How Do You Know Which Model Is Nested Within Which in Sem?

ANOVA > Nested Model

Contents:

1. What is a Nested Model?
2. Nested Factors
3. Nested ANOVA
• Nested vs. Crossed

What is a Nested Model?

Very simply, "nested" ways that 1 model is a subset of some other. For example, take a model for pregnancy outcomes that includes four categorical independent variables:

• Age,
• Weight,
• Pre-existing weather,
• Hereditary factors.

Several smaller models can exist derived from this main one, and each is "nested" inside the main model. For example:

1. Historic period and weight,
2. Weight and pre-existing conditions,
3. Age and hereditary factors.

Basically, if you lot tin become i model by constraining parameters of another, those models are nested. For example, the set of normal distribution models contains an infinite number of nested models, including normal distributions with means of 0, one, or 99.

Uses

Nested models are used for several statistical tests and analyses, including multiple regression, likelihood-ratio tests, conjoint analysis, and independent of irrelevant alternatives (IIA). While the above definition should give you a general sense of what a nested model is, the definition gets a bit more technical depending on where you lot are using is. For example:

In multiple regression and structural equation modeling (SEM), the idea is the aforementioned — that one model is nested inside another. More technically, both models must accept identical terms and one of the models must have one or more extra terms. For example:

• y = β₀ + β₁x₁ + β_iix_ii + 10
• y = β₀ + β₁10_i + β_iix₂ + β_iiiten_i10₂ + x

The larger model is chosen the full model and the smaller model is called the reduced model.

Caution: non all nested models are as obvious as the ones I have highlighted above. Rigdon (1999) suggest caution when deciding to analyze nested models because of this fact. At the time of writing there isn't any software that can clarify if two unlike structural models are similar (Bentler & Satorra).

Nested Factors

Nested factors 'fit within each other'.

As a reminder, a "factor" is a set of observed variables that have similar response patterns. 2 factors A and B are nested if there is an entirely dissimilar fix of values of B for every value of A.

As an example, let'due south say factor "A" is family and factor "B" is children. A child tin be Simpson or Lawson, just non both. Bill, Frank, or Ellis are Simpson; Jace, Renee, or Polly are Lawson. These two factors (family/child) are nested because whatever given child exists in merely ane family. In more than formal terms, nosotros say that every value of B exists for 1 and only i value of A.

Examples of Nested Factors.

Imagine a product tester needs to test lead and arsenic levels in canned broiled beans produced by a certain well-known brand. He might visit three unlike factories, test two different batches in each factory, and open five cans per batch. Manufactory, Batch #, and Can ID# are iii different variables, simply they are nested. Every tin exists only in one detail batch, and each batch exists in only one factory.

Or suppose a class of children was surveyed on their favorite snack which was either sweet or salty (factor A). Some said they preferred savory snacks, others said they liked sweetness snacks better. Among the children who liked savory snacks were specific brands (factor B). Some children preferred Cheez-its(ane), nachos(2), spicy popcorn(3) or Slim Jims(iv). Other children preferred sweet snacks: ice foam(1), chocolate(2), fruit(3) and candy(4).

The ii factors A and B are nested. Each of the snack variables is included in exactly one of the sweet/savory variable distinctions.

Notation for Nested Factors

The subscript j(i) indicates that the cistron indexed past j is nested in the cistron indexed by i. In the above snack example, yous could alphabetize A with i = 1, 2 and B with j = 1, 2, 3 ,four. Notation that fifty-fifty though the indexing of B (brand) is repeated across the ii instances of A (sweet or salty), the actual values of B are unlike for both. For example, 1(2) (preference for nachos) is not the aforementioned every bit 2(ii) (preference for chocolate).

Determining if Factors are Nested

Sometimes it isn't immediately obvious whether or non factors are nested. The easiest way to bank check is to make a table; if  every value of B is nonzero for only 1 value of A, B is nested in A.

In the table above, numbers were randomly assigned to each value of each variable. Columns with no data (nothing everywhere) were deleted; For case, all instances of B = 5 are in A = 2. The column A = 1, B = 5 is empty, then is not included in the table.

What is Nested ANOVA?

A nested ANOVA (also called a hierarchical ANOVA) is an extension of a uncomplicated ANOVA for experiments where each grouping is divided into two or more random subgroups. Information technology tests to see if there is variation between groups, or inside nested subgroups of the attribute variable. Y'all should use nested ANOVA when you have:

• I measurement variable,
• 2 or more nested nominal variables (factors).

Examples

Allow'southward say you wanted to investigate the wage gap between men and women. You also remember that height affects wages (which is true — meet The Atlantic's story on Why Tall People Make More) as does obesity (also true: see Forbes' story The Price of Obesity). Your factors or levels (sex activity, summit, weight) are nested within each other. For example, "weight" is non a standalone cistron — it'south nested nether male/female. The following image shows the hierarchical model:

In the following example, 5 unlike seedlings have been sampled from 5 unlike flowers in two dissimilar fields A and B:

Model I and Model II in Nested ANOVA

A model I ANOVA (also called a fixed-effects model) is where the treatments are stock-still by the experimenter. For example, if yous are comparing how unlike weights affect health you might choose specific weight ranges. If a nested ANOVA has a highest level of Model I, it's chosen a mixed model nested ANOVA.

Model II ANOVAs are where the treatments are random and not stock-still. For example, instead of the researcher choosing weights, they would be chosen at random. If a nested ANOVA has a highest level of model Ii, information technology'due south chosen a pure model II nested ANOVA.

Nested vs. Crossed Designs

While nested models can be represented by a purely hierarchical graph — such as the ones above — crossed models involve some crossover between the levels of independent variable. An example of a pure crossed model is where two groups of students are taught different ways to solve math bug by teacher A and teacher B. Every bit all students in both groups are exposed to teacher A's methods and teacher B'due south methods, the model is crossed. If information technology was nested, one grouping of students would simply feel one teacher'south methods.

Crossed designs are preferable, because they are improve at detecting differences between groups than nested models. Nevertheless, it may not be possible to ever use crossed models — some experiments necessitate the used of nested models.

In many experiments, it may not exist clear if your model is nested or crossed — and in some cases y'all might have a combination of both. Figuring out if your design is nested or non can be challenging. Drawing a hierarchical graph like the ones higher up tin can help.

The above instance of the wage gap betwixt men and women would be crossed instead of nested if it was possible for the factor levels to cantankerous over. For example, if a man could be both short and alpine, or normal weight and overweight. While this is theoretically possible (for example, y'all could take twins, i of whom is overweight and one is normal weight), in this case the scenario is not crossed.

References

Bentler, P. & Satorra, A. (2010). Testing Model Nesting and Equivalence. Psychol Methods. 2010 Jun; 15(two): 111–123. Retrieved 9/19/2016 from http://world wide web.ncbi.nlm.nih.gov/pmc/articles/PMC2929578/.
Carriquiry, A. Multiple Regression. Retrieved 9/19/2016 from: http://www.public.iastate.edu/~alicia/stat328/Multiple%20regression%xx-%20nested%20models.pdf
Doncaster & Davey (2007). Assay of Variance and Covariance: How to Cull and Construct Models for the Life Sciences. Cambridge University Press.
Purdue Statistics. A Await at Nested Factors. Retrieved September 17, 2017 from: http://www.stat.purdue.edu/~bacraig/notes1/topic19.pdf
Rigdon, E.Due east. (1999). Using the Friedman method of ranks for model comparison in structural equation modeling. Structural equation modeling, 6(three), 219-232

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