Lecture 8- factor analysis
steps in factor analysis
1.Assess strength of the correlation among the variables/items 2.Extract Factors 3. Factor rotation and interpretation
what is a factor
A factor is a hypothetical variable that is assumed to underlie a group of highly correlated items The greater the loading of variables/items on a factor, the more that factor explains relationships between those variables. An item's 'loading' on a particular factor is the correlation of that item with the factor in question. As the factor loading is a correlation coefficient, it ranges from -1 to 1 Squaring the loading gives the amount of variance on that factor explained by the item in question.
inductive reasoning
A1 is B A2 is B A3 is B -------------- All As are B Generalise from a finite number of observations to a whole class of possible observation Francis Bacon (1561 - 1626)
deductive reasoning
All As are B All Bs are C -------------------- All As are C If the premises are true, then the conclusion necessarily follows
item analysis v factor analysis
Assumption of item analysis that data are unidimensional (all testing same underlying variable ) But this may not be the case Item analysis may miss underlying factors
types of factor analysis
Exploratory FA is used to highlight factors within a set of responses Confirmatory FA is used to test whether a data set fits a pre-existing pattern of factors (this is not available via SPSS) FA is usually used when conducting research that involves the use of questionnaires and psychometric tests
goals of factor analysis
FA is more theoretical To understand the psychological structures and processes underlying the responses. The goal of FA and is to have as clear a solution as possible, i.e. that each item loads highly on one factor and not on the others. Then have to decide what these factors are in the light of past research and theory Having extracted the factors and discovered the loading of each item on the factors, we have to define the factors and give them a name The main goal of EFA is to generate rudimentary explanatory theories in order to explain covariational data patterns. As a preliminary to this goal, it is noted that EFA functions as a data analytic method that contributes to the detection of empirical regularities
why might an item load on 2 factors
If a variable or item loads on two factors, this might be because there is a genuine loading on the two factors, or the item is ambiguous and should not really be there in the first place
Step 1: Assess strength of the correlation among the variables/items- correlation matrix
Inspect the correlation matrix for evidence of a number of correlation co-efficients greater than r = 0.3. and smaller than 0.9 If factorable then numerous pairs are significant. The more above this level the more suitable the analysis will be. If nothing correlated- no factors as factors rely on inter-correlation of items
Step 2: Extract Factors
It involves determining the smallest number of factors that can be used. There are two techniques that can be used to assist in the decision concerning the number of factors to retain: a. Kaiser's (1960) criterion or the Eigenvalue factor rule. Using this rule only factors with an eigenvalue of 1.0 or more are retained. The Eigenvalue of a factor represents the amount of the total variance explained by that factor. b. Cattell's Scree test which involves plotting each of the eigenvalues of the factors and inspecting the plot to find a point at which the shape of the curve changes direction and becomes horizontal.
what is factor analysis
It is a data reduction statistical technique It takes large set of variables and reduces it using a smaller set of factors/components that are largely independent of one another It does this by looking for groups among the inter-correlations of a set of variables Factors/Components are thought to reflect underlying processes that have created the correlations among variables. multivariate technique for identifying whether the correlations between a set of observed variables stem from their relationship to one or more latent variables in the data, each of which takes the form of a linear model. f several variables/items correlate highly, they might measure aspects of a common underlying dimension. These dimensions are called
what do we need for factorability to be assumes
KMO larger than 6 Bartletts test sig
Step 3: Factor rotation and Interpretation
Once the number of factors have been determined, the next step is to try and interpret them. To assist in this process the factors are rotated. It presents a pattern of loadings in a manner that is easier to interpret. want loadings greater then 0.6/0.7 SPSS does not label or interpret the factors for us. It just tell us which variables (or items) cluster together Available methods are varimax, direct oblimin, quartimax, equamax, or promax
how is factor analysis different to other stats
Other stats tend to look at the differences between means or the correlations between variables - here we are using correlations to find out what the variables are
which types of rotations are used
Quite often researchers will run both types of rotation and report that which gives the clearest results. When deciding whether an item makes an important contribution to a factor, there are really no hard and fast rules - it's really a matter of convention. Usually a loading of 0.4 or greater is considered important, whereas one of 0.2 or less can be ignored. Some (e.g. Kline) suggest at a loading of 0.3 and above be considered important.
labelling factors
The labelling of the factors is a qualitative process and should be based on the data as well as relevant theory and research
exploratory factor analysis
There are two stages to factor analysis 1. Extraction - this determines how many factors underlie the data 2. Rotation - this determines what the loading of each item is on each factor that has been extracted in the previous step. There are several types of extraction process, but one of the most commonly used is called principal components analysis. SPSS output gives us 'eigenvalues', which we use to decide how many factors there are.
rotation
There are two types of rotation: 1. orthogonal - this type of rotation assumes that each factor is unique and has no shared associations. This tends to be used when testing a theoretical model that specifies independent factors (e.g. varimax). 2. oblique - this is more often used and determines the relationship of factors to one another rather than assuming independence (e.g. oblimin).
when to use factor analysis
To understand the structure of a set of variables/items To construct a Questionnaire to measure an underlying variable / construct To reduce a data set to a more manageable size while retaining as much of the original information as possible It is very different from other statistical techniques The quality of analysis depends upon the quality of the data
Step 1: Assess strength of the correlation among the variables/items- stats measured generated by spss
Two statistical measures are generated by spss to help assess the factoriability of the data. 1. Bartlett's Test of Sphericity which tests the hypothesis that correlations in the matrix of variables are zero. It should be significant (p<0.05). 2. Kaiser-Meyer-Olkin (KMO) measure of sampling adequacy (Kaiser, 1970, 1974) index which ranges between 0-1; with .6 suggested as the minimum value for a good analysis (usually greater than 0.6 is sig)
kaisers criterion
a method of extraction in factor analysis based on the idea of retaining factors with associated eigenvalues greater than 1. This method appears to be accurate when the number of variables in the analysis is less than 30 and the resulting communalities (after extraction) are all greater than 0.7, or when the sample size exceeds 250 and the average communality is greater than or equal to 0.6
Abductive Reasoning
concluding something is true by testing hypotheses with evidence "abduction consists in studying the facts and devising a theory to explain them" The surprising fact, C, is observed But if A were true, C would be a matter of course Hence, there is reason to suspect that A is true Abductive inference depends on background knowledge Charles Sanders Peirce (1839-1914)
when we inspect the correlation matrix what are we looking for
correlations larger that .3 and less than .9
what do we look for on rotated component matrix
loadings greater than 0.6/0.7
