# Regression analysis

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In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships among variables. It includes many techniques for modeling and analyzing several variables, when the focus is on the relationship between a dependent variable and one or more independent variables (or 'predictors'). More specifically, regression analysis helps one understand how the typical value of the dependent variable (or 'criterion variable') changes when any one of the independent variables is varied, while the other independent variables are held fixed.

A statistical model is a mathematical model that embodies a set of statistical assumptions concerning the generation of sample data. A statistical model represents, often in considerably idealized form, the data-generating process.

Estimation theory is a branch of statistics that deals with estimating the values of parameters based on measured empirical data that has a random component. The parameters describe an underlying physical setting in such a way that their value affects the distribution of the measured data. An estimator attempts to approximate the unknown parameters using the measurements.

## Contents

Most commonly, regression analysis estimates the conditional expectation of the dependent variable given the independent variables – that is, the average value of the dependent variable when the independent variables are fixed. Less commonly, the focus is on a quantile, or other location parameter of the conditional distribution of the dependent variable given the independent variables. In all cases, a function of the independent variables called the regression function is to be estimated. In regression analysis, it is also of interest to characterize the variation of the dependent variable around the prediction of the regression function using a probability distribution. A related but distinct approach is Necessary Condition Analysis [1] (NCA), which estimates the maximum (rather than average) value of the dependent variable for a given value of the independent variable (ceiling line rather than central line) in order to identify what value of the independent variable is necessary but not sufficient for a given value of the dependent variable.

In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value – the value it would take “on average” over an arbitrarily large number of occurrences – given that a certain set of "conditions" is known to occur. If the random variable can take on only a finite number of values, the “conditions” are that the variable can only take on a subset of those values. More formally, in the case when the random variable is defined over a discrete probability space, the "conditions" are a partition of this probability space.

In statistics and probability quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile than the number of groups created. Thus quartiles are the three cut points that will divide a dataset into four equal-sized groups. Common quantiles have special names: for instance quartile, decile. The groups created are termed halves, thirds, quarters, etc., though sometimes the terms for the quantile are used for the groups created, rather than for the cut points.

In statistics, a location family is a class of probability distributions that is parametrized by a scalar- or vector-valued parameter , which determines the "location" or shift of the distribution. Formally, this means that the probability density functions or probability mass functions in this class have the form

Regression analysis is widely used for prediction and forecasting, where its use has substantial overlap with the field of machine learning. Regression analysis is also used to understand which among the independent variables are related to the dependent variable, and to explore the forms of these relationships. In restricted circumstances, regression analysis can be used to infer causal relationships between the independent and dependent variables. However this can lead to illusions or false relationships, so caution is advisable.

A prediction, or forecast, is a statement about a future event. A prediction is often, but not always, based upon experience or knowledge. There is no universal agreement about the exact difference between the two terms; different authors and disciplines ascribe different connotations.

Forecasting is the process of making predictions of the future based on past and present data and most commonly by analysis of trends. A commonplace example might be estimation of some variable of interest at some specified future date. Prediction is a similar, but more general term. Both might refer to formal statistical methods employing time series, cross-sectional or longitudinal data, or alternatively to less formal judgmental methods. Usage can differ between areas of application: for example, in hydrology the terms "forecast" and "forecasting" are sometimes reserved for estimates of values at certain specific future times, while the term "prediction" is used for more general estimates, such as the number of times floods will occur over a long period.

Machine learning (ML) is the scientific study of algorithms and statistical models that computer systems use to perform a specific task without using explicit instructions, relying on patterns and inference instead. It is seen as a subset of artificial intelligence. Machine learning algorithms build a mathematical model based on sample data, known as "training data", in order to make predictions or decisions without being explicitly programmed to perform the task. Machine learning algorithms are used in a wide variety of applications, such as email filtering and computer vision, where it is difficult or infeasible to develop a conventional algorithm for effectively performing the task.

Many techniques for carrying out regression analysis have been developed. Familiar methods such as linear regression and ordinary least squares regression are parametric, in that the regression function is defined in terms of a finite number of unknown parameters that are estimated from the data. Nonparametric regression refers to techniques that allow the regression function to lie in a specified set of functions, which may be infinite-dimensional.

In statistics, linear regression is a linear approach to modeling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression. For more than one explanatory variable, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.

In statistics, ordinary least squares (OLS) is a type of linear least squares method for estimating the unknown parameters in a linear regression model. OLS chooses the parameters of a linear function of a set of explanatory variables by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the given dataset and those predicted by the linear function.

Parametric statistics is a branch of statistics which assumes that sample data come from a population that can be adequately modeled by a probability distribution that has a fixed set of parameters. Conversely a non-parametric model differs precisely in that the parameter set is not fixed and can increase, or even decrease, if new relevant information is collected.

The performance of regression analysis methods in practice depends on the form of the data generating process, and how it relates to the regression approach being used. Since the true form of the data-generating process is generally not known, regression analysis often depends to some extent on making assumptions about this process. These assumptions are sometimes testable if a sufficient quantity of data is available. Regression models for prediction are often useful even when the assumptions are moderately violated, although they may not perform optimally. However, in many applications, especially with small effects or questions of causality based on observational data, regression methods can give misleading results. [2] [3]

Data collection is the process of gathering and measuring information on targeted variables in an established system, which then enables one to answer relevant questions and evaluate outcomes. Data collection is a component of research in all fields of study including physical and social sciences, humanities, and business. While methods vary by discipline, the emphasis on ensuring accurate and honest collection remains the same. The goal for all data collection is to capture quality evidence that allows analysis to lead to the formulation of convincing and credible answers to the questions that have been posed.

In statistics, an effect size is a quantitative measure of the magnitude of a phenomenon. Examples of effect sizes are the correlation between two variables, the regression coefficient in a regression, the mean difference, or even the risk with which something happens, such as how many people survive after a heart attack for every one person that does not survive. For most types of effect size, a larger absolute value always indicates a stronger effect, with the main exception being if the effect size is an odds ratio. Effect sizes complement statistical hypothesis testing, and play an important role in power analyses, sample size planning, and in meta-analyses. They are the first item (magnitude) in the MAGIC criteria for evaluating the strength of a statistical claim. Especially in meta-analysis, where the purpose is to combine multiple effect sizes, the standard error (S.E.) of the effect size is of critical importance. The S.E. of the effect size is used to weigh effect sizes when combining studies, so that large studies are considered more important than small studies in the analysis. The S.E. of the effect size is calculated differently for each type of effect size, but generally only requires knowing the study's sample size (N), or the number of observations in each group.

Causality is efficacy, by which one process or state, a cause, contributes to the production of another process or state, an effect, where the cause is partly responsible for the effect, and the effect is partly dependent on the cause. In general, a process has many causes, which are also said to be causal factors for it, and all lie in its past. An effect can in turn be a cause of, or causal factor for, many other effects, which all lie in its future. Some writers have held that causality is metaphysically prior to notions of time and space.

In a narrower sense, regression may refer specifically to the estimation of continuous response (dependent) variables, as opposed to the discrete response variables used in classification. [4] The case of a continuous dependent variable may be more specifically referred to as metric regression to distinguish it from related problems. [5]

In machine learning and statistics, classification is the problem of identifying to which of a set of categories (sub-populations) a new observation belongs, on the basis of a training set of data containing observations whose category membership is known. Examples are assigning a given email to the "spam" or "non-spam" class, and assigning a diagnosis to a given patient based on observed characteristics of the patient. Classification is an example of pattern recognition.

## History

The earliest form of regression was the method of least squares, which was published by Legendre in 1805, [6] and by Gauss in 1809. [7] Legendre and Gauss both applied the method to the problem of determining, from astronomical observations, the orbits of bodies about the Sun (mostly comets, but also later the then newly discovered minor planets). Gauss published a further development of the theory of least squares in 1821, [8] including a version of the Gauss–Markov theorem.

The term "regression" was coined by Francis Galton in the nineteenth century to describe a biological phenomenon. The phenomenon was that the heights of descendants of tall ancestors tend to regress down towards a normal average (a phenomenon also known as regression toward the mean). [9] [10] For Galton, regression had only this biological meaning, [11] [12] but his work was later extended by Udny Yule and Karl Pearson to a more general statistical context. [13] [14] In the work of Yule and Pearson, the joint distribution of the response and explanatory variables is assumed to be Gaussian. This assumption was weakened by R.A. Fisher in his works of 1922 and 1925. [15] [16] [17] Fisher assumed that the conditional distribution of the response variable is Gaussian, but the joint distribution need not be. In this respect, Fisher's assumption is closer to Gauss's formulation of 1821.

In the 1950s and 1960s, economists used electromechanical desk "calculators" to calculate regressions. Before 1970, it sometimes took up to 24 hours to receive the result from one regression. [18]

Regression methods continue to be an area of active research. In recent decades, new methods have been developed for robust regression, regression involving correlated responses such as time series and growth curves, regression in which the predictor (independent variable) or response variables are curves, images, graphs, or other complex data objects, regression methods accommodating various types of missing data, nonparametric regression, Bayesian methods for regression, regression in which the predictor variables are measured with error, regression with more predictor variables than observations, and causal inference with regression.

## Regression models

Regression models involve the following parameters and variables:

• The unknown parameters, denoted as ${\displaystyle \beta }$, which may represent a scalar or a vector.
• The independent variables, ${\displaystyle X}$.
• The dependent variable, ${\displaystyle Y}$.

In various fields of application, different terminologies are used in place of dependent and independent variables.

A regression model relates ${\displaystyle Y}$ to a function of ${\displaystyle X}$ and ${\displaystyle \beta }$.

${\displaystyle Y\approx f(X,\beta ).}$

The approximation is usually formalized as ${\displaystyle \operatorname {E} (Y|X)=f(X,\beta )}$. To carry out regression analysis, the form of the function ${\displaystyle f}$ must be specified. Sometimes the form of this function is based on knowledge about the relationship between ${\displaystyle Y}$ and ${\displaystyle X}$ that does not rely on the data. If no such knowledge is available, a flexible or convenient form for ${\displaystyle f}$ is chosen.

Assume now that the vector of unknown parameters ${\displaystyle \beta }$ is of length ${\displaystyle k}$. In order to perform a regression analysis the user must provide information about the dependent variable ${\displaystyle Y}$:

• If ${\displaystyle N}$ data points of the form ${\displaystyle (Y,X)}$ are observed, where ${\displaystyle N, most classical approaches to regression analysis cannot be performed: since the system of equations defining the regression model is underdetermined, there are not enough data to recover ${\displaystyle \beta }$.
• If exactly ${\displaystyle N=k}$ data points are observed, and the function ${\displaystyle f}$ is linear, the equations ${\displaystyle Y=f(X,\beta )}$ can be solved exactly rather than approximately. This reduces to solving a set of ${\displaystyle N}$ equations with ${\displaystyle N}$ unknowns (the elements of ${\displaystyle \beta )}$, which has a unique solution as long as the ${\displaystyle X}$ are linearly independent. If ${\displaystyle f}$ is nonlinear, a solution may not exist, or many solutions may exist.
• The most common situation is where ${\displaystyle N>k}$ data points are observed. In this case, there is enough information in the data to estimate a unique value for ${\displaystyle \beta }$ that best fits the data in some sense, and the regression model when applied to the data can be viewed as an overdetermined system in ${\displaystyle \beta }$.

In the last case, the regression analysis provides the tools for:

1. Finding a solution for unknown parameters ${\displaystyle \beta }$ that will, for example, minimize the distance between the measured and predicted values of the dependent variable ${\displaystyle Y}$ (also known as method of least squares).
2. Under certain statistical assumptions, the regression analysis uses the surplus of information to provide statistical information about the unknown parameters ${\displaystyle \beta }$ and predicted values of the dependent variable ${\displaystyle Y}$.

### Necessary number of independent measurements

Consider a regression model which has three unknown parameters, ${\displaystyle \beta _{0}}$, ${\displaystyle \beta _{1}}$, and ${\displaystyle \beta _{2}}$. Suppose an experimenter performs 10 measurements all at exactly the same value of independent variable vector ${\displaystyle X}$ (which contains the independent variables ${\displaystyle X_{1}}$, ${\displaystyle X_{2}}$, and ${\displaystyle X_{3}}$). In this case, regression analysis fails to give a unique set of estimated values for the three unknown parameters; the experimenter did not provide enough information. The best one can do is to estimate the average value and the standard deviation of the dependent variable ${\displaystyle Y}$. Similarly, measuring at two different values of ${\displaystyle X}$ would give enough data for a regression with two unknowns, but not for three or more unknowns.

If the experimenter had performed measurements at three different values of the independent variable vector ${\displaystyle X}$, then regression analysis would provide a unique set of estimates for the three unknown parameters in ${\displaystyle \beta }$.

In the case of general linear regression, the above statement is equivalent to the requirement that the matrix ${\displaystyle X^{\top }X}$ is invertible.

When the number of measurements, ${\displaystyle N}$, is larger than the number of unknown parameters, ${\displaystyle k}$, and the measurement errors ${\displaystyle \epsilon _{i}}$ are normally distributed then the excess of information contained in ${\displaystyle (N-k)}$ measurements is used to make statistical predictions about the unknown parameters. This excess of information is referred to as the degrees of freedom of the regression.

## Underlying assumptions

Classical assumptions for regression analysis include:

• The sample is representative of the population for the inference prediction.
• The error is a random variable with a mean of zero conditional on the explanatory variables.
• The independent variables are measured with no error. (Note: If this is not so, modeling may be done instead using errors-in-variables model techniques).
• The independent variables (predictors) are linearly independent, i.e. it is not possible to express any predictor as a linear combination of the others.
• The errors are uncorrelated, that is, the variance–covariance matrix of the errors is diagonal and each non-zero element is the variance of the error.
• The variance of the error is constant across observations (homoscedasticity). If not, weighted least squares or other methods might instead be used.

These are sufficient conditions for the least-squares estimator to possess desirable properties; in particular, these assumptions imply that the parameter estimates will be unbiased, consistent, and efficient in the class of linear unbiased estimators. It is important to note that actual data rarely satisfies the assumptions. That is, the method is used even though the assumptions are not true. Variation from the assumptions can sometimes be used as a measure of how far the model is from being useful. Many of these assumptions may be relaxed in more advanced treatments. Reports of statistical analyses usually include analyses of tests on the sample data and methodology for the fit and usefulness of the model.

Independent and dependent variables often refer to values measured at point locations. There may be spatial trends and spatial autocorrelation in the variables that violate statistical assumptions of regression. Geographic weighted regression is one technique to deal with such data. [19] Also, variables may include values aggregated by areas. With aggregated data the modifiable areal unit problem can cause extreme variation in regression parameters. [20] When analyzing data aggregated by political boundaries, postal codes or census areas results may be very distinct with a different choice of units.

## Linear regression

In linear regression, the model specification is that the dependent variable, ${\displaystyle y_{i}}$ is a linear combination of the parameters (but need not be linear in the independent variables). For example, in simple linear regression for modeling ${\displaystyle n}$ data points there is one independent variable: ${\displaystyle x_{i}}$, and two parameters, ${\displaystyle \beta _{0}}$ and ${\displaystyle \beta _{1}}$:

straight line: ${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\varepsilon _{i},\quad i=1,\dots ,n.\!}$

In multiple linear regression, there are several independent variables or functions of independent variables.

Adding a term in ${\displaystyle x_{i}^{2}}$ to the preceding regression gives:

parabola: ${\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i}+\beta _{2}x_{i}^{2}+\varepsilon _{i},\ i=1,\dots ,n.\!}$

This is still linear regression; although the expression on the right hand side is quadratic in the independent variable ${\displaystyle x_{i}}$, it is linear in the parameters ${\displaystyle \beta _{0}}$, ${\displaystyle \beta _{1}}$ and ${\displaystyle \beta _{2}.}$

In both cases, ${\displaystyle \varepsilon _{i}}$ is an error term and the subscript ${\displaystyle i}$ indexes a particular observation.

Returning our attention to the straight line case: Given a random sample from the population, we estimate the population parameters and obtain the sample linear regression model:

${\displaystyle {\widehat {y}}_{i}={\widehat {\beta }}_{0}+{\widehat {\beta }}_{1}x_{i}.}$

The residual, ${\displaystyle e_{i}=y_{i}-{\widehat {y}}_{i}}$, is the difference between the value of the dependent variable predicted by the model, ${\displaystyle {\widehat {y}}_{i}}$, and the true value of the dependent variable, ${\displaystyle y_{i}}$. One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals, SSR:

${\displaystyle SSR=\sum _{i=1}^{n}e_{i}^{2}.\,}$

Minimization of this function results in a set of normal equations, a set of simultaneous linear equations in the parameters, which are solved to yield the parameter estimators, ${\displaystyle {\widehat {\beta }}_{0},{\widehat {\beta }}_{1}}$.

In the case of simple regression, the formulas for the least squares estimates are

${\displaystyle {\widehat {\beta }}_{1}={\frac {\sum (x_{i}-{\bar {x}})(y_{i}-{\bar {y}})}{\sum (x_{i}-{\bar {x}})^{2}}}}$
${\displaystyle {\widehat {\beta }}_{0}={\bar {y}}-{\widehat {\beta }}_{1}{\bar {x}}}$

where ${\displaystyle {\bar {x}}}$ is the mean (average) of the ${\displaystyle x}$ values and ${\displaystyle {\bar {y}}}$ is the mean of the ${\displaystyle y}$ values.

Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:

${\displaystyle {\hat {\sigma }}_{\varepsilon }^{2}={\frac {SSR}{n-2}}.\,}$

This is called the mean square error (MSE) of the regression. The denominator is the sample size reduced by the number of model parameters estimated from the same data, ${\displaystyle (n-p)}$ for ${\displaystyle p}$ regressors or ${\displaystyle (n-p-1)}$ if an intercept is used. [21] In this case, ${\displaystyle p=1}$ so the denominator is ${\displaystyle n-2}$.

The standard errors of the parameter estimates are given by

${\displaystyle {\hat {\sigma }}_{\beta _{1}}={\hat {\sigma }}_{\varepsilon }{\sqrt {\frac {1}{\sum (x_{i}-{\bar {x}})^{2}}}}}$
${\displaystyle {\hat {\sigma }}_{\beta _{0}}={\hat {\sigma }}_{\varepsilon }{\sqrt {{\frac {1}{n}}+{\frac {{\bar {x}}^{2}}{\sum (x_{i}-{\bar {x}})^{2}}}}}={\hat {\sigma }}_{\beta _{1}}{\sqrt {\frac {\sum x_{i}^{2}}{n}}}.}$

Under the further assumption that the population error term is normally distributed, the researcher can use these estimated standard errors to create confidence intervals and conduct hypothesis tests about the population parameters.

### General linear model

In the more general multiple regression model, there are ${\displaystyle p}$ independent variables:

${\displaystyle y_{i}=\beta _{1}x_{i1}+\beta _{2}x_{i2}+\cdots +\beta _{p}x_{ip}+\varepsilon _{i},\,}$

where ${\displaystyle x_{ij}}$ is the ${\displaystyle i}$-th observation on the ${\displaystyle j}$-th independent variable. If the first independent variable takes the value 1 for all ${\displaystyle i}$, ${\displaystyle x_{i1}=1}$, then ${\displaystyle \beta _{1}}$ is called the regression intercept.

The least squares parameter estimates are obtained from ${\displaystyle p}$ normal equations. The residual can be written as

${\displaystyle \varepsilon _{i}=y_{i}-{\hat {\beta }}_{1}x_{i1}-\cdots -{\hat {\beta }}_{p}x_{ip}.}$

The normal equations are

${\displaystyle \sum _{i=1}^{n}\sum _{k=1}^{p}x_{ij}x_{ik}{\hat {\beta }}_{k}=\sum _{i=1}^{n}x_{ij}y_{i},\ j=1,\dots ,p.\,}$

In matrix notation, the normal equations are written as

${\displaystyle \mathbf {(X^{\top }X){\hat {\boldsymbol {\beta }}}={}X^{\top }Y} ,\,}$

where the ${\displaystyle ij}$ element of ${\displaystyle \mathbf {X} }$ is ${\displaystyle x_{ij}}$, the ${\displaystyle i}$ element of the column vector ${\displaystyle Y}$ is ${\displaystyle y_{i}}$, and the ${\displaystyle j}$ element of ${\displaystyle {\hat {\boldsymbol {\beta }}}}$ is ${\displaystyle {\hat {\beta }}_{j}}$. Thus ${\displaystyle \mathbf {X} }$ is ${\displaystyle n\times p}$, ${\displaystyle Y}$ is ${\displaystyle n\times 1}$, and ${\displaystyle {\hat {\boldsymbol {\beta }}}}$ is ${\displaystyle p\times 1}$. The solution is

${\displaystyle \mathbf {{\hat {\boldsymbol {\beta }}}=(X^{\top }X)^{-1}X^{\top }Y} .\,}$

### Diagnostics

Once a regression model has been constructed, it may be important to confirm the goodness of fit of the model and the statistical significance of the estimated parameters. Commonly used checks of goodness of fit include the R-squared, analyses of the pattern of residuals and hypothesis testing. Statistical significance can be checked by an F-test of the overall fit, followed by t-tests of individual parameters.

Interpretations of these diagnostic tests rest heavily on the model assumptions. Although examination of the residuals can be used to invalidate a model, the results of a t-test or F-test are sometimes more difficult to interpret if the model's assumptions are violated. For example, if the error term does not have a normal distribution, in small samples the estimated parameters will not follow normal distributions and complicate inference. With relatively large samples, however, a central limit theorem can be invoked such that hypothesis testing may proceed using asymptotic approximations.

### Limited dependent variables

Limited dependent variables, which are response variables that are categorical variables or are variables constrained to fall only in a certain range, often arise in econometrics.

The response variable may be non-continuous ("limited" to lie on some subset of the real line). For binary (zero or one) variables, if analysis proceeds with least-squares linear regression, the model is called the linear probability model. Nonlinear models for binary dependent variables include the probit and logit model. The multivariate probit model is a standard method of estimating a joint relationship between several binary dependent variables and some independent variables. For categorical variables with more than two values there is the multinomial logit. For ordinal variables with more than two values, there are the ordered logit and ordered probit models. Censored regression models may be used when the dependent variable is only sometimes observed, and Heckman correction type models may be used when the sample is not randomly selected from the population of interest. An alternative to such procedures is linear regression based on polychoric correlation (or polyserial correlations) between the categorical variables. Such procedures differ in the assumptions made about the distribution of the variables in the population. If the variable is positive with low values and represents the repetition of the occurrence of an event, then count models like the Poisson regression or the negative binomial model may be used.

## Nonlinear regression

When the model function is not linear in the parameters, the sum of squares must be minimized by an iterative procedure. This introduces many complications which are summarized in Differences between linear and non-linear least squares.

## Interpolation and extrapolation

Regression models predict a value of the Y variable given known values of the X variables. Prediction within the range of values in the dataset used for model-fitting is known informally as interpolation. Prediction outside this range of the data is known as extrapolation. Performing extrapolation relies strongly on the regression assumptions. The further the extrapolation goes outside the data, the more room there is for the model to fail due to differences between the assumptions and the sample data or the true values.

It is generally advised [ citation needed ] that when performing extrapolation, one should accompany the estimated value of the dependent variable with a prediction interval that represents the uncertainty. Such intervals tend to expand rapidly as the values of the independent variable(s) moved outside the range covered by the observed data.

For such reasons and others, some tend to say that it might be unwise to undertake extrapolation. [23]

However, this does not cover the full set of modeling errors that may be made: in particular, the assumption of a particular form for the relation between Y and X. A properly conducted regression analysis will include an assessment of how well the assumed form is matched by the observed data, but it can only do so within the range of values of the independent variables actually available. This means that any extrapolation is particularly reliant on the assumptions being made about the structural form of the regression relationship. Best-practice advice here[ citation needed ] is that a linear-in-variables and linear-in-parameters relationship should not be chosen simply for computational convenience, but that all available knowledge should be deployed in constructing a regression model. If this knowledge includes the fact that the dependent variable cannot go outside a certain range of values, this can be made use of in selecting the model – even if the observed dataset has no values particularly near such bounds. The implications of this step of choosing an appropriate functional form for the regression can be great when extrapolation is considered. At a minimum, it can ensure that any extrapolation arising from a fitted model is "realistic" (or in accord with what is known).

## Power and sample size calculations

There are no generally agreed methods for relating the number of observations versus the number of independent variables in the model. One rule of thumb conjectured by Good and Hardin is ${\displaystyle N=m^{n}}$, where ${\displaystyle N}$ is the sample size, ${\displaystyle n}$ is the number of independent variables and ${\displaystyle m}$ is the number of observations needed to reach the desired precision if the model had only one independent variable. [24] For example, a researcher is building a linear regression model using a dataset that contains 1000 patients (${\displaystyle N}$). If the researcher decides that five observations are needed to precisely define a straight line (${\displaystyle m}$), then the maximum number of independent variables the model can support is 4, because

${\displaystyle {\frac {\log 1000}{\log 5}}=4.29.}$

## Other methods

Although the parameters of a regression model are usually estimated using the method of least squares, other methods which have been used include:

## Software

All major statistical software packages perform least squares regression analysis and inference. Simple linear regression and multiple regression using least squares can be done in some spreadsheet applications and on some calculators. While many statistical software packages can perform various types of nonparametric and robust regression, these methods are less standardized; different software packages implement different methods, and a method with a given name may be implemented differently in different packages. Specialized regression software has been developed for use in fields such as survey analysis and neuroimaging.

## Related Research Articles

The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems, i.e., sets of equations in which there are more equations than unknowns. "Least squares" means that the overall solution minimizes the sum of the squares of the residuals made in the results of every single equation.

In statistics, the Gauss–Markov theorem states that in a linear regression model in which the errors are uncorrelated, have equal variances and expectation value of zero, the best linear unbiased estimator (BLUE) of the coefficients is given by the ordinary least squares (OLS) estimator, provided it exists. Here "best" means giving the lowest variance of the estimate, as compared to other unbiased, linear estimators. The errors do not need to be normal, nor do they need to be independent and identically distributed. The requirement that the estimator be unbiased cannot be dropped, since biased estimators exist with lower variance. See, for example, the James–Stein estimator or ridge regression.

In statistics, the logistic model is used to model the probability of a certain class or event existing such as pass/fail, win/lose, alive/dead or healthy/sick. This can be extended to model several classes of events such as determining whether an image contains a cat, dog, lion, etc... Each object being detected in the image would be assigned a probability between 0 and 1 and the sum adding to one.

The general linear model or multivariate regression model is a statistical linear model. It may be written as

In statistics, nonlinear regression is a form of regression analysis in which observational data are modeled by a function which is a nonlinear combination of the model parameters and depends on one or more independent variables. The data are fitted by a method of successive approximations.

In statistics, the coefficient of determination, denoted R2 or r2 and pronounced "R squared", is the proportion of the variance in the dependent variable that is predictable from the independent variable(s).

In statistics, econometrics, epidemiology and related disciplines, the method of instrumental variables (IV) is used to estimate causal relationships when controlled experiments are not feasible or when a treatment is not successfully delivered to every unit in a randomized experiment. Intuitively, IVs are used when an explanatory variable of interest is correlated with the error term, in which case ordinary least squares and ANOVA give biased results. A valid instrument induces changes in the explanatory variable but has no independent effect on the dependent variable, allowing a researcher to uncover the causal effect of the explanatory variable on the dependent variable.

In statistics, omitted-variable bias (OVB) occurs when a statistical model leaves out one or more relevant variables. The bias results in the model attributing the effect of the missing variables to the estimated effects of the included variables.

In statistics, simple linear regression is a linear regression model with a single explanatory variable. That is, it concerns two-dimensional sample points with one independent variable and one dependent variable and finds a linear function that, as accurately as possible, predicts the dependent variable values as a function of the independent variables. The adjective simple refers to the fact that the outcome variable is related to a single predictor.

In statistics, a fixed effects model is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are considered as random variables. In many applications including econometrics and biostatistics a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population. Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity.

In statistics, generalized least squares (GLS) is a technique for estimating the unknown parameters in a linear regression model when there is a certain degree of correlation between the residuals in a regression model. In these cases, ordinary least squares and weighted least squares can be statistically inefficient, or even give misleading inferences. GLS was first described by Alexander Aitken in 1934.

Omnibus tests are a kind of statistical test. They test whether the explained variance in a set of data is significantly greater than the unexplained variance, overall. One example is the F-test in the analysis of variance. There can be legitimate significant effects within a model even if the omnibus test is not significant. For instance, in a model with two independent variables, if only one variable exerts a significant effect on the dependent variable and the other does not, then the omnibus test may be non-significant. This fact does not affect the conclusions that may be drawn from the one significant variable. In order to test effects within an omnibus test, researchers often use contrasts.

The topic of heteroscedasticity-consistent (HC) standard errors arises in statistics and econometrics in the context of linear regression and time series analysis. These are also known as Eicker–Huber–White standard errors, to recognize the contributions of Friedhelm Eicker, Peter J. Huber, and Halbert White.

In statistics, the variance inflation factor (VIF) is the quotient of the variance in a model with multiple terms by the variance of a model with one term alone. It quantifies the severity of multicollinearity in an ordinary least squares regression analysis. It provides an index that measures how much the variance of an estimated regression coefficient is increased because of collinearity.

In statistics, principal component regression (PCR) is a regression analysis technique that is based on principal component analysis (PCA). Typically, it considers regressing the outcome on a set of covariates based on a standard linear regression model, but uses PCA for estimating the unknown regression coefficients in the model.

In statistics, a sum of squares due to lack of fit, or more tersely a lack-of-fit sum of squares, is one of the components of a partition of the sum of squares of residuals in an analysis of variance, used in the numerator in an F-test of the null hypothesis that says that a proposed model fits well. The other component is the pure-error sum of squares.

In statistics, polynomial regression is a form of regression analysis in which the relationship between the independent variable x and the dependent variable y is modelled as an nth degree polynomial in x. Polynomial regression fits a nonlinear relationship between the value of x and the corresponding conditional mean of y, denoted E(y |x), and has been used to describe nonlinear phenomena such as the growth rate of tissues, the distribution of carbon isotopes in lake sediments, and the progression of disease epidemics. Although polynomial regression fits a nonlinear model to the data, as a statistical estimation problem it is linear, in the sense that the regression function E(y | x) is linear in the unknown parameters that are estimated from the data. For this reason, polynomial regression is considered to be a special case of multiple linear regression.

Linear least squares (LLS) is the least squares approximation of linear functions to data. It is a set of formulations for solving statistical problems involved in linear regression, including variants for ordinary (unweighted), weighted, and generalized (correlated) residuals. Numerical methods for linear least squares include inverting the matrix of the normal equations and orthogonal decomposition methods.

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