Error degrees of freedom

These two quantities are often referred to as the error sum of squares and the error degrees of freedom, respectively. The former divided by the latter is a statistical value that is the best estimate of the variance, a2, common to all k populations ... 

Reilly (1970) gave an improved criterion that the next event be designed to maximize the expectation. R, of information gain rather than the upper bound D. His expression for R with a known is included in GREG-PLUS and extended to unknown a by including a posterior probability density p((j s,t e) based on a variance estimate with Oe error degrees of freedom. The extended R function thus obtained is an expectation over the posterior distributions of y and cr. 

An integer list of the experimental error degrees of freedom assigned to the elements of VAR for each response block b. 

Fitting errors, degrees of freedom, number of parameters and model complexity... 

In total we have 16x4=64 degrees of freedom. If we remain with the basic form of model (21.8) then we could, for example, simply treat the data as 16 replicated measurements on each of four doses and lit a common intercept a and common slope jS. This is model A of Table 21.1 and has two model degrees of freedom, leaving 62 error degrees of freedom. Because we have made no attempt to incorporate the information that measurements are made repeatedly on the same subjects, all of the variation between subjects must be contained in these 62 degrees of freedom. Effectively we have the sort of analysis which would be appropriate for a trial in which 64 subjects were split at random into four different doses. If we make a minimal attempt to recognize that four data values are provided by every subject and fit a model with a different intercept for each patient but a common slope, we then have model B. In model C, a quadratic parameter, y, is fitted as well as a cubic parameter, 8, so that the model is of the form... 

This sum, when divided by the number of data points minus the number of degrees of freedom, approximates the overall variance of errors. It is a measure of the overall fit of the equation to the data. Thus, two different models with the same number of adjustable parameters yield different values for this variance when fit to the same data with the same estimated standard errors in the measured variables. Similarly, the same model, fit to different sets of data, yields different values for the overall variance. The differences in these variances are the basis for many standard statistical tests for model and data comparison. Such statistical tests are discussed in detail by Crow et al. (1960) and Brownlee (1965). 

However, in many applications the essential space cannot be reduced to only one degree of freedom, and the statistics of the force fluctuation or of the spatial distribution may appear to be too poor to allow for an accurate determination of a multidimensional potential of mean force. An example is the potential of mean force between two ions in aqueous solution the momentaneous forces are two orders of magnitude larger than their average which means that an error of 1% in the average requires a simulation length of 10 times the correlation time of the fluctuating force. This is in practice prohibitive. The errors do not result from incorrect force fields, but they are of a statistical nature even an exact force field would not suffice. 

The errors in the present stochastic path formalism reflect short time information rather than long time information. Short time data are easier to extract from atomically detailed simulations. We set the second moment of the errors in the trajectory - [Pg.274]

The distribution of the /-statistic (x — /ji)s is symmetrical about zero and is a function of the degrees of freedom. Limits assigned to the distance on either side of /x are called confidence limits. The percentage probability that /x lies within this interval is called the confidence level. The level of significance or error probability (100 — confidence level or 100 — a) is the percent probability that /X will lie outside the confidence interval, and represents the chances of being incorrect in stating that /X lies within the confidence interval. Values of t are in Table 2.27 for any desired degrees of freedom and various confidence levels. 

A second way to work with the data in Table 14.7 is to treat the results for each analyst separately. Because the repeatability for any analyst is influenced by indeterminate errors, the variance, s, of the data in each column provides an estimate of O rand- A better estimate is obtained by pooling the individual variances. The result, which is called the within-sample variance (s ), is calculated by summing the squares of the differences between the replicates for each sample and that sample s mean, and dividing by the degrees of freedom. 

Let us take as an example some of the reaction series listed in Table IX, e.g. the oxidation of the 2-methylmercaptobenzothiazoles. The calculations are summarized in Table X, which is self-explanatory. In these calculations the deviations from regression were used as measure of error, but, when duplicate determinations are available, additional degrees of freedom for replication are obtainable, and should be used as measure of error. 

The two-sided confidence intervals for the coefficients b and b, w hen and are random variables having t distributions with (n - 2) degrees of freedom and error variances of... 

Variance The mean square of deviations, or errors, of a set of observations the sum of square deviations, or errors, of individual observations with respect to their arithmetic mean divided by the number of observations less one (degree of freedom) the square of the standard deviation, or standard error. 

In the graphs of log k, versus log k2, independent errors of the same magnitude in both directions can be anticipated. Hence, they can be represented by the usual circles. That is, if this radius equals the standard error 6 (or 26), it means that the actual value is situated inside with a probability of. 393 (or. 865) according to the distribution with two degrees of freedom (204). The circles in Figures 4 and 7 correspond to an error of 5% in k. 

Table 1.3. Critical Student s t-Factors for the One- and Two-Sided Cases for Three Values of the Error Probability p and 7 Degrees of Freedom f...

Assignments A-D K F T P scratchpad variables index degrees of freedom Student s t probability of error... 

Using the Student s t factors and the number of degrees of freedom, calculate the probabilities p that correlations are due to chance alone (error probabilities) these are interpreted as follows ... 

Statistical testing of model adequacy and significance of parameter estimates is a very important part of kinetic modelling. Only those models with a positive evaluation in statistical analysis should be applied in reactor scale-up. The statistical analysis presented below is restricted to linear regression and normal or Gaussian distribution of experimental errors. If the experimental error has a zero mean, constant variance and is independently distributed, its variance can be evaluated by dividing SSres by the number of degrees of freedom, i.e. 

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