Log-linear models

In the linear model, concentration-effect relationships are described by the following equation  

Although the linear model will predict no effect when drug concentrations are zero, it cannot predict a maximum effect. Therefore, for many effects, this model is only applicable over a narrow concentration range. At low drugs concentrations ( ECso), the slope will approach the value of Emax/ECso- 

FIGURE 18.16 Theophylline pharmacodynamics in patients with asthma. Effect, which was measured as improvement in forced expiratory volume in 1 second (FEVI), is related to the serum drug level in six patients, who were studied after placebo and tliree incremental doses of theophylline. An Emax model is fit to the concentration-effect data. Based on this analysis, a therapeutic range of 10-20 pg/mL was proposed (shaded area). (Adapted from data published by Mitenko PA, Ogilvie RI. N Engl J Med 1973 289 600-3.) 

Friedman HS, Kokkinakis DM, Pluda J, Friedman AH, Cokgor I, Haglund MM et al. Phase I trial of 

O -beitzylguanine for patients undergoing surgery for malignant glioma. J Clin Oncol 1998 16 3570-5. 

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L. A. Goodman, Some useful extensions of the usual correspondence analysis approach and the usual log-linear models approach in the analysis of contingency tables. Int. Statistical Rev., 54 (1986) 243-309. 

The log-linear model (LLM) is closely related to correspondence factor analysis (CFA). Both methods pursue the same objective, i.e. the analysis of the association (or correspondence) between the rows and columns of a contingency table. In CFA this can be obtained by means of double-closure of the data in LLM this is achieved by means of double-centring of the logarithmic data. 

Fig. 32.11. Log-linear model (LLM) biplot computed from the data in Table 32.10. Conventions are the same as in Fig. 32.10. The areas of circles (representing years) and of squares (representing categories) are made proportional to the row- and column-totals in Table 32.10.

Advanced Log-Linear Models Using SAS0 by Daniel Zelterman... 

The estimates found for Zellner and Revankar s model were. 254 and. 882, respectively, so these are quite similar. For the simple log-linear model, the corresponding values are. 2790 and. 927. 

For the model in Exercise 3, test the hypothesis that X = 0 using a Wald test, a likelihood ratio test, and a Lagrange multiplier test. Note, the restricted model is the Cobb-Douglas, log-linear model. 

Finally, to compute the Lagrange Multiplier statistic, we regress the residuals from the log-linear regression on a constant, InK, lnZ, and ( i 2)(I yK + AjirZ) where the coefficients are those from the log-linear model (.27898 and. 92731). The R1 in this regression is. 23001, so the Lagrange multiplier statistic is LM = nR2 = 25(.23001) = 5.7503. All three statistics suggest the same conclusion, the hypothesis should be rejected. 

Commonly used descriptor variables for QSARs involving redox reactions include substituent constants (o), ionization potential, electron affinity, energy of the highest occupied molecular orbital (EHOMO)or lowest unoccupied molecular orbital (ELUMO), one-electron reduction or oxidation potential (E1), and half-wave potential (E1/2)- One descriptor variable (D), fit to a log-linear model, is usually sufficient to describe a redox property of P. Such a QSAR will have the form... 

Where Eo denotes the effect without medication (= baseline), m is the slope and C the drug (active metabolite) concentration. A more frequently used model is the log-linear model where a linear relationship between the effect and the logarithm of the concentration is assumed ... 

Pharmacodynamic models mathematically relate a drug s pharmacological effect to its concentration at the effect site. Examples of the types of pharmacodynamic models that have been employed include the fixed-effect model/ maximum-effect models (Emax and sigmoid Emax)/ and linear and log-linear models (11). Unlike pharmacokinetic modelS/ pharmacodynamic models are time independent. However these models can be linked to pharmacokinetic modelS/ as discussed in Chapter 19. 

Log-Linear Model. If the effect is measured over a large concentration range, the relationship between effect and concentration may appear curvilinear. The log-linear model is given in Eq. (3)... 

For non-polar solutes, or those solutes that are more non-polar than the cosolvent of choice, solubilization generally follows the log-linear model of Eq. (18). The degree of solubilization is directly related to the polarity of the solute. From Eq. (20) it has been shown that the solubilization slope, octanol-water partition coefficient (logAio/w)- Fig- 6 shows data for a series of hydrocortisones in propylene glycol. As log Mo/w increases it follows that the solubilization slopes increase. Li and Yalkowsky have also illustrated this relationship for non-polar solutes in ethanol-water... 

The solubility of drugs in aqueous mixed solvents often exhibits a maximum in the curve solubility versus mixed solvent composition. This enhancement in solubility often greatly exceeds the solubilities not only in water, which is quite natural, but also in nonaqueous cosolvents. Such a dependence could not be explained by simple equations like the log-linear model for the solubility in a mixed solvent (Yalkowsky and Roseman, 1981)... 

Finally, it should be noted that the log-linear model holds for mixed cosolvent systems in the following form ... 

A straightforward and reliable approach for selecting cosolvents and predicting their solubilization effects on dmgs that requires little or no experimental data, and thus minimal time and drug, is the log-linear model proposed by Yalkowsky and coworkers." This model describes an exponential increase in a nonpolar drug s solubility with a linear increase in cosolvent concentration. The relationship is described by ... 

Alvarez-Nunez, F.A., Pinsuwan, S., Lerkpulsawad, S. and Yalkowsky, S.H. (1998) Effect of the most common co-solvents upon the extent of solubilization of some drugs log-linear model. AAPS Annual Meeting, San Francisco, CA, USA. Volume 1 (4), Poster 2433. 

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