f(x) = {[(+)/2]}/[(/2) *(/2)]*(/)/2 *

x(/2)-1 * {1+[(/)*x]}-(+)/2

0 x <
= 1, 2, ...,    = 1, 2, ...

where
,   are the degrees of freedom
        (gamma) is the Gamma function.

The animation above shows various tail areas (p-values) for an F distribution with both degrees of freedom equal to 10.

FACT. FACT is a classification tree program developed by Loh and Vanichestakul (1988) that is a precursor of the QUEST program. For discussion of the differences of FACT from other classification tree programs, see A Brief Comparison of Classification Tree Programs.

Factor Analysis. The main applications of factor analytic techniques are: (1) to reduce the number of variables and (2) to detect structure in the relationships between variables, that is to classify variables. Therefore, factor analysis is applied as a data reduction or (exploratory) structure detection method (the term factor analysis was first introduced by Thurstone, 1931).

For example, suppose we want to measure people's satisfaction with their lives. We design a satisfaction questionnaire with various items; among other things we ask our subjects how satisfied they are with their hobbies (item 1) and how intensely they are pursuing a hobby (item 2). Most likely, the responses to the two items are highly correlated with each other. Given a high correlation between the two items, we can conclude that they are quite redundant.

One can summarize the correlation between two variables in a scatterplot. A regression line can then be fitted that represents the "best" summary of the linear relationship between the variables. If we could define a variable that would approximate the regression line in such a plot, then that variable would capture most of the "essence" of the two items. Subjects' single scores on that new factor, represented by the regression line, could then be used in future data analyses to represent that essence of the two items. In a sense we have reduced the two variables to one factor.

Factor Analysis is an exploratory method; for information in Confirmatory Factor Analysis, see the Structural Equation Modeling chapter.

For more information on Factor Analysis, see the Factor Analysis chapter.

Feature Selection. One of preliminary stages in the process of data mining applicable when the data set includes more variables than could be included (or would be efficient to include) in the actual model building phase (or even in initial exploratory operations).

Fisher LSD. This post hoc test (or multiple comparison test) can be used to determine the significant differences between group means in an analysis of variance setting. The Fisher LSD test is considered to be one of the least conservative post hoc tests (for a detailed discussion of different post hoc tests, see Winer, Michels, & Brown (1991). For more details, see the General Linear Models chapter. See also, Post Hoc Comparisons. For a discussion of statistical significance, see Elementary Concepts.

Fixed Effects (in ANOVA). The term fixed effects in the context of analysis of variance is used to denote factors in an ANOVA design with levels that are deliberately arranged by the experimenter, rather than randomly sampled from an infinite population of possible levels (those factors are called random effects). For example, if one were interested in conducting an experiment to test the hypothesis that higher temperature leads to increased aggression, one would probably expose subjects to moderate or high temperatures and then measure subsequent aggression. Temperature would be a fixed effect in this experiment, because the levels of temperature of interest to the experimenter were deliberately set, or fixed, by the experimenter.

A simple criterion for deciding whether or not an effect in an experiment is random or fixed is to ask how one would select (or arrange) the levels for the respective factor in a replication of the study. For example, if one wanted to replicate the study described in this example, one would choose the same levels of temperature from the population of levels of temperature. Thus, the factor "temperature" in this study would be a fixed factor. If instead, one's interest is in how much of the variation of aggressiveness is due to temperature, one would probably expose subjects to a random sample of temperatures from the population of levels of different temperatures. Levels of temperature in the replication study would likely be different from the levels of temperature in the first study, thus temperature would be considered a random effect.

See also, Analysis of Variance and Variance Components and Mixed Model ANOVA/ANCOVA.

Free Parameter. A numerical value in a structural model (see Structural Equation Modeling) that is part of the model, but is not fixed at any particular value by the model hypothesis. Free parameters are estimated by the program using iterative methods. Free parameters are indicated in the PATH1 language with integers placed between dashes on an arrow or a wire. For example, the following paths both have the free parameter 14.

(F1)-14->[X1]

(e1)-14-(e1)

If two different coefficients have the same free parameter number, as in the above example, then both will of necessity be assigned the same numerical value. Simple equality constraints on numerical coefficients are thus imposed by assigning them the same free parameter number.

Frequency Tables (One-way Tables). Frequency or one-way tables represent the simplest method for analyzing categorical (nominal) data (see also Elementary Concepts). They are often used as one of the exploratory procedures to review how different categories of values are distributed in the sample. For example, in a survey of spectator interest in different sports, we could summarize the respondents' interest in watching football in a frequency table as follows:

STATISTICA BASIC STATS	FOOTBALL: "Watching football"
Category	Count	Cumulatv Count	Percent	Cumulatv Percent
ALWAYS : Always interested USUALLY : Usually interested SOMETIMS: Sometimes interested NEVER : Never interested Missing	39 16 26 19 0	39 55 81 100 100	39.00000 16.00000 26.00000 19.00000 0.00000	39.0000 55.0000 81.0000 100.0000 100.0000

The table above shows the number, proportion, and cumulative proportion of respondents who characterized their interest in watching football as either (1) Always interested, (2) Usually interested, (3) Sometimes interested, or (4) Never interested.

For more information, see the Frequency Tables section of the Basic Statistics chapter.

Function Minimization Algorithms. Algorithms used (e.g., in Nonlinear Estimation) to guide the search for the minimum of a function. For example, in the process of nonlinear estimation, the currently specified loss function is being minimized.

g2 Inverse. A g2 inverse is a generalized inverse of a rectangular matrix of values A that satisfies both

AA`A=A

and

A`AA`=A

The g2 inverse is used to find a solution to the normal equations in the general linear model; refer to the General Linear Models chapter for additional details.

See also matrix singularity, matrix inverse.

Gain Chart. See Lift chart.

Gamma Distribution. The Gamma distribution (the term first used by Weatherburn, 1946) is defined as:

f(x) = (x/b)^c-1 * e^(-x/b) * [1/b (c)]
0 x, b > 0, c > 0

where
  (gamma) is the Gamma function
b     is the scale parameter
a     is the so-called shape parameter
e     is the base of the natural logarithm, sometimes called Euler's e (2.71...)

The animation above shows the gamma distribution as the shape parameter changes from 1 to 6.

Gaussian Distribution. The normal distribution - a bell-shaped function.

Gauss-Newton Method. The Gauss-Newton method is a class of methods for solving nonlinear least-squares problems. In general, this method makes use of the Jacobian matrix J of first-order derivatives of a function F to find the vector of parameter values x that minimizes the residual sums of squares (sum of squared deviations of predicted values from observed values). An improved and efficient version of the method is the so-called Levenberg-Marquardt algorithm. For a detailed discussion of this class of methods, see Dennis & Schnabel (1983).

General ANOVA/MANOVA. The purpose of analysis of variance (ANOVA) is to test for significant differences between means by comparing (i.e., analyzing) variances. More specifically, by partitioning the total variation into different sources (associated with the different effects in the design), we are able to compare the variance due to the between-groups (or treatments) variability with that due to the within-group (treatment) variability. Under the null hypothesis (that there are no mean differences between groups or treatments in the population), the variance estimated from the within-group (treatment) variability should be about the same as the variance estimated from between-groups (treatments) variability.

For more information, see the ANOVA/MANOVA chapter.

General Linear Model. The general linear model is a generalization of the linear regression model, such that effects can be tested (1) for categorical predictor variables, as well as for effects for continuous predictor variables and (2) in designs with multiple dependent variables as well as in designs with a single dependent variable.

For an overview of the general linear model see the General Linear Models chapter.

Generalization (in Neural Networks). The ability of a neural network to make accurate predictions when faced with data not drawn from the original training set (but drawn from the same source as the training set).

Generalized Additive Models. Generalized Additive Models are generalizations of generalized linear models. In generalized linear models, the transformed dependent variable values are predicted from (is linked to) a linear combination of predictor variables; the transformation is referred to as the link function; also, different distributions can be assumed for the dependent variable values. An example of a generalized linear model is the Logit Regression model, where the dependent variable is assumed to be binomial, and the link function is the logit transformation. In generalized additive models, the linear function of the predictor values is replaced by an unspecified (non-parametric) function, obtained by applying a scatterplot smoother to the scatterplot of partial residuals (for the transformed dependent variable values).

See also, Hastie and Tibshirani, 1990, or Schimek, 2000.

Generalized Inverse. A generalized inverse (denoted by a superscript of -) of a rectangular matrix of values A is any matrix that satisfies

A^-AA=A

A generalized inverse of a nonsingular matrix is unique and is called the regular matrix inverse.

See also matrix singularity, matrix inverse.

Generalized Linear Model. The generalized linear model is a generalization of the linear regression model such that (1) nonlinear, as well as linear, effects can be tested (2) for categorical predictor variables, as well as for continuous predictor variables, using (3) any dependent variable whose distribution follows several special members of the exponential family of distributions (e.g., gamma, Possion, binomial, etc.), as well as for any normally-distributed dependent variable.

For an overview of the generalized linear model see the Generalized Linear Models chapter.

Generalized Regression Neural Network (GRNN). A type of neural network using kernel-based approximation to perform regression. One of the so-called Bayesian networks (Speckt, 1991; Patterson, 1996; Bishop, 1995).

Genetic Algorithm. A search algorithm which locates optimal binary strings by processing an initially random population of strings using artificial mutation, crossover and selection operators, in an analogy with the process of natural selection (Goldberg, 1989).

See also, Neural Networks.

Genetic Algorithm Input Selection. Application of a genetic algorithm to determine an "optimal" set of input variables, by constructing binary masks which indicate which inputs to retain and which to discard (Goldberg, 1989). This method is implemented in STATISTICA Neural Networks and can be used as part of a model building process where variables identified as the most "relevant" (in STATISTICA Neural Networks) are then used in a traditional model building stage of the analysis (e.g., using a linear regression or nonlinear estimation method).

Geometric Distribution. The geometric distribution (the term first used by Feller, 1950) is defined as:

f(x) = p*(1-p)^x

where
p     is the probability that a particular event (e.g., success) will occur

Geometric Mean. The Geometric Mean is a "summary" statistic useful when the measurement scale is not linear; it is computed as:

G = (x₁*x₂*...*x_n)^1/n

where
n     is the sample size.

Gibbs Sampler. The Gibbs sampler is a popular method used for MCMC (Markov chain Monte Carlo) analyses. It provides an elegant way for sampling from the joint distributions of multiple variables, by applying the notion that: to sample from a joint distribution just sample repeatedly from its one-dimensional conditionals given whatever you've seen at the time.

For example, the values from the joint distribution of two random variables, X and Y, can be easily simulated by the Gibbs sampler that uses their conditional distributions rather than their joint distribution. Starting with an arbitrary choice of X and Y, X is simulated from the conditional distribution of X, given Y, and Y is simulated from conditional distribution of Y, given X. Alternating between two conditional distributions, in the subsequent steps, generates a sample from the correct joint distribution of X and Y; the approximation gets better and better as the length of the Gibbs sampler path increases.

r(x)=Bc^x,    for x £ 0, B > 0, c £ 1

is often used in survival analysis. See Johnson, Kotz, Blakrishnan (1995) for additional details.

Gradient. In Structural Equation Modeling the gradient is the vector of first partial derivatives of the discrepancy function with respect to the parameter values. At a local or global minimum, the discrepancy function should be at the bottom of a "valley," where all first partial derivatives are zero, so the elements of the gradient should all be near to zero when a minimum is obtained.

The elements of the gradient, by themselves, can, on occasion, be somewhat unreliable as indicators of when convergence has occurred, especially when the model fit is not good, and the discrepancy function value itself is quite large. For this reason, the gradient is not employed as a convergence criterion by this program.

Gradient Descent. Optimization techniques for non-linear functions (e.g. the error function of a neural network as the weights are varied) which attempt to move incrementally to successively lower points in search space, in order to locate a minimum.

Gradual Permanent Impact. In Time Series, the gradual permanent impact pattern implies that the increase or decrease due to the intervention is gradual, and that the final permanent impact becomes evident only after some time. This type of intervention can be summarized by the expression:

Impact _t = * Impact _t-1 +
(for all t time of impact, else = 0).

Note that this impact pattern is defined by the two parameters (delta) and (omega). If is near 0 (zero), then the final permanent amount of impact will be evident after only a few more observations; if is close to 1, then the final permanent amount of impact will only be evident after many more observations. As long as the d parameter is greater than 0 and less than 1 (the bounds of system stability), the impact will be gradual and result in an asymptotic change (shift) in the overall mean by the quantity:

Asymptotic change in level = /(1-)

Grouping (or Coding) Variable. A grouping (or coding) variable is used to identify group membership for individual cases in the data file. Typically, the grouping variable is categorical (i.e., contains either discrete values, e.g., 1, 2, 3, ...,

Group	Score 1	Score 2
1 3 2 2	383.5 726.4 843.7 729.9	4568.4 6752.3 5384.7 6216.9

or a few text values, e.g., MALE, FEMALE)

Group	Score 1	Score 2
MALE FEMALE FEMALE MALE	383.5 726.4 843.7 729.9	4568.4 6752.3 5384.7 6216.9

and the values are referred to as codes (they can be integer values or integer values with text value equivalents).

Groupware. Software intended to enable a group of users on a network to collaborate on specific projects. Groupware may provide services for communication (such as e-mail), collaborative document development, analysis, reporting, statistical data analysis, scheduling, or tracking. Documents may include text, images, or any other forms of information (e.g., multimedia). See also Enterprise-Wide Systems.