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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", [11] i.e. the sum of leverage scores. One way to help to conceptualize this is to consider a simple smoothing matrix like a Gaussian blur, used to mitigate data noise. In ...
For the statistic t, with ν degrees of freedom, A(t | ν) is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t ≥ 0). It can be easily calculated from the cumulative distribution function F ν (t) of the t distribution:
Illustration of the Kolmogorov–Smirnov statistic. The red line is a model CDF, the blue line is an empirical CDF, and the black arrow is the KS statistic.. In statistics, the Kolmogorov–Smirnov test (also K–S test or KS test) is a nonparametric test of the equality of continuous (or discontinuous, see Section 2.2), one-dimensional probability distributions.
The chi-square distribution has (k − c) degrees of freedom, where k is the number of non-empty bins and c is the number of estimated parameters (including location and scale parameters and shape parameters) for the distribution plus one. For example, for a 3-parameter Weibull distribution, c = 4.
The hypothesis that a data set in a regression analysis follows the simpler of two proposed linear models that are nested within each other. Multiple-comparison testing is conducted using needed data in already completed F-test, if F-test leads to rejection of null hypothesis and the factor under study has an impact on the dependent variable. [1]
Suppose that we take a sample of size n from each of k populations with the same normal distribution N(μ, σ 2) and suppose that ¯ is the smallest of these sample means and ¯ is the largest of these sample means, and suppose S 2 is the pooled sample variance from these samples. Then the following random variable has a Studentized range ...
has the Studentized range distribution for n groups and ν degrees of freedom. In applications, the x i are typically the means of samples each of size m, s 2 is the pooled variance, and the degrees of freedom are ν = n(m − 1). The critical value of q is based on three factors: α (the probability of rejecting a true null hypothesis)
Informally, a statistical model can be thought of as a statistical assumption (or set of statistical assumptions) with a certain property: that the assumption allows us to calculate the probability of any event. As an example, consider a pair of ordinary six-sided dice. We will study two different statistical assumptions about the dice.