Fixed-effects analysis

In a later analysis, however, Graversen (2004) calls into question the previously measured effect on private job training. The analysis shows very big variations in the effect for different individuals, and it is probably the strongest cash benefit recipients, who are able to find a job by themselves, that are offered private job training. The analysis was carried out as an advanced regression analysis, in which the employment effect for different types of individuals in different types of activation was compared. This is why the analysis cannot be compared with the above presented fixed-effect analysis. 

Nevertheless, analyses as carried out by statisticians wedded to the type III philosophy show signs of many concessions to a type II approach. For example, it is a common habit to combine small centres for analysis. This, of course, down-weights their influence on the final result, thus producing an answer more like the type II approach. Furthermore, for meta-analyses, nobody uses an analysis which weights the trials equally (Senn, 2000). It is true that a random effects analysis (see below) is sometimes advocated, but where a fixed effects analysis is employed it is essentially a type II analysis which is used. A similar concession is made when fitting baselines and baseline-by-treatment interactions. (The way type II and type III analyses behave where continuous outcomes... 

Power Analysis for ANOVA Designs can be used to calculate sample size for one and two-way factorial designs with fixed effects http //evall.crc.uiuc.e du/ fp o wer. html/... 

In Chapter 4 Aidan Hollis examines three proposals in considerable detail. The first is an Advanced Purchase Commitment by sponsors, who offer an explicit subsidy in advance for innovative products. The subsidy offer includes a fixed-dollar amount per unit as well as a commitment to purchase a specific number of units at that price. The second proposal is that sponsors pay annual rewards based on the therapeutic effectiveness of innovative drugs. The third approach is to offer a patent extension on patented products to pharmaceutical companies if they successfully developed a vaccine for a disease such as HIV/AIDS that is highly prevalent, particularly in some low-income countries. Hollis concludes that the third approach is an extremely inefficient way to reward innovation. By contrast, the second approach could correct the market failure directly by rewarding innovative drugs according to their therapeutic effectiveness, which is measurable by cost-effectiveness analysis, a topic discussed later in greater detail in Chapters 10 and 11. 

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 panel data models estimated in Example 21.5.1, neither the logit nor the probit model provides a framework for applying a Hausman test to determine whether fixed or random effects is preferred. Explain. (Hint Unlike our application in the linear model, the incidental parameters problem persists here.) Look at the two cases. Neither case has an estimator which is consistent in both cases. In both cases, the unconditional fixed effects effects estimator is inconsistent, so the rest of the analysis falls apart. This is the incidental parameters problem at work. Note that the fixed effects estimator is inconsistent because in both models, the estimator of the constant terms is a function of 1/T. Certainly in both cases, if the fixed effects model is appropriate, then the random effects estimator is inconsistent, whereas if the random effects model is appropriate, the maximum likelihood random effects estimator is both consistent and efficient. Thus, in this instance, the random effects satisfies the requirements of the test. In fact, there does exist a consistent estimator for the logit model with fixed effects - see the text. However, this estimator must be based on a restricted sample observations with the sum of the ys equal to zero or T muust be discarded, so the mechanics of the Hausman test are problematic. This does not fall into the template of computations for the Hausman test. 

In addition it is now time to think about the two assumption models, or types of analysis of variance. ANOVA type 1 assumes that all levels of the factors are included in the analysis and are fixed (fixed effect model). Then the analysis is essentially interested in comparing mean values, i.e. to test the significance of an effect. ANOVA type 2 assumes that the included levels of the factors are selected at random from the distribution of levels (random effect model). Here the final aim is to estimate the variance components, i.e. the variance fractions with respect to total variance caused by the samples taken or the measurements made. In that case one is well advised to ensure balanced designs, i.e. equally occupied cells in the above scheme, because only then is the estimation process straightforward. 

Analysis of variance in general serves as a statistical test of the influence of random or systematic factors on measured data (test for random or fixed effects). One wants to test if the feature mean values of two or more classes are different. Classes of objects or clusters of data may be given a priori (supervised learning) or found in the course of a learning process (unsupervised learning see Section 5.3, cluster analysis). In the first case variance analysis is used for class pattern confirmation. 

For the characterization of the selected test area it is necessary to investigate whether there is significant variation of heavy metal levels within this area. Univariate analysis of variance is used analogously to homogeneity characterization of solids [DANZER and MARX, 1979]. Since potential interactions of the effects between rows (horizontal lines) and columns (vertical lines in the raster screen) are unimportant to the problem of local inhomogeneity as a whole, the model with fixed effects is used for the two-way classification with simple filling. The basic equation of the model, the mathematical fundamentals of which are formulated, e.g., in [WEBER, 1986 LOHSE et al., 1986] (see also Sections 2.3 and 3.3.9), is ... 

These results were analyzed using both Analysis of Variance and analysis of covariance with the change In temperature during the run, used as the covarlate. Statistically, this Is a fixed-effect model except for the covariate which Is random. Analyses were also carried out on the Individual samples, but the conclusions and residual mean squares were essentially the same as for the samples combined. 

Meta-analysis of association studies between DAOA and schizophrenia under fixed-effects model... 

A failure mode and effects analysis (also known as failure mode and criticality analysis) examines a high-risk process in advance of an error to detect potential problems. The problems can then be fixed before an error occurs. It is used to discover the potential risk in a product or system. It involves examining a product or system to identify all the ways in which it might fail and allows for a proactive approach to fixing problems before they occur. 

Assume an experiment in which a group of subjects selected to represent a spectrum of severity of some condition (e.g., renal insufficiency) is given a dose of drug, and drug concentrations are measured in blood samples collected at intervals after dosing. The structural kinetic models used when performing a population analysis do not differ at all from those used for analysis of data from an individual patient. One still needs a model for the relationship of concentration to dose and time, and this relationship does not depend on whether the fixed-effect parameter changes... 

NONMEM is a one-stage analysis that simultaneously estimates mean parameters, fixed-effect parameters, interindividual variability, and residual random effects. The fitting routine makes use of the EES method. A global measure of goodness of fit is provided by the objective function value based on the final parameter estimates, which, in the case of NONMEM, is minus twice the log likelihood of the data (1). Any improvement in the model would be reflected by a decrease in the objective function. The purpose of adding independent variables to the model, such as CLqr in Equation 10.7, is usually to explain kinetic differences between individuals. This means that such differences were not explained by the model prior to adding the variable and were part of random interindividual variability. Therefore, inclusion of additional variables in the model is warranted only if it is accompanied by a decrease in the estimates of the intersubject variance and, under certain circumstances, the intrasubject variance. 

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... 

The main statistical issue is the choice between fixed effects and random effects models. Fleiss describes and discusses the two approaches in detail. Peto argues for the former as being assumption-free, as it is based just on the studies being considered at the time of analysis. This assumes that the same true statement effect underlies the apparent effect seen in each trial, study to study variation being due to sampling error. On the other... 

Precision components are defined at three levels reproducibility, intermediate precision, and repeatability. Reproducibility is the variability of the method between laboratories or facilities. However, as a laboratory is not randomly selected from a large population of facilities, laboratory is a fixed effect. Consequently, the assessment of reproducibility is a question of comparing the average results between laboratories. Additionally, the variation observed within laboratory should be compared to ensure that laboratory does not have an effect either on the average result of the method or on the variability of the method. To assess reproducibility, conduct the same set of validation experiments within each laboratory and compare both the accuracy results and the precision results. If the differences are meaningful, analysis of variance (ANOVA) tests can be conducted to determine whether there is a statistically significant laboratory effect on the mean or on the variance of the method. For simplicity, the validation discussed within this chapter will not consider reproducibility and only one laboratory is considered. 

The experimental design selected, as well as the type of factors in the design, dictates the statistical model to be used for data analysis. As mentioned previously, fixed effects influence the mean value of a response, while random effects influence the variance. In this validation, the model has at least one fixed effect of the overall average response and the intermediate precision components are random effects. When a statistical model has both fixed effects and random effects it is called a mixed effects model. 

Accuracy is estimated from the fixed effects components of the model. If the overall mean is the only fixed effect, then accuracy is reported as the estimate of the overall mean accuracy with a 95% confidence interval. As the standard error will be calculated from a variance components estimate including intermediate precision components and repeatability, the degrees of freedom can be calculated using Satterthwaite s approximation (6). The software program SAS has a procedure for mixed model analysis (PROC MIXED) PROC MIXED has an option to use Satterthwaite s degrees of freedom in calculating the confidence interval for the mean accuracy. An example program and output is shown later for the example protocol. 

Note that this is a fixed effect ANOYA, as the instances of the factor are confined to specific values, that is, the method of analysis is being chosen by the analyst. 

AUC(0—oo) and Cmax are presented in Table 6.5. Two subjects did not return to the clinic and did not complete the study. Hence, these subjects had only data from Period 1. Natural-log transformed AUC(0—oo) and Cmax were used as the dependent variables. The analysis of variance consisted of sequence, treatment, and period as fixed effects. Subjects nested within sequence were treated as a random effect using a random intercept model. The model was fit using REML. Table 6.6 presents the results. The 90% Cl for the ratio of treatment means for both AUC(0—oo) and Cmax were entirely contained within the interval 80-125%. Hence, it was concluded that food had no effect on the pharmacokinetics of the drug. 

On the other hand, it is sometimes seen in the literature that the estimation of the random effects are not of interest, but are treated more as nuisance variables in an analysis. In this case, the analyst is more interested in the fixed effects and their estimation. This view of random effects characterization is rather narrow because in order to precisely estimate the fixed effects in a model, the random effects have to be properly accounted for. Too few random effects in a model leads to biased estimates of the fixed effects, whereas too many random effects lead to overly large standard errors (Altham, 1984). 

One of the most basic questions in any mixed effects model analysis is which parameters should be treated as fixed and which are random. As repeatedly mentioned in the chapter on Linear Mixed Effects Models, an overparameterized random effects matrix can lead to inefficient estimation and poor estimates of the standard errors of the fixed effects, whereas too restrictive a random effects matrix may lead to invalid and biased estimation of the mean response profile (Altham, 1984). In a data rich situation where there are enough observations per subject to obtain individual parameter estimates, i.e., each subject can be fit individually using... 

For example, Bonate (2003) in a PopPK analysis of an unnamed drug performed an influence analysis on 40 subjects from a Phase 1 study. Forty (40) new data sets were generated, each one having a different subject removed. The model was refit to each data set and the results were standardized to the original parameter estimates. Figure 7.18 shows the influence of each subject on four of the fixed effect model parameters. Subject 19 appeared to show influence over clearance, intercompartmental clearance, and peripheral volume. Based on this, the subject was removed from the analysis and original model refit the resultant estimates were considered the final parameter estimates. 

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