A nested model is a simpler version of a more complex statistical model, created by removing or restricting one or more of its parameters. Two models are “nested” when the smaller model is a special case of the larger one. If you can take the bigger model and set certain parameters to zero, or force certain parameters to equal each other, and arrive at the smaller model, those two models are nested.
This concept matters because it gives you a principled way to test whether those extra parameters actually improve the model’s ability to explain your data, or whether the simpler version does just as well.
How Nesting Works in Practice
Think of nesting like Russian dolls. The smaller model fits entirely inside the larger one. The larger model (often called the “full” model) has more free parameters, while the smaller model (the “restricted” or “reduced” model) locks some of those parameters to fixed values or constrains them.
The most common way to create a nested model is parameter nesting: a free parameter in the full model gets fixed in the restricted model, or a free parameter gets folded into a constraint that reduces the total number of freely estimated values. For example, suppose you have a regression model that predicts exam scores using study hours and sleep quality. The full model has coefficients for both predictors. If you set the sleep-quality coefficient to zero, you get a simpler model with only study hours. That simpler model is nested within the full model, because you can recover it by applying a specific restriction.
This works in the other direction too. You can start with a simple model and ask whether adding a parameter meaningfully improves the fit. The key requirement is that both models must be fit to the same dataset, and one model’s parameters must be a strict subset of the other’s.
Testing Whether the Extra Complexity Helps
The whole point of identifying nested models is so you can formally compare them. The most widely used tool for this is the likelihood ratio test. It works by calculating how much better the full model fits the data compared to the restricted model, then checking whether that improvement is large enough to justify the added complexity.
The test statistic follows a chi-squared distribution, with degrees of freedom equal to the difference in free parameters between the two models. If your full model has 5 free parameters and your restricted model has 3, the test uses 2 degrees of freedom. A statistically significant result means the restricted model fits meaningfully worse, so the extra parameters earn their place.
Information criteria like AIC and BIC offer an alternative approach. AIC penalizes model complexity by 2 units per additional parameter. A difference of less than 2 AIC units between models is generally considered too small to favor one over the other. In fact, comparing two nested models that differ by a single term using AIC is functionally equivalent to running a likelihood ratio test with a significance threshold of about p = 0.157, which is more liberal than the conventional p = 0.05. This means AIC is more willing to keep extra parameters than a standard hypothesis test would be.
The Role of Parsimony
Nested model comparisons are rooted in a statistical version of Occam’s razor: if a simpler explanation fits the data just as well as a complex one, prefer the simpler one. Bayesian approaches formalize this idea. A more complex model has to spread its probability more thinly across possible outcomes, so if the data are compatible with both the simple and complex versions, the simpler model gets higher support automatically.
This isn’t just philosophical tidiness. Overly complex models tend to fit noise in the data rather than genuine patterns, which makes their predictions worse on new data. Nested model testing gives you a structured way to trim unnecessary complexity while keeping the parameters that genuinely matter.
Nested Models in Structural Equation Modeling
One of the most prominent uses of nested model comparisons is in structural equation modeling (SEM), particularly for testing measurement invariance. Measurement invariance asks whether a psychological or behavioral construct, like anxiety or job satisfaction, means the same thing across different groups or time points.
This is tested through a sequence of increasingly restrictive nested models. First, you check whether the basic structure of the model holds across groups (configural invariance). Then you constrain the factor loadings to be equal across groups and compare this more restrictive model to the first one (metric invariance). Next, you constrain the item intercepts to be equal as well (scalar invariance). Each step adds constraints, making each successive model nested within the previous one.
At each stage, you compare the nested models using a chi-squared difference test or by examining the change in fit indices like CFI. If the more constrained model fits significantly worse, the equality assumption fails at that level, and you know the construct isn’t being measured equivalently across groups. If the fit holds, you can move to the next level of constraint. The entire logic depends on the models being properly nested, so that any difference in fit can be attributed specifically to the imposed constraints.
Nested vs. Non-Nested Models
Not all model comparisons involve nesting. Two models are non-nested when neither one can be obtained by restricting the other. For instance, a model predicting income from education level and a completely separate model predicting income from geographic region are non-nested, because you can’t get from one to the other by setting parameters to zero or adding constraints.
This distinction matters because the likelihood ratio test only works for nested models. When models are non-nested, you need different tools. Information criteria like AIC and BIC can compare any set of models fit to the same data, nested or not, which is part of why they’re so popular. Specialized tests like the Vuong test also exist specifically for non-nested comparisons.
If you’re unsure whether two models are nested, ask one question: can I transform the larger model into the smaller one purely by fixing or constraining its parameters? If yes, they’re nested. If you’d need to swap out variables or change the model’s fundamental structure, they’re not.