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Measuring abstract concepts: Latent Variables and Factor Analysis.

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1 Measuring abstract concepts: Latent Variables and Factor Analysis

2 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?) Correlations show how accurately you can predict the score on a second variable if you are told the first. Correlations suggest that there may be some underlying or abstract connection: growth, for example.

3 Multiple dimensions We can also show correlations among 3 variables (e.g. the 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 (body size). Mathematically, we can also summarize correlations between more than 3 dimensions (but I can’t draw it) Head circum. arm leg

4 Indicators Latent trait We interpret the inter-correlation of indicators as resulting from the latent trait Indicators correlate because they reflect a common (latent) trait

5 Latent, or underlying variables The idea of an underlying theme that is not directly observable is familiar: – The true efficacy of a drug is estimated by observing patients in a drug trial. The individual data points vary from person to person but allow us to estimate the true efficacy. – If you want to interview a candidate to assess their suitability for a job you typically ask a number of questions and combine subjective judgments from several interviewers.

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

7 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. ??

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

9 Principal Components analysis Translates a set 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.

10 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. Specific variance (red circles) sometimes raises and sometimes lowers the score, so they cancel out if you have enough items. +-+-

11 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

12 Dimensionality & Rotation. The principal component is that which accounts for the most variance; this depends on the conceptual space 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.

13 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). But it depends on how the axes are rotated; 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

14 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

15 Factor Loadings and Item Validity In the second example, the latent variable is more strongly reflected in the item; the item 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 item or indicator. The green segment represents the contribution of other latent variables; the red section shows all other sources of variance (error, etc). 100% of item variance

16 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.

17 Solution with rotated axes 1 Anxiety items Depression items Using factor 1 alone = general mental health factor? 1 Using 2 factors clarifies different constructs, but neither explains substantial variance, at least using orthogonal axes. 2 Anxiety factor Depression factor

18 To rotate or not to rotate? Dimensions are traditionally shown perpendicular to each other: independent & uncorrelated (measures of distinct 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 share characteristics and are not completely distinct. They also often co-occur, so are correlated; the axes could therefore be rotated obliquely to resolve the maximum variance (next slide). But this becomes data-driven, rather than measuring conceptually distinct constructs.

19 Oblique rotation Allow the axes to correlate Resolves more variance But does not create conceptually independent entities. What are the arguments for and against this approach? Anxiety factor Depression factor

20 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 HBM = Health Belief Model SUS = perceived susceptibility SER = seriousness of disease BEN = benefits of taking action BAR = barriers to acting CTA = cues to action

21 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. The values all rise over time, so 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.

22 Questions to debate Would you use a 1- or a 2-factor solution for anxiety & depression questions? – What sort of rotation? – Does your choice depend on the purpose of the study? 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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