Measuring abstract concepts: - PowerPoint Presentation

1. Measuring abstract concepts: Latent Variables and Factor Analysis

2. Correlation as the shape of an ellipse of plotted pointsooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooooHigh 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 secondvariable if you are told the first. Correlations suggest that theremay be some underlying or abstract connection: growth, for example.

3. Multiple dimensionsWe can also show correlations among 3variables (e.g. the lengthof your arm & leg, and headcircumference). If they are correlated, the diagram becomes an ellipsoid. It has a central axis runningthrough it, forming a singlesummary indicator of thelatent variable (body size). Mathematically, we can also summarizecorrelations betweenmore than 3 dimensions(but I can’t draw it)Headcircum.armleg

4. IndicatorsLatent traitWe interpret the inter-correlationof indicators as resulting from thelatent traitIndicators correlate becausethey reflect a common (latent) trait

5. Latent, or underlying variablesThe 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 approachϕψsocialAbstract concept, e.g. HealthConceptualmodelSelection ofindicators(sampling)Scoringsysteme.g. ϕ x 2 + Ψ x 1.2 + social

7. Possible hierarchiesϕψsocialϕψsocialHealthHealthIn 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) variablesIndicator scoresHealth(Probability model)For variables likeincome & expenditureswe can give a relatively fixed modelFor health thereis more variationbetween people,so a less precisemodel. IncomeExpenditureon sportsactivities

9. Principal Components analysisTranslates 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 tryingto measure)Unique variance in thisitem (irrelevant biasin the measurement)Spearman’s 1904 core idea: each item contains somecommon (shared) variance plus some specific variance. Specific variance (red circles) sometimes raises and sometimeslowers the score, so they cancel out if you have enough items. +-

11. One principal componentRed lines show scores on 8 tests as vectorsCosine of angles between them represent correlations: if 2 vectors overlap the correlation is perfect (Cosine 0° = 1.0)Principal component 1resolves most of the variance in the 8 measures: it’s the bestfit, 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 distancebetween 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. PrincipalaxisBBʹCCʹSecondaxis

14. Thurstone’s 1930 multi-factor idea: each item containssome 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 tryingto measure)Unique variance in thisitem (irrelevant biasin the measurement)Second theme inthe measurement

15. Factor Loadings and Item ValidityIn 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 variableto 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-factorsolution (here related toconcepts in the HealthBelief Model)Source: K.S. Lewis, PhD thesis“An examination of the HealthBelief Model when applied toDiabetes Mellitus”University of Sheffield, 1994.

17. Solution with rotated axes1AnxietyitemsDepressionitemsUsing factor 1 alone = general mental health factor?1Using 2 factors clarifies different constructs,but neither explains substantial variance,at least using orthogonal axes.2AnxietyfactorDepressionfactor

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: as they are both are both facets of mental distress, they could be summarized along a single factorperhaps it is diagnostically useful to keep anxiety & depression conceptually distinct: 2 orthogonal factors. If so, our indicators are not terrible good (low variance explained)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 rotationAllow the axes to correlateResolves more varianceBut does not create conceptually independent entities.What are the arguments for and against this approach?AnxietyfactorDepressionfactor

20. An example of turningprincipal componentsanalysis results intolinear modeling (LISREL):The Health Belief Model.Source: Cao Z-J, Chen Y, Wang S-M. BMC Public Health 2014, 14: 26HBM = Health Belief ModelSUS = perceived susceptibilitySER = seriousness of diseaseBEN = benefits of taking actionBAR = barriers to actingCTA = 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 debateWould 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?