On indirect measurement of health based on survey data. Responses to health related questions (items) Y 1,..,Y k A unidimensional latent health state

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1 On indirect measurement of health based on survey data Responses to health related questions (items) Y 1,..,Y k A unidimensional latent health state A scaling model: P(Y 1,..,Y k ;α, ) α = item difficulties = other item parameters Measurement is provided by estimates of : 1 k Standard error of measurement = s.e.( ˆ ) ˆ f(y,..y, ˆ, ˆ) 1

2 Assumptions shared by all standard scaling models 1 Unidimensionality Local independence P(Y 1,..,Y k ;α, ) = i P(Y i ;α i, i ) No differential item functioning 2

3 Assumptions shared by all standard scaling models 2 Monotonicity expected item scores are increasing functions of health state Item difficulty = the value where the expected items score is equal to half the maximum score 3

4 Three types of psychometric scaling models Structural equation models for confirmatory factor analysis IRT models Rasch models All models share the same fundamental properties and all models may provide person measures by estimates of To many researchers, the Rasch model is the preferred scaling model. To others it is far too simple. Why is that? 4

5 Three views on scaling models As a statistical model Inference Estimation of parameters Tests of data-model fit As an IRT model Realistic descriptions of response behavior As a measurement model Properties of measurements Validity, objectivity and reliability The Rasch model is special from all three viewpoints 5

6 Statistical inference The Rasch model is the only scaling model with a sufficient score Data from Rasch models is simpler to work with. Item parameters can be estimated in ways that do not depend on assumptions on the distribution of persons. Many more ways to check the data-model fit. Properties of person estimates (measures) are known exactly and do not depend on assumptions that the number of items increase towards infinity. Adjustment for model errors are (to some degree) possible 6

7 The IRT/SEM view point Realistic descriptions of response behavior Response probabilities depend on item parameters P(Y i =y ) = f(y,, i, i ) i, represents item difficulty/location and i represents other properties A serious scaling model require that all parameters are meaningful and interpretable 7

8 The IRT/SEM view point Realistic descriptions of response behavior Item responses are homogenous if the expected score of an easier item is larger than the expected score of a more difficult item for all persons Item difficulty = the value of where P(Y i =y ) = Max(Y i )/2 The Rasch model is the only known IRT model with homogenous item responses 8

9 The Rasch model Item homogeneity Walking one flight of stairs (PF5) is easier than bending and kneeling (PF6) at all levels of physical functioning 9

10 The two parameter model weaker item discrimination (PF6) Low level of : kneeling is easier than walking one flight of stairs High level of : kneeling is more difficult than walking one flight of stairs 10

11 Measurement properties The person measure IRT & SEM models: 1 k Rasch models: ˆ ˆ g(s, ˆ, ˆ ) g(y,..,y, ˆ, ˆ) where S=Σ i Y i Estimates are easily calculated on incomplete sets of responses Person parameters on interval scales if response probabilities depend on the difference between person and item parameters P(Y i =y ) = f(y, - i, i ) All IRT and SEM models satisfy this condition 11

12 Requirements of person estimates No bias Person measurement is objective if the estimates of person parameters do not depend in a systematic way on the choice of items and on the distribution and sampling of the persons The Rasch model is the only model providing objective measures Small standard error The Rasch model is the only model where the standard error of measurement can be calculated exactly 12

13 Practical issues Item analysis by Rasch models tests all the fundamental properties of the model: Unidimensionality Local independence No DIF Item fit to the response probabilities 13

14 What to do when the fit fails? Multidimensionality Partition into subscales Purification (elimination of misfitting items) Rewrite or replace misfitting items Uniform DIF Partition into subscales Uniform local response dependence Use composite item scores 14

15 Log linear Rasch model DIF and local response dependence 15

16 ICC curves for PF3 Uniform DIF 16

17 ICC curves for PF4 & PF6 Local dependence Rasch Local dependence (PF4) 17

18 Summary The Rasch model has a long list of useful properties that are unique to the model. The Rasch model is the most parsimonious scaling model and to those who favor the principle that you should always look for the most parsimonious model it is the first choice, but it also means that we have to be very careful during the analysis of data-model fit. The list of available fit-statistics is much larger for the Rasch model than for the other models. There are simple ways to modify the Rasch model to take care of DIF and local dependence. 18