Modeling random effects model

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

Lee Y, Nelder JA (2001) Hierarchical generahsed linear models A synthesis of generalised linear models, random-effect models and structured dispersions. Biometrika 88 987-1006. 

In discussion of meta-analysis, there is often much attention given to the random effects model versus the fixed effects model. Random effects models assume that the true treatment effects of the individual trials represent a random sample from some population. The random effects model estimates the population mean of the treatment effects and accounts for the variation in the observed effects. It is sometimes stated that the fixed effects model assumes that the individual trial effects are constant. However, this is not a necessary assumption. An alternative view is that the fixed effects model estimates the mean of the true treatment effects of the trials in the meta-analysis. Senn (2000) discussed the analogy with center effects in multicenter trials. In safety, random effects models may be problematic because of the need to estimate between-trial effects with sparse data. Additionally, the random effects model is less statistically powerful than the fixed effects model, albeit the hypotheses are different. In the fixed effects model, the variance estimate should account for trial effect differences either through stratification, conditioning, or modeling of fixed effects. 

Fixed effect model Random effects model Heterogeneity /-squared = 0%, tau-squared = 0,/ = 0.8727... 

P. Lansky, M. Weiss. Modeling heterogeneity of particles and random effects in drug dissolution. Pharm. 

Hoffmann, D., Kringle, R. Two-sided tolerance intervals for balanced and unbalanced random effects models. J. Biopharm. Stat., 15, 2005, 283-293. 

We usually seek to distinguish between two possibilities (a) the null hypothesis—a conjecture that the observed set of results arises simply from the random effects of uncontrolled variables and (b) the alternative hypothesis (or research hypothesis)—a trial idea about how certain factors determine the outcome of an experiment. We often begin by considering theoretical arguments that can help us decide how two rival models yield nonisomorphic (i.e., characteristically different) features that may be observable under a certain set of imposed experimental conditions. In the latter case, the null hypothesis is that the observed differences are again haphazard outcomes of random behavior, and the alternative hypothesis is that the nonisomorphic feature(s) is (are) useful in discriminating between the two models. 

Data used to describe variation are ideally representative of some population of risk assessment interest. Representativeness was a focus of an earlier workshop on selection of distributions (USEPA 1998). The role of problem formulation is emphasized. In case of representativeness issues, some adjustment of the data may be possible, perhaps based on a mechanistic or statistical model. Statistical random-effects models may be useful in situations where the model includes distributions among as well as within populations. However, simple approaches may be adequate, depending on the assessment tier, such as an attempt to characterize quantitatively the consequences of assuming the data to be representative. 

With regard to relevant statistical methodologies, it is possible to dehne 2 situations, which can be termed a meta-analysis context and a shrinkage estimation context. Similar statistical models, in particular random-effects models, may be applicable in both situations. However, the results of such a model will be used somewhat differently. 

Methods of statistical meta-analysis may be useful for combining information across studies. There are 2 principal varieties of meta-analytic estimation (Normand 1995). In a hxed-effects analysis the observed variation among estimates is attributable to the statistical error associated with the individual estimates. An important step is to compute a weighted average of unbiased estimates, where the weight for an estimate is computed by means of its standard error estimate. In a random-effects analysis one allows for additional variation, beyond statistical error, making use of a htted random-effects model. 

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. 

There is a growing literature that addresses the transferability of a study s pooled results to subgroups. Approaches include evaluation of the homogeneity of different centers and countries results use of random effects models to borrow information from the pooled results when deriving center-specific or country-specific estimates direct statistical inference by use of net monetary benefit regression and use of decision analysis. 

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 a Model II ANOVA (random effect model) the result can be decomposed as yij=iu+Aj+eij, where Aj represents a normally distributed variable with mean zero and variance a]j. In this model one is not interested in a specific effect due to a certain level of the factor, but in the general effect of all levels on the variance. That effect is considered to be normally distributed. Since the effects are random it is of no interest to estimate the magnitude of these random effects for any one group, or the differences from group to group. What can be done is to estimate their contribution [Pg.141]

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. 

The algorithm used is attributed to J. B. J. Read. For many manipulations on large matrices it is only practical for use with a fairly large computer. The data are arranged in two matrices by sample i and nuclide j one matrix, V, contains the amount of each nuclide in each sample the other matrix, E, contains the variances of these numbers, as estimated from counting statistics, agreement between replicate analyses, and known analytical errors. It is also possible to add an arbitrary term Fik to each variance to account for random effects between samples not considered in the model this is usually done in terms of an additional fractional error. Zeroes are inserted for missing data in cases in which not all nuclides were measured in every sample. 

In order to estimate the random effects model, we need some additional parameter estimates. The group means are y x... 

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. 

To estimate the variance components for the random effects model, we also computed the group means regression. The sum of squared residuals from the LSDV estimator is 444,288. The sum of squares from the group means regression is 22382.1. The estimate of a,.2 is 444,288/93 = 4777.29. The estimate of a 2 is 22,382.1/2 - (1/20)4777.29 = 10,952.2. The model is then reestimated by FGLS using these estimates ... 

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. 

Unbalanced design for random effects. Suppose that the random effects model of Section 13.4 is to be estimated with a panel in which the groups have different numbers of observations. Let 7j be the number of observations in group i. 

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. 

Suppose that A is an nxn matrix of the form A = (l-pl) + pii, where i is a column of Is and 0 < p < 1. Write out the format of A explicitly for n = 4. Find all of the characteristic roots and vectors of A. (Hint There are only two distinct characteristic roots, which occur with multiplicity 1 and n-1. Every c of a certain type is a characteristic vector of A.) For an application which uses a matrix of this type, see Section 14.5 on the random effects model. 

In such statistically designed experiments one wants to exclude the random effects of a limited number of features by varying them systematically, i.e. by variation of the so-called factors. At the same time the order in which the experiments are performed should be randomized to avoid systematic errors in experimentation. In another basic type of experiment, sequential experiments, the set-up of an experiment depends on the results obtained from previous experiments. For help in deciding which design is preferable, see Section 3.6. In principle, statistical design is one recommendation of how to perform the experiments. The design should always be based on an exact question or on a working hypothesis. These in turn are often based on models. 

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. 

The interoccasion variability (IOV) or intraindividual variability [11] arises when a parameter of the model, e.g. CL, varies within a subject between study occasions. The term occasion can be defined arbitrarily, usually logical intervals for an occasion are chosen, e.g. each dosing interval in multiple dose studies or each treatment period of a cross-over study can be defined as an occasion. To assess the IOV of a specific parameter more than one measurement per individual has to be available per occasion. The IOV can be implemented in the random effect model as described in the following ... 

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