeanne Brooks‐Gunn

  14 shared

• Guanglei Hong

  12 shared

Awards & honors

• American Educational Research Association award for Distingu…
• Member of the National Academy of Sciences
• Member of the American Academy of Arts and Sciences