Fixed effect model

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. 

The fixed-effect pharmacodynamic model is a simple model that relates drug concentration to a pharmacological effect that is either present or is absent/ such as sleep/ or is a defined cutoff for a continuous effect/ such as diastolic blood pressure 90 mm Hg in a patient with hypertension. The specific pharmacological effect is present when the drug concentration is greater than a threshold level required to produce 

Although the maximum-effect pharmacodynamic models are empirically based/ they do incorporate the concept of a maximum effect predicted by the drug-receptor interactions described earlier. The Hill equation/ which takes the same form as the equation describing drug effect as a function of receptor occupancy/ relates a continuous drug effect to the drug concentration at the effect site as shown  

FIGURE 18.15 Sigmoid Emax pharmacodynamic model relating drug effect to the drug concentration at the effect site. The three curves show the effect of the exponential Hill (H) constant n on the slope of the sigmoid curves. 

The Emax model is a simpler form of the sigmoid Emax model, with a slope factor n = 1, so that 

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The PD models fall under two categories graded or quantal of fixed-effect model. Graded refers to a continuous response at different concentrations, whereas the quantal model would evaluate discrete response such as dead or alive, desired or undesired and are almost invariably clinical end points. 

Lag periods Fixed-effect model Random-effect model Hausman test... 

The Hausman test was used to test the null hypothesis that the coefficients estimated by the efficient random-effect model are the same as the ones estimated by the consistent fixed-effect model. If this null hypothesis cannot be rejected (insignificant P-value in general, it is larger than 0.05), then the random-effect model is more appropriate. 

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. 

The subordinate level of a nested ANOVA is always Model II (random effect model). The highest level of classification of a nested ANOVA may be Model I (fixed effect model) or Model II. If it is Model II it is called a pure Model II nested ANOVA. If the highest level is Model I it is called a mixed model nested ANOVA. 

Use the data in Section 13.9.7 (these are the Grunfeld data) to fit the random and fixed effects models. There are five films and 20 years of data for each. Use the F, LM, and/or Hausman statistics to detennine which model, the fixed or random effects model, is preferable for these data. 

A two way fixed effects model Suppose the fixed effects model is modified to include a time specific dummy variable as well as an individual specific variable. Then, yit = a, + y, + P x + At every observation, the individual- and time-specific dummy variables sum to one, so there are some redundant coefficients. The discussion in Section 13.3.3 shows one way to remove the redundancy. Another useful way to do this is to include an overall constant and to drop one of the time specific and one of the time-dummy variables. The model is, thus, ylt = 5 + (a, - aj) + (y, - y,) + P x + e . (Note that the respective time or individual specific variable is zero when t or i equals one.) Ordinary least squares estimates of P can... 

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. 

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. 

Vector of parameter, describing the fixed effect model... 

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

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. 

Ferrlols-Llsart Alos-Aliminana, 1996 [140] 18 Fixed-effects model OR 0.60 (0.40-0.86)... 

Should we use fixed-effect models or random-effect models ... 

In practice, confidence limits for random-effect models will be wider than for fixed-effect models and this is a more realistic representation of the true uncertainties if we are interested in prediction. This may accord with our intuition in the sense that in borderline cases where the result was otherwise significant, a large centre-by-treatment interaction might cause us to doubt that it was genuine. 

The technique is based on the fact that the double contrast of standard minus placebo subtracted from that of experimental minus standard eliminates the effect of the standard. One is thus left with an indirectly established contrast of experimental versus placebo. Assuming a so-called fixed-effects model, its variance is equal to the sum of the variances of the two contrasts used in its construction and it therefore follows that, however large the current trial, its variance cannot be lower that that pertaining to the (historical) comparison of the standard treatment with placebo. 

An almost identical issue with respect to centres was touched upon when discussing multicentre trials. Within drug development, however, the almost uniform practice is to fit fixed-effect models for treatments within centres. (Elsewhere it is not unusual to ignore centres altogether ) Outside of drug development, and in the context of meta-analyses, there has been an extremely vigorous debate between the proponents of fixed-effect and random-effect models for treatment effects within trials. 

The principal disadvantage is that they can be difficult to fit. (Although sometimes it can be impossible to fit a fixed-effects model.) They also run into conceptual difficulties. The meaning and nature of the second-level parameters is not always clear and often... 

Random-effect model. A term which is used in at least two rather different senses by statisticians in the context of drug development. (1) A model for which more than one term is assumed random but the treatment effect is assumed fixed. (All statistical models, including so-called fixed ones have at least one error term which is random.) (2) A model in which the treatment effect itself is assumed to vary randomly from unit to unit. For balanced designs, random-effect models of the first sort can lead to identical inferences to fixed-effect models. Even for balanced designs, random-effect models of the second sort will not. 

Let us consider a single covariate and one qualitative factor in a fixed effects model. The basic model in regression is... 

A fixed effect model, also known as a quantal effect model, relates a certain drug concentration with the statistical likelihood of a predefined, fixed effect to be present... 

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