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# Explained: Regression analysis

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❶Life is usually simple, when you know only one or two techniques.

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.

The earliest form of regression was the method of least squares , which was published by Legendre in , [7] and by Gauss in Gauss published a further development of the theory of least squares in , [9] 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. This assumption was weakened by R. Fisher in his works of and In this respect, Fisher's assumption is closer to Gauss's formulation of In the s and s, economists used electromechanical desk "calculators" to calculate regressions.

Before , it sometimes took up to 24 hours to receive the result from one regression. 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.

In various fields of application , different terminologies are used in place of dependent and independent variables. 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. This excess of information is referred to as the degrees of freedom of the regression.

Classical assumptions for regression analysis include:. 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. With aggregated data the modifiable areal unit problem can cause extreme variation in regression parameters.

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

One method of estimation is ordinary least squares. This method obtains parameter estimates that minimize the sum of squared residuals , SSR:. Under the assumption that the population error term has a constant variance, the estimate of that variance is given by:. This is called the mean square error MSE of the regression. The standard errors of the parameter estimates are given by. 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.

The residual can be written as. 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 , 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. 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.

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. However, this does not cover the full set of modeling errors that may be made: 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.

There are no generally agreed methods for relating the number of observations versus the number of independent variables in the model. Although the parameters of a regression model are usually estimated using the method of least squares, other methods which have been used include:.

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.

Glossary of artificial intelligence. List of datasets for machine-learning research Outline of machine learning. See simple linear regression for a derivation of these formulas and a numerical example. For a derivation, see linear least squares. For a numerical example, see linear regression. List of statistical packages. Curve fitting Estimation Theory Forecasting Fraction of variance unexplained Function approximation Generalized linear models Kriging a linear least squares estimation algorithm Local regression Modifiable areal unit problem Multivariate adaptive regression splines Multivariate normal distribution Pearson product-moment correlation coefficient Quasi-variance Prediction interval Regression validation Robust regression Segmented regression Signal processing Stepwise regression Trend estimation.

One of the best trick to find out which technique to use, is by checking the family of variables i. In this article, I discussed about 7 types of regression and some key facts associated with each technique. Hi Sunil, Really a nice article for understanding the regression models. Especially for novice like me who are stepping into Analytic. Hi Sunil Thanks for posting this. Very nice summary on a technique used so often but underutilised when looking at the different forms available.

You wouldnt be interested in doing something similar for classification techniques.. Thanks Tom…you can refer article on most common machine learning algorithms http: Here I have discussed various types of classification algorithms like decision tree, random forest, KNN, Naive Bayes….

The difference given between linear regression and multiple regression needs correction. It did help me broaden my perspective regarding the regression techniques specially ElasticNet ,but still it would be nice to elucidate upon the differences between l1 and l2 regularization techniques.

Though it could be incorporated into a new article I think. If I print from IE, the only browser allowed on my network, all the ads and hypertext links cover the article text; you cannot read the article. I had suggested having a feature where you use a button to convert the article to a PDF, which can them be printed without the ads and hypertext. You did in once, then stopped. Read this article to understand the effect of interaction in detail. Hi sunil, The article seems very interesting.

Please can you let me know how can we implement Forward stepwise Regression in python as we dont have any inbuilt lib for it. Thanks fo the guide. And it is performed by making several successive real regression technics linear, polynomial, ridge or lasso…. Are there any specific types of regression techniques which can be used for a time series stationary data?

Very nice article, crisp n neat! Sunil, Great feeling to get a modern insight to what I learnt 35 years ago. Professional practicing today may have several question to clarify. Compliment to you for such a vast subject so lucidly worded and explained.. What fascinated me most, is you mention of a tutor teaching students in an institute — if outcome is continuous use linear and if it is binary, use logistics.

What I want to ask is as under:. In case of multiple independent variables, we can go with forward selection, backward elimination and step wise approach for selection of most significant independent variables. Please let me know where to get little details on these? Amazing article, broadens as once seemingly narrow concept and gives food for thought. This is an awesome article. This is a mix of different techniques with different characteristics, all of which can be used for linear regression, logistic regression or any other kind of generalized linear model.

Polynomial is just using transformations of the variables, but the model is still linear in the beta parameters. Thus it is still linear regression. This is a concept that bewilders a lot of people. Stepwise is just a method of building a model by adding and removing variables based on the F statistic. Hence, they are useful for other models that are distinct from regression, like SVMs. To be technical, different regression models would be plain linear, logistic, multinomial, poisson, gamma, Cox, etc.

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## Main Topics

Regression Analysis Regression analysis is a quantitative research method which is used when the study involves modelling and analysing several variables, where the relationship includes a dependent variable and one or more independent variables.

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Regression analysis. It sounds like a part of Freudian psychology. In reality, a regression is a seemingly ubiquitous statistical tool appearing in legions of scientific papers, and regression analysis is a method of measuring the link between two or more phenomena.