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**SOME IMPORTANT STATISTICS HOMEWORK TOPICS:**

Regression analysis is one of the most effective statistical methods of identifying and estimating the relationship between a dependent variable and one or more independent variables. To understand this in simple terms, a regression analysis estimates how a change in an independent variable causes change in the dependent variable. It helps an analyst determine how the variables affect each other and identify those that have the most and the least impact on an event or a topic of study. With the application of a set of tools and procedures, it assesses the strength of the relationship among the variables and allows the analyst or researcher to model future relationships.

Regression analysis has two important and distinct applications: **predicting and forecasting future trends and results**, and **to analyze the causal relationship between dependent and independent variables**.

The key terms in a statistical regression analysis are:

**Dependent variable:**A**dependent variable**is the main factor that the analyst or researcher wants to understand and make predictions about. Dependent variables are also called response or outcome variables.**Independent variables:**Independent variables are the factors that the analyst believes to influence a change in the behavior of the dependent variable. Independent variables are also referred to as features, explanatory variables, covariates or predictors.

Regression analysis can be applied in almost all disciplines of study involving the analysis of data. Its application can be seen across the fields of research in science, economics, finance and humanities. It describes the relationship among the variables by fitting a line through the observed data on a scatterplot. There are three main types of regression analysis—simple linear, multiple linear and nonlinear regression. A straight line is used in linear regression models while a curved line is used in nonlinear models.

The most commonly used regression models are simple linear and multiple linear regression. It is important to understand here that a regression analysis by itself can only be used to determine the relationship among variables. To use any regression method for predictive purposes or to determine the extent of a causal relationship, the researcher must explain why the existing relationship has the ability to predict outcomes in a given context or why the relationship can be interpreted as causal. Before performing a regression analysis using any of the linear models, a number of assumptions are to be considered by the analyst.

Regression analysis using linear models is based on six important assumptions.

**These are:**

The sample represents the entire population.

A linear relationship between the slope and the intercept exists for the dependent and independent variables.

The independent variable/s is/are not random.

Value of the error (residual) is always zero.

Value of the error remains constant of all the observations in the dataset.

Value of the error is non-correlated for all the included observations.

Error values always follow a normal distribution.

Simple linear regression is a statistical model used to analyze the relationship between one dependent and only one independent variable. It is used when the researcher wants to know: how strong the relationship between the variables is, and the value of the dependent variable corresponding to a specific value of the independent variable.

**The following equation is used to express a simple linear regression:Y= a + bX + ε**

**where**:

“Y” is the dependent variable,

“X” is the independent variable,

“a” is the intercept,

“b” is the slope or the regression coefficient, and

“ε” is the error or residual of the estimate.

Simple linear regression determines the line that best fits through the plotted data by finding the regression coefficient (b) which is capable of minimizing the total error (e) of the model. While a simple linear regression can be performed by hand, it is a very tedious and time consuming process. This is why most analysts use statistical software such as “R” to analyze the data quickly and accurately.

Multiple linear regression analysis is in most ways similar to simple linear regression analysis. The only difference between the two is that while simple linear regression analysis uses only one independent variable, multiple linear regression analysis uses multiple independent variables in the model.

**It is mathematically represented as: Y= a + bX1 + cX2 + dX3 + ε**

Y is the dependent variable,

X1, X2 and X3 are the independent or explanatory variables,

“a” is the intercept,

“b”, “c” and “d’ are the slopes or the regression coefficients, and

“ε” is the error or residual of the estimate.

Multiple linear regression analysis is based on the same conditions and assumptions as used in a simple linear regression model. However, because multiple linear regression analysis uses multiple independent variables, there is one additional assumption made for this model. This assumption is of non-co linearity. The model assumes that there is none or minimal correlation among the independent variables. This assumption is important because it will be extremely difficult to estimate the extent of relationships between the dependent variable and each of the independent variables if they are highly correlated to each other.

Nonlinear regression is based on a mathematical model in which the regression model is used to examine and demonstrate a nonlinear relationship between the dependent variable and one or more independent variables. A curved line is generated in the model which displays nonlinearity of the relationship between the variables. The nonlinear model provides for greater flexibility and is capable of creating a line that suits the given scenario in the best way possible. The ultimate goal of a nonlinear model is to reduce the sum of squares to the least value possible using iterative numerical procedures. While a nonlinear regression generates highly accurate results, the process of performing the analysis is quite complex. It can therefore be applied to a wide range of complex scenarios—from understanding the growth of population as time progresses to assessing the effect of investor behavior on stock market returns.

**The mathematical expression for a simple nonlinear regression is: Y= f(X, β) + ε**

“Y” is the dependent or response variable,

“X” is the independent variable or vector of predictors (denoted by P),

“β” is the vector of parameters (denoted by k),

“f” is the regression function which is known, and

“ε” is the error or residual.

**The mathematical expression for multiple nonlinear regression analysis is: Yi = h [Xi(1) , Xi(2), … , Xi(m) ; Ѳ1, Ѳ2, …, Ѳp] + ε i**

“Yi” is the dependent or response variable,

“h” is the model function,

“X” is the independent variable or input,

“Ѳ” is the parameter that has to be estimated, and

“ε” is the model error or residual.

Since all the parameters in the multiple nonlinear models can be analyzed to determine if they are linear or nonlinear, the given function “Yi” can include both linear and nonlinear parameters. The function “h” is also considered in these models because the analyst cannot write it as linear for the parameters. This function is ultimately deduced from the expression.

It should be noted here that the term ‘nonlinear” is used to describe parameters rather than the independent variables. There are almost infinite possible ways for describing the “deterministic” aspect of a nonlinear model. Therefore, such a model provides greater number of opportunities to make valid statistical inferences.

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