Factor: Definition, Meaning, and Examples
As a verb, “factor” means to account for or include something as part of an analysis or calculation. “Factor” is a versatile term with meanings that vary depending on the context. It is commonly used to describe elements or influences that contribute to outcomes in fields such as mathematics, science, and everyday language. As a direct object, it’s usually accompanied by the verbs soltar, decir, tirar, all meaning spill (to express) in this context. It is highly common in academic, technical, and everyday conversations, particularly when identifying contributors or influences. A “factor” refers to an element, circumstance, or condition that plays a role in producing a particular result.
Arguments contrasting PCA and EFA
So 1, 2, 3, 4, 6 and 12 are all factors of 12And -1, -2, -3, -4, -6 and -12 also, because multiplying negatives makes a positive. The slightly slower hard courts, humid conditions and its slot as the final major in a busy season have been contributing factors to six different champions in the past seven years.
Statistical model
For instance, the parallel analysis may suggest 5 factors while Velicer’s MAP suggests 6, so the researcher may request both 5 and 6-factor solutions and discuss each in terms of their relation to external data and theory. For this reason, Brown (2009) recommends using factor analysis when theoretical ideas about relationships between variables exist, whereas PCA should be used if the goal of the researcher is to explore patterns in their data. Image factoring is based on the correlation matrix of predicted variables rather than actual variables, where each variable is predicted from the others using multiple regression. By placing a prior distribution over the number of latent factors and then applying Bayes’ theorem, Bayesian models can return a probability distribution over the number of latent factors. This has been modeled using the Indian buffet process,23 but can be modeled more simply by placing any discrete prior (e.g. a negative binomial distribution) on the number of components. Alpha factoring is based on maximizing the reliability of factors, assuming variables are randomly sampled from a universe of variables.
Orthogonal methods
Common factor analysis, also called principal factor analysis (PFA) or principal axis factoring (PAF), seeks the fewest factors which can account for the common variance (correlation) of a set of variables. The observable data that go into factor analysis would be 10 scores of each of the 1000 students, a total of 10,000 numbers. The factor loadings and levels of the two kinds of intelligence of each student must be inferred from the data. Since any rotation of a solution is also a solution, this makes interpreting the factors difficult.
Derived terms
If the solution factors are allowed to be correlated (as in ‘oblimin’ rotation, for example), then the corresponding mathematical model uses skew coordinates rather than orthogonal coordinates. Raymond Cattell was a strong advocate of factor analysis and psychometrics and used Thurstone’s multi-factor theory to explain intelligence. The word “factor” originates from the Latin term factor, meaning “a doer or maker.” Its roots lie in the verb facere, which means “to make or do,” reflecting its action-oriented meaning. Factor regression model is a combinatorial model of factor model and regression model; or alternatively, it can be viewed as the hybrid factor model,5 whose factors are partially known.
Rotations can be orthogonal or oblique; oblique rotations allow the factors to correlate.24 This increased flexibility means that more rotations are possible, some of which may be better at achieving a specified goal. However, this can also make the factors more difficult to interpret, as some information is “double-counted” and included multiple times in different components; some factors may even appear to be near-duplicates of each other. Researchers wish to avoid such subjective or arbitrary criteria for factor retention as “it made sense to me”. A number of objective methods have been developed to solve this problem, allowing users to determine an appropriate range of solutions to investigate.7 However these different methods often disagree with one another as to the number of factors that ought to be retained.
- There is no specification of dependent variables, independent variables, or causality.
- Common factor analysis, also called principal factor analysis (PFA) or principal axis factoring (PAF), seeks the fewest factors which can account for the common variance (correlation) of a set of variables.
- The MinRes algorithm is particularly suited to this problem, but is hardly the only iterative means of finding a solution.
- The mean values of the factors must also be constrained to be zero, from which it follows that the mean values of the errors will also be zero.
- By choosing a different basis for the same principal components – that is, choosing different factors to express the same correlation structure – it is possible to create variables that are more easily interpretable.
- However, this can also make the factors more difficult to interpret, as some information is “double-counted” and included multiple times in different components; some factors may even appear to be near-duplicates of each other.
- The output of PCA maximizes the variance accounted for by the first factor first, then the second factor, etc.
- Its versatility makes it an essential word in mathematics, science, and everyday language.
In this particular example, if we do not know beforehand that the two types of intelligence are uncorrelated, then we cannot interpret the two factors as the two different types of intelligence. Even if they are uncorrelated, we cannot tell which factor corresponds to verbal intelligence and which corresponds to mathematical intelligence without an outside argument. Two students assumed to have identical degrees of verbal and mathematical intelligence may have different measured aptitudes in astronomy because individual aptitudes differ from average aptitudes (predicted above) and because of measurement error itself. Such differences make up what is collectively called the “error” — a statistical term that means the amount by which an individual, as measured, differs from what is average for or predicted by his or her levels of intelligence (see errors and residuals in statistics). By choosing a different basis for the same principal components – that is, choosing different factors to express the same correlation structure – it is possible to create variables that are more easily interpretable. The output of PCA maximizes the variance accounted for by the first factor first, then the second factor, etc.
This is equivalent to minimizing the off-diagonal components of the error covariance which, in the model equations have expected values of zero. With the advent of high-speed computers, the minimization problem can be solved iteratively with adequate speed, and the communalities are calculated in the process, rather than being needed beforehand. The MinRes algorithm is particularly suited to this problem, but is hardly the only iterative means of finding a solution. The analysis will isolate the underlying factors that explain the data using a matrix of associations.52 Factor analysis is an interdependence technique.
The degree of correlation between the initial raw score and the final factor score is called a factor loading. A common rationale behind factor analytic methods is that the information gained about the interdependencies between observed variables can be used later to reduce the set of variables in a dataset. Factor analysis is commonly used in psychometrics, personality psychology, biology, marketing, product management, operations research, finance, and machine learning. It may help to deal with data sets where there are large numbers of observed variables that are thought to reflect Factor Definition a smaller number of underlying/latent variables.
Analysis
A disadvantage of this procedure is that most items load on the early factors, while very few items load on later variables. This makes interpreting the factors by reading through a list of questions and loadings difficult, as every question is strongly correlated with the first few components, while very few questions are strongly correlated with the last few components. Large values of the communalities will indicate that the fitting hyperplane is rather accurately reproducing the correlation matrix. The mean values of the factors must also be constrained to be zero, from which it follows that the mean values of the errors will also be zero.