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Measuring abstract concepts: Latent Variables and Factor Analysis

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Correlation as the shape of an ellipse of plotted points o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o o oo o High correlation (people’s arm & leg lengths?) Lower correlation (arm length & body weight?) No correlation (arm length & income?) Correlation shows how accurately you can predict the score on a second variable if you are told the first. It suggests that there may be some underlying connection: growth, for example.

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Multiple dimensions We can show correlations among 3 variables (e.g. length of your arm & leg, and head circumference). If they are correlated, the diagram becomes an ellipsoid. It has a central axis running through it, forming a single summary indicator of the latent variable (size). Mathematically, we can also summarize correlations between more than 3 dimensions (but I can’t draw it) Head circum. arm leg

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General measurement approach ϕψsocial Health Conceptual model Selection of indicators (sampling) Scoring system e.g. ϕ x 2 +Ψ x 1.2 + soc.

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Possible hierarchies ϕψsocial ϕψ Health In a multi-level construct we need to specify how the different levels relate to each other. This comes entirely from a conceptual approach: there is no empirical way to assert one or the other model. ??

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Indicators Latent trait Indicators inter-correlate because they all reflect the latent trait Indicators correlate because they reflect a common (latent) trait

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Modeling the link between manifest (measured) and latent (inferred) variables Income Expenditure Indicator scores Health(Probability model) For variables like income & expenditure we can give a relatively fixed model For health there is more variation between people, so a less precise model.

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Principal Components analysis Translates a complex system of correlations between many variables into fewer underlying dimensions (or ‘principal components’). Developed by Charles Spearman in 1904 to identify a simpler underlying structure in large matrices of correlations between measures of mental abilities. Later greatly misused in ‘defining’ intelligence.

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Common variance (what we are trying to measure) Unique variance in this item (irrelevant bias in the measurement) Spearman’s 1904 core idea: each item contains some common (shared) variance plus some specific variance. The latter (red circles) sometimes raises and sometimes lowers the score, so they cancel out if you have enough items. +-+-

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One principal component Red lines show scores on 8 tests as vectors Cosine of angles between them represent correlations: if 2 vectors overlap the correlation is perfect (Cosine 0° = 1.0) Principal component 1 resolves most of the variance in the 8 measures: it’s the best fit, or grand average. 1

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Dimensionality & Rotation. The principal component is that which accounts for the most variance; this depends on the conceptual shape of the latent trait being measured. For Chile, one dimension will account for most of the variance in distance between cities; for HK a more complex model is required. To find the dominant dimension with the maximal variation, axes need to be rotated.

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Variance ‘explained’ Here 2 vectors, B & C, are only partially correlated. Resolving power of the principal component is shown by comparing length of the vector (B or C) and its projection onto the axis (Bʹ, C ʹ) Here, axis 1 ‘explains’ more variance for B than for C (Bʹ > C ʹ) A second (horizontal) component may be required for C: axis 2 resolves much of the variance in C, but very little for B. Principal axis B Bʹ C Cʹ Second axis

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Thurstone’s 1930 multi-factor idea: each item contains some common variance plus several types of unique variance. The latter (colored circles) can compose an additional factor being measured, or just random ‘error’. +-+- Common variance (what we are trying to measure) Unique variance in this item (irrelevant bias in the measurement) Second theme in the measurement

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Factor Loadings and Validity In the second example, the latent variable is more strongly reflected in the item; it has a higher loading on the variable and is a purer indicator of the underlying variable. The blue rectangle represents the contribution of the latent variable to the indicator. The green segment represents the contribution of other latent variables; the red section shows all other sources of variance (error, etc).

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Example of a two-factor solution (here related to concepts in the Health Belief Model) Source: K.S. Lewis, PhD thesis “An examination of the Health Belief Model when applied to Diabetes Mellitus” University of Sheffield, 1994.

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Solution with rotated axes 1 Anxiety items Depression items Using factor 1 alone = general mental health factor? 1 Using 2 factors clarifies different groups, but neither explains substantial variance 2 Anxiety factor Depression factor

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To rotate or not to rotate? Dimensions are traditionally shown perpendicular to each other: independent & uncorrelated (measures of different things should not be confounded). Applied to example of anxiety & depression there are various options: 1.as they are both are both facets of mental distress, they could be summarized along a single factor 2.perhaps it is diagnostically useful to keep anxiety & depression conceptually distinct: 2 orthogonal factors. If so, our indicators are not terrible good (low variance explained) 3.anxiety & depression often co-occur, so in reality are correlated; the axes could therefore be rotated obliquely to resolve the maximum variance (next slide)

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Oblique rotation Allow the axes to correlate Resolves more variance But does not create conceptually independent entities Do you like this approach? Anxiety factor Depression factor

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An example of turning principal components analysis results into linear modeling (LISREL), The Health Belief Model. Source: Cao Z-J, Chen Y, Wang S-M. BMC Public Health 2014, 14:26

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Cautions to ponder… Correlations between measures do not prove that they record anything concrete. Test scores may or may not result from (or be caused by) the underlying factor. The principal component is a mathematical abstraction; it may not represent anything real – (correlate your age for successive years with the population of Mexico, the weight of your pet turtle, the price of cheese and the distance between any 2 galaxies: this will produce a strong principal component). – Rotating the axes causes the principal component to disappear, so it has no reality We cannot declare that a factor represents an underlying reality (intelligence or health, etc.) unless we have clear evidence from other sources.

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Questions to debate Would you use a 1- or a 2-factor solution for anxiety & depression questions? – What sort of rotation? What type of evidence could demonstrate that your presumed health measures really do measure health? Should we ever use oblique rotation?

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Copyright (c) Bani K. Mallick1 STAT 651 Lecture #18.

Copyright (c) Bani K. Mallick1 STAT 651 Lecture #18.

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