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[Stephaine Zitzow]

A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data (and similar data from a larger population). A statistical model represents, often in considerably idealized form, the data-generating process.[1] When referring specifically to probabilities, the corresponding term is probabilistic model. All statistical hypothesis tests and all statistical estimators are derived via statistical models. More generally, statistical models are part of the foundation of statistical inference. A statistical model is usually specified as a mathematical relationship between one or more random variables and other non-random variables. As such, a statistical model is "a formal representation of a theory" (Herman Adr quoting Kenneth Bollen).[2]

Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-sided dice. We will study two different statistical assumptions about the dice.

The first statistical assumption constitutes a statistical model: because with the assumption alone, we can calculate the probability of any event. The alternative statistical assumption does not constitute a statistical model: because with the assumption alone, we cannot calculate the probability of every event. In the example above, with the first assumption, calculating the probability of an event is easy. With some other examples, though, the calculation can be difficult, or even impractical (e.g. it might require millions of years of computation). For an assumption to constitute a statistical model, such difficulty is acceptable: doing the calculation does not need to be practicable, just theoretically possible.

A statistical model is a special class of mathematical model. What distinguishes a statistical model from other mathematical models is that a statistical model is non-deterministic. Thus, in a statistical model specified via mathematical equations, some of the variables do not have specific values, but instead have probability distributions; i.e. some of the variables are stochastic. In the above example with children's heights, ε is a stochastic variable; without that stochastic variable, the model would be deterministic. Statistical models are often used even when the data-generating process being modeled is deterministic. For instance, coin tossing is, in principle, a deterministic process; yet it is commonly modeled as stochastic (via a Bernoulli process). Choosing an appropriate statistical model to represent a given data-generating process is sometimes extremely difficult, and may require knowledge of both the process and relevant statistical analyses. Relatedly, the statistician Sir David Cox has said, "How [the] translation from subject-matter problem to statistical model is done is often the most critical part of an analysis".[4]

In this example, the dimension, k, equals 2. As another example, suppose that the data consists of points (x, y) that we assume are distributed according to a straight line with i.i.d. Gaussian residuals (with zero mean): this leads to the same statistical model as was used in the example with children's heights. The dimension of the statistical model is 3: the intercept of the line, the slope of the line, and the variance of the distribution of the residuals. (Note the set of all possible lines has dimension 2, even though geometrically, a line has dimension 1.)

Parametric models are by far the most commonly used statistical models. Regarding semiparametric and nonparametric models, Sir David Cox has said, "These typically involve fewer assumptions of structure and distributional form but usually contain strong assumptions about independencies".[7]

Two statistical models are nested if the first model can be transformed into the second model by imposing constraints on the parameters of the first model. As an example, the set of all Gaussian distributions has, nested within it, the set of zero-mean Gaussian distributions: we constrain the mean in the set of all Gaussian distributions to get the zero-mean distributions. As a second example, the quadratic model

In both those examples, the first model has a higher dimension than the second model (for the first example, the zero-mean model has dimension 1). Such is often, but not always, the case. As an example where they have the same dimension, the set of positive-mean Gaussian distributions is nested within the set of all Gaussian distributions; they both have dimension 2.

Comparing statistical models is fundamental for much of statistical inference. Konishi & Kitagawa (2008, p. 75) state: "The majority of the problems in statistical inference can be considered to be problems related to statistical modeling. They are typically formulated as comparisons of several statistical models." Common criteria for comparing models include the following: R2, Bayes factor, Akaike information criterion, and the likelihood-ratio test together with its generalization, the relative likelihood.

The science of statistics is the study of how to learn from data. It helps you collect the right data, perform the correct analysis, and effectively present the results with statistical knowledge. Statistical modeling is key to making scientific discoveries, data-driven decisions, and predictions.

It is crucial to evaluate the quality of the analyses that others present to you, considering how critical data-based decisions and opinions have become. There is more to statistics than just numbers and facts. Instead, it's a collection of knowledge and procedures that reliably let you learn from data.

Statistical modeling helps you differentiate between reasonable and dubious conclusions based on quantitative evidence. Analyses and predictions made by statisticians are highly trustworthy. A statistician can help investigators avoid various analytical traps along the way.

The statistical modeling process is a way of applying statistical analysis to datasets in data science. The statistical model involves a mathematical relationship between random and non-random variables.

Data gathering is the foundation of statistical modeling. The data may come from the cloud, spreadsheets, databases, or other sources. There are two categories of statistical modeling methods used in data analysis. These are:

In the supervised learning model, the algorithm uses a labeled data set for learning, with an answer key the algorithm uses to determine accuracy as it trains on the data. Supervised learning techniques in statistical modeling include:

Regression model: A predictive model designed to analyze the relationship between independent and dependent variables. The most common regression models are logistical, polynomial, and linear. These models determine the relationship between variables, forecasting, and modeling.

Classification model: An algorithm analyzes and classifies a large and complex set of data points. Common models include decision trees, Naive Bayes, the nearest neighbor, random forests, and neural networking models.

In the unsupervised learning model, the algorithm is given unlabeled data and attempts to extract features and determine patterns independently. Clustering algorithms and association rules are examples of unsupervised learning. Here are two examples:

Reinforcement learning: This technique involves training the algorithm to iterate over many attempts using deep learning, rewarding moves that result in favorable outcomes, and penalizing activities that produce undesired effects.

Statistics and machine learning (ML) differ primarily in their purposes. You can build ML models for predicting the future by making accurate predictions without explicit programming, while statistical models can explain the relationship between variables.

However, some statistical models are inaccurate because of their inability to capture complex relationships between data, even if they can predict. ML predictions are more accurate, but they are also more challenging to understand and explain.

In statistical models, probabilistic models for the data and variables are interpreted and identified, such as the effects of predictor variables. A statistical model establishes the magnitude and significance of relationships between variables and their scale. Models based on machine learning are more empirical.

Even though data scientists are usually responsible for developing algorithms and models, analysts may also use statistical models in their work from time to time. As a result, analysts seeking to excel should gain a solid grasp of the factors that contribute to the success of these models.

Companies and organizations are leveraging statistical modeling to make predictions based on data to keep pace with the explosive growth of machine learning and artificial intelligence. The following are some benefits of understanding statistical modeling.

A data analyst needs a comprehensive understanding of all the statistical models available. You should identify which model is most appropriate for your data and which model best addresses the question at hand.

Raw data is rarely ready for analysis. Data must be clean before conducting accurate and viable research. The cleanup process usually involves organizing the collected information and removing "bad or incomplete data" from the sample.

To build a good statistical model, you need to explore and understand the data. If the data is not good enough, you can't draw any meaningful inferences. Knowing how different statistical models work and how they leverage data will enable you to determine what data is most relevant to the questions you are trying to answer.

Most organizations require data analysts to present their findings to two different audiences. First, the business team is not interested in the details of your analysis but wants to know the main conclusions. There is a second group of people often interested in the granular details. These people often require a summary of your broad findings and an explanation of how you reached them.

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