Survey

The statistical nature of the survey

The statistical nature of the survey is to draw a representative sample from the population, then conduct statistical analysis on this sample to infer certain characteristics of the population. In other words, the survey is a typical application of inferential statistics, where observations and analyses of the sample are used to estimate the parameters of the population.

In this process, survey design and statistical analysis need to address the following two core issues:

1. How to ensure the representativeness of the sample through appropriate sample design (such as random sampling, stratified sampling, etc.).

2. How to handle sampling errors, specifically how to estimate the difference between the sample and the population.

True Value

In statistics, the true value typically refers to the actual parameter of the population, such as the population mean, population proportion, etc. In survey research, the true value is a theoretical concept and is not directly observable. The purpose of the survey is to estimate the true value of the population through a sample, so we use the estimate to approximate the true value.

However, due to issues with the representativeness of the survey sample and the existence of sampling errors, the survey results are often merely an estimate of the true value, accompanied by a certain confidence interval and margin of error. Therefore, the true value is the target that the survey aims to approximate, but in practice, only an estimate close to the true value can be obtained.

Law of Large Numbers and Latent Variable

The Law of Large Numbers (LLN) is a fundamental theorem in statistics that states that as the sample size increases, the sample statistic (such as the sample mean) will approach the population parameter (such as the population mean). In surveys, the applicability of the Law of Large Numbers depends on the following factors:

1. Random Sampling Assumption: The Law of Large Numbers applies in the context of random sampling, where each individual has an equal and independent probability of being selected.    

2. Sufficient Sample Size: The Law of Large Numbers requires a sufficiently large sample size. The larger the sample size, the closer the sample statistic will be to the population parameter (true value). 

3. Unbiased Estimation: The design of the survey and estimation methods must ensure that the sample statistic is an unbiased estimate of the population parameter. In other words, the design of the survey method must not systematically favor certain individuals; otherwise, even with a large sample size, the results may not approach the true value.

In our study, attitudes towards Love, Marriage, Fertility and Parenting are multidimensional concepts that encompass various aspects such as the ideal number of children, timing of childbirth, responsibility for childbearing, and family roles. Each specific aspect (such as attitudes toward the ideal number of children, support for reproductive policies, etc.) can be encoded and quantified in different ways, and each can have its own "true value." The aspect measured by each question can be considered a specific, individual true value. For example, a respondent's rating of the ideal number of children can be estimated using statistics such as the mean, variance, etc., to approximate the "true value" of this aspect.

Overall, the concept of attitudes towards childbirth is composed of multiple specific aspects. In this case, the overall "true value" of attitudes towards childbirth can be seen as the combined representation of the true values of these specific aspects.

In psychometrics, methods such as factor analysis and structural equation modeling (SEM) are typically used to reveal the underlying common factors behind these observed variables. The goal of these models is to identify the latent true value that influences the overall attitude towards childbirth, thereby combining multiple observed variables into a composite index, which is treated as a latent variable. We consider attitudes towards love, marriage, fertility, and parenting as the weighted average or some combination of multiple specific true values (attitude scores for each aspect). This composite result can be seen as the overall "true value" of the concept of attitudes towards childbirth.

In the context we are discussing, if we treat the concept of attitudes towards childbirth as a multidimensional latent variable, then this "true value" can be represented as a vector, where each element corresponds to a specific dimension of the attitudes towards childbirth (e.g., ideal number of children, attitudes towards the timing of childbirth, etc.). A simpler perspective is that if there is some complex relationship between these dimensions, this "true value" can also be represented as a matrix, where the elements may represent the associations or correlations between different dimensions. From a statistical standpoint, this representation is similar to the mean vector or covariance matrix in multivariate statistical analysis, or the factor loading matrix in factor analysis.

Actually, the Law of Large Numbers applies not only to single scalars but also to estimates in the form of vectors and matrices. In multidimensional cases, we typically study the mean or covariance structure of a multivariate random variable.

The Law of Large Numbers holds in this context as well, known as the multidimensional law of large numbers or the vector form of the strong law of large numbers. The basic idea is that, when the sample size is large enough, the sample mean vector will approach the population mean vector, or the sample covariance matrix will approach the population covariance matrix.

Proof

Based on our discussion above, we hereby give the proof of the reasonability of our survey segmentation: