Degrees of freedom (statistics)

degrees of freedomdegree of freedomEffective degrees of freedomstatistical degrees of freedomd.f.degrees of freedom in statisticsdegrees-of-freedomEffective degree of freedomeffective number of degrees of freedomResidual effective degrees of freedom
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.wikipedia
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Chi-squared distribution

chi-squaredchi-square distributionchi square distribution
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 chi-squared and other distributions that arise in associated statistical testing problems.
In probability theory and statistics, the chi-square distribution (also chi-squared or χ 2 -distribution) with k degrees of freedom is the distribution of a sum of the squares of k independent standard normal random variables.

Statistical parameter

parametersparameterparametrization
Estimates of statistical parameters can be based upon different amounts of information or data.
For example, the family of chi-squared distributions can be indexed by the number of degrees of freedom: the number of degrees of freedom is a parameter for the distributions, and so the family is thereby parameterized.

Student's t-distribution

Student's ''t''-distributiont-distributiont''-distribution
While Gosset did not actually use the term 'degrees of freedom', he explained the concept in the course of developing what became known as Student's t-distribution.
where \nu is the number of degrees of freedom and \Gamma is the gamma function.

Nu (letter)

nuΝGreek letter nu
In equations, the typical symbol for degrees of freedom is ν (lowercase Greek letter nu).

Analysis of variance

ANOVAanalysis of variance (ANOVA)corrected the means
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 number of degrees of freedom DF can be partitioned in a similar way: one of these components (that for error) specifies a chi-squared distribution which describes the associated sum of squares, while the same is true for "treatments" if there is no treatment effect.

Student's t-test

t-testt''-testStudent's ''t''-test
Likewise, the one-sample t-test statistic,
degrees of freedom.

Errors and residuals

residualserror termresidual
are residuals that may be considered estimates of the errors X i − μ.
The sum of squares of the statistical errors, divided by σ 2, has a chi-squared distribution with n degrees of freedom:

Least squares

least-squaresmethod of least squaresleast squares method
An example which is only slightly less simple is that of least squares estimation of a and b in the model
The denominator, n − m, is the statistical degrees of freedom; see effective degrees of freedom for generalizations.

Ordinary least squares

OLSleast squaresOrdinary least squares regression
Many non-standard regression methods, including regularized least squares (e.g., ridge regression), linear smoothers, smoothing splines, and semiparametric regression are not based on ordinary least squares projections, but rather on regularized (generalized and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures.
: The numerator, n−p, is the statistical degrees of freedom.

Residual sum of squares

sum of squared residualssum of squares of residualsresidual sum-of-squares
In the example above, the residual sum-of-squares is

Restricted randomization

nested datasplit plotNested factors
In some complicated settings, such as unbalanced split-plot designs, the sums-of-squares no longer have scaled chi-squared distributions.
Since there are 8 degrees of freedom for the subplot error term, this MSE can be used to test each effect that involves current.

F-distribution

F distributionF''-distributionF'' distribution
If there is no difference between population means this ratio follows an F distribution with 2 and 3n − 3 degrees of freedom.

Random variable

random variablesrandom variationrandom
are random variables each with expected value μ, and let
This is a chi-squared distribution with one degree of freedom.

Welch–Satterthwaite equation

Pooled degree of freedomPooled degrees of freedomSatterthwaite approximation
For the regression effective degrees of freedom, appropriate definitions can include the trace of the hat matrix, 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 statistics and uncertainty analysis, the Welch–Satterthwaite equation is used to calculate an approximation to the effective degrees of freedom of a linear combination of independent sample variances, also known as the pooled degrees of freedom, corresponding to the pooled variance.

Tikhonov regularization

ridge regressionregularizeda squared regularizing function
Many non-standard regression methods, including regularized least squares (e.g., ridge regression), linear smoothers, smoothing splines, and semiparametric regression are not based on ordinary least squares projections, but rather on regularized (generalized and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures.
where is the residual sum of squares, and \tau is the effective number of degrees of freedom.

Goodness of fit

goodness-of-fitfitgoodness-of-fit test
The effective degrees of freedom of the fit can be defined in various ways to implement goodness-of-fit tests, cross-validation, and other statistical inference procedures.
The chi-squared distribution has (k − c) degrees of freedom, where k is the number of non-empty cells and c is the number of estimated parameters (including location and scale parameters and shape parameters) for the distribution plus one.

Confidence region

3-dimensional uncertainty ellipsoiderror ellipseline of variation
The more general formulation of effective degree of freedom would result in a more realistic estimate for, e.g., the error variance σ 2, which in its turn scales the unknown parameters' a posteriori standard deviation; the degree of freedom will also affect the expansion factor necessary to produce an error ellipse for a given confidence level.
Further, F is the quantile function of the F-distribution, with p and \nu = n - p degrees of freedom, \alpha is the statistical significance level, and the symbol X^\prime means the transpose of X.

Reduced chi-squared statistic

reduced chi-squaredMean square weighted deviationChi-squared per degree of freedom
It is defined as chi-squared per degree of freedom:

Projection matrix

hat matrixannihilator matrixobservation matrix
:where \hat{y} 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 other models such as LOESS that are still linear in the observations \mathbf{y}, the projection matrix can be used to define the effective degrees of freedom of the model.

Semiparametric regression

regressionsemiparametric modelingsingle and multiple index models
Many non-standard regression methods, including regularized least squares (e.g., ridge regression), linear smoothers, smoothing splines, and semiparametric regression are not based on ordinary least squares projections, but rather on regularized (generalized and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures.

Generalized least squares

feasible generalized least squaresgeneralizedgeneralized (correlated)
Many non-standard regression methods, including regularized least squares (e.g., ridge regression), linear smoothers, smoothing splines, and semiparametric regression are not based on ordinary least squares projections, but rather on regularized (generalized and/or penalized) least-squares, and so degrees of freedom defined in terms of dimensionality is generally not useful for these procedures.

Sample size determination

sample sizeSampling sizessample
All the parameters in the equation are in fact the degrees of freedom of the number of their concepts, and hence, their numbers are subtracted by 1 before insertion into the equation.

Generalized chi-squared distribution

Generalized chi-square distribution
The residual sum-of-squares \|y-Hy\|^2 has a generalized chi-squared distribution, and the theory associated with this distribution provides an alternative route to the answers provided above.
* Degrees of freedom (statistics)#Alternative

Statistics

statisticalstatistical analysisstatistician
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.