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# Statistical Model

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 Sub Topics Statistics is an important and a very special branch of mathematics. It is concerned about the gathering, organizing, observing, calculating, analyzing, interpreting and forecasting numerical (usually large) data. Statistics involves various different formulae that are useful in performing statistical researches and surveys. There are different types of variables used in statistics. At times, it is required to find the relationships among the variables of a research. It is also needed that that these relations are expressed in the form of equations. These mathematical equation used in statistical processes are known as statistical models. These models describe the connection between two or more variables and also do formulate the relation. In this article, we are going to learn about the statistical models, their various types and their applications.

## Definition

The statistical model may be defined as the mathematical equation that are formulated in the form of relationships between variables. A statistical model illustrates how a set of random variables is related to another set of random variables.

We can define this concept more mathematically as below :
A statistical model is represented as the ordered pair (X , P) ; in which X denotes the set of all possible observations, while P refers to the set of probability distributions on X. It is to be assumed that there are distinct elements in P generating the observed data.With the help of statistical inferences, the researcher becomes enable to make statements the element or elements that are likely to be correct or true.

## Types

There are different types of statistical models used in various statistical processes. The statistical models are classified on the basis of number and types of variables and equations used in the process.
These models can be broadly categorized as:
1) The complete models
2) The incomplete models
complete model does have the number of variables equal to the number of equations ; while an incomplete model does not have number of variables same as number of equations.
Few most frequently used statistical models are :
1) General Linear Model (having only continuous dependent variables, such as logistic regression.
2) Structural Equation Model
3) Multilevel Model, etc.

## Linear Statistical Models

A linear statistical model has been named so because it has linear relationships among the variables. Let us assume that there are n independent observations having independent variables $x_{i}=x_{1},x_{2},...,x_{n}$ ; also the dependent variable as y that assigns the numeric response in it. Also assume that $\alpha_{i}=\alpha_{1}, \alpha_{2},..., \alpha_{n}$ be the parameters for the model. Then, the linear statistical model can be expressed as the following general form -
$y$ = $\alpha_{1} x_{1}+ \alpha_{2} x_{2}+...+ \alpha_{n} x_{n}$
The dependent variable in a linear models is usually a continuous variable.The generalized linear statistical models are the frameworks for modeling a response or a resultant variable y. This type of model is used while modelling positive data on a very large scale. The examples of generalized linear models are lognormal distribution, skewed distribution, Poisson distribution, Bernoulli or binomial distribution, categorical distribution etc.

## Statistical Forecasting Models

The forecasting is a process that includes the making of statements about the experiments or events whose real and actual outcomes have not yet been analyzed. In statistics, the forecasting is a very useful concept. Statistical forecasting can be done using different methods and models. These models attempt to identify the factors which might influence the variables that are being forecast. Such as - forecasting of climatic and weather conditions may improve the model of availability and predicted sale of umbrellas.

The models used in the forecast are known as statistical forecasting models. These models also take previous relationships of the variables into consideration.

For example: if the variables have been linearly related for a long time, they may be considered to be related linearly for the forecast also. In such case, it would not be necessary to understand the reasons for this relationship.

## How to Build a Statistical Model

In order to build a statistical model, one needs to remember the following points:

1) Data Gathering : Data should be collected from a proper and authoritative source so that our the uncertainty is to be reduced from the information about something on interest.

2) Descriptive Methods : The descriptive methods and graphs are to be utilized in order to summarize the factuals of the data we have.

3) Thinking about Predictors : The predictors and the variables are required to be thought and imagined about a model. For an example of model of socio-economic status of a person, the potential sets of predictors and variables may include - the demographics such as age, sex economic status, mental health i.e. any diagnoses of mental illness or history of alcoholism, psychological health such as stress or depression, the social condition like - isolation, number of friends, connection with the family.
By creating each set separately, one is able to build theoretically meaningful models.

4) Building of model and Interpreting the Results
A statistical model should be such an expressive one that it tells us a story. Emphasis at the coefficients; have a look at the relationships between coefficients and control variables. It should be quite easier to interpret and forecast the result by having just a look at the model.

## Example

$H$ = $\alpha_{0}+ \alpha_{1} age + \lambda$
$\alpha_{0}$ is an intercept.
$\alpha_{1}$ is a parameter such that the age is multiplied by it to get the prediction of the height.
$\lambda$ be an error term.