My Dear
Factor analysis use with dimension reduction (as in exploratory factor analysis) while factoral analysis use with design of experiments (as in MANOVA).
Dealing the muti-variate analysis for dimension reduction is Factor Analysis. Factorial analysis may be related to Design of Experiments, once dealing with number of factors at different level in analysing or designing the comprative experiment.
My Dear
Factor analysis use with dimension reduction (as in exploratory factor analysis) while factoral analysis use with design of experiments (as in MANOVA).
Both are correct. Factor analysis is used to reduce a number of observed varibles into a few hidden ones. Factorial analysis refers to Two Way ANOVA.
adding to Amer's answer:
not MANOVA - "factorial" it is not used when several response variables are analyzed, but when several (categorical) predictors are used, and that refers to multi-way ANOVA (there is also multi-way MANOVA, though).
adding to Mohammad's answer:
not only for two-way but generally for multi-way ANOVA.
adding my own 2 pence:
in "factor analysis", "factor" is a variable calculated from several other variables. As said that is a tool to reduce the dimension of many original variables in a few "factors". In "factorial analysis", "factors" refers to (often categorical) predictors. The mentioned ANOVAs represent one aspect of analyzing such "(multi-)factorial models" (models with (many) factors as predictor variables). (Multiple-)Regression analysis, ANCOVAs, and the entire class of generalized (non-)linear models are tools of a "factorial analysis". The analysis is typically only then called "factorial" when there is more than one predictor, but technically analyses with only one predictor are "factorial" analyses, too.
Factor analysis is correct: 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 (from Wikipedia). There is no "factorial analysis" per se in statistics. There is (a) Analysis of factorial experiments; (b) factorial-designs analysis; (c) analysis of variance of factorial designs; ...
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