Here is a detailed explanation of how Akaike's (1974) work on statistical model identification applies to multivariate methods in sample surveys.

Akaike's Information Criterion (AIC)

In his seminal 1974 paper, "A new look at the statistical model identification," Hirotugu Akaike introduced the Akaike Information Criterion (AIC).1 This was a groundbreaking development in statistical modeling, providing a new way to select the best model from a set of candidates.2

The core idea behind AIC is to balance the trade-off between the goodness of fit of a model and its complexity.3 A model that fits the data very well but has a large number of parameters (is complex) may be "overfitting" the data, meaning it captures random noise rather than the underlying pattern. Such a model will likely perform poorly when used to make predictions on new data.4 Conversely, a very simple model may not be flexible enough to capture the true underlying relationships in the data.

The AIC formula is:
AIC = 2k - 2ln(L)5

Where:
●	k is the number of parameters in the model.
●	L is the maximized value of the likelihood function for the model.
When comparing a set of models, the one with the lowest AIC value is considered to be the best.6 The AIC penalizes models for having more parameters (the 2k term), thus favoring more parsimonious models, while also rewarding models that fit the data well (a higher likelihood L results in a lower AIC).7

Application to Multivariate Methods in Sample Surveys

Multivariate methods in sample surveys involve analyzing the relationships between multiple variables that have been collected from a sample of a larger population.8 These methods can include:

●	Multivariate regression: Modeling a single response variable using multiple predictor variables.9

●	Structural equation modeling: Testing complex relationships between multiple variables.10

●	Cluster analysis: Grouping individuals based on multiple characteristics.
When applying these methods, researchers often have several competing models that could potentially explain the relationships in the data. For example, in multivariate regression, a key question is which predictor variables to include in the final model.11 This is where model selection criteria like AIC become crucial.

However, a significant challenge arises when applying standard statistical methods, including AIC, to data from complex sample surveys. These surveys rarely use simple random sampling. Instead, they often employ design features such as:
●	Stratification: Dividing the population into subgroups (strata) and sampling independently from each.12

●	Clustering: Sampling groups (clusters) of individuals, such as households or schools.
●	Unequal probabilities of selection: Some individuals having a higher or lower chance of being included in the sample.
These design features mean that the standard assumption of independent and identically distributed observations, which underlies the derivation of the classic AIC, is violated.

Adapting AIC for Complex Survey Data

To address the challenges of complex survey data, statisticians have developed modified versions of AIC. A key contribution in this area is the work of Lumley and Scott (2015), who proposed a survey-specific version of AIC.13

Their approach involves adjusting the penalty term of the AIC to account for the survey design. This is often done by incorporating a design effect (deff). The design effect is a measure of how much the variance of an estimate is inflated due to the complex survey design compared to a simple random sample of the same size.14

The general idea is to use a pseudo-likelihood function that is weighted to account for the unequal probabilities of selection. The penalty term in the AIC is then adjusted to reflect the effective sample size, which is often smaller than the nominal sample size due to the design effect.
For multivariate models, the application of a survey-adjusted AIC allows researchers to:
●	Compare different models for the relationships between multiple variables while properly accounting for the survey design.
●	Select the most parsimonious model that adequately fits the data, avoiding the inclusion of unnecessary variables that might be statistically significant only due to the large sample sizes of some surveys.
●	Obtain more reliable and valid inferences about the relationships between variables in the population from which the sample was drawn.
In summary, while Akaike's original formulation of AIC is not directly applicable to data from complex sample surveys, the fundamental principle of balancing model fit and complexity remains highly relevant. By adapting the AIC to account for survey design features like stratification, clustering, and unequal weighting, researchers can effectively use this powerful tool for model selection in the context of multivariate analysis of survey data.15 This ensures that the chosen models are not only statistically sound but also appropriately reflect the complexities of the data collection process.