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Random Effects
In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are drawn from a hierarchy of different populations whose differences relate to that hierarchy. A random effects model is a special case of a mixed model. Contrast this to the biostatistics definitions, as biostatisticians use "fixed" and "random" effects to respectively refer to the population-average and subject-specific effects (and where the latter are generally assumed to be unknown, latent variables). Qualitative description Random effect models assist in controlling for unobserved heterogeneity when the heterogeneity is constant over time and not correlated with independent variables. This constant can be removed from longitudinal data through differencing, since taking a first difference will remove any time invariant components of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expected Mean Square
In statistics, expected mean squares (EMS) are the expected values of certain statistics arising in partitions of sums of squares in the analysis of variance (ANOVA). They can be used for ascertaining which statistic should appear in the denominator in an F-test for testing a null hypothesis that a particular effect is absent. Definition When the total corrected sum of squares in an ANOVA is partitioned into several components, each attributed to the effect of a particular predictor variable, each of the sums of squares in that partition is a random variable that has an expected value. That expected value divided by the corresponding number of degrees of freedom is the expected mean square for that predictor variable. Example The following example is from ''Longitudinal Data Analysis'' by Donald Hedeker and Robert D. Gibbons. Donald Hedeker, Robert D. Gibbons. ''Longitudinal Data Analysis.'' Wiley Interscience. 2006. pp. 21–24 Each of ''s'' treatments (one of which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Statistics
Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model to be studied. Populations can be diverse groups of people or objects such as "all people living in a country" or "every atom composing a crystal". Statistics deals with every aspect of data, including the planning of data collection in terms of the design of statistical survey, surveys and experimental design, experiments.Dodge, Y. (2006) ''The Oxford Dictionary of Statistical Terms'', Oxford University Press. When census data cannot be collected, statisticians collect data by developing specific experiment designs and survey sample (statistics), samples. Representative sampling as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grand Average
The grand mean or pooled mean is the average of the means of several subsamples, as long as the subsamples have the same number of data points. For example, consider several lots, each containing several items. The items from each lot are sampled for a measure of some variable and the means of the measurements from each lot are computed. The mean of the measures from each lot constitutes the subsample mean. The mean of these subsample means is then the grand mean. Example Suppose there are three groups of numbers: group A has 2, 6, 7, 11, 4; group B has 4, 6, 8, 14, 8; group C has 8, 7, 4, 1, 5. The mean of group A = (2+6+7+11+4)/5 = 6, The mean of group B = (4+6+8+14+8)/5 = 8, The mean of group C = (8+7+4+1+5)/5 = 5, Therefore, the grand mean of all numbers = (6+8+5)/3 = 6.333. Everitt, B. S. (2006). The Cambridge Dictionary of Statistics (3 ed.). Cambridge University Press. Application Suppose one wishes to determine which states in America have the tallest men. To do s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conditional Variance
In probability theory and statistics, a conditional variance is the variance of a random variable given the value(s) of one or more other variables. Particularly in econometrics, the conditional variance is also known as the scedastic function or skedastic function. Conditional variances are important parts of autoregressive conditional heteroskedasticity (ARCH) models. Definition The conditional variance of a random variable ''Y'' given another random variable ''X'' is :\operatorname(Y, X) = \operatorname\Big(\big(Y - \operatorname(Y\mid X)\big)^\mid X\Big). The conditional variance tells us how much variance is left if we use \operatorname(Y\mid X) to "predict" ''Y''. Here, as usual, \operatorname(Y\mid X) stands for the conditional expectation of ''Y'' given ''X'', which we may recall, is a random variable itself (a function of ''X'', determined up to probability one). As a result, \operatorname(Y, X) itself is a random variable (and is a function of ''X''). Explanation, r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariance Estimation
In statistics, sometimes the covariance matrix of a multivariate random variable is not known but has to be estimated. Estimation of covariance matrices then deals with the question of how to approximate the actual covariance matrix on the basis of a sample from the multivariate distribution. Simple cases, where observations are complete, can be dealt with by using the sample covariance matrix. The sample covariance matrix (SCM) is an unbiased and efficient estimator of the covariance matrix if the space of covariance matrices is viewed as an extrinsic convex cone in R''p''×''p''; however, measured using the intrinsic geometry of positive-definite matrices, the SCM is a biased and inefficient estimator. In addition, if the random variable has a normal distribution, the sample covariance matrix has a Wishart distribution and a slightly differently scaled version of it is the maximum likelihood estimate. Cases involving missing data, heteroscedasticity, or autocorrelated residuals ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
MINQUE
In statistics, the theory of minimum norm quadratic unbiased estimation (MINQUE) was developed by C. R. Rao. Its application was originally to the problem of heteroscedasticity and the estimation of variance components in random effects model In statistics, a random effects model, also called a variance components model, is a statistical model where the model parameters are random variables. It is a kind of hierarchical linear model, which assumes that the data being analysed are dra ...s. The theory involves three stages: :*defining a general class of potential estimators as quadratic functions of the observed data, where the estimators relate to a vector of model parameters; :*specifying certain constraints on the desired properties of the estimators, such as unbiasedness; :*choosing the optimal estimator by minimising a "norm" which measures the size of the covariance matrix of the estimators. {{stats-stub, date=August 2016 References Estimation theory Statistical de ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fixed Effects
In statistics, a fixed effects model is a statistical model in which the model parameters are fixed or non-random quantities. This is in contrast to random effects models and mixed models in which all or some of the model parameters are random variables. In many applications including econometrics and biostatistics a fixed effects model refers to a regression model in which the group means are fixed (non-random) as opposed to a random effects model in which the group means are a random sample from a population. Generally, data can be grouped according to several observed factors. The group means could be modeled as fixed or random effects for each grouping. In a fixed effects model each group mean is a group-specific fixed quantity. In panel data where longitudinal observations exist for the same subject, fixed effects represent the subject-specific means. In panel data analysis the term fixed effects estimator (also known as the within estimator) is used to refer to an estimator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hierarchical Linear Modeling
Multilevel models (also known as hierarchical linear models, linear mixed-effect model, mixed models, nested data models, random coefficient, random-effects models, random parameter models, or split-plot designs) are statistical models of parameters that vary at more than one level. An example could be a model of student performance that contains measures for individual students as well as measures for classrooms within which the students are grouped. These models can be seen as generalizations of linear models (in particular, linear regression), although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available. Multilevel models are particularly appropriate for research designs where data for participants are organized at more than one level (i.e., nested data). The units of analysis are usually individuals (at a lower level) who are nested within contextual/aggregate units (at a higher level ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Small Area Estimation
Small area estimation is any of several statistical techniques involving the estimation of parameters for small sub-populations, generally used when the sub-population of interest is included in a larger survey. The term "small area" in this context generally refers to a small geographical area such as a county. It may also refer to a "small domain", i.e. a particular demographic within an area. If a survey has been carried out for the population as a whole (for example, a nation or statewide survey), the sample size within any particular small area may be too small to generate accurate estimates from the data. To deal with this problem, it may be possible to use additional data (such as census records) that exists for these small areas in order to obtain estimates. One of the more common small area models A model is an informative representation of an object, person or system. The term originally denoted the plans of a building in late 16th-century English, and derived via Fren ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bühlmann Model
In credibility theory, a branch of study in actuarial science, the Bühlmann model is a random effects model (or "variance components model" or hierarchical linear model) used to determine the appropriate premium for a group of insurance contracts. The model is named after Hans Bühlmann who first published a description in 1967. Model description Consider ''i'' risks which generate random losses for which historical data of ''m'' recent claims are available (indexed by ''j''). A premium for the ''i''th risk is to be determined based on the expected value of claims. A linear estimator which minimizes the mean square error is sought. Write * ''X''ij for the ''j''-th claim on the ''i''-th risk (we assume that all claims for ''i''-th risk are independent and identically distributed) * \scriptstyle =\frac\sum_^X_ for the average value. * \Theta_i - the parameter for the distribution of the i-th risk * m(\vartheta)= \operatorname E\left \Theta_i = \vartheta\right /math> * \Pi=\opera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simpler Formula But Positive Bias
Simplicity is the state or quality of being simple. Something easy to understand or explain seems simple, in contrast to something complicated. Alternatively, as Herbert A. Simon suggests, something is simple or complex depending on the way we choose to describe it. In some uses, the label "simplicity" can imply beauty, purity, or clarity. In other cases, the term may suggest a lack of nuance or complexity relative to what is required. The concept of simplicity is related to the field of epistemology and philosophy of science (e.g., in Occam's razor). Religions also reflect on simplicity with concepts such as divine simplicity. In human lifestyles, simplicity can denote freedom from excessive possessions or distractions, such as having a simple living style. Some other information In some contextual uses, "simplicity" can imply beauty, purity, or clarity. In other cases, the term may have negative connotations, as when referring to people as simpletons. In philosophy ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
