Freedom, degrees of, in statistics

These methods of test require a knowledge of what is known as the number of degrees of freedom. In statistical terms this is the number of independent values necessary to determine the statistical quantity. Thus a sample of n values has n degrees of freedom, whilst the sum (x — x)2 is considered to have n — 1 degrees of freedom, as for any defined value of x only n — 1 values can be freely assigned, the nth being automatically defined from the other values. 

The number of degree of freedom in statistical analysis is the number of independent elements used in the computation of that statistic. 

When experimental data are not available, methods of estimation based on statistical mechanics are employed (7,19). Classical kinetic theory suggests a contribution to CP of S R for each translational degree of freedom in the molecule, a contribution of S R for each axis of rotation, and of R for each vibrational degree of freedom. A cmde estimate of CP for small molecules can be obtained which neglects vibrational degrees of freedom ... 

Analysis and prediction of side-chain conformation have long been predicated on statistical analysis of data from protein structures. Early rotamer libraries [91-93] ignored backbone conformation and instead gave the proportions of side-chain rotamers for each of the 18 amino acids with side-chain dihedral degrees of freedom. In recent years, it has become possible to take account of the effect of the backbone conformation on the distribution of side-chain rotamers [28,94-96]. McGregor et al. [94] and Schrauber et al. [97] produced rotamer libraries based on secondary structure. Dunbrack and Karplus [95] instead examined the variation in rotamer distributions as a function of the backbone dihedrals ( ) and V /, later providing conformational analysis to justify this choice [96]. Dunbrack and Cohen [28] extended the analysis of protein side-chain conformation by using Bayesian statistics to derive the full backbone-dependent rotamer libraries at all... 

This expression has a formal character and has to be complemented with a prescription for its evaluation. A priori, we can vary the values of the fields independently at each point in space and then we deal with uncountably many degrees of freedom in the system, in contrast with the usual statistical thermodynamics as seen above. Another difference with the standard statistical mechanics is that the effective Hamiltonian has to be created from the basic phenomena that we want to investigate. However, a description in terms of fields seems quite natural since the average of fields gives us the actual distributions of particles at the interface, which are precisely the quantities that we want to calculate. In a field-theoretical approach we are closer to the problem under consideration than in the standard approach and then we may expect that a simple Hamiltonian is sufficient to retain the main features of the charged interface. A priori, we have no insurance that it... 

The main difference between the Z-test and the /-test is that the Z-statistic is based on a known standard deviation, a, while the /-statistic uses the sample standard deviation, s, as an estimate of a. With the assumption of normally distributed data, the variance sample variance, v2 as n gets large. It can be shown that the /-test is equivalent to the Z-test for infinite degrees-of-freedom. In practice, a large sample is usually considered n > 30. 

Recall, the standard deviation of the added noise in Y was lxlO-3. It is reached approximately after the removal of 3 sets of eigenvectors (at t=4). Note that, from a strictly statistical point of view, it is not quite appropriate to use Matlab s std function for the determination of the residual standard deviation since it doesn t properly take into account the gradual reduction in the degrees of freedom in the calculation of R. But it is not our intention to go into the depths of statistics here. For more rigorous statistical procedures to determine the number of significant factors, we refer to the relevant chemometrics literature on this topic. 

Phase space theory (PST) has been widely used for estimation of rates and energy partitioning for ion dissociations. It can be considered within the framework of transition-state theory as the limiting case of a loose transition state, in which all product degrees of freedom are statistically fully accessible at the transition state. As such, it is expected to give an upper limit for dissociation rates and to be best suited to barrierless dissociations involving reaction coordinates with simple bond cleavage character. 

It was assumed that the experimental uncertainties followed Gaussian statistics with equal standard deviation a for all points. Then the standard deviation was determined as ct = (x /v), where v is the number of degrees of freedom in the fit. v is equal to the number of the experimental points less the number of parameters used in the minimization. The best fit with the Frumkin prediction has one free parameter (j6) and gives p =- 2.051 and CT = 0.41 mN/m. The standard deviation in surface tension is small, indicating that the fit with the Frumkin prediction is statistically significant. Similar best fits are obtained for the other surfactants of the homologue series of... 

This resembles the square of the standard /-ratio for testing the hypothesis that p — 0. It would be exactly that save for the absence of a degrees of freedom correction in v. However, since we have not estimated p with x in fact, LM is exactly the square of a standard normal variate divided by a chi-squared variate over its degrees of freedom. Thus, in this model, LM is exactly an F statistic with 1 degree of freedom in the numerator and n degrees of freedom in the denominator. 

Statistical manuals usually list degrees of freedom in this column with valuesthat are equal tothe samplesize minus one... 

The development of quantum theory, particularly of quantum mechanics, forced certain changes in statistical mechanics. In the development of the resulting quantum statistics, the phase space is divided into cells of volume hf. where h is the Planck constant and / is the number of degrees of freedom. In considering the permutations of the molecules, it is recognized that the interchange of two identical particles does not lead to a new state. With these two new ideas, one arrives at the Bose-Einstein statistics. These statistics must be further modified for particles, such as electrons, to which the Pauli exclusion principle applies, and the Fermi-Dirac statistics follow. 

The arithmetic value of the F-criterion calculated by formula (2.160), is compared to its tabular value (Table E) for the chosen significance level and the associated degrees of freedom in order to check the statistical significance of the difference between the lack of jit variance and reproducibility variance. When this difference is sta-... 

Analogously to f, Fhas to reach a certain size to attain statistical significance. This size is dictated by the associated degrees of freedom in each instance, and in turn the degrees of freedom are dictated by the total number of subjects participating in the study. To attain significance, F must always be greater than 1, which means... 

Mean centering changes the number of degrees of freedom in a principal component model from k to k + 1. This affects the number of degrees of freedom used in some statistical equations that are described later. 

In these models, the potential energy function is based on the molecular mechanics all-atom force field and includes the bond, angle, dihedral and non-bonded energy terms. The parameterization is based on the statistical analysis of sets of experimental structures. If a variable q describes a degree of freedom in the system (e.g., bond distances, angles, dihedrals) then, P(q), the probability distribution associated with this degree of freedom, is related to the potential of mean force, W(q), by the following equation... 

In statistical analysis it is necessary to use a quantity called degrees of freedom, designated henceforth as d.f. This quantity allows for a mathematical correction of the data for constraints placed upon the data. In this case, in the calculation of the estimated standard deviation, the number of observations n is fixed and the estimated standard deviation is calculated from the mean. Only (n - 1) of the observations or sample terms can be varied and the last term is fixed by X and n. Thus, there are only ( — 1) degrees of freedom in estimating the standard deviation from a sample of the population data. 

The statistical average over the electronic degrees of freedom in Eq. [15] is equivalent, in the Drude model, to integration over the induced dipole moments pg and py. The Hamiltonian H, is quadratic in the induced dipoles, and the trace can be calculated exactly as a functional integral over the fluctuating fields pg and The resulting solute-solvent interaction energy... 

The most common error in application of this method lies in a lack of appreciation of the minimum statistical requirements involved. Thus, one needs to have about five well-chosen compounds for every variable term in a Hansch analysis in order to feel confident about the results. For example, an equation such as Equation 2 above should be derived from 10 or more compounds, and one such as Equation 3, from 15 or more examples. A smaller number of examples per term may lead to useful results, but one cannot often support these results by statistics. A frequent abuse is seen when a large number of variable terms are used in a complex equation (four or more terms) which was derived from only 10 or 12 examples. The statistician would prefer to have 15 to 20 more compounds than the degrees of freedom in the resulting equation not often is this luxury met. 

This is especially true because there will be an inherent direct improvement in the fit in any case, on account of additional variables being added in the simulation. It is therefore useful to measure the fit quality using a reduced chi-squared statistic (f ) that weights the quality of the fit relative to the number of degrees of freedom in the data, as outlined in equation (7). ... 

---