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The distribution above is sometimes referred to as the tau distribution; [2] it was first derived by Thompson in 1935. [3] When ν = 3, the internally studentized residuals are uniformly distributed between and +. If there is only one residual degree of freedom, the above formula for the distribution of internally studentized residuals doesn't ...
However, the fact that a sample standard deviation is used, rather than the unknown population standard deviation, complicates the mathematics of finding the probability distribution of a Studentized statistic.
In regression analysis, the distinction between errors and residuals is subtle and important, and leads to the concept of studentized residuals. Given an unobservable function that relates the independent variable to the dependent variable – say, a line – the deviations of the dependent variable observations from this function are the ...
The studentized range distribution function arises from re-scaling the sample range R by the sample standard deviation s, since the studentized range is customarily tabulated in units of standard deviations, with the variable q = R ⁄ s. The derivation begins with a perfectly general form of the distribution function of the sample range, which ...
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)
In the same volume Fisher contributed applications of Student's t-distribution to regression analysis. [3] Although introduced by others, Studentized residuals are named in Student's honour because, like the problem that led to Student's t-distribution, the idea of adjusting for estimated standard deviations is central to that concept. [7]
Thus, for low leverage points, DFFITS is expected to be small, whereas as the leverage goes to 1 the distribution of the DFFITS value widens infinitely. For a perfectly balanced experimental design (such as a factorial design or balanced partial factorial design), the leverage for each point is p/n, the number of parameters divided by the ...
Diagram showing the cumulative distribution function for the normal distribution with mean (μ) 0 and variance (σ 2) 1. These numerical values "68%, 95%, 99.7%" come from the cumulative distribution function of the normal distribution. The prediction interval for any standard score z corresponds numerically to (1 − (1 − Φ μ,σ 2 (z)) · 2).
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