F-Vogt.qxd 1/21/2005 7:24 PM Page 117 F F (a) Uppercase F, the statistic that is computed when conducting an *analy- sis of variance. See *F distribution, *F ratio. (b) Lowercase italicized f, the usual symbol for *frequency in a table depicting a *frequency distribution. (c) Lowercase f, the symbol for *function, as in Y = f(X). (d) Lowercase f, symbol for *sampling fraction. Face Validity Logical or conceptual validity; so called because it is a form of validity determined by whether, “on the face of it,” a measure appears to make sense. In determining face validity, one often asks expert judges whether the measure seems to them to be valid. See *prima facie evidence. Fact A piece of information believed to be true or to describe something real. Compare *objective. Factor (a) In *analysis of variance, an *independent variable. (b) In *factor analysis, a cluster of related variables that are a distinguishable component of a larger group of variables. See also *latent variable. (c) A number by which another number is multiplied, as in the statement: Real estate values increased by a factor of three, meaning that they tripled. (d) In mathemat- ics, a number that divides exactly into another number. For example, 1, 2, and 4 are factors of 8, since when you divide each of them into 8, the result (quotient) is a whole number. See *factoring. Factor Analysis (FA) Any of several methods of analysis that enable researchers to reduce a large number of *variables to a smaller number of variables, or *factors, or *latent variables. A factor is a set of variables, such as items on a survey, that can be conceptually and statistically related or grouped together. Factor analysis is done by finding patterns among the variations in the values of several variables; a cluster of highly intercorre- lated variables is a factor. *Exploratory factor analysis was the original 117 F-Vogt.qxd 1/21/2005 7:24 PM Page 118 118 Factor Equations type. *Confirmatory factor analysis developed later and is generally considered more theoretically advanced. *Principal components analysis F is sometimes regarded as a form of factor analysis, although the mathemat- ical models on which they are based are different. While each method has strong advocates, the two techniques tend to produce similar results, especially when the number of variables is large. For example, factor analysis is often used in survey research to see if a long series of questions can be grouped into shorter sets of questions, each of which describes an aspect or factor of the phenomena being studied. See *index (definition c). Factor Equations In *factor analysis, equations analogous to *regression equations describing the regression of observed (*manifest) variables on unobserved (*latent) variables. Factor equations have no *intercept, or rather, the intercept is fixed at zero. Factorial Said of a whole number (positive integer) multiplied by each of the whole numbers smaller than itself. It is usually indicated by an excla- mation point. Factorials are used extensively when calculating probabilities because they indicate the number of different ways individuals or other units of analysis can be ordered. See *permutation. For example, 5 factorial, or 5!, means: 5 × 4 × 3 × 2 × 1 = 120. Therefore, 5 individuals can be ordered in 120 different ways. Factorial Experiments or Designs Research designs with two or more *categorical *independent variables (*factors), each studied at two or more *levels. The goal of factorial designs is to determine whether the factors combine to produce *interaction effects; if the treatments do not influence one another, their combined effects can be gotten simply by studying them one at a time and adding the separate effects. See *analysis of variance, *main effect. For example, a study of the effects of a drug administered at three doses or levels (factor 1) on male and female (factor 2) subjects’ psychological states would be a factorial design. There would be an interaction effect between the drug and gender if, say, at some level(s) the drug had a differ- ent effect on men and women. Factorial Plot A graph of two or more *independent variables (factors) in which nonparallel lines for the different factors indicate the presence of *interaction. See *disordinal interaction and *interaction effect for examples. Factorial Table A table showing the influence of two or more *independent variables on a *dependent variable. See *crossbreak for an example. Factoring Breaking a number into its *factors, that is, breaking it into parts which, when multiplied together, equal the number. F-Vogt.qxd 1/21/2005 7:24 PM Page 119 Fallibilism 119 For example, 24 can be factored into 2 × 12, 3 × 8, and 4 × 6. Each of these numbers (2, 3, 4, 6, 8, and 12) is a factor of 24. F Factor Loadings The *correlations between each *variable and each factor in a *factor analysis. They are analogous to *regression (*slope) coeffi- cients. The higher the loading, the closer the association of the item with the group of items that make up the factor. Loadings of less than 0.3 or 0.4 are generally not considered meaningful. See *factor equation, *structural coefficient. Factor Pattern Matrix In *factor analysis, a table showing the *loadings of each variable on each factor. The pattern matrix is used when the *rotation is *oblique. See *factor structure matrix. Factor Rotation Any of several methods in *factor analysis by which the researcher attempts (by the *transformation of *loadings) to relate the cal- culated factors to theoretical entities. This is done differently depending upon whether the factors are believed to be correlated (oblique) or uncor- related (orthogonal). Factor Score The score of an individual on a factor after the factor has been identified by *factor analysis. Factor Structure Matrix In *factor analysis, a table showing the *loadings of each variable on each factor. The structure matrix is used when the *rota- tion is *orthogonal. See *factor pattern matrix. Failure In *probability theory, one of the two ways a *Bernoulli trial can turn out. For example, in flipping a coin, tails might be called *success and heads “failure.” While the designations are mostly arbitrary, the term failure is reserved for occasions when the predicted event does not occur. Fail Safe N In *meta-analysis, the number of studies confirming the *null hypothesis that would be necessary to change the results of a meta-analytic study that found a statistically significant relationship. Fallacy of Composition An error of reasoning made by assuming that some trait or characteristic of an individual must also describe the individual’s group. Compare *ecological fallacy. For example, if you knew one or several white males who were unsym- pathetic to affirmative action, it would be a fallacy of composition to assume that all white males were equally unsympathetic. Fallibilism The philosophical doctrine, developed by C. S. Peirce, to the effect that all knowledge claims, scientific or otherwise, are always falli- ble or open to question. See *falsificationism, *Heisenberg’s uncertainty principle. F-Vogt.qxd 1/21/2005 7:24 PM Page 120 120 False Alarm False Alarm See *signal detection theory. F False Negative Said of a test that wrongly indicates the absence of a condi- tion—for example, the test shows that you don’t have cancer when in fact you do. Originating in medical testing, the term has broader use today. Compare *false positive, *Type II error, *specificity. False Positive Said of a test that wrongly indicates the presence of a condi- tion—for example, the test shows that you do have cancer when in fact you do not. Originating in medical testing, the term has broader use today. Compare *false negative, *Type I error, *sensitivity. Falsifiability Said of theories that can be subject to tests that could prove them to be false. Those who believe in *falsificationism contend that only falsifiable theories are truly scientific. Falsificationism The doctrine, originating with Karl Popper, that we can only refute (“falsify”) theories; we can never confirm them. A good theory is one that we have tried—repeatedly, but unsuccessfully—to disprove or falsify. Compare *null hypothesis. Familywise Error Rate The probability that a *Type I error has been committed in research involving *multiple comparisons. “Family” in this context means group or set of “related” statistical tests. Also called “exper- imentwise error.” For example, if you set your *alpha level at 0.05 and make three com- parisons using the same data, the probability of familywise error is roughly .15 (.05 + .05 + .05 = .15). One way around this problem is to lower the alpha level to, say, .01. But this increases the probability of *Type II error. A better alternative is the *Scheffe test. See *Bonferroni technique, *post hoc comparison. Fan Spread When the scores or values for two groups grow further apart with the passing of time, plotting this on a graph results in a figure resem- bling a fan and is called a fan spread. See *interaction effect. FASEM Factor Analytic Structural Equation Modeling. See *structural equation modeling, *analysis of covariance structures, *LISREL. The pro- liferation of names for identical or highly similar techniques is due to the relative newness of the techniques and, perhaps, to the desire of *software companies to come up with distinctive brand names. F Distribution A *theoretical distribution used to study *population *vari- ances. It is the distribution of the *ratio of two *independent variables, each of which has been divided by its *degrees of freedom. See *F ratio, *chi- squared distribution. The distribution is perhaps most widely used in or closely associated with *analysis of variance, where it is used to assign *p values.