Linear mixed model

McLean, R.A. and Sanders, W.L. Approximating degrees of freedom for standard errors in mixed linear models. In Proceedings of the Statistical Computing Section. American Statistical Association, New Orleans, 1988, pp. 50-59. 

Results for the various binary mixed surfactant systems are shown in figures 1-7. Here, experimental results for the surface tension at the cmc (points) for the mixtures are compared with calculated results from the nonideal mixed monolayer model (solid line) and results for the ideal model (dashed line). Calculations of the surface tension are based on equation 17 with unit activity coefficients for the ideal case and activity coefficients determined using the net interaction 3 (from the mixed micelle model) and (equations 12 and 13) in the nonideal case. In these calculations the area per mole at the surface for each pure component, tOj, is obtained directly from the slope of the linear region in experimental surface tension data below the cmc (via equation 5) and the maximum surface pressure, from the linear best fit of... 

In a linear well-mixed reactor model the flushing rate is kv = 0.5 h-1, the total reaction rate constant of a specific chemical, ot =1.5 h-1. What is the retention factor of the reactor for the considered chemical, that is, what percentage of the chemical is reacting in the reactor How does this percentage change when the input of the chemical is doubled ... 

Figure 3. Relationship between leaf area (A), epidermal cell density (B), stomatal density (C) and stomatal index (D) versus altitude for Nothofagus solandri leaves growing on the slope of Mt. Ruapehu, New Zealand (collected in 1999). Black diamonds indicate the mean of ten counting fields on each leaf, white squares are the averages of five to eight leaves per elevation, with error bars of 1 S.E.M. Nested mixed-model ANOVA with a general linear model indicates significant differences for all factors (p = 0.000). Averages per elevation were used for regression analysis A. y = -0.0212 + 73.1 R2 = 0.276 p = 0.147. B. y = 1.70 + 3122 R2 = 0.505 p = 0.048. C. y = 0.164 + 360 R2 = 0.709 p = 0.009. D. linear (dashed) y = 0.004 + 9.33 R2 = 0.540 p = 0.038 non-linear (solid) y = 0.00001 2 - 0.0206 + 21.132 R2 = 0.770.

Table 1. Stomatal density (SD), epidermal cell density (ED) and stomatal index (SI) of sun and shade leaves of Nothofagus solandri var. cliffortioides. Sun and shade leaves were collected at three localities (Fig. 2) Horrible Bog (HOR), Kawatiri Junction (KJ) and St. Arnaud (SA). Values are means of five leaves per light level (seven counts per leaf). The complete data set (total) was analyzed with a nested mixed-model ANOVA based on a general linear model, for comparisons within the individual localities a fully nested ANOVA was used.

Table 2. Stomatal density (SD), Epidermal cell density (ED) and stomatal index (SI) of modern Quercus kelloggii leaves, assigned to light regime during growth by degree of undulation, and p-values from a pairwise comparison using a nested mixed-model ANOVA based on a general linear model.

A population PK evaluation of patients from the safety and efficacy trials can be used to assess the impact of renal function on the disposition of a drug. Special care must be taken that patients with severe renal impairment are adequately represented in the population. The population PK approach assess the impact of various covariates on the disposition of a drug. Non linear mixed effects modeling may be used to model the relationship between various covariates and pharmacokinetic parameters. CLcr as a measure of renal function may be one of the covariates. This type of approach has it advantageous as it involves assessment of the effect of renal impairment on the PK in the target population. 

Hayashi W, Kinoshita H, Yukawa E, Higuchi S. Pharmacokinetic analysis of subcutaneous erythropoietin administration with non-linear mixed effect model including endogenous production. Br J Clin Pharmacol 1998 46 11-9. 

The non-linear mixed effects model is the most widely used method and has proven to be very useful for continuous measures of drug effect, categorical response data, and survival-type data. The nonlinear mixed-effects modeling software (NONMEM) introduced by Sheiner and Beal is one of the most commonly used programs for population analysis. A detailed review of software for performing population PK/PD analysis is available. ... 

The first attempt at estimating interindividual pharmacokinetic variability without neglecting the difficulties (data imbalance, sparse data, subject-specific dosing history, etc.) associated with data from patients undergoing drug therapy was made by Sheiner et al. " using the Non-linear Mixed-effects Model Approach. The vector 9 of population characteristics is composed of all quantities of the first two moments of the distribution of the parameters the mean values (fixed effects), and the elements of the variance-covariance matrix that characterize random effects.f " " ... 

Most of the non-linear mixed-effects modeling methods estimate the parameters by means of ML. The probability of the data under the model is written as a function of the model parameters, and parameter estimates are chosen to maximize this probability. This amounts to asserting that the best parameter estimates are those that render the observed data more probable than they would be under any other set of parameters. 

First-Order (NONMEM) Method. The first nonlinear mixed-effects modeling program introduced for the analysis of large pharmacokinetic data was NONMEM, developed by Beal and Sheiner. In the NONMEM program, linearization of the model in the random effects is effected by using the first-order Taylor series expansion with respect to the random effect variables r], and Cy. This software is the only program in which this type of linearization is used. The jth measurement in the ith subject of the population can be obtained from a variant of Eq. (5) as follows ... 

Steimer, J.L. Mallet, A. Gohnard, J.L. Boisvieux, J.F. Alternative approaches to the estimation of population pharmacokinetic parameters comparison with the non-linear mixed effects model. Drug. Metab. Rev. 1984, 15 (1-2), 265-292. 

Mentre, F. Geomeni, R. A two-step iterative algorithm for estimation in non-linear mixed-effect models with an evaluation in population pharmacokinetics. J. Biopharm. Stat. 1995, 5, 141-158. 

Davidian, M. Gallant, A.R. The non-linear mixed effects model with a smooth random effects density. In Institute of Statistics Mimeo Series No. 2206 North Carolina State University Raleigh, NC, 1992. 

Fattinger, K.E. Sheiner, L.B. Verotta, D. A new method to explore the distribution of interindividual random effects in non-linear mixed effects models. Biometrics 1995, 51, 1236-1251. 

Retout, S. Mentre, E. Bruno, R. Eisher information matrix for non-linear mixed-effects models evaluation... 

Bremicker, J. F., Papalambros, P. Y., and Loh, H. T. Solution of Mixed-Discrete Structural Optimization with a New Sequential Linearization Model, Comput. Struct. 37,451-461 (1990). 

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)... 

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