Concept in mathematical modeling, statistical modeling and experimental sciences
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 the scope of the experiment in question.[a] In this sense, some common independent variables are time, space, density, mass, fluid flow rate,[1][2] and previous values of some observed value of interest (e.g. human population size) to predict future values (the dependent variable).[3]
Of the two, it is always the dependent variable whose variation is being studied, by altering inputs, also known as regressors in 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 single variable calculus, a function is typically graphed with the horizontal axis representing the independent variable and the vertical axis representing the dependent variable. In this function, y is the dependent variable and x is the independent variable.
In pure mathematics
In mathematics, a function is a rule for taking an input (in the simplest case, a number or set of numbers)[5] and providing an output (which may also be a number).[5] 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.[6] The most common symbol for the input is x, and the most common symbol for the output is y; the function itself is commonly written y = f(x).[6][7]
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.[8] Functions with multiple outputs are often referred to as vector-valued functions.
In the simple stochasticlinear modelyi = a + bxi + ei the term yi is the ith value of the dependent variable and xi is the ith value of the independent variable. The term ei is known as the "error" and contains the variability of the dependent variable not explained by the independent variable.[citation needed]
With multiple independent variables, the model is yi = a + bxi,1 + bxi,2 + ... + bxi,n + ei, where n is the number of independent variables.[citation needed]
In statistics, more specifically in linear regression, a scatter plot of data is generated with X as the independent variable and Y as the dependent variable. This is also called a bivariate dataset, (x1, y1)(x2, y2) ...(xi, yi). The simple linear regression model takes the form of Yi = a + Bxi + Ui, for i = 1, 2, ... , n. In this case, Ui, ... ,Un are independent random variables. This occurs when the measurements do not influence each other. Through propagation of independence, the independence of Ui implies independence of Yi, even though each Yi has a different expectation value. Each Ui has an expectation value of 0 and a variance of σ2.[9] Expectation of Yi Proof:[9]
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.[9]
In an experiment, the variable manipulated by an experimenter is something that is proven to work, called an independent variable.[10] The dependent variable is the event expected to change when the independent variable is manipulated.[11]
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 a role as regular variable[12] 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.
"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.[20][21] If the independent variable is referred to as an "explanatory variable" then the term "response variable" is preferred by some authors for the dependent variable.[14][20][21]
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".[14] In economics endogenous variables are usually referencing the target.
"Explained variable" is preferred by some authors over "dependent variable" when the quantities treated as "dependent variables" may not be statistically dependent.[22] 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.[22]
An example is provided by the analysis of trend in sea level by Woodworth (1987). Here 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 the covariate.
Antonym pairs
independent
dependent
input
output
regressor
regressand
predictor
predicted
explanatory
explained
exogenous
endogenous
manipulated
measured
exposure
outcome
feature
label or target
Other variables
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 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 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 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:
Subject variables, which are the characteristics of the individuals being studied that might affect their actions. These variables include age, gender, health status, mood, background, etc.
Blocking variables or experimental variables are characteristics of the persons conducting the experiment which might influence how a person behaves. Gender, the presence of racial discrimination, language, or other factors may qualify as such variables.
Situational variables are features of the environment in which the study or research was conducted, which have a bearing on the outcome of the experiment in a negative way. Included are the air temperature, level of activity, lighting, and time of day.
In modelling, variability that is not covered by the independent variable is designated by and is known as the "residual", "side effect", "error", "unexplained share", "residual variable", "disturbance", or "tolerance".
Examples
Effect of fertilizer on plant growths:
In a study measuring the influence of different quantities of fertilizer on plant growth, the independent variable would be the amount of fertilizer used. The dependent variable would be the growth in height or mass of the plant. The controlled variables would be the type of plant, the type of fertilizer, the amount of sunlight the plant gets, the size of the pots, etc.
Effect of drug dosage on symptom severity:
In a study of how different doses of a drug affect the severity of symptoms, a researcher could compare the frequency and intensity of symptoms when different doses are administered. Here the independent variable is the dose and the dependent variable is the frequency/intensity of symptoms.
Effect of temperature on pigmentation:
In measuring the amount of color removed from beetroot samples at different temperatures, temperature is the independent variable and amount of pigment removed is the dependent variable.
Effect of sugar added in a coffee:
The taste varies with the amount of sugar added in the coffee. Here, the sugar is the independent variable, while the taste is the dependent variable.
↑ Even if the existing dependency is invertible (e.g., by finding the inverse function when it exists), the nomenclature is kept if the inverse dependency is not the object of study in the experiment.
Related Research Articles
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In statistical modeling, regression analysis is a set of statistical processes for estimating the relationships between a dependent variable and one or more error-free independent variables. The most common form of regression analysis is linear regression, in which one finds the line that most closely fits the data according to a specific mathematical criterion. For example, the method of ordinary least squares computes the unique line that minimizes the sum of squared differences between the true data and that line. For specific mathematical reasons, this allows the researcher to estimate the conditional expectation 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 or estimate the conditional expectation across a broader collection of non-linear models.
In statistics and in particular in regression analysis, a design matrix, also known as model matrix or regressor matrix and often denoted by X, is a matrix of values of explanatory variables of a set of objects. Each row represents an individual object, with the successive columns corresponding to the variables and their specific values for that object. The design matrix is used in certain statistical models, e.g., the general linear model. It can contain indicator variables that indicate group membership in an ANOVA, or it can contain values of continuous variables.
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In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression model by the principle of least squares: minimizing the sum of the squares of the differences between the observed dependent variable in the input dataset and the output of the (linear) function of the independent variable. Some sources consider OLS to be linear regression.
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In statistics and in machine learning, a linear predictor function is a linear function of a set of coefficients and explanatory variables, whose value is used to predict the outcome of a dependent variable. This sort of function usually comes in linear regression, where the coefficients are called regression coefficients. However, they also occur in various types of linear classifiers, as well as in various other models, such as principal component analysis and factor analysis. In many of these models, the coefficients are referred to as "weights".
In statistics, linear regression is a model that estimates the linear relationship between a scalar response and one or more explanatory variables. 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 a single dependent variable.
↑ Anton, Howard, Irl C. Bivens, and Stephen Davis. Calculus Single Variable. John Wiley & Sons, 2012. Section 0.1
↑ Larson, Ron, and Bruce Edwards. Calculus. Cengage Learning, 2009. Section 13.1
1 2 3 Dekking, Frederik Michel (2005), A modern introduction to probability and statistics: understanding why and how, Springer, ISBN1-85233-896-2, OCLC783259968
↑ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN0-19-920613-9 (entry for "independent variable")
1 2 3 Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms, OUP. ISBN0-19-920613-9 (entry for "regression")
↑ Gujarati, Damodar N.; Porter, Dawn C. (2009). "Terminology and Notation". Basic Econometrics (Fifth internationaled.). New York: McGraw-Hill. p.21. ISBN978-007-127625-2.
↑ Wooldridge, Jeffrey (2012). Introductory Econometrics: A Modern Approach (Fifthed.). Mason, OH: South-Western Cengage Learning. pp.22–23. ISBN978-1-111-53104-1.
↑ Last, John M., ed. (2001). A Dictionary of Epidemiology (Fourthed.). Oxford UP. ISBN0-19-514168-7.
↑ Everitt, B. S. (2002). The Cambridge Dictionary of Statistics (2nded.). Cambridge UP. ISBN0-521-81099-X.
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