Fixed and random effects

The fixed effects model considers the studies that have been combined as the totality of all the studies conducted. An alternative approach considers the collection of studies included in the meta-analysis as a random selection of the studies that have been conducted or a random selection of those that could have been conducted. This results in a slightly changed methodology, termed the random effects model The mathematics for the two models is a little different and the reader is referred to Fleiss (1993), for example, for further details. The net effect, however, of using a random effects model is to produce a slightly more conservative analysis with wider confidence intervals. 

In the remainder of this section we will concentrate on the fixed effects approach, which is probably the more common and appropriate approach, within the pharmaceutical setting. 

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The F and LM statistics are not useful for comparing the fixed and random effects models. The Hausman statistic can be used. The value appears above. Since the Hausman statistic is small (only 3.14 with two degrees of freedom), we conclude that the GLS estimator is consistent. The statistic would be large if the two estimates were significantly different. Since they are not, we conclude that the evidence favors the random effects model. 

In most models developed for pharmacokinetic and pharmacodynamic data it is not possible to obtain a closed form solution of E(yi) and var(y ). The simplest algorithm available in NONMEM, the first-order estimation method (FO), overcomes this by providing an approximate solution through a first-order Taylor series expansion with respect to the random variables r i,Kiq, and Sij, where it is assumed that these random effect parameters are independently multivariately normally distributed with mean zero. During an iterative process the best estimates for the fixed and random effects are estimated. The individual parameters (conditional estimates) are calculated a posteriori based on the fixed effects, the random effects, and the individual observations using the maximum a posteriori Bayesian estimation method implemented as the post hoc option in NONMEM [10]. 

Population pharmacodynamic data, i.e., observed 24-hour efficacy scores were modeled as a function of individual predicted 24-hour steady state AUCs. Various pharmacodynamic models were explored including linear, Emax, and sigmoidal Emax models. Fixed and random-effect parameters were used to describe the PK/PD relationship. The results of the model development are presented in Table 7. 

Prior knowledge and various hypotheses are condensed into models. NONMEM determines the parameter vector including fixed and random effects of each model using the maximum likelihood algorithm. NONMEM uses each model to predict the observed data set and selects the best PPK parameter vector minimising the deviation between model prediction and observed data. Comparing model fits by the criteria discussed in the section Evaluation should decide which hypothesis is the most likely. As a general rule, the model should be as simple as possible and the number of parameters should be at a minimum. 

Population pharmacokinetic analysis provides not only an opportunity to estimate variability, but also to explain it. Variability is usually characterized in terms of fixed and random effects. The fixed effects are the population average values of pharmacokinetic parameters, which may in turn be a function of patient characteristics discussed above. The random effects... 

Common features among the three different classes of models and their implementation within the S-Plus environment come into light during the analysis of the examples in particular, the syntax for defining the fixed and random effects in the models, as well as methods for extracting estimates from fitted objects. All data sets discussed in this chapter are fictitious that is, they are generated by simulation. The reader is encouraged to experiment with the code provided in Appendix 4.1 to explore alternative scenarios. 

Step 4. Estimate the posterior model and obtain fixed and random effects from the model. This is a plausible model from the posterior distribution of the dependent variables obtained in step 3. 

The control stream/skeleton data set pairs used to simulate the first PK example, the weight change example, and the multiple mixture seizure count example can be found in the book s ftp site in the following files C22. TXT/skewdata.txt, C23. TXT/MiXDATAl. TXT, and C24. TXT/MIXDATA2. TXT, respectively. For each of these control stream/data set skeleton pairs the control stream simulates a new data set with identical structure to the skeleton, but with the DV simulated based on the fixed and random effects parameters in the control stream. 

To further delineate a random effect from a fixed effect, suppose a researcher studied the effect of a drug on blood pressure in a group of patients. Ignoring for a moment the specifics of how one measures blood pressure or quantifies the drug effect, if the researcher was interested in only those patients used in the study, then those patients would be considered a fixed effect. If, however, the researcher wanted to make generalizations to the patient population, and the patients that were used in the experiment were a random sample from the patient population, then patients would be considered a random effect. With that distinction now made, any linear model that contains both fixed and random effects is a linear mixed effects model. 

The last step is model reduction. Fixed and random effects that appear to be nonsignificant can now be removed, keeping in mind the boundary issue with the LRT. Once the final model is identified, it is refit using REML to obtain the unbiased estimates of the variance parameters. 

Incorporating Fixed and Random Effects into the Structural Model... 

A true PPC requires sampling from the posterior distribution of the fixed and random effects in the model, which is typically not known. A complete solution then usually requires Markov Chain Monte Carlo simulation, which is not easy to implement. Luckily for the analyst, Yano, Sheiner, and Beal (2001) showed that complete implementation of the algorithm does not appear to be necessary since fixing the values of the model parameters to their final values obtained using maximum likelihood resulted in PPC distributions that were as good as the full-blown Bayesian PPC distributions. In other words, using a predictive check resulted in distributions that were similar to PPC distributions. Unfortunately they also showed that the PPC is very conservative and not very powerful at detecting model misspecification. 

What is the difference between fixed- and random-effect estimators ... 

Finally, it is perhaps useful to draw attention to Bayesian analogues of frequentist fixed-and random-effect models, not least because more and more statisticians are using Bayesian methods to estimate treatment effects. 

Other comments regarding fixed- and random-effect models will be found in Chapter 14 and an excellent discussion of various issues including choice of design will be found in Steimer et al. (1994). 

Since much of the testing being done today, and for the foreseeable future, will involve scoring of a product characteristic, the AOV becomes an essential resource in support of data analysis and interpretation. Since there are many AOV models, one needs to be familiar with those most appropriate for sensory data for example, the AOV mixed model (fixed and random effects) with replication is appropriate. Other features should allow for the ability to test the main effect by interaction when interaction is significant, and so forth. Finally, one needs to be cautious when using software that allows for exclusion of some data but without providing the details of what was excluded. Procrustes analysis is one such system. See Huitson (1989) for more discussion on this topic when applied to sensory data. The problem with any computation that removes some data is an assumption that data are an aberration when it may not. How does one know that this does or does not represent a unique... 

ANOVA can also be used in situations where there is more than one source of random variation. Consider, for example, the purity testing of a barrelful of sodium chloride. Samples are taken from different parts of the barrel chosen at random and replicate analyses performed on these samples, in addition to the random error in the measurement of the purity, there may also be variation in the purity of the samples from different parts of the barrel. Since the samples were chosen at random, this variation will be random and is thus sometimes known as a random-effect factor. Again, ANOVA can be used to separate and estimate the sources of variation. Both types of statistical analysis described above, i.e. where there is one factor, either controlled or random, in addition to the random error in measurement, are known as one-way ANOVA. The arithmetical procedures are similar in the fixed- and random-effect factor cases examples of the former are given in this chapter and of the latter in the next chapter, where sampling is considered in more detail. More complex situations in which there are two or more factors, possibly interacting with each other, are considered in Chapter 7. 

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