2 Factor Analysis A data reduction technique designed to represent a wide range of attributes on a smaller number of dimensions. Aim is to identify groups of variables which are relatively homogeneous. Groups of related variables are called factors.
3 Purposes 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.
4 Conceptual Model for a Factor Analysis with a Simple Model Factor 1 Factor 2 Factor 3 e.g., 12 items testing might actually tap only 3 underlying factors
5 Conceptual Model for Factor Analysis (with cross-loadings)
6 Common Factor Model It is suggested that X 1, X 2, and X 3 are functions of two underlying factors, F 1 and F 2. It is assumed that each X variables are linearly related to the two factors as shown in the following model. X 1 = β 11 F 1 + β 12 F 2 + e 1 X 2 = β 21 F 1 + β 22 F 2 + e 2 X 3 = β 31 F 1 + β 32 F 2 + e 3 The error terms e 1, e 2, and e 3, serve to indicate that the hypothesized relationships are not exact. In the vocabulary of factor analysis, the parameters β ij are referred to asfactorloadings. For example, β 12 is the factor loading of variable X 1 on factor F 2.
7 Expected Structure of Loadings It is expected that the loadings have roughly the structure as shown in the table. Loading (F1) β i1 Loading (F2) β i2 X1 + 0 X2 0 + X3 0 + Of course, the zeros in the preceding table are not expected to be exactly equal to zero. By `0' we mean approximately equal to zero and by `+' a positive number substantially different from zero.
8 Model Assumptions A1: The error terms e i are independent of one another and E(e i ) = 0 and Var(e i ) = σ i2, A2:The unobservable factors are independent of one another. It is also assumed that the factors and error terms are independent. As for the factor means and variances, the assumption is that the factors are standardized: E(F j ) = 0 and Var(F j ) = 1. It is an assumption made for convenience; since the factors are unobservable, we might as well think of them as measured in standardized form.
9 Implications of Assumptions The variance of X i from the model can be expressed as Var(X i ) = Var(F 1 ) + Var(F 2 )+ Var(e i ) = + + We see that the variance of X i consists of two parts: ( + ) and. The first part is called communality of the variable. It is the part of Var(X i ) explained by the common factors F 1 and F 2. The second part is called specific variance of the variable. It is the part of Var(X i ) unable to explain by the common factors. The covariance of any two observable variables, X i and X j, from the model can be expressed as Cov(X i, X j ) = β i1 β j1 Var(F 1 )+ β i2 β j2 Var(F 2 ) = β i1 β j1 +β i2 β j2
10 History of Factor Analysis? Invented by Spearman (1904) Usage hampered by onerousness of hand calculation Since the advent of computers, usage has thrived, esp. to develop: Theory e.g., determining the structure of personality Practice e.g., development of 10,000s+ of psychological screening and measurement tests
11 Assumption Testing Factorability It is important to check the factorability of the correlation matrix (i.e., how suitable is the data for factor analysis?) Check correlation matrix for correlations Check the anti-image matrix for diagonals Check measures of sampling adequacy (MSAs) Bartlett s KMO
12 Rule of thumb: Measures of Sampling Adequacy Are there several correlations over.3? Are the diagonals of anti-image matrix >.5? Is Bartlett s test significant? Is KMO >.5?
13 Assumption Testing Factorability (Correlation and partial correlation) Medium effort, reasonably accurate Examine the diagonals on the anti-image correlation matrix to assess the sampling adequacy of each variable Variables with diagonal anti-image correlations of less that.5 should be excluded from the analysis they lack sufficient correlation with other variables
14 Assumption Testing Factorability (Bartlett s and KMO measure) Sampling Adequacy predicts whether the data you have collected are likely to "factor well" based on correlation and partial correlation and this is measured by the Kaiser- Meyer-Olkin (KMO) statistic Quickest method, but least reliable Global diagnostic indicators - correlation matrix is factorable if: Bartlett s test of sphericity is significant and/or (Null: no correlation among the variables(unit R matrix) Kaiser-Mayer Olkin (KMO) measure of sampling adequacy >.5
16 Communalities The proportion of variance in each variable which can be explained by the factors Also called the explained variation due to factor. Communalities range between 0 and 1 High communalities (>.5) show that the factors extracted explain most of the variance in the variables being analysed. Low communalities (<.5) mean there is considerable variance unexplained by the factors extracted
17 Eigen Values EV = sum of squared correlations for each factor EV = overall strength of relationship between a factor and the variables Successive EVs have lower values Eigen values over 1 are stable
18 Explained Variance A good factor solution is one that explains the most variance with the fewest factors Realistically happy with 50-75% of the variance explained
19 Example: interpreting the communality Variable (1) Variance (2) Loadings of F 1 (3) Loadings of F 2 (4) Communality (5) % explained (6) = 100 (5)/(2) Finance 1, Marketing Policy Overall The loadings on F 1 are relatively large for marketing and policy but close to zero for finance. On the contrary, the loadings on F 2 are relatively large for finance but relatively low for marketing and policy. This solution supports the expectation. F 1 could be interpreted as verbal ability, and F 2 as quantitative ability.
20 Assessment of the First Solution based on R The communalities show that the factor model explains nearly 100%, 99.5%, and 99.5% respectively of the observed variance of finance, marketing and policy grades. Overall, the two factors explain 99.65% of the sum of all observed variances. The sum of squared loadings on F 1 can be interpreted as the contribution of F 1, and that on F 2 as the contribution of F 2 in explaining the sum of the observed variances. In our example F 1 explains about /3 or 66%, and F 2 about 33.7% of the sum of the observed variances. Theoretically, the sum of squared loadings, , is the largest eigenvalue of R and the loadings on F1 constitute the corresponding eigenvector. the sum of squared loadings, , is the second largest eigenvalue of R and the loadings on F2 constitute the corresponding eigenvector.
22 How Many Factors? A subjective process. Seek to explain maximum variance using fewest factors, considering: 1. Theory what is predicted/expected? 2. Eigen Values > 1? (Kaiser s criterion) 3. Scree Plot where does it drop off? 4. Factors must be able to be meaningfully interpreted & make theoretical sense?
23 Cattell & Jaspers (1967) suggest that the number of factors be taken as the number of eigenvalues immediately before the straight line begins.
24 Scree Plot A bar graph of Eigen Values Depicts the amount of variance explained by each factor. Look for point where additional factors fail to add appreciably to the cumulative explained variance. 1st factor explains the most variance Last factor explains the least amount of variance
25 Factor Rotation Factor loadings are not unique. There exist an infinite sets of factor loadings yielding the same theoretical dispersion matrix. The process of obtaining a new set of loadings with some specific objective is called factor rotation. Orthogonal (Varimax) Oblimin
26 Factor loading stages In practice, FA can be carried out in two stages. In the first stage, one set of loadings is estimated. These loadings may not agree with the prior expectations, or may not lend themselves to a reasonable interpretation. In the second stage, the first set of factor loadings are "rotated" in an effort to arrive at another set that are more consistent with prior expectations or more easily interpretable. variables with cross-loading shall be omitted from the further analysis.
27 How do I eliminate items? A subjective process, but consider: Size of main loading (min=.5) Size of cross loadings (max=.3?) Eliminate 1 variable at a time, then re-run, before deciding which/if any items to eliminate next Number of items already in the factor More items in a factor -> greater reliability Minimum = 3 Maximum = unlimited
28 Factor Analysis: an example suppose that an automobile company asked a large number of questions about different vehicles. Consider how the different items (features) might be more parsimoniously represented by just a few constructs (factors). - Ideally, interval data (e.g., a rating on a k- point scale), regarding the perceptions of consumers are required regarding a number of features
32 Cumulative percent of variance explained. We are looking for an eigenvalue above 1.0.
35 Expensive Exciting Luxury Distinctive Not Conservative Not Family Not Basic Appeals to Others Attractive Looking Trend Setting Reliable Latest Features Trust
36 What shall these components be called? Expensive Exciting Luxury Distinctive Not Conservative Not Family Not Basic Appeals to Others Attractive Looking Trend Setting Reliable Latest Features Trust
37 EXCLUSIVE TRENDY RELIABLE Expensive Exciting Luxury Distinctive Not Conservative Not Family Not Basic Appeals to Others Attractive Looking Trend Setting Reliable Latest Features Trust
43 Practical session : using SPSS Step 1: Open the data file, for example, Example.SAV Step 2: Click on sequentially: Analyze Data Reduction Factor. Step 3: Move the three variables X1, X2 & X3 - from Source to Variable box
44 Step 4: Click on Descriptives. Activate Coefficients Significance levels KMO and Bartlett s test of sphericity Anti-image Click on Continue. This will produce correlation matrix and significance of correlations, sampling adequacy and test of sphericity. Step 5: Click on Extraction. Activate Correlation Matrix Unrotated factor solution Eigenvalues greater than 1 Click on Continue. This will produce loadings from correlation matrix and the number of factors is same as the number of eigenvalues greater than 1.
45 Step 6: Click on Rotation. Activate Varimax Rotated Solution Click on Continue Step 7: Click on OK SPSS will produce 8 tables as outputs with table titles 1. Correlation Matrix 2. KMO & Bartlett s Test 3. Anti-image Matrices 4. Communalities 5. Total Variance Explained 6. Component Matrix 7. Rotated Component Matrix 8. Component Transformation Matrix
46 Composite Factor Values Frequently, FA is not an end in itself but an intermediate step on the way to further analysis of the data. In such case we may require the composite values of each factor based on original/standardized data. are generated through three techniques. In recent years, the composite values Surrogated variables (A surrogated variable of a factor is a single variable that has the highest factor loading) Summated scale (The values of several variables defining a factor are summed and their total or average scores are considered) Factor scores (computer generated scores available under Scores of the main FA procedure). There are three methods Regression, Bartlett, and Anderson-Rubin.
47 Advantages & Disadvantages of the Techniques Surrogate Variables Factor Scores Summated Scales Source: Hair et al Advantages Simple to administer and interpret Represent all variables through loadings Best method for complete data reduction By default orthogonal Compromise between the surrogate variable and factor score options Reduce measurement error Represent multiple facets of a concept Disadvantages Does not represent all facets of a factor Prone to measurement error Interpretation more difficult because all variables contribute through loadings Include only the variables that load highly on the factor and exclude those having little or marginal impact Not necessarily orthogonal Require extensive analysis of reliability and validity
48 Judging Practical Significance of FA In interpreting factors, a decision must be made regarding the factor loadings. A factor loading is the correlation of the variable and the factor, the squared loading is the amount of the variable's total variation accounted for by the factor. Thus, a 0.3 loading translates to 9 per cent explanation; and a 0.5 loading denotes that 25% of the variation is accounted for by the factor. The loading must exceed 0.7 for the factor to account for 50% of the variation of the variable. Thus larger the absolute size of the factor loading, the more improvement the loading in interpreting the factor matrix using the practical significance as the criteria, we can assess the loadings as follows. Factor loadings in the range of ±0.3 to ±0.4 are considered to meet the minimal level for interpretation of structure Absolute value of loading 0.5 or greater are considered practically significant Absolute value of loading 0.7 or greater are considered indicative of well-defined structure and are the goal of any FA
49 Some Relations Among Output Values A number of relations exist among outputs, which help us to understand and interpret outputs better. The major relations are the followings when input matrix is p p correlation matrix. 1. Sum of all eigenvalues = p = total variance of p standardized variables. 2. Sum of squared factor loadings for the j th factor =λ j = j th largest eigenvalue 3. λ j = amount of variance the j th factor explains 4. λ j /p = proportion of variance explained by the j th factor 5. Sum of squared factor loadings for the i th variable = i th communality 6. i th communality = proportion of the variance of the i th standardized variable explained by the common factor model 7. (i, j) th factor loading is the correlation between the i th variable and the j th factor
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