Obtaining the P value

Pretty well all statistical packages will include a P value as part of their out-put for the f-test. It was included as the last line in the generic output in Table 6.2, although its meaning was not explained at that stage. 

Whatever package is used, the theophylline clearance data from the previous chapter should produce a P value of 0.001. This can be read as  

If rifampicin actually had no effect on theophylline clearance, there would be only a 0.1 per cent (1 in 1000) chance that our samples would suggest an effect as great as (or greater than) the difference we actually observed. 

As this data had already been declared significant on the basis of the 95 per cent Cl for the size of the treatment effect, we would expect the P value to agree with this. The achieved P is indeed less than 0.05 and therefore significant. 

The data provided significant evidence of a change in clearance (P = 0.001) 

---

Obtain the p-value by adding up the probabilities associated with the calculated value of the test statistic and more extreme values when the null hypothesis is true. This will correspond to adding up the probabilities associated with the observed signal or more extreme (larger) values of that signal. 

To obtain the p-value we now need to add up all of the probabilities associated with values of the test statistic at least as big as the value observed, 7.75 in our... 

The former procedure has been used for correlating the reactivities towards electrophiles of monosubstituted thiophenes. Good cr+ plots have been obtained. The p-values for the different reactions in thiophene and benzene have been tabulated (Table 2) (71AHC(13)235, 72IJS(C)(7)6l). 

Although it is difficult to obtain the p value of NAD", it can be reasonably assumed that this compound is adsorbed on negatively charged electrodes with either its nicotinamide or adenine moiety in... 

In a second regression, the standard deviation of each group of six measurements was regressed and Table 2 was obtained. The p-value and adjusted R squared indicate a significant regressed model that explains 56.0% of the variation present. According to the model, mold temperature significantly affects the standard deviation of... 

To obtain the different values of p, it is only hecessary to produce as many independent equations as there are components in the mixture and, if the mixture has n components, to solve a system of n equations having n unknowns. Individual analysis is now possible for mixtures having a few components but even gasoline has more than 200 It soon becomes unrealistic to have ail the sensitivity coefficients necessary for analysis in this case, 200. ... 

To obtain the G2 value of Eq we add five corrections to the starting energy, [MP4/6-31 lG(d,p)] and then add the zero point energy to obtain the ground-state energy from the energy at the bottom of the potential well. In Pople s notation these additive terms are... 

Good to excellent Hammett plots were obtained using substituent constants (see Figure 2.6). Surprisingly, literature examples of good Hammett correlations of stability constants are rare The p-values are shown in Table 2.7. 

For solvolysis carried out at 25°C, the correlation gives the following values for p -6.15 for series 262 and -6.68 for series 263. The p values obtained for five other heterocycles and for benzene appear to be related... 

Indicates the type of a-value ((t+, a or a ) used in the reaction series from which the p-value needed was obtained. 

These constants, K toK/, may be estimated by use of the Hammett equation. Estimation of 1 and K 4 involves application of the methods outlined in Section II, A, i.e., application of substituent constants for and N+H to the Hammett equation for the acid-base equilibria of benzoic acids. Estimation of A2 and involves application of the method used in Section III,A, i.e., the p-value for the basicity of substituted pyridines, with cr-values for COOH and COO . Provided the necessary a- and p-values are known, this procedure permits the calculation of four independent, or virtually independent, estimates of Krp. A check on the method is available from the relationships shown in Eq. (16) which is readily obtained by multiplication of Eq. (12) and (14) and of Eq. (13) and (15). 

To calculate —APf it is therefore necessary to evaluate e/d and obtain the corresponding value of

head loss due to friction hf as ... 

The same procedure is then adopted for obtaining the optimum value of P,2 and hence ... 

James also carried out a treatment of the lithium molecule-ion with use of 25 orbitals and explicit consideration of the four K electrons, obtaining the energy value —0.304 e. v. at r = 2.98 A. (this being not necessarily the minimum point of the curve). Our curve for an 5 bond similarly gives a very small energy value, — 0.22 e. v., at this value of r. The calculations described above indicate that the principal source of inaccuracy in these treatments is the assumption that the bonds are 5 bonds, and that the consideration of s-p hybridization makes a great improvement. 

The value of p was set at 1.00 for ionization of XC6H4CO2H in water at 25°C. The parameters a and Cp values were then calculated for each group (for a group X, a is different for the meta and para positions). Once a set of a values was obtained, p values could be obtained for other reactions from the rates of just two X-substituted compounds, if the a values of the X groups were known (in practice, at least four well-spaced values are used to calculate p because of experimental error and because the treatment is not exact). With the p value thus calculated and the known ct values for other groups, rates can be predicted for reactions that have not yet been run. 

A sinusoidal plot of grf>2 vs.

crystal plane gives another set of Ks that depend on other combinations of the gy, eventually enough data are obtained to determine the six independent values of gy (g is a symmetric matrix so that gy = gy,). The g-matrix is then diagonalized to obtain the principal values and the transformation matrix, elements of which are the direction cosines of the g-matrix principal axes relative to the crystal axes. An analogous treatment of the effective hyperfine coupling constants leads to the principal values of the A2-matrix and the orientation of its principal axes in the crystal coordinate system. 

For an ideal Nernstian reaction, the peak potentials of the cathodic and anodic sweeps will be the same, and equal to E0. The width of the cathodic (or anodic) wave at half peak height, A 1/2, can be found by replacing / by /p/2 = — [n2/WTe0ty8] in equation (2.39) and so obtaining the two values for 0. Using equation (2.36) then gives a AE1/2 of c. 0.09/nV, at 298 K. In practice, A 1/2 is never as small as this ideal value as a result of adsorbate adsorbate interactions these intermolecular interactions cause a smearing out of the observed redox potential, and a full treatment of this and other complications can be found in the standard texts referred to at the end of chapter 1. 

Because of the absence of any obvious reference value, the p -value of — 3.1 is not readily discussed in terms of charge magnitude or brominebridging at the rate-limiting transition states. For alkene hydration, it is now accepted that the intermediates are carbocations (20). The corresponding structure-reactivity relationship (21) is obtained by using o and [Pg.244]

The question of bridged and/or open intermediates has been considered in Section 4, where the data on kinetic substituent effects were discussed with the help of the multipathway scheme (Scheme 7) to determine the relative importance of bromonium and carbocation paths. It is not straightforward to obtain significant p-values for each of them from the complex pa relationship (34) and (35) corresponding to this scheme for an a,/f-Ri,R2... 

---