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78-412: It is mostly used as a way of describing a sample population and as an independent variable . Occasionally it is used as a dependent variable . MACS levels are stable over time and so they can be used as part of a prognosis for individuals. Although MACS was not designed for adults, it has been used with a good measure of reliability in young adult populations ranging in ages from 18-24. Although it has

156-405: A function is a rule for taking an input (in the simplest case, a number or set of numbers) and providing an output (which may also be a number). A symbol that stands for an arbitrary input is called an independent variable , while a symbol that stands for an arbitrary output is called a dependent variable . The most common symbol for the input is x , and the most common symbol for the output

234-444: A regression analysis as independent variables, may aid a researcher with accurate response parameter estimation, prediction , and goodness of fit , but are not of substantive interest to the hypothesis under examination. For example, in a study examining the effect of post-secondary education on lifetime earnings, some extraneous variables might be gender, ethnicity, social class, genetics, intelligence, age, and so forth. A variable

312-478: A statistical context. In an experiment, any variable that can be attributed a value without attributing a value to any other variable is called an independent variable. Models and experiments test the effects that the independent variables have on the dependent variables. Sometimes, even if their influence is not of direct interest, independent variables may be included for other reasons, such as to account for their potential confounding effect. In mathematics,

390-410: A common value for the given predictor variable. This is the only interpretation of "held fixed" that can be used in an observational study . The notion of a "unique effect" is appealing when studying a complex system where multiple interrelated components influence the response variable. In some cases, it can literally be interpreted as the causal effect of an intervention that is linked to the value of

468-522: A good level of reliability when used for children between 2 and 5 years of age, there is less evidence for using it with children younger than 2. Unlike the GMFCS, there are no age bands for the MACS. Assessment is typically done by asking questions of the parent or therapist of the child to see where the child fits. MACS has had some studies demonstrating good to excellent inter-rater reliability . As of 2015,

546-576: A group of predictor variables, say, { x 1 , x 2 , … , x q } {\displaystyle \{x_{1},x_{2},\dots ,x_{q}\}} , a group effect ξ ( w ) {\displaystyle \xi (\mathbf {w} )} is defined as a linear combination of their parameters where w = ( w 1 , w 2 , … , w q ) ⊺ {\displaystyle \mathbf {w} =(w_{1},w_{2},\dots ,w_{q})^{\intercal }}

624-942: A linear regression model assumes that the relationship between the dependent variable y and the vector of regressors x is linear . This relationship is modeled through a disturbance term or error variable ε —an unobserved random variable that adds "noise" to the linear relationship between the dependent variable and regressors. Thus the model takes the form y i = β 0 + β 1 x i 1 + ⋯ + β p x i p + ε i = x i T β + ε i , i = 1 , … , n , {\displaystyle y_{i}=\beta _{0}+\beta _{1}x_{i1}+\cdots +\beta _{p}x_{ip}+\varepsilon _{i}=\mathbf {x} _{i}^{\mathsf {T}}{\boldsymbol {\beta }}+\varepsilon _{i},\qquad i=1,\ldots ,n,} where denotes

702-491: A penalized version of the least squares cost function as in ridge regression ( L -norm penalty) and lasso ( L -norm penalty). Use of the Mean Squared Error (MSE) as the cost on a dataset that has many large outliers, can result in a model that fits the outliers more than the true data due to the higher importance assigned by MSE to large errors. So, cost functions that are robust to outliers should be used if

780-407: A predictor variable. However, it has been argued that in many cases multiple regression analysis fails to clarify the relationships between the predictor variables and the response variable when the predictors are correlated with each other and are not assigned following a study design. Numerous extensions of linear regression have been developed, which allow some or all of the assumptions underlying

858-652: A role as regular variable or feature variable. Known values for the target variable are provided for the training data set and test data set, but should be predicted for other data. The target variable is used in supervised learning algorithms but not in unsupervised learning. Depending on the context, an independent variable is sometimes called a "predictor variable", "regressor", "covariate", "manipulated variable", "explanatory variable", "exposure variable" (see reliability theory ), " risk factor " (see medical statistics ), " feature " (in machine learning and pattern recognition ) or "input variable". In econometrics ,

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936-419: A single dependent variable. In linear regression, the relationships are modeled using linear predictor functions whose unknown model parameters are estimated from the data . Most commonly, the conditional mean of the response given the values of the explanatory variables (or predictors) is assumed to be an affine function of those values; less commonly, the conditional median or some other quantile

1014-400: A study design, the comparisons of interest may literally correspond to comparisons among units whose predictor variables have been "held fixed" by the experimenter. Alternatively, the expression "held fixed" can refer to a selection that takes place in the context of data analysis. In this case, we "hold a variable fixed" by restricting our attention to the subsets of the data that happen to have

1092-464: Is y ; the function itself is commonly written y = f ( x ) . It is possible to have multiple independent variables or multiple dependent variables. For instance, in multivariable calculus , one often encounters functions of the form z = f ( x , y ) , where z is a dependent variable and x and y are independent variables. Functions with multiple outputs are often referred to as vector-valued functions . In mathematical modeling ,

1170-477: Is a model that estimates the linear relationship between a scalar response ( dependent variable ) and one or more explanatory variables ( regressor or independent variable ). A model with exactly one explanatory variable is a simple linear regression ; a model with two or more explanatory variables is a multiple linear regression . This term is distinct from multivariate linear regression , which predicts multiple correlated dependent variables rather than

1248-417: Is a framework for modeling response variables that are bounded or discrete. This is used, for example: Generalized linear models allow for an arbitrary link function , g , that relates the mean of the response variable(s) to the predictors: E ( Y ) = g − 1 ( X B ) {\displaystyle E(Y)=g^{-1}(XB)} . The link function is often related to

1326-476: Is a generalization of simple linear regression to the case of more than one independent variable, and a special case of general linear models, restricted to one dependent variable. The basic model for multiple linear regression is for each observation i = 1 , … , n {\textstyle i=1,\ldots ,n} . In the formula above we consider n observations of one dependent variable and p independent variables. Thus, Y i

1404-577: Is a meaningful effect. It can be accurately estimated by its minimum-variance unbiased linear estimator ξ ^ A = 1 q ( β ^ 1 ′ + β ^ 2 ′ + ⋯ + β ^ q ′ ) {\textstyle {\hat {\xi }}_{A}={\frac {1}{q}}({\hat {\beta }}_{1}'+{\hat {\beta }}_{2}'+\dots +{\hat {\beta }}_{q}')} , even when individually none of

1482-435: Is a special group effect with weights w 1 = 1 {\displaystyle w_{1}=1} and w j = 0 {\displaystyle w_{j}=0} for j ≠ 1 {\displaystyle j\neq 1} , but it cannot be accurately estimated by β ^ 1 ′ {\displaystyle {\hat {\beta }}'_{1}} . It

1560-551: Is a weight vector satisfying ∑ j = 1 q | w j | = 1 {\textstyle \sum _{j=1}^{q}|w_{j}|=1} . Because of the constraint on w j {\displaystyle {w_{j}}} , ξ ( w ) {\displaystyle \xi (\mathbf {w} )} is also referred to as a normalized group effect. A group effect ξ ( w ) {\displaystyle \xi (\mathbf {w} )} has an interpretation as

1638-695: Is also not a meaningful effect. In general, for a group of q {\displaystyle q} strongly correlated predictor variables in an APC arrangement in the standardized model, group effects whose weight vectors w {\displaystyle \mathbf {w} } are at or near the centre of the simplex ∑ j = 1 q w j = 1 {\textstyle \sum _{j=1}^{q}w_{j}=1} ( w j ≥ 0 {\displaystyle w_{j}\geq 0} ) are meaningful and can be accurately estimated by their minimum-variance unbiased linear estimators. Effects with weight vectors far away from

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1716-571: Is called confounding or omitted variable bias ; in these situations, design changes and/or controlling for a variable statistical control is necessary. Extraneous variables are often classified into three types: In modelling, variability that is not covered by the independent variable is designated by e I {\displaystyle e_{I}} and is known as the " residual ", "side effect", " error ", "unexplained share", "residual variable", "disturbance", or "tolerance". Linear regression In statistics , linear regression

1794-417: Is captured by x j . In this case, including the other variables in the model reduces the part of the variability of y that is unrelated to x j , thereby strengthening the apparent relationship with x j . The meaning of the expression "held fixed" may depend on how the values of the predictor variables arise. If the experimenter directly sets the values of the predictor variables according to

1872-412: Is extraneous only when it can be assumed (or shown) to influence the dependent variable . If included in a regression, it can improve the fit of the model . If it is excluded from the regression and if it has a non-zero covariance with one or more of the independent variables of interest, its omission will bias the regression's result for the effect of that independent variable of interest. This effect

1950-754: Is meaningful when the latter is. Thus meaningful group effects of the original variables can be found through meaningful group effects of the standardized variables. In Dempster–Shafer theory , or a linear belief function in particular, a linear regression model may be represented as a partially swept matrix, which can be combined with similar matrices representing observations and other assumed normal distributions and state equations. The combination of swept or unswept matrices provides an alternative method for estimating linear regression models. A large number of procedures have been developed for parameter estimation and inference in linear regression. These methods differ in computational simplicity of algorithms, presence of

2028-400: Is minimized. For example, it is common to use the sum of squared errors ‖ ε ‖ 2 2 {\displaystyle \|{\boldsymbol {\varepsilon }}\|_{2}^{2}} as a measure of ε {\displaystyle {\boldsymbol {\varepsilon }}} for minimization. Consider a situation where a small ball is being tossed up in

2106-410: Is preferred by some authors for the dependent variable. Depending on the context, a dependent variable is sometimes called a "response variable", "regressand", "criterion", "predicted variable", "measured variable", "explained variable", "experimental variable", "responding variable", "outcome variable", "output variable", "target" or "label". In economics endogenous variables are usually referencing

2184-401: Is probable. Group effects provide a means to study the collective impact of strongly correlated predictor variables in linear regression models. Individual effects of such variables are not well-defined as their parameters do not have good interpretations. Furthermore, when the sample size is not large, none of their parameters can be accurately estimated by the least squares regression due to

2262-433: Is regressed on C . It is often used where the variables of interest have a natural hierarchical structure such as in educational statistics, where students are nested in classrooms, classrooms are nested in schools, and schools are nested in some administrative grouping, such as a school district. The response variable might be a measure of student achievement such as a test score, and different covariates would be collected at

2340-461: Is still assumed, with a matrix B replacing the vector β of the classical linear regression model. Multivariate analogues of ordinary least squares (OLS) and generalized least squares (GLS) have been developed. "General linear models" are also called "multivariate linear models". These are not the same as multivariable linear models (also called "multiple linear models"). Various models have been created that allow for heteroscedasticity , i.e.

2418-496: Is strongly correlated with other predictor variables, it is improbable that x j {\displaystyle x_{j}} can increase by one unit with other variables held constant. In this case, the interpretation of β j {\displaystyle \beta _{j}} becomes problematic as it is based on an improbable condition, and the effect of x j {\displaystyle x_{j}} cannot be evaluated in isolation. For

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2496-423: Is the i observation of the dependent variable, X ij is i observation of the j independent variable, j = 1, 2, ..., p . The values β j represent parameters to be estimated, and ε i is the i independent identically distributed normal error. In the more general multivariate linear regression, there is one equation of the above form for each of m > 1 dependent variables that share

2574-449: Is the least squares estimator of β j ′ {\displaystyle \beta _{j}'} . In particular, the average group effect of the q {\displaystyle q} standardized variables is which has an interpretation as the expected change in y ′ {\displaystyle y'} when all x j ′ {\displaystyle x_{j}'} in

2652-431: Is used. Like all forms of regression analysis , linear regression focuses on the conditional probability distribution of the response given the values of the predictors, rather than on the joint probability distribution of all of these variables, which is the domain of multivariate analysis . Linear regression is also a type of machine learning algorithm , more specifically a supervised algorithm, that learns from

2730-412: The β j ′ {\displaystyle \beta _{j}'} can be accurately estimated by β ^ j ′ {\displaystyle {\hat {\beta }}_{j}'} . Not all group effects are meaningful or can be accurately estimated. For example, β 1 ′ {\displaystyle \beta _{1}'}

2808-413: The q {\displaystyle q} variables via testing H 0 : ξ A = 0 {\displaystyle H_{0}:\xi _{A}=0} versus H 1 : ξ A ≠ 0 {\displaystyle H_{1}:\xi _{A}\neq 0} , and (3) characterizing the region of the predictor variable space over which predictions by

2886-490: The multicollinearity problem. Nevertheless, there are meaningful group effects that have good interpretations and can be accurately estimated by the least squares regression. A simple way to identify these meaningful group effects is to use an all positive correlations (APC) arrangement of the strongly correlated variables under which pairwise correlations among these variables are all positive, and standardize all p {\displaystyle p} predictor variables in

2964-580: The transpose , so that x i β is the inner product between vectors x i and β . Often these n equations are stacked together and written in matrix notation as where Fitting a linear model to a given data set usually requires estimating the regression coefficients β {\displaystyle {\boldsymbol {\beta }}} such that the error term ε = y − X β {\displaystyle {\boldsymbol {\varepsilon }}=\mathbf {y} -\mathbf {X} {\boldsymbol {\beta }}}

3042-555: The MACS is used worldwide except in Africa. It is not recommended to use the MACS to detect change. The widespread adoption of the GMFCS inspired the development of the MACS. Alternative classification systems used for children with CP include: ABILHAND, AHA, CHEQ, CPQOL, House, MUUL, PedsQLCP, and SHUEE. A version of the test for children under the age of four years old, the Mini-MACS, was developed in 2016. It has similar tiers to

3120-486: The MACS, with descriptions that are more relevant for the toddler age group, and has good inter-rater reliability. Independent variable A variable is considered dependent if it depends on an independent variable . Dependent variables are studied under the supposition or demand that they depend, by some law or rule (e.g., by a mathematical function ), on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in

3198-416: The air and then we measure its heights of ascent h i at various moments in time t i . Physics tells us that, ignoring the drag , the relationship can be modeled as where β 1 determines the initial velocity of the ball, β 2 is proportional to the standard gravity , and ε i is due to measurement errors. Linear regression can be used to estimate the values of β 1 and β 2 from

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3276-458: The basic model to be relaxed. The simplest case of a single scalar predictor variable x and a single scalar response variable y is known as simple linear regression . The extension to multiple and/or vector -valued predictor variables (denoted with a capital X ) is known as multiple linear regression , also known as multivariable linear regression (not to be confused with multivariate linear regression ). Multiple linear regression

3354-401: The central role of the linear predictor β ′ x as in the classical linear regression model. Under certain conditions, simply applying OLS to data from a single-index model will consistently estimate β up to a proportionality constant. Hierarchical linear models (or multilevel regression ) organizes the data into a hierarchy of regressions, for example where A is regressed on B , and B

3432-450: The centre are not meaningful as such weight vectors represent simultaneous changes of the variables that violate the strong positive correlations of the standardized variables in an APC arrangement. As such, they are not probable. These effects also cannot be accurately estimated. Applications of the group effects include (1) estimation and inference for meaningful group effects on the response variable, (2) testing for "group significance" of

3510-586: The centred y {\displaystyle y} and x j ′ {\displaystyle x_{j}'} be the standardized x j {\displaystyle x_{j}} . Then, the standardized linear regression model is Parameters β j {\displaystyle \beta _{j}} in the original model, including β 0 {\displaystyle \beta _{0}} , are simple functions of β j ′ {\displaystyle \beta _{j}'} in

3588-607: The classroom, school, and school district levels. Errors-in-variables models (or "measurement error models") extend the traditional linear regression model to allow the predictor variables X to be observed with error. This error causes standard estimators of β to become biased. Generally, the form of bias is an attenuation, meaning that the effects are biased toward zero. In a multiple linear regression model parameter β j {\displaystyle \beta _{j}} of predictor variable x j {\displaystyle x_{j}} represents

3666-399: The covariate. A variable may be thought to alter the dependent or independent variables, but may not actually be the focus of the experiment. So that the variable will be kept constant or monitored to try to minimize its effect on the experiment. Such variables may be designated as either a "controlled variable", " control variable ", or "fixed variable". Extraneous variables, if included in

3744-419: The data strongly influence the performance of different estimation methods: A fitted linear regression model can be used to identify the relationship between a single predictor variable x j and the response variable y when all the other predictor variables in the model are "held fixed". Specifically, the interpretation of β j is the expected change in y for a one-unit change in x j when

3822-504: The dataset has many large outliers . Conversely, the least squares approach can be used to fit models that are not linear models. Thus, although the terms "least squares" and "linear model" are closely linked, they are not synonymous. Given a data set { y i , x i 1 , … , x i p } i = 1 n {\displaystyle \{y_{i},\,x_{i1},\ldots ,x_{ip}\}_{i=1}^{n}} of n statistical units ,

3900-460: The dependent variable (and variable of most interest) was the annual mean sea level at a given location for which a series of yearly values were available. The primary independent variable was time. Use was made of a covariate consisting of yearly values of annual mean atmospheric pressure at sea level. The results showed that inclusion of the covariate allowed improved estimates of the trend against time to be obtained, compared to analyses which omitted

3978-399: The dependent variable not explained by the independent variable. With multiple independent variables, the model is y i = a + b x i ,1 + b x i ,2 + ... + b x i,n + e i , where n is the number of independent variables. In statistics, more specifically in linear regression , a scatter plot of data is generated with X as the independent variable and Y as

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4056-441: The dependent variable. This is also called a bivariate dataset, ( x 1 , y 1 )( x 2 , y 2 ) ...( x i , y i ) . The simple linear regression model takes the form of Y i = a + B x i + U i , for i = 1, 2, ... , n . In this case, U i , ... , U n are independent random variables. This occurs when the measurements do not influence each other. Through propagation of independence,

4134-440: The distribution of the response, and in particular it typically has the effect of transforming between the ( − ∞ , ∞ ) {\displaystyle (-\infty ,\infty )} range of the linear predictor and the range of the response variable. Some common examples of GLMs are: Single index models allow some degree of nonlinearity in the relationship between x and y , while preserving

4212-514: The errors for different response variables may have different variances . For example, weighted least squares is a method for estimating linear regression models when the response variables may have different error variances, possibly with correlated errors. (See also Weighted linear least squares , and Generalized least squares .) Heteroscedasticity-consistent standard errors is an improved method for use with uncorrelated but potentially heteroscedastic errors. The Generalized linear model (GLM)

4290-427: The expected change in y {\displaystyle y} when variables in the group x 1 , x 2 , … , x q {\displaystyle x_{1},x_{2},\dots ,x_{q}} change by the amount w 1 , w 2 , … , w q {\displaystyle w_{1},w_{2},\dots ,w_{q}} , respectively, at

4368-470: The group effect also reduces to an individual effect. A group effect ξ ( w ) {\displaystyle \xi (\mathbf {w} )} is said to be meaningful if the underlying simultaneous changes of the q {\displaystyle q} variables ( x 1 , x 2 , … , x q ) ⊺ {\displaystyle (x_{1},x_{2},\dots ,x_{q})^{\intercal }}

4446-452: The independence of U i implies independence of Y i , even though each Y i has a different expectation value. Each U i has an expectation value of 0 and a variance of σ . Expectation of Y i Proof: The line of best fit for the bivariate dataset takes the form y = α + βx and is called the regression line. α and β correspond to the intercept and slope, respectively. In an experiment ,

4524-403: The individual effect of x j {\displaystyle x_{j}} . It has an interpretation as the expected change in the response variable y {\displaystyle y} when x j {\displaystyle x_{j}} increases by one unit with other predictor variables held constant. When x j {\displaystyle x_{j}}

4602-400: The information in x j , so that once that variable is in the model, there is no contribution of x j to the variation in y . Conversely, the unique effect of x j can be large while its marginal effect is nearly zero. This would happen if the other covariates explained a great deal of the variation of y , but they mainly explain variation in a way that is complementary to what

4680-445: The labelled datasets and maps the data points to the most optimized linear functions that can be used for prediction on new datasets. Linear regression was the first type of regression analysis to be studied rigorously, and to be used extensively in practical applications. This is because models which depend linearly on their unknown parameters are easier to fit than models which are non-linearly related to their parameters and because

4758-543: The least squares estimated model are accurate. A group effect of the original variables { x 1 , x 2 , … , x q } {\displaystyle \{x_{1},x_{2},\dots ,x_{q}\}} can be expressed as a constant times a group effect of the standardized variables { x 1 ′ , x 2 ′ , … , x q ′ } {\displaystyle \{x_{1}',x_{2}',\dots ,x_{q}'\}} . The former

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4836-404: The measured data. This model is non-linear in the time variable, but it is linear in the parameters β 1 and β 2 ; if we take regressors x i  = ( x i 1 , x i 2 )  = ( t i , t i ), the model takes on the standard form Standard linear regression models with standard estimation techniques make a number of assumptions about the predictor variables,

4914-472: The model so that they all have mean zero and length one. To illustrate this, suppose that { x 1 , x 2 , … , x q } {\displaystyle \{x_{1},x_{2},\dots ,x_{q}\}} is a group of strongly correlated variables in an APC arrangement and that they are not strongly correlated with predictor variables outside the group. Let y ′ {\displaystyle y'} be

4992-511: The other covariates are held fixed—that is, the expected value of the partial derivative of y with respect to x j . This is sometimes called the unique effect of x j on y . In contrast, the marginal effect of x j on y can be assessed using a correlation coefficient or simple linear regression model relating only x j to y ; this effect is the total derivative of y with respect to x j . Care must be taken when interpreting regression results, as some of

5070-428: The regressors may not allow for marginal changes (such as dummy variables , or the intercept term), while others cannot be held fixed (recall the example from the introduction: it would be impossible to "hold t i fixed" and at the same time change the value of t i ). It is possible that the unique effect be nearly zero even when the marginal effect is large. This may imply that some other covariate captures all

5148-399: The relationship between the set of dependent variables and set of independent variables is studied. In the simple stochastic linear model y i = a + b x i + e i the term y i is the i th value of the dependent variable and x i is the i th value of the independent variable. The term e i is known as the "error" and contains the variability of

5226-552: The response variable y is still a scalar. Another term, multivariate linear regression , refers to cases where y is a vector, i.e., the same as general linear regression . The general linear model considers the situation when the response variable is not a scalar (for each observation) but a vector, y i . Conditional linearity of E ( y ∣ x i ) = x i T B {\displaystyle E(\mathbf {y} \mid \mathbf {x} _{i})=\mathbf {x} _{i}^{\mathsf {T}}B}

5304-755: The response variable and their relationship. Numerous extensions have been developed that allow each of these assumptions to be relaxed (i.e. reduced to a weaker form), and in some cases eliminated entirely. Generally these extensions make the estimation procedure more complex and time-consuming, and may also require more data in order to produce an equally precise model. The following are the major assumptions made by standard linear regression models with standard estimation techniques (e.g. ordinary least squares ): Violations of these assumptions can result in biased estimations of β , biased standard errors, untrustworthy confidence intervals and significance tests. Beyond these assumptions, several other statistical properties of

5382-420: The same set of explanatory variables and hence are estimated simultaneously with each other: for all observations indexed as i = 1, ... , n and for all dependent variables indexed as j = 1, ... , m . Nearly all real-world regression models involve multiple predictors, and basic descriptions of linear regression are often phrased in terms of the multiple regression model. Note, however, that in these cases

5460-611: The same time with other variables (not in the group) held constant. It generalizes the individual effect of a variable to a group of variables in that ( i {\displaystyle i} ) if q = 1 {\displaystyle q=1} , then the group effect reduces to an individual effect, and ( i i {\displaystyle ii} ) if w i = 1 {\displaystyle w_{i}=1} and w j = 0 {\displaystyle w_{j}=0} for j ≠ i {\displaystyle j\neq i} , then

5538-423: The scope of the experiment in question. In this sense, some common independent variables are time , space , density , mass , fluid flow rate , and previous values of some observed value of interest (e.g. human population size) to predict future values (the dependent variable). Of the two, it is always the dependent variable whose variation is being studied, by altering inputs, also known as regressors in

5616-422: The standardized model. A group effect of { x 1 ′ , x 2 ′ , … , x q ′ } {\displaystyle \{x_{1}',x_{2}',\dots ,x_{q}'\}} is and its minimum-variance unbiased linear estimator is where β ^ j ′ {\displaystyle {\hat {\beta }}_{j}'}

5694-431: The standardized model. The standardization of variables does not change their correlations, so { x 1 ′ , x 2 ′ , … , x q ′ } {\displaystyle \{x_{1}',x_{2}',\dots ,x_{q}'\}} is a group of strongly correlated variables in an APC arrangement and they are not strongly correlated with other predictor variables in

5772-448: The statistical properties of the resulting estimators are easier to determine. Linear regression has many practical uses. Most applications fall into one of the following two broad categories: Linear regression models are often fitted using the least squares approach, but they may also be fitted in other ways, such as by minimizing the " lack of fit " in some other norm (as with least absolute deviations regression), or by minimizing

5850-469: The strongly correlated group increase by ( 1 / q ) {\displaystyle (1/q)} th of a unit at the same time with variables outside the group held constant. With strong positive correlations and in standardized units, variables in the group are approximately equal, so they are likely to increase at the same time and in similar amount. Thus, the average group effect ξ A {\displaystyle \xi _{A}}

5928-433: The target. "Explained variable" is preferred by some authors over "dependent variable" when the quantities treated as "dependent variables" may not be statistically dependent. If the dependent variable is referred to as an "explained variable" then the term "predictor variable" is preferred by some authors for the independent variable. An example is provided by the analysis of trend in sea level by Woodworth (1987) . Here

6006-400: The term "control variable" is usually used instead of "covariate". "Explanatory variable" is preferred by some authors over "independent variable" when the quantities treated as independent variables may not be statistically independent or independently manipulable by the researcher. If the independent variable is referred to as an "explanatory variable" then the term "response variable"

6084-453: The variable manipulated by an experimenter is something that is proven to work, called an independent variable. The dependent variable is the event expected to change when the independent variable is manipulated. In data mining tools (for multivariate statistics and machine learning ), the dependent variable is assigned a role as target variable (or in some tools as label attribute ), while an independent variable may be assigned

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