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Within statistics, Local independence is the underlying assumption of latent variable models. The observed items are conditionally independent of each other given an individual score on the latent variable(s). This means that the latent variable explains why the observed items are related to one another. This can be explained by the following example.
Local independence can be explained by an example of Lazarsfeld and Henry (1968). Suppose that a sample of 1000 people was asked whether they read journals A and B. Their responses were as follows:
| Read A | Did not read A | Total | |
| Read B | 260 | 140 | 400 |
| Did not read B | 240 | 360 | 600 |
| Total | 500 | 500 | 1000 |
One can easily see that the two variables (reading A and reading B) are strongly related, and thus dependent on each other. Readers of A tend to read B more often (52%) than non-readers of A (28%). If reading A and B were independent, then the formula P(A&B) = P(A)×P(B) would hold. But 260/1000 isn't 400/1000 × 500/1000. Thus, reading A and B are statistically dependent on each other.
If the analysis is extended to also look at the education level of these people, the following tables are found.
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Again, if reading A and B were independent, then P(A&B) = P(A)×P(B) would hold separately for each education level. And, in fact, 240/500 = 300/500×400/500 and 20/500 = 100/500×100/500. Thus if a separation is made between people with high and low education backgrounds, there is no dependence between readership of the two journals. That is, reading A and B are independent once educational level is taken into consideration. The educational level 'explains' the difference in reading of A and B. If educational level is never actually observed or known, it may still appear as a latent variable in the model.
| This article includes a list of references, related reading or external links, but its sources remain unclear because it lacks inline citations .(June 2012) |
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied.
Psychological statistics is application of formulas, theorems, numbers and laws to psychology. Statistical Methods for psychology include development and application statistical theory and methods for modeling psychological data. These methods include psychometrics, Factor analysis, Experimental Designs, and Multivariate Behavioral Research. The article also discusses journals in the same field.
Unsupervised learning is a type of algorithm that learns patterns from untagged data. The hope is that through mimicry, which is an important mode of learning in people, the machine is forced to build a compact internal representation of its world and then generate imaginative content from it. In contrast to supervised learning where data is tagged by an expert, e.g. as a "ball" or "fish", unsupervised methods exhibit self-organization that captures patterns as probability densities or a combination of neural feature preferences. The other levels in the supervision spectrum are reinforcement learning where the machine is given only a numerical performance score as guidance, and semi-supervised learning where a smaller portion of the data is tagged. Two broad methods in Unsupervised Learning are Neural Networks and Probabilistic Methods.
Factor analysis is a statistical method used to describe variability among observed, correlated variables in terms of a potentially lower number of unobserved variables called factors. For example, it is possible that variations in six observed variables mainly reflect the variations in two unobserved (underlying) variables. Factor analysis searches for such joint variations in response to unobserved latent variables. The observed variables are modelled as linear combinations of the potential factors plus "error" terms, hence factor analysis can be thought of as a special case of errors-in-variables models.
Readability is the ease with which a reader can understand a written text. In natural language, the readability of text depends on its content and its presentation. Researchers have used various factors to measure readability, such as:
Social research is a research conducted by social scientists following a systematic plan. Social research methodologies can be classified as quantitative and qualitative.
Analysis of covariance (ANCOVA) is a general linear model which blends ANOVA and regression. ANCOVA evaluates whether the means of a dependent variable (DV) are equal across levels of a categorical independent variable (IV) often called a treatment, while statistically controlling for the effects of other continuous variables that are not of primary interest, known as covariates (CV) or nuisance variables. Mathematically, ANCOVA decomposes the variance in the DV into variance explained by the CV(s), variance explained by the categorical IV, and residual variance. Intuitively, ANCOVA can be thought of as 'adjusting' the DV by the group means of the CV(s).
In psychometrics, item response theory (IRT) is a paradigm for the design, analysis, and scoring of tests, questionnaires, and similar instruments measuring abilities, attitudes, or other variables. It is a theory of testing based on the relationship between individuals' performances on a test item and the test takers' levels of performance on an overall measure of the ability that item was designed to measure. Several different statistical models are used to represent both item and test taker characteristics. Unlike simpler alternatives for creating scales and evaluating questionnaire responses, it does not assume that each item is equally difficult. This distinguishes IRT from, for instance, Likert scaling, in which "All items are assumed to be replications of each other or in other words items are considered to be parallel instruments" (p. 197). By contrast, item response theory treats the difficulty of each item as information to be incorporated in scaling items.
Dependent and Independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand that they depend, by some law or rule, on the values of other variables. Independent variables, in turn, are not seen as depending on any other variable in the scope of the experiment in question. In this sense, some common independent variables are time, space, density, mass, fluid flow rate, and previous values of some observed value of interest to predict future values.
In statistics, Gibbs sampling or a Gibbs sampler is a Markov chain Monte Carlo (MCMC) algorithm for obtaining a sequence of observations which are approximated from a specified multivariate probability distribution, when direct sampling is difficult. This sequence can be used to approximate the joint distribution ; to approximate the marginal distribution of one of the variables, or some subset of the variables ; or to compute an integral. Typically, some of the variables correspond to observations whose values are known, and hence do not need to be sampled.
In probability theory, conditional independence describes situations wherein an observation is irrelevant or redundant when evaluating the certainty of a hypothesis. Conditional independence is usually formulated in terms of conditional probability, as a special case where the probability of the hypothesis given the uninformative observation is equal to the probability without. If is the hypothesis, and and are observations, conditional independence can be stated as an equality:
This glossary of statistics and probability is a list of definitions of terms and concepts used in the mathematical sciences of statistics and probability, their sub-disciplines, and related fields. For additional related terms, see Glossary of mathematics.
The Rasch model, named after Georg Rasch, is a psychometric model for analyzing categorical data, such as answers to questions on a reading assessment or questionnaire responses, as a function of the trade-off between (a) the respondent's abilities, attitudes, or personality traits and (b) the item difficulty. For example, they may be used to estimate a student's reading ability or the extremity of a person's attitude to capital punishment from responses on a questionnaire. In addition to psychometrics and educational research, the Rasch model and its extensions are used in other areas, including the health profession, agriculture, and market research because of their general applicability.
In statistics, latent variables are variables that are not directly observed but are rather inferred from other variables that are observed. Mathematical models that aim to explain observed variables in terms of latent variables are called latent variable models. Latent variable models are used in many disciplines, including psychology, demography, economics, engineering, medicine, physics, machine learning/artificial intelligence, bioinformatics, chemometrics, natural language processing, econometrics, management and the social sciences.
In statistics, a latent class model (LCM) relates a set of observed multivariate variables to a set of latent variables. It is a type of latent variable model. It is called a latent class model because the latent variable is discrete. A class is characterized by a pattern of conditional probabilities that indicate the chance that variables take on certain values.
A latent variable model is a statistical model that relates a set of observable variables to a set of latent variables.
Multilevel models 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, although they can also extend to non-linear models. These models became much more popular after sufficient computing power and software became available.
In statistics, missing data, or missing values, occur when no data value is stored for the variable in an observation. Missing data are a common occurrence and can have a significant effect on the conclusions that can be drawn from the data.
Differential item functioning (DIF) is a statistical characteristic of an item that shows the extent to which the item might be measuring different abilities for members of separate subgroups. Average item scores for subgroups having the same overall score on the test are compared to determine whether the item is measuring in essentially the same way for all subgroups. The presence of DIF requires review and judgment, and it does not necessarily indicate the presence of bias. DIF analysis provides an indication of unexpected behavior of items on a test. An item does not display DIF if people from different groups have a different probability to give a certain response; it displays DIF if and only if people from different groups with the same underlying true ability have a different probability of giving a certain response. Common procedures for assessing DIF are Mantel-Haenszel, item response theory (IRT) based methods, and logistic regression.
In statistics, linear regression is a linear approach for modelling the relationship between a scalar response and one or more explanatory variables. The case of one explanatory variable is called simple linear regression; for more than one, the process is called multiple linear regression. This term is distinct from multivariate linear regression, where multiple correlated dependent variables are predicted, rather than a single scalar variable.