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2 Factor analysis A statistical method used to describe variability among observed variables in terms of fewer unobserved variables called factors The observed variables are modeled as linear combinations of the factors plus error terms The information gained about the interdependencies can be used later to reduce the set of variables in a dataset Related to principal component analysis (PCA) PCA performs a variance-maximizing rotation of the variable space, i.e. takes into account all variability in the variables Factor analysis estimates how much of the variability is due to common factors

3 Factor analysis Variables Factors X 1 Factor 1 X 2 X 3 X 4 Factor 2 Factor k < p X p

4 Factor analysis - an example: Financial ratios Variables DSales DAssets Factors Growth EBIT-% ROI ROE Profitability CF/Sales Equity Ratio QR Solidity CR

5 Types/purposes of factor analysis Exploratory factor analysis Used to uncover the underlying structure of a relatively large set of variables A priori assumption is that any indicator may be associated with any factor No prior theory, factor loadings are used to intuit the factor structure of the data Confirmatory factor analysis Seeks to determine if the number of factors and the loadings of the measured variables on them confirm to what is expected on the basis of a preestablished theory

6 Factor analysis with SPSS Analyze Dimension Reduction Factor Extraction method, several alternatives e.g. Principal Components (the most common) Maximum Likelihood Number of factors Statistically defined (based on eigenvalues) Used defined (Fixed) when prior assumption on factor structure Rotation in order to extract a clearer factor pattern, several alternatives, e.g. Varimax Oblimin

7 Confirmatory Factor analysis - an example: Financial Ratios for Finnish listed companies 9 variables DSales, DAssets, EBIT-%, ROI, ROE, Cash Flow(Operations)/Sales, Equity Ratio, Quick Ratio Current Ratio Fixed number of factors: 3 Predefined assumption on three factors: Growth, Profitability and Solidity Extraction method: Principal Components Analysis Rotation method: Varimax

8 Factor analysis: Component matrix (Factor loadings) Component DSales (%),231,835,422 Dassets (%),212,829,450 EBIT-%,900,138 -,248 CF(Oper)/Sales,669,271 -,155 ROI,765,122 -,408 ROE,665,213 -,459 Equity Ratio,689 -,422,247 Quick Ratio,701 -,431,429 Current Ratio,645 -,488,435

9 Factor loadings Called Component loadings in PCA Correlation coefficients between the variables (rows) and factors (columns) Values between -1 and 1 The larger the absolute value of the factor loading, the stronger the connection between the variable and the factor Analogous to Pearson's r, the squared factor loading is the percent of variance in that indicator variable explained by the factor For example: 0,701 2 = 0,491 = 49,1 % of the variability in Quick Ratio is explained by the first common factor

11 Factor analysis: Varimax-rotated component matrix Component DSales (%),132 -,055,953 DAssets (%),100 -,048,960 EBIT-%,869,344,128 CF(Oper)/Sales,671,183,248 ROI,875,177,003 ROE,834,037,031 Equity Ratio,274,795 -,086 Quick Ratio,173,911,011 Current Ratio,111,911 -,042

12 Factor analysis - an example: Financial ratios for Finnish listed companies The three pre-assumed factors Growth, Profitability and Solidity - may be clearly identified in the rotated component matrix For example Growth is represented by component 3 combining the major part of ratios DSales and DAssets with minor influences from the other seven variables In the same manner Profitability is represented by component 1 and Solidity by component 2 The component matrix may be further transformed into a Component score coefficient matrix to be used to create new ratios describing the factors

13 Factor analysis: Communalities Initial Extraction DSales (%) 1,000,928 DAssets (%) 1,000,934 EBIT-% 1,000,890 CF(Oper)/Sales 1,000,545 ROI 1,000,766 ROE 1,000,698 Equity Ratio 1,000,715 Quick Ratio 1,000,861 Current Ratio 1,000,843

14 Communalities The communalities for a variable are computed by taking the sum of the squared loadings for that variable May be interpreted as multiple R 2 values for regression models predicting the variables of interest from the factors The sum of the squared factor loadings for all factors for a given variable (row) is the variance in that variable accounted for by all the factors For example 86.1 % of the variation in Quick Ratio is explained by the three common factors, 13.9 % is left unexplained

15 Communalities... One assessment of how well the model is doing can be obtained from the communalities Values close to one indicate that the model explains most of the variation for the variables Adding up the communality values for individual variables gives the Total communality of the model In the example case we have total communality of Dividing total communality by the number of variables gives the percentage of variation explained in the model In the example case 7.182/9 = 79.8 %

16 SPSS: Total Variance Explained Component Total Initial Eigenvalues % of Variance Cumulative % Extraction Sums of Squared Loadings Total % of Variance Cumulative % 1 3,765 41,832 41,832 3,765 41,832 41, ,139 23,764 65,595 2,139 23,764 65, ,277 14,184 79,780 1,277 14,184 79,780 4,708 7,871 87, ,103 1, ,000 Even an Explorative Factor Analysis with the default eigenvalue limit 1,0 in SPSS would have resulted in extractíng three factors Total variance explained by the three factor model %

17 Factor analysis: Computing factor scores The observed nine ratios for Alma Media 2005 were DSales DAssets EBIT-% Cash Flow(Oper.)/Sales ROI ROE Equity Ratio Quick Ratio Current Ratio

18 Factor analysis: Computing factor scores The nine variables may be summarized in three new variables Profitability, Solidity and Growth by multiplying the observed ratio values with component scores: Profitability = ( ) ( ) = Solidity = Growth =

19 Alternative rotation methods: Orthogonal rotations Varimax Quartimax An orthogonal alternative which minimizes the number of factors needed to explain each variable Generates often a general factor on which most variables are loaded to a high or medium degree Creates a factor structure usually not helpful to the research purpose Equimax A compromise between Varimax and Quartimax criteria

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