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73-790: Electoral Calculus is a political consultancy and pollster, known for its political forecasting website that attempts to predict future United Kingdom general election results. It uses MRP (Multi-level Regression and Post-stratification) to combine national factors and local demographics. Electoral Calculus was founded and is run by Martin Baxter, who was a financial analyst specialising in mathematical modelling . The Electoral Calculus website includes election data, predictions and analysis. It has separate sections for elections in Scotland and in Northern Ireland . The election predictions are based around

146-480: A least squares model with k {\displaystyle k} distinct parameters, one must have N ≥ k {\displaystyle N\geq k} distinct data points. If N > k {\displaystyle N>k} , then there does not generally exist a set of parameters that will perfectly fit the data. The quantity N − k {\displaystyle N-k} appears often in regression analysis, and

219-412: 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

292-495: A dependent variable and a collection of independent variables in a fixed dataset. To use regressions for prediction or to infer causal relationships, respectively, a researcher must carefully justify why existing relationships have predictive power for a new context or why a relationship between two variables has a causal interpretation. The latter is especially important when researchers hope to estimate causal relationships using observational data . The earliest form of regression

365-594: A mechanism of political forecasting. Prediction markets show very accurate forecasts of an election outcome. One example is the Iowa Electronic Markets . In a study, 964 election polls were compared with the five US presidential elections from 1988 to 2004. Berg et al. (2008) showed that the Iowa Electronic Markets topped the polls 74% of the time. However, damped polls have been shown to top prediction markets. Comparing damped polls to forecasts of

438-597: 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

511-410: A particular form for the relation between Y and X is another source of uncertainty. 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

584-465: A reasonable approximation for the statistical process generating the data. Once researchers determine their preferred statistical model , different forms of regression analysis provide tools to estimate the parameters β {\displaystyle \beta } . For example, least squares (including its most common variant, ordinary least squares ) finds the value of β {\displaystyle \beta } that minimizes

657-451: A seat-by-seat basis. The models are explained in detail on the web site. Across the 12 general elections from 1992 to 2024, the site correctly predicted the party to win the most seats in all but one (1992). They also correctly predicted the outcome, that is, the party winning a majority or a hung parliament, in six elections (majorities in 1997, 2001, 2005, 2015, 2017 (by a majority of only 3), 2019, 2024; hung parliament for 2010). In 2004,

730-428: A small range of shorthand phrases. These include: Forecasting can involve skin-in-the-game crowdsourcing via prediction markets on the theory that people more honestly evaluate and express their true perception with money at stake. However, individuals with a large economic or ego investment in the outcome of a future election may be willing to sacrifice economic gain in order to alter public perception of

803-668: A term in x i 2 {\displaystyle x_{i}^{2}} to the preceding regression gives: This is still linear regression; although the expression on the right hand side is quadratic in the independent variable x i {\displaystyle x_{i}} , it is linear in the parameters β 0 {\displaystyle \beta _{0}} , β 1 {\displaystyle \beta _{1}} and β 2 . {\displaystyle \beta _{2}.} In both cases, ε i {\displaystyle \varepsilon _{i}}

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876-454: A version of the Gauss–Markov theorem . The term "regression" was coined by Francis Galton in the 19th 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 ). For Galton, regression had only this biological meaning, but his work

949-430: Is n × 1 {\displaystyle n\times 1} , and β ^ {\displaystyle {\hat {\boldsymbol {\beta }}}} is p × 1 {\displaystyle p\times 1} . The solution is 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

1022-428: Is a function ( regression function ) of X i {\displaystyle X_{i}} and β {\displaystyle \beta } , with e i {\displaystyle e_{i}} representing an additive error term that may stand in for un-modeled determinants of Y i {\displaystyle Y_{i}} or random statistical noise: Note that

1095-428: Is a set of statistical processes for estimating the relationships between a dependent variable (often called the outcome or response variable, or a label in machine learning parlance) and one or more error-free independent variables (often called regressors , predictors , covariates , explanatory variables or features ). The most common form of regression analysis is linear regression , in which one finds

1168-479: 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

1241-481: Is an error term and the subscript i {\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: The residual , e i = y i − y ^ i {\displaystyle e_{i}=y_{i}-{\widehat {y}}_{i}} ,

1314-619: Is available, a flexible or convenient form for f {\displaystyle f} is chosen. For example, a simple univariate regression may propose f ( X i , β ) = β 0 + β 1 X i {\displaystyle f(X_{i},\beta )=\beta _{0}+\beta _{1}X_{i}} , suggesting that the researcher believes Y i = β 0 + β 1 X i + e i {\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{i}+e_{i}} to be

1387-482: 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. 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

1460-445: 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. One method conjectured by Good and Hardin is N = m n {\displaystyle N=m^{n}} , where N {\displaystyle N}

1533-523: Is done to help forecast the votes of the political parties – for example, Democrats and Republicans in the US. The information helps their party's next presidential candidate forecast the future. Most models include at least one public opinion variable, a trial heat poll, or a presidential approval rating. Bayesian statistics can also be used to estimate the posterior distributions of the true proportion of voters that will vote for each candidate in each state, given both

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1606-461: Is for the model to fail due to differences between the assumptions and the sample data or the true values. A prediction interval that represents the uncertainty may accompany the point prediction. 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. The assumption of

1679-416: Is important to note that there must be sufficient data to estimate a regression model. For example, suppose that a researcher has access to N {\displaystyle N} rows of data with one dependent and two independent variables: ( Y i , X 1 i , X 2 i ) {\displaystyle (Y_{i},X_{1i},X_{2i})} . Suppose further that

1752-424: Is primarily used for two conceptually distinct purposes. First, regression analysis is widely used for prediction and forecasting , where its use has substantial overlap with the field of machine learning . Second, in some situations regression analysis can be used to infer causal relationships between the independent and dependent variables. Importantly, regressions by themselves only reveal relationships between

1825-431: Is referred to as the degrees of freedom in the model. Moreover, to estimate a least squares model, the independent variables ( X 1 i , X 2 i , . . . , X k i ) {\displaystyle (X_{1i},X_{2i},...,X_{ki})} must be linearly independent : one must not be able to reconstruct any of the independent variables by adding and multiplying

1898-708: Is that the dependent variable, y i {\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 n {\displaystyle n} data points there is one independent variable: x i {\displaystyle x_{i}} , and two parameters, β 0 {\displaystyle \beta _{0}} and β 1 {\displaystyle \beta _{1}} : In multiple linear regression, there are several independent variables or functions of independent variables. Adding

1971-405: Is the difference between the value of the dependent variable predicted by the model, y ^ i {\displaystyle {\widehat {y}}_{i}} , and the true value of the dependent variable, y i {\displaystyle y_{i}} . One method of estimation is ordinary least squares . This method obtains parameter estimates that minimize

2044-431: Is the sample size, n {\displaystyle n} is the number of independent variables and m {\displaystyle m} is the number of observations needed to reach the desired precision if the model had only one independent variable. For example, a researcher is building a linear regression model using a dataset that contains 1000 patients ( N {\displaystyle N} ). If

2117-598: Is typically done with statistical and spreadsheet software packages on computers as well as on handheld scientific and graphing calculators . In practice, researchers first select a model they would like to estimate and then use their chosen method (e.g., ordinary least squares ) to estimate the parameters of that model. Regression models involve the following components: In various fields of application , different terminologies are used in place of dependent and independent variables . Most regression models propose that Y i {\displaystyle Y_{i}}

2190-446: Is used. In this case, p = 1 {\displaystyle p=1} so the denominator is n − 2 {\displaystyle n-2} . 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

2263-461: The j {\displaystyle j} element of β ^ {\displaystyle {\hat {\boldsymbol {\beta }}}} is β ^ j {\displaystyle {\hat {\beta }}_{j}} . Thus X {\displaystyle \mathbf {X} } is n × p {\displaystyle n\times p} , Y {\displaystyle Y}

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2336-401: 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

2409-473: The conditional expectation (or population average value ) of the dependent variable when the independent variables take on a given set of values. Less common forms of regression use slightly different procedures to estimate alternative location parameters (e.g., quantile regression or Necessary Condition Analysis ) or estimate the conditional expectation across a broader collection of non-linear models (e.g., nonparametric regression ). Regression analysis

2482-419: The population parameters . In the more general multiple regression model, there are p {\displaystyle p} independent variables: where x i j {\displaystyle x_{ij}} is the i {\displaystyle i} -th observation on the j {\displaystyle j} -th independent variable. If the first independent variable takes

2555-456: The Iowa Electronic Markets, Erikson and Wlezien (2008) showed that the damped polls outperform all markets or models. According to a 2020 study, election forecasting "increases [voters'] certainty about an election's outcome, confuses many, and decreases turnout. Furthermore, we show that election forecasting has become prominent in the media, particularly in outlets with liberal audiences, and show that such coverage tends to more strongly affect

2628-515: The United States was first brought to the attention of the wider public by Nate Silver and his FiveThirtyEight website in 2008 . Currently, there are many competing models trying to predict the outcome of elections in the United States, the United Kingdom , and elsewhere. In a national or state election, macroeconomic conditions , such as employment, new job creation, the interest rate, and

2701-544: 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 denominator is the sample size reduced by the number of model parameters estimated from the same data, ( n − p ) {\displaystyle (n-p)} for p {\displaystyle p} regressors or ( n − p − 1 ) {\displaystyle (n-p-1)} if an intercept

2774-913: The candidate who is ahead." Other types of forecasting include forecasting models designed to predict the outcomes of international relations or bargaining events. One notable example is the expected utility model developed by American political scientist Bruce Bueno de Mesquita, which solves for the Bayesian Perfect Equilibria outcome of unidimensional policy events, with numerous applications including international conflict and diplomacy. Various implementations of political science forecasting tools have become increasingly common in political science, and numerous other Bayesian models exist with their components increasingly detailed in scientific literature. Ranked voting requires polling ranked preferences to predict winners. Psephology Regression analysis In statistical modeling , regression analysis

2847-416: The case of simple regression, the formulas for the least squares estimates are where x ¯ {\displaystyle {\bar {x}}} is the mean (average) of the x {\displaystyle x} values and y ¯ {\displaystyle {\bar {y}}} is the mean of the y {\displaystyle y} values. Under

2920-409: The choice of how to model e i {\displaystyle e_{i}} within geographic units can have important consequences. The subfield of econometrics is largely focused on developing techniques that allow researchers to make reasonable real-world conclusions in real-world settings, where classical assumptions do not hold exactly. In linear regression, the model specification

2993-413: The class of linear unbiased estimators. Practitioners have developed a variety of methods to maintain some or all of these desirable properties in real-world settings, because these classical assumptions are unlikely to hold exactly. For example, modeling errors-in-variables can lead to reasonable estimates independent variables are measured with errors. Heteroscedasticity-consistent standard errors allow

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3066-1110: The data equally well: any combination can be chosen that satisfies Y ^ i = β ^ 0 + β ^ 1 X 1 i + β ^ 2 X 2 i {\displaystyle {\hat {Y}}_{i}={\hat {\beta }}_{0}+{\hat {\beta }}_{1}X_{1i}+{\hat {\beta }}_{2}X_{2i}} , all of which lead to ∑ i e ^ i 2 = ∑ i ( Y ^ i − ( β ^ 0 + β ^ 1 X 1 i + β ^ 2 X 2 i ) ) 2 = 0 {\displaystyle \sum _{i}{\hat {e}}_{i}^{2}=\sum _{i}({\hat {Y}}_{i}-({\hat {\beta }}_{0}+{\hat {\beta }}_{1}X_{1i}+{\hat {\beta }}_{2}X_{2i}))^{2}=0} and are therefore valid solutions that minimize

3139-405: The data. To carry out regression analysis, the form of the function f {\displaystyle f} must be specified. Sometimes the form of this function is based on knowledge about the relationship between Y i {\displaystyle Y_{i}} and X i {\displaystyle X_{i}} that does not rely on the data. If no such knowledge

3212-408: The data. Whether the researcher is intrinsically interested in the estimate β ^ {\displaystyle {\hat {\beta }}} or the predicted value Y i ^ {\displaystyle {\hat {Y_{i}}}} will depend on context and their goals. As described in ordinary least squares , least squares is widely used because

3285-520: The employment of scientific techniques on data about the United Kingdom's electoral geography. Up to 2017, it used a modified uniform national swing , and it took account of national polls and trends but excluded local issues. Since 2019, they have used MRP (Multi-Level Regression and Post-Stratification) methods to make their election predictions. Their model uses demographic, past voting behaviour and geographic data to estimate predicted vote shares on

3358-416: The estimate from the true (unknown) parameter value that generated the data. Using this estimate, the researcher can then use the fitted value Y i ^ = f ( X i , β ^ ) {\displaystyle {\hat {Y_{i}}}=f(X_{i},{\hat {\beta }})} for prediction or to assess the accuracy of the model in explaining

3431-582: The estimated function f ( X i , β ^ ) {\displaystyle f(X_{i},{\hat {\beta }})} approximates the conditional expectation E ( Y i | X i ) {\displaystyle E(Y_{i}|X_{i})} . However, alternative variants (e.g., least absolute deviations or quantile regression ) are useful when researchers want to model other functions f ( X i , β ) {\displaystyle f(X_{i},\beta )} . It

3504-476: 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's assumptions. Although examination of the residuals can be used to invalidate

3577-461: The independent variables X i {\displaystyle X_{i}} are assumed to be free of error. This important assumption is often overlooked, although errors-in-variables models can be used when the independent variables are assumed to contain errors. The researchers' goal is to estimate the function f ( X i , β ) {\displaystyle f(X_{i},\beta )} that most closely fits

3650-532: The inflation rate are also considered. Combining poll data lowers the forecasting mistakes of a poll. Poll damping is when incorrect indicators of public opinion are not used in a forecast model. For instance, early in the campaign, polls are poor measures of the future choices of voters. The poll results closer to an election are a more accurate prediction. Campbell shows the power of poll damping in political forecasting. Political scientists and economists oftentimes use regression models of past elections. This

3723-429: The likely outcome of an election prior to election day—a positive perception of a favoured candidate is widely depicted as helping to "energize" voter turnout in support of that candidate when voting begins. When the prognosis derived from the election market itself becomes instrumental in determining voter turnout or voter preference leading up to an election, the valuation derived from the market becomes less reliable as

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3796-423: The line (or a more complex linear combination ) that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line (or hyperplane ) that minimizes the sum of squared differences between the true data and that line (or hyperplane). For specific mathematical reasons (see linear regression ), this allows the researcher to estimate

3869-440: The normal equations are written as where the i j {\displaystyle ij} element of X {\displaystyle \mathbf {X} } is x i j {\displaystyle x_{ij}} , the i {\displaystyle i} element of the column vector Y {\displaystyle Y} is y i {\displaystyle y_{i}} , and

3942-507: 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

4015-439: The outcomes of elections. These forecasts are derived from theories and empirical evidence about what matters to voters when they make electoral choices. The forecast models typically rely on a few predictors in highly aggregated form, with an emphasis on phenomena that change in the short-run, such as the state of the economy, so as to offer maximum leverage for predicting the result of a specific election. Election forecasting in

4088-437: The output of regression as a meaningful statistical quantity that measures real-world relationships, researchers often rely on a number of classical assumptions . These assumptions often include: A handful of conditions are sufficient for the least-squares estimator to possess desirable properties: in particular, the Gauss–Markov assumptions imply that the parameter estimates will be unbiased , consistent , and efficient in

4161-418: The polling data available and the previous election results for each state. Each poll can be weighted based on its age and its size, providing a highly dynamic forecasting mechanism as Election day approaches. http://electionanalytics.cs.illinois.edu/ is an example of a site that employs such methods. When discussing the likelihood of a particular electoral outcome, political forecasters tend to use one of

4234-440: 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. Modern regression analysis

4307-427: The remaining independent variables. As discussed in ordinary least squares , this condition ensures that X T X {\displaystyle X^{T}X} is an invertible matrix and therefore that a unique solution β ^ {\displaystyle {\hat {\beta }}} exists. By itself, a regression is simply a calculation using the data. In order to interpret

4380-872: The researcher decides that five observations are needed to precisely define a straight line ( m {\displaystyle m} ), then the maximum number of independent variables ( n {\displaystyle n} ) the model can support is 4, because 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

4453-800: The researcher wants to estimate a bivariate linear model via least squares : Y i = β 0 + β 1 X 1 i + β 2 X 2 i + e i {\displaystyle Y_{i}=\beta _{0}+\beta _{1}X_{1i}+\beta _{2}X_{2i}+e_{i}} . If the researcher only has access to N = 2 {\displaystyle N=2} data points, then they could find infinitely many combinations ( β ^ 0 , β ^ 1 , β ^ 2 ) {\displaystyle ({\hat {\beta }}_{0},{\hat {\beta }}_{1},{\hat {\beta }}_{2})} that explain

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4526-404: 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

4599-530: The site was listed by The Guardian as one of the "100 most useful websites", being "the best" for predictions. With reference to the 2010 United Kingdom general election , it was cited by journalists Andrew Rawnsley and Michael White of The Guardian . John Rentoul of The Independent referred to the site after the election. Political forecasting People have long been interested in predicting election outcomes. Quotes of betting odds on papal succession appear as early as 1503, when such wagering

4672-407: The structural form of the regression relationship. 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

4745-420: The sum of squared residuals , SSR : 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, β ^ 0 , β ^ 1 {\displaystyle {\widehat {\beta }}_{0},{\widehat {\beta }}_{1}} . In

4818-438: The sum of squared residuals . To understand why there are infinitely many options, note that the system of N = 2 {\displaystyle N=2} equations is to be solved for 3 unknowns, which makes the system underdetermined . Alternatively, one can visualize infinitely many 3-dimensional planes that go through N = 2 {\displaystyle N=2} fixed points. More generally, to estimate

4891-482: The sum of squared errors ∑ i ( Y i − f ( X i , β ) ) 2 {\displaystyle \sum _{i}(Y_{i}-f(X_{i},\beta ))^{2}} . A given regression method will ultimately provide an estimate of β {\displaystyle \beta } , usually denoted β ^ {\displaystyle {\hat {\beta }}} to distinguish

4964-626: The time." Before the advent of scientific polling in 1936, betting odds in the United States correlated strongly to vote results. Since 1936, opinion polls have been a basic part of political forecasting. With the advent of statistical techniques, electoral data have become increasingly easy to handle. It is no surprise, then, that election forecasting has become a big business, for polling firms, news organizations, and betting markets as well as academic students of politics. Academic scholars have constructed models of voting behavior to forecast

5037-471: The value 1 for all i {\displaystyle i} , x i 1 = 1 {\displaystyle x_{i1}=1} , then β 1 {\displaystyle \beta _{1}} is called the regression intercept . The least squares parameter estimates are obtained from p {\displaystyle p} normal equations. The residual can be written as The normal equations are In matrix notation,

5110-444: The variance of e i {\displaystyle e_{i}} to change across values of X i {\displaystyle X_{i}} . Correlated errors that exist within subsets of the data or follow specific patterns can be handled using clustered standard errors, geographic weighted regression , or Newey–West standard errors, among other techniques. When rows of data correspond to locations in space,

5183-469: Was already considered "an old practice." Political betting also has a long history in Great Britain. As one prominent example, Charles James Fox, the late-eighteenth-century Whig statesman, was known as an inveterate gambler. His biographer, George Otto Trevelyan, noted that"(f)or ten years, from 1771 onwards, Charles Fox betted frequently, largely, and judiciously, on the social and political occurrences of

5256-468: Was later extended by Udny Yule and Karl Pearson to a more general statistical context. 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. 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

5329-452: Was the method of least squares , which was published by Legendre in 1805, and by Gauss in 1809. 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, including

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