Students Are Not Independent: A Multilevel Modeling Case Study

The research question

Suppose a researcher wants to answer:

After accounting for students' prior achievement, is a school-level resource index associated with end-of-year achievement, and does the association between prior achievement and the outcome vary across schools?

This question contains two levels.

That distinction matters because multilevel analysis permits predictors to operate at different levels rather than treating student and contextual characteristics as if they were interchangeable. Tabachnick and Fidell (2013), for example, describe student-level predictors such as motivation or study time alongside classroom- and school-level predictors such as teacher characteristics, school poverty, school type, and school size.

The hierarchy of observations

The synthetic study contains 1,138 students from 40 schools, with 22–35 students observed per school.

For clarity, the primary analysis uses a two-level structure:

A fuller education study could contain three levels—students within classrooms within schools. Both Tabachnick and Fidell (2013) and Lovric (2011) explicitly use pupils/students, classrooms, and schools as examples of multilevel structures. Lovric (2011) also describes extensions in which students are nested in classrooms that are themselves nested in schools.

The two-level version is used here because it makes the dependence problem and model interpretation transparent without adding a level that is unnecessary for the synthetic research question.

Why students within a school may be correlated

Ordinary linear regression typically treats residual errors from different observations as independent. That assumption becomes questionable when observations share a context.

Students in the same classroom may influence one another and may therefore be more alike than students from different classrooms; students in the same school can likewise be more similar than students from different schools. Multilevel models are specifically designed to represent this type of dependence rather than requiring independent errors across all student observations (Tabachnick & Fidell, 2013).

Lovric (2011) describes the same principle more generally: when cluster means differ, observations within a cluster tend to be more homogeneous than observations from different clusters. The intraclass correlation coefficient can quantify this within-cluster resemblance.

Why ordinary regression is not enough

The researcher's initial model is:

Achievement = intercept + prior achievement + school resources + error

The formula looks reasonable. The problem is the error structure.

If all 1,138 students are treated as independent, the analysis effectively acts as though the school-level resource information has been independently replicated more than a thousand times. But there are only 40 independently varying school contexts in this synthetic study.

Tabachnick and Fidell (2013) warn that analyzing individual observations without accounting for their hierarchical structure can inflate the Type I error rate because the analysis uses degrees of freedom that are not truly independent. They also distinguish this problem from the opposite mistake of aggregating everything to the group level, which can discard individual-level information and reduce power.

The problem is therefore not simply that ordinary regression coefficients must always be numerically different. Ignoring clustering can particularly distort uncertainty—standard errors, confidence intervals, and hypothesis tests—even when a coefficient estimate itself looks plausible. Frey's encyclopedia entry on hierarchical linear modeling similarly emphasizes that HLM provides parameter estimates and standard errors appropriate for clustered observations (Frey, 2022).

Synthetic comparison

The generated data illustrate the point:

The school-resource coefficient barely changes: approximately 3.1 achievement points per one-unit increase in the synthetic resource index.

Its estimated standard error, however, is about twice as large in the multilevel model.

These particular numbers are synthetic and should not be generalized. They demonstrate why the independence decision affects inferential precision.

Start with an empty model and calculate the ICC

Before adding predictors, an intercept-only multilevel model can be used to partition outcome variability into between-school and within-school components. A random-intercept baseline model also provides the variance components required for an intraclass correlation coefficient (Frey, 2022).

In this synthetic dataset, the fitted empty model produced:

For a two-level random-intercept model, the ICC can be expressed as:

ICC = between-school variance / (between-school variance + within-school variance)

giving:

ICC = 23.88 / (23.88 + 104.34) = 0.186

The synthetic ICC therefore indicates that about 18.6% of the unconditional variation in achievement is between schools, while the remainder is within schools.

The ICC is not merely a descriptive curiosity. It measures the magnitude of clustering or dependency in the outcome; larger within-cluster resemblance means a stronger departure from treating all observations as independent (Frey, 2022; Lovric, 2011; Tabachnick & Fidell, 2013).

Separate within-level and between-level predictors

A major advantage of multilevel modeling is that the model can preserve the level at which a predictor actually varies (Tabachnick & Fidell, 2013; Lovric, 2011).

In this case:

The distinction changes interpretation.

Those are not the same kind of comparison.

Multilevel modeling was developed precisely for situations in which student, classroom, teacher, school, or other contextual variables contribute at different levels of a hierarchy (Lovric, 2011).

Random intercepts: schools can start at different levels

The first useful multilevel extension gives each school its own intercept.

A readable version is:

Achievement_ij = β₀ + β₁(Prior achievement_ij) + β₂(Resources_j) + u_0j + e_ij

where:

• i identifies a student;
• j identifies a school;
• β₀ is the overall intercept;
• β₁ and β₂ are fixed regression coefficients;
• u_0j is the school-specific random intercept;
• e_ij is the student-level residual.

The random intercept represents remaining differences among school mean outcomes after accounting for predictors in the model. Random-intercept models therefore allow schools to differ in their baseline outcome levels rather than forcing every school onto one common regression line (Frey, 2022; Lovric, 2011).

This is already a substantial improvement over pretending that every student's residual is unrelated to every other student's residual.

Random slopes: the student-level relationship can vary across schools

The next question is whether the association between prior achievement and end-of-year achievement is identical in every school.

A random-slope model relaxes that restriction:

Achievement_ij = β₀ + β₁(Prior achievement_ij) + β₂(Resources_j) + u_0j + u_1j(Prior achievement_ij) + e_ij

Now u_1j allows the prior-achievement slope to vary across schools.

This is directly supported by the multilevel framework. Tabachnick and Fidell (2013) describe multilevel models as allowing both intercepts and predictor slopes to vary across higher-level units. Frey (2022) likewise distinguishes random-intercept models from random-coefficient models in which a Level 1 relationship varies across groups.

In the synthetic random-slope model, the estimated variance of the school-specific prior-achievement slopes was 1.47. The model therefore represents some between-school heterogeneity in the prior-achievement relationship rather than imposing exactly the same slope on every school.

Interpreting the fitted model

The synthetic random-slope model produced the following fixed effects:

Student-level interpretation

Holding the school resource index constant, a student whose prior achievement is one standard deviation higher is predicted to score about 4.27 points higher on the end-of-year outcome on average.

The phrase on average matters because the random slope allows that relationship to differ among schools.

School-level interpretation

Holding the included student-level prior-achievement measure constant, a one-unit higher school resource index is associated with approximately 3.10 points higher expected achievement in this synthetic model.

Random-effects interpretation

The school random intercept represents residual differences in average achievement among schools after adjustment for the predictors.

The random slope represents residual differences among schools in the strength of the prior-achievement relationship.

Multilevel models can therefore address two related questions: whether schools differ in outcome level and whether a student-level relationship itself varies across schools (Tabachnick & Fidell, 2013; Frey, 2022).

Why levels should not be blurred in interpretation

Hierarchical data create interpretive risks as well as statistical ones.

A relationship estimated among schools should not automatically be attributed to individual students, and an individual-level relationship should not automatically be interpreted as a school-level relationship. Tabachnick and Fidell (2013) discuss these errors as ecological and atomistic fallacies. Lovric (2011) likewise links the development of multilevel analysis to the problem of incorrectly transferring associations from one aggregation level to another.

Those statements operate at different levels and should remain separate.

What if classrooms were also observed?

Suppose the real study recorded each student's classroom as well as school.

The hierarchy could become:

Level 1: students → Level 2: classrooms → Level 3: schools

A three-level model could then include random effects corresponding to classroom and school variation. Both Lovric (2011) and Tabachnick and Fidell (2013) explicitly describe education examples in which pupils or students are nested within classrooms and classrooms within schools.

That extension could matter if classmates share a teacher or classroom environment above and beyond the broader school context. But the additional level should be supported by the actual sampling and organizational structure. The present case therefore stays with two levels rather than pretending to have classroom information that was not generated.

Analogous multilevel structures occur outside education. Lovric (2011) discusses combinations such as patients, doctors, hospitals, and other organizational contexts, while Tabachnick and Fidell (2013) describe higher-level units more generally as groups, clusters, or contexts.

Practical checks before interpreting the model

Multilevel modeling solves the specific problem of representing hierarchy, but fitting a mixed model does not make every other modeling issue disappear.

A practical review should ask:

1. Does the hierarchy match the study design? Students should be assigned to the schools or classrooms that actually define the clustering in the data.
2. Do predictors have the correct level? A student characteristic and a school characteristic should not be interpreted as though they vary at the same level.
3. Is there enough information to estimate the proposed model? Complex random-effect structures require sufficient data, particularly at higher levels; small datasets can create estimation and convergence problems (Tabachnick & Fidell, 2013).
4. Are the random effects answering a research question? A random intercept represents between-cluster variation in outcome level, while a random slope represents between-cluster variation in a predictor–outcome relationship (Frey, 2022; Tabachnick & Fidell, 2013).
5. Are coefficient interpretations kept at the appropriate level? Individual- and group-level associations should not be exchanged.
6. Are model assumptions and diagnostics examined? Multilevel linear models still make distributional and variance assumptions about residual and random components, so estimated coefficients, variance components, and model behavior require scrutiny rather than automatic acceptance (Lovric, 2011; Tabachnick & Fidell, 2013).

What this case teaches

The researcher's original regression was not unreasonable because regression itself was inappropriate. It was incomplete because its error structure contradicted the design.

There were 1,138 student records, but there were only 40 school contexts. Students within the same school could share sources of variation, and the synthetic ICC confirmed appreciable between-school variation.

The multilevel analysis made that structure explicit. It separated a student-level predictor from a school-level predictor, allowed school mean achievement to vary through random intercepts, and allowed the prior-achievement relationship to vary through random slopes. Multilevel models are built precisely to represent these multiple sources of variation in hierarchical data (Lovric, 2011; Tabachnick & Fidell, 2013).

Most importantly, the comparison showed why dependence is an inferential issue. In the synthetic example, ordinary and multilevel regression produced similar school-resource coefficients, but ordinary regression reported much smaller uncertainty. Treating clustered students as independent made the evidence appear more precise than the hierarchy justified.

Limitations

1. Every observation and result in this case is synthetic. The coefficients illustrate methodology rather than educational effects.
2. The case uses only two levels. Many education datasets contain students within classrooms within schools, repeated measurements within students, or other structures. The appropriate hierarchy must follow the actual design (Lovric, 2011; Tabachnick & Fidell, 2013).
3. The example uses a continuous outcome and a relatively simple linear specification. It should not be treated as a template for every educational outcome.
4. The random-slope specification is deliberately limited to one student-level predictor. More complicated random structures require enough information to estimate them reliably and should be justified by the research question and design (Tabachnick & Fidell, 2013).
5. Multilevel modeling accounts for dependence but does not convert an observational association into a causal effect. The school-resource coefficient in this case remains an adjusted association within the synthetic design.

Conclusion

The central question in clustered education data is not simply, How many student rows do we have?

It is also, Which observations share a classroom, school, or other context?

When students are nested within schools, ordinary regression can treat correlated observations as though they provide fully independent information. Multilevel modeling instead represents the hierarchy directly, distinguishes student-level from school-level predictors, partitions variation across levels, and can allow both intercepts and slopes to vary across clusters (Frey, 2022; Lovric, 2011; Tabachnick & Fidell, 2013).

References

Frey, B. B. (Ed.). (2022). The SAGE encyclopedia of research design (2nd ed.). SAGE Publications.

Lovric, M. (Ed.). (2011). International encyclopedia of statistical science. Springer. https://doi.org/10.1007/978-3-642-04898-2

Tabachnick, B. G., & Fidell, L. S. (2013). Using multivariate statistics (6th ed.). Pearson.