The p value

One shortcoming of the hypothesis testing approach is the arbitrary choice of a value for a. Depending upon our risk tolerance for committing a type 1 error, the conventional value of 0.05 may not be acceptable. Another way to convey the extremeness of the resulting test statistic is to report a p value. 

A p value is the probability that the result obtained or one more extreme (in favor of the alternate) would be observed by chance alone. We know from the definition of the critical region that a value of the test statistic t —2.26 or t 2.26 would have occurred with probability 

The p values can be estimated from a table of values from the appropriate t distribution (for example, by finding the tail areas associated with a particular value of the test statistic). More commonly, however, statistical software is used for all statistical analyses and p values are included in the results. The following is a helpful way to interpret p values  

It is not uncommon for results of hypothesis tests to be represented simply by the p value. However, it is not a wise practice to rely solely on them. Recall that increasing the sample size reduces the standard error, which increases the size of the test statistic and therefore reduces the p value. This serves as a reminder that it is not just the statistical significance of the result (that is, the p value) that counts. The clinical relevance of the size of the effect (for example, the 

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The fact that good correlations are observed with d" rather than with a, is indicative of a strong infiuence of the substituent through a direct resonance interaction with a positive charge in the reacting system. The p-values are positive, which is expected for substituted dienophiles in a normal electron demand Diels-Alder reaction. Furthermore, the p-values do not exceed unity and are not significantly different from literature values reported for the uncatalysed reaction. It is tempting to... 

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. 

As anticipated, the complexation is characterised by negative p-values, indicating that the binding process is favoured by electron donating substituents. The order of the p-values for complexation of the different Lewis-acids again follows the Irving-Williams series. 

The effect of substituents on the rate of the reaction catalysed by different metal ions has also been studied Correlation with resulted in perfectly linear Hammett plots. Now the p-values for the four Lewis-acids are of comparable magnitude and do not follow the Irving-Williams order. Note tlrat the substituents have opposing effects on complexation, which is favoured by electron donating substituents, and reactivity, which is increased by electron withdrawirg substituents. The effect on the reactivity is clearly more pronounced than the effect on the complexation equilibrium. 

Streitwieser pointed out that the eorrelation whieh exists between relative rates of reaetion in deuterodeprotonation, nitration, and ehlorination, and equilibrium eonstants for protonation in hydrofluorie aeid amongst polynuelear hydroearbons (ef. 6.2.3) constitutes a relationship of the Hammett type. The standard reaetion is here the protonation equilibrium (for whieh p is unity by definition). For eon-venience he seleeted the i-position of naphthalene, rather than a position in benzene as the referenee position (for whieh o is zero by definition), and by this means was able to evaluate /) -values for the substitutions mentioned, and cr -values for positions in a number of hydroearbons. The p -values (for protonation equilibria, i for deuterodeprotonation, 0-47 for nitration, 0-26 and for ehlorination, 0-64) are taken to indieate how elosely the transition states of these reaetions resemble a cr-eomplex. 

The applicability of the two-parameter equation and the constants devised by Brown to electrophilic aromatic substitutions was tested by plotting values of the partial rate factors for a reaction against the appropriate substituent constants. It was maintained that such comparisons yielded satisfactory linear correlations for the results of many electrophilic substitutions, the slopes of the correlations giving the values of the reaction constants. If the existence of linear free energy relationships in electrophilic aromatic substitutions were not in dispute, the above procedure would suffice, and the precision of the correlation would measure the usefulness of the p+cr+ equation. However, a point at issue was whether the effect of a substituent could be represented by a constant, or whether its nature depended on the specific reaction. To investigate the effect of a particular substituent in different reactions, the values for the various reactions of the logarithms of the partial rate factors for the substituent were plotted against the p+ values of the reactions. This procedure should show more readily whether the effect of a substituent depends on the reaction, in which case deviations from a hnear relationship would occur. It was concluded that any variation in substituent effects was random, and not a function of electron demand by the electrophile. ... 

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... 

The reactions were run with methyl iodide in DMSO at 25 °C. Logfor pyridine-2-carbonitrile is also equal to 0. In order to combine two sets of results, the author assumes that the p value for each reaction is the same. 

The numerical values of the terms a and p are defined by specifying the ionization of benzoic acids as the standard reaction to which the reaction constant p = 1 is assigned. The substituent constant, a, can then be determined for a series of substituent groups by measurement of the acid dissociation constant of the substituted benzoic acids. The a values so defined are used in the correlation of other reaction series, and the p values of the reactions are thus determined. The relationship between Eqs. (4.12) and (4.14) is evident when the Hammett equation is expressed in terms of fiee energy. For the standard reaction, o%K/Kq = ap. Thus,... 

Example 4.3. The p value for alkaline saponification of methyl esters of substituted benzoic acids is 2.38, and the rate constant for saponification of methyl benzoate under the conditions of interest is 2 x 10 s . Calculate the rate constant for the hydrolysis... 

Another example of enhanced sensitivity to substituent effects in the gas phase can be seen in a comparison of the gas-phase basicity for a series of substituted acetophenones and methyl benzoates. It was foimd that scnsitivtiy of the free energy to substituent changes was about four times that in solution, as measured by the comparison of A( for each substituent. The gas-phase data for both series were correlated by the Yukawa-Tsuno equation. For both series, the p value was about 12. However, the parameter r" ", which reflects the contribution of extra resonance effects, was greater in the acetophenone series than in the methyl benzoate series. This can be attributed to the substantial resonance stabilization provided by the methoxy group in the esters, which diminishes the extent of conjugation with the substituents. 

Match the p values with the appropriate reactions. Explain your reasoning. Reaction constants -h2.45, -hO.75, —2.39, —7.29... 

The acidity of various substituted acetophenones has been measured in DMSO. Would you expect the p value for a Hammett correlation to be positive or negative Would you expect the best correlation with a, or [Pg.442]

This would disperse the positive charge over several atoms and diminish the sensitivity of the reaction to substituent effects. The p values that are observed are consistent with this interpretation. Whereas p is —3.25 for acetal hydrolysis, it is only —1.9 for hemiacetal hydrolysis. ... 

The extraordinanly high polaiity of the fluonnated alcohols reflects their strong hydrogen-bonding capability [54] The P values of 10 2 for CF3CH2OH and 11 08 for (CF3)2CHOH compare with 10 6 and 12 1 for 50% formic acid and water, respectively [39]... 

FIGURE 4.17 A plot of chemical shifts versus pH for the carbons of lysine. Changes in chemical shift are most pronounced for atoms near the titrating groups. Note the correspondence between the p. values and the particular chemical shift changes. All chemical shifts are defined relative to tetramethylsilane (TMS). (From Suprcnant, H., ct at., 1980. 

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

Interestingly enough, the p-value is very similar to that applicable to the acid-base equilibria of phenols, which is a reaction in which both reactant and product are isoelectronic with the corresponding species in the reaction under consideration. 

Using these assumptions and conventions, Imoto and co-workers have correlated a number of series of reactions of thiophenes and furans. The reactions studied are the acid-base equilibria pK values) and the acid catalyzed methylations (thiophenes only) of thiophene-and furan-carboxylic acids and the alkaline hydrolyses of their ethyl esters the side-chain bromination of the a-acetylthiophenes, and the a-mercuration of thiophenes and the polarographic half-wave potentials of the methyl esters of thiophene- and furan-carboxylic acids and of nitrothiophenes. The pK values were determined and the ester hydrolyses studied for all three substitution orientations in the thiophene series. For the 4-R-2-Y and 5-R-2-Y series, the p-values do not appear significantly different and the data could probably be combined into a single series unfortunately, however, no limits of accuracy are reported for the p-values, and some of the raw data are not readily available so recalculation is not easily possible. For the 5-R-3-Y series the p-values deviate considerably from the other values however, whereas they are higher for the pK values, they are lower for the ester hydrolyses, and it is possible that the differences are neither systematic nor significant. 

The authors have compared in every case the p-values for the heterocyclic series (ph) with the corresponding p-value for the benzene series (Pb) under identical conditions. The ratios ph/pb f e thiophene series vary between 0.63 and 1.34, with the two extreme values applying to the 5-R-3-Y systems. For the other systems the variation is between 0.83 and 1.20. with an average value of 0.99. The three furan-system values, all for 5-R-2-Y, are more divergent, between 0.89 and 1.39, with an average of 1.15, apparently slightly higher, than in the thiophene system. 

Imoto and co-workers have also studied the pA values of substituted thiazolecarboxylic acids and the alkaline hydrolysis of their ethyl esters, each in three relative positions (2-B-4-Y, 2-B-5-Y, and 5-II-2-Y). In the case of the pA values, the p-values are far from constant, varying from 0.83 to 2,35, This variation is likely to be due to the intervention of tautomeric equilibria and of hydrogen bonds. The /3-ratios for the three sets of ester hydrolyses are roughly constant (0,61-0.73), and, assuming that the introduction of two heteroatoms leads to cumulative (multiplicative) effects on the transmission, this result is of the same order of magnitude as the product of the and values discussed above, i.e. 1.0 and 0.6, respectively. The lowest value for the pA (0,83) for the 2-R-5-COOH series is also of the same order of magnitude. All the available reaction constants are summarized in Table VI. 

The first reaction series to be considered are the basicities of the various quinolines. Baciocchi and Illuminati have demonstrated that the pK values of quinolines substituted in the B-ring follow the Hammett equation well if ApK, i.e., the difference between the pK values of substituted and unsubstituted compounds, is plotted against a, the quinoline points fall on the same line as the pyridine points, as shown in Fig. 5, so that the p-values for the two series are identical. 

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). 

As a first approximation, within a given family of nucleophilic reagents, such as amines, basicity changes are mainly responsible for differences in nucleophilic power. The p values of some of the more familiar amines together with the rate constants for some of their reactions with chloroheteroaromatic compounds are shown... 

Thus, a methyl group placed at the site of hydration decreases the proportion of the hydrated species and, hence, shifts both the ultraviolet spectra (cf. Fig. 2A and B) and the ionization constant of the substance towards normality. A valuable means for locating the site of hydration, therefore, is to introduce a methyl group in various likely places until the anomalous spectrum is lost and the spectrum of the predominantly anhydrous species restored. The effect of such a methyl group on the pjfiT value is also revealing because a decrease in the amount of the hydrated species causes a decrease in the p value,... 

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