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In statistics, the number of degrees of freedom is the number of values in the final calculation of a statistic that are free to vary.^{[1]}
The number of independent ways by which a dynamic system can move without violating any constraint imposed on it, is called degree of freedom. In other words, the degree of freedom can be defined as the minimum number of independent coordinates that can specify the position of the system completely.
Estimates of statistical parameters can be based upon different amounts of information or data. The number of independent pieces of information that go into the estimate of a parameter is called the degrees of freedom. In general, the degrees of freedom of an estimate of a parameter is equal to the number of independent scores that go into the estimate minus the number of parameters used as intermediate steps in the estimation of the parameter itself (i.e., the sample variance has N1 degrees of freedom, since it is computed from N random scores minus the only 1 parameter estimated as intermediate step, which is the sample mean).^{[2]}
Mathematically, degrees of freedom is the number of dimensions of the domain of a random vector, or essentially the number of 'free' components (how many components need to be known before the vector is fully determined).
The term is most often used in the context of linear models (linear regression, analysis of variance), where certain random vectors are constrained to lie in linear subspaces, and the number of degrees of freedom is the dimension of the subspace. The degrees of freedom are also commonly associated with the squared lengths (or "sum of squares" of the coordinates) of such vectors, and the parameters of chisquared and other distributions that arise in associated statistical testing problems.
While introductory textbooks may introduce degrees of freedom as distribution parameters or through hypothesis testing, it is the underlying geometry that defines degrees of freedom, and is critical to a proper understanding of the concept. Walker (1940)^{[3]} has stated this succinctly as "the number of observations minus the number of necessary relations among these observations."
In equations, the typical symbol for degrees of freedom is (lowercase Greek letter nu). In text and tables, the abbreviation "d.f." is commonly used. R.A. Fisher used n to symbolize degrees of freedom but modern usage typically reserves n for sample size.
A common way to think of degrees of freedom is as the number of independent pieces of information available to estimate another piece of information. More concretely, the number of degrees of freedom is the number of independent observations in a sample of data that are available to estimate a parameter of the population from which that sample is drawn. For example, if we have two observations, when calculating the mean we have two independent observations; however, when calculating the variance, we have only one independent observation, since the two observations are equally distant from the mean.
In fitting statistical models to data, the vectors of residuals are constrained to lie in a space of smaller dimension than the number of components in the vector. That smaller dimension is the number of degrees of freedom for error.
Perhaps the simplest example is this. Suppose
are random variables each with expected value μ, and let
be the "sample mean." Then the quantities
are residuals that may be considered estimates of the errors X_{i} − μ. The sum of the residuals (unlike the sum of the errors) is necessarily 0. If one knows the values of any n − 1 of the residuals, one can thus find the last one. That means they are constrained to lie in a space of dimension n − 1. One says that "there are n − 1 degrees of freedom for errors."
An only slightly less simple example is that of least squares estimation of a and b in the model
where x_{i} are given, but e_{i} and hence Y_{i} are random. Let and be the leastsquares estimates of a and b. Then the residuals
are constrained to lie within the space defined by the two equations
One says that there are n − 2 degrees of freedom for error.
Note about notation: the capital letter Y is used in specifying the model, while lowercase y in the definition of the residuals; that is because the former are hypothesized random variables and the latter are actual data.
We can generalise this to multiple regression involving p parameters and covariates (e.g. p − 1 predictors and one mean), in which case the cost in degrees of freedom of the fit is p.
Geometrically, the degrees of freedom can be interpreted as the dimension of certain vector subspaces. As a starting point, suppose that we have a sample of n independent normally distributed observations,
This can be represented as an ndimensional random vector:
Since this random vector can lie anywhere in ndimensional space, it has n degrees of freedom.
Now, let be the sample mean. The random vector can be decomposed as the sum of the sample mean plus a vector of residuals:
The first vector on the righthand side is constrained to be a multiple of the vector of 1's, and the only free quantity is . It therefore has 1 degree of freedom.
The second vector is constrained by the relation . The first n − 1 components of this vector can be anything. However, once you know the first n − 1 components, the constraint tells you the value of the nth component. Therefore, this vector has n − 1 degrees of freedom.
Mathematically, the first vector is the orthogonal, or leastsquares, projection of the data vector onto the subspace spanned by the vector of 1's. The 1 degree of freedom is the dimension of this subspace. The second residual vector is the leastsquares projection onto the (n − 1)dimensional orthogonal complement of this subspace, and has n − 1 degrees of freedom.
In statistical testing applications, often one isn't directly interested in the component vectors, but rather in their squared lengths. In the example above, the residual sumofsquares is
If the data points are normally distributed with mean 0 and variance , then the residual sum of squares has a scaled chisquared distribution (scaled by the factor ), with n − 1 degrees of freedom. The degreesoffreedom, here a parameter of the distribution, can still be interpreted as the dimension of an underlying vector subspace.
Likewise, the onesample ttest statistic,
follows a Student's t distribution with n − 1 degrees of freedom when the hypothesized mean is correct. Again, the degreesoffreedom arises from the residual vector in the denominator.
The demonstration of the t and chisquared distributions for onesample problems above is the simplest example where degreesoffreedom arise. However, similar geometry and vector decompositions underlie much of the theory of linear models, including linear regression and analysis of variance. An explicit example based on comparison of three means is presented here; the geometry of linear models is discussed in more complete detail by Christensen (2002).^{[4]}
Suppose independent observations are made for three populations, , and . The restriction to three groups and equal sample sizes simplifies notation, but the ideas are easily generalized.
The observations can be decomposed as
where are the means of the individual samples, and is the mean of all 3n observations. In vector notation this decomposition can be written as
The observation vector, on the lefthand side, has 3n degrees of freedom. On the righthand side, the first vector has one degree of freedom (or dimension) for the overall mean. The second vector depends on three random variables, , and . However, these must sum to 0 and so are constrained; the vector therefore must lie in a 2dimensional subspace, and has 2 degrees of freedom. The remaining 3n − 3 degrees of freedom are in the residual vector (made up of n − 1 degrees of freedom within each of the populations).
In statistical testing problems, one usually isn't interested in the component vectors themselves, but rather in their squared lengths, or Sum of Squares. The degrees of freedom associated with a sumofsquares is the degreesoffreedom of the corresponding component vectors.
The threepopulation example above is an example of oneway Analysis of Variance. The model, or treatment, sumofsquares is the squared length of the second vector,
with 2 degrees of freedom. The residual, or error, sumofsquares is
with 3(n1) degrees of freedom. Of course, introductory books on ANOVA usually state formulae without showing the vectors, but it is this underlying geometry that gives rise to SS formulae, and shows how to unambiguously determine the degrees of freedom in any given situation.
Under the null hypothesis of no difference between population means (and assuming that standard ANOVA regularity assumptions are satisfied) the sums of squares have scaled chisquared distributions, with the corresponding degrees of freedom. The Ftest statistic is the ratio, after scaling by the degrees of freedom. If there is no difference between population means this ratio follows an F distribution with 2 and 3n − 3 degrees of freedom.
In some complicated settings, such as unbalanced splitplot designs, the sumsofsquares no longer have scaled chisquared distributions. Comparison of sumofsquares with degreesoffreedom is no longer meaningful, and software may report certain fractional 'degrees of freedom' in these cases. Such numbers have no genuine degreesoffreedom interpretation, but are simply providing an approximate chisquared distribution for the corresponding sumofsquares. The details of such approximations are beyond the scope of this page.
Several commonly encountered statistical distributions (Student's t, ChiSquared, F) have parameters that are commonly referred to as degrees of freedom. This terminology simply reflects that in many applications where these distributions occur, the parameter corresponds to the degrees of freedom of an underlying random vector, as in the preceding ANOVA example. Another simple example is: if are independent normal random variables, the statistic
follows a chisquared distribution with n−1 degrees of freedom. Here, the degrees of freedom arises from the residual sumofsquares in the numerator, and in turn the n−1 degrees of freedom of the underlying residual vector .
In the application of these distributions to linear models, the degrees of freedom parameters can take only integer values. The underlying families of distributions allow fractional values for the degreesoffreedom parameters, which can arise in more sophisticated uses. One set of examples is problems where chisquared approximations based on effective degrees of freedom are used. In other applications, such as modelling heavytailed data, a t or F distribution may be used as an empirical model. In these cases, there is no particular degrees of freedom interpretation to the distribution parameters, even though the terminology may continue to be used.
Many regression methods, including ridge regression, linear smoothers and smoothing splines are not based on ordinary least squares projections, but rather on regularized (generalized and/or penalized) leastsquares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures. However, these procedures are still linear in the observations, and the fitted values of the regression can be expressed in the form
where is the vector of fitted values at each of the original covariate values from the fitted model, y is the original vector of responses, and H is the hat matrix or, more generally, smoother matrix.
For statistical inference, sumsofsquares can still be formed: the model sumofsquares is ; the residual sumofsquares is . However, because H does not correspond to an ordinary leastsquares fit (i.e. is not an orthogonal projection), these sumsofsquares no longer have (scaled, noncentral) chisquared distributions, and dimensionally defined degreesoffreedom are not useful.
The effective degrees of freedom of the fit can be defined in various ways to implement goodnessoffit tests, crossvalidation and other inferential procedures. Here one can distinguish between regression effective degrees of freedom and residual effective degrees of freedom.
Regarding the former, appropriate definitions can include the trace of the hat matrix,^{[5]} tr(H), the trace of the quadratic form of the hat matrix, tr(H'H), the form tr(2H – H H'), or the Satterthwaite approximation, tr(H'H)^{2}/tr(H'HH'H). In the case of linear regression, the hat matrix H is X(X 'X)^{−1}X ', and all these definitions reduce to the usual degrees of freedom. Notice that
the regression (not residual) degrees of freedom in linear models are "the sum of the sensitivities of the fitted values with respect to the observed response values",^{[6]} i.e., the sum of leverage scores.
There are corresponding definitions of residual effective degreesoffreedom (redf), with H replaced by I − H. For example, if the goal is to estimate error variance, the redf would be defined as tr((I − H)'(I − H)), and the unbiased estimate is (with ),
or:^{[7]}^{[8]}^{[9]}
The last approximation above^{[8]} reduces the computational cost from O(n^{2}) to only O(n). In general the numerator would be the objective function being minimized; e.g., if the hat matrix includes an observation covariance matrix, Σ, then becomes .
Note that unlike in the original case, noninteger degrees of freedom are allowed, though the value must usually still be constrained between 0 and n.
Consider, as an example, the knearest neighbour smoother, which is the average of the k nearest measured values to the given point. Then, at each of the n measured points, the weight of the original value on the linear combination that makes up the predicted value is just 1/k. Thus, the trace of the hat matrix is n/k. Thus the smooth costs n/k effective degrees of freedom.
As another example, consider the existence of nearly duplicated observations. Naive application of classical formula, n − p, would lead to overestimation of the residuals degree of freedom, as if each observation were independent. More realistically, though, the hat matrix H = X(X ' Σ^{−1} X)^{−1}X ' Σ^{−1} would involve an observation covariance matrix Σ indicating the nonzero correlation among observations. The more general formulation of effective degree of freedom would result in a more realistic estimate for, e.g., the error variance σ^{2}.
Similar concepts are the equivalent degrees of freedom in nonparametric regression,^{[10]} the degree of freedom of signal in atmospheric studies,^{[11]}^{[12]} and the noninteger degree of freedom in geodesy.^{[13]}^{[14]}
This section requires expansion. (August 2013) 
The residual sumofsquares has a generalized chisquared distribution, and the theory associated with this distribution^{[15]} provides an alternative route to the answers provided above.
