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3. Harmonization of multiple observational studies:Non-identical ordered scales October 2014 Edwin R. van den Heuvel Professor of Statistics e.r.v.d.heuvel@tue.nl Department of Mathematics and Computer Science Eindhoven University of Technology

4. Content • Introduction • Content Equivalence • Motivating Example • Principles • General Model • Analysis Motivating Example

5. Introduction • Epidemiology is concerned with finding associations between risk factors and health outcomes • For some diseases or some risk factors large cohort studies are required • Such research can only be performed by combining several already existing population studies, e.g. • LifeLines in the Netherlands • HUNT in Norway • CORA in Germany

6. Introduction • Multiple cohort studies may not measure the same characteristic with the same instruments • Physical Activity • Memory • Quality of Life • Gene expression • Income • Combining individual participants data is a therefore challenge, variables can not just simply pooled

7. Introduction • Example Smoking • Study 1: Have you ever smoked (Yes/No)? • Study 2: Have you smoked the last year (Yes/No)? • Examples Income • Study 1: What is your gross income per months? • Study 2: Indicate your yearly salary category: <15000, [15000-30000], ….., ≥100000 • Variables from different studies • May not contain the exact same information • Require some form of manipulation to make them equivalent

8. Content Equivalence Retrospective harmonization: • Core variables are • The primary units of interest used in a statistical analysis • Depends strongly on the research question • A study can be viewed harmonized if the assessment items in that study can be used to generate a ‘valid’ equivalent to the required core variable • The goal is to achieve comparability or content equivalence • Heterogeneity between studies that would be caused by groups of individuals should not be eliminated

9. Content Equivalence DataShaper: • Provides an overview and algorithms for core variables that can be harmonized from different studies (http://www.datashaper.org/) • Example smoking: • The core variable could be “smoked last year” • Study 2 provide direct information • Study 1 requires an additional item or question: if and when individuals stopped smoking • Example income • Income variable in study 1 will most likely be reduced to a categorical variable that matches the categories from the income variable in study 2

10. Content Equivalence • Complex constructs can not be harmonized with the deterministic algorithms • Cognition (e.g. memory) • Physical activity • Etc. • Complex constructs are typically formulated from items of questionnaires: ordered scales • Currently, pooling individual data from different studies is performed on the basis of linear scaling methods: • Z-scores: scales are standardized within studies with the same mean and standard deviations • C-scores: scales are standardized within studies with respect to a reference or control group available in all studies

11. Content Equivalence • There is only limited statistical literature on harmonizing constructs or scales • Two publications provided a latent variable model for combining the items from questionnaires • Van Buurenet al (2005) • Bauer and Hussong (2009) • They use bridge items to connect the different studies with each other • They essentially assume that the observed items provide adequate information on the same construct • The current literature does not describe concepts and criteria for harmonization of complex constructs

12. Motivating Example • Three Canadian studies: • CCHS: Canadian Community Health Survey 10,263 older adults (≥ 65 years) across Canada • CSHA: Canadian Study of Healthy Aging 130,000 participants (≥ 45 years) from 10 Canadian provinces • NuAge: Quebec Longitudinal Study on Nutrition and Aging 1,793 older adults (68-82 years) living in Quebec • Measurements of memory on individuals • Health Utility Index (HUI): an indirect measure of memory • Rey Auditory Verbal Learning Test (RAVLT): short-term memory • Buschke Cued Recall Procedure (BCRP): free and cued memory • Memory was only observed on subsets of participants in the Canadian studies

13. Motivating Example • Rey Auditory Verbal Learning Test • Participants are read a list of 15-item words • They should listen carefully and are then asked to recall them • The number of correct words are counted • BUSCHKE Cued Recall Procedure: • Participants are shown a picture sheet with four images of objects belonging to four different categories • Participants need to point out and name the objects • The sheet is taken away and the examiner ensures that the words are properly encoded • After a distraction activity the participants are asked to recall the words: both free recall and cued recall • CSHA used 12 words and NuAge used 16 words • Can we harmonize memory from the three studies?

14. Principles Length measurements: • USA measures length in inches and Europe measures length in centimeters • Harmonization of length requires a change from one unit to the other unit • The relation is linear: 1 inch = 2.54 cm • This relation has varied many times in the past until it was settled only in 1959 (Astinet al., 1959) Principle 1: Existence of calibration model There exist a one-to-one mathematical relationship between the different true variables

15. Principles Measurement error: • Measurements are not without noise • Measurement errors for length are relative small: • Error variation: RSD (%) = 0.1% • Subject variations: RSD (%) = 5.5% • The intraclass correlation coefficient between length in inches and centimeters on one person is ICC > 0.99 Principle 2: Predictability An individual measurement in one variable can be predicted precisely by the measurement of the other variable

16. Principles Length is universal: • For length measurements there exist one unique reference standard: • Meter in Paris (although the definition is currently artifact free) • The measurement errors are • Small across the whole range of relevant outcomes • Almost independent from any characteristic (e.g. length, time, measurement devices, etc.) Principle 3: Invariance The calibration model and predictability are invariant or consistent across the world

17. General Model • Assume measurements Y1,i and Y2,i on subject i are obtained with instruments 1 and 2 • Assume that the measurements try to capture the latent variables Z1,i and Z2,i • The conditional distribution of Yhi given Zhi is given by • It describes the measurement uncertainty given the true value • Let Xi be a vector of p covariates for subject i • The conditional distribution of Zhi given Xhi is given by • It describes the distribution of the latent variable in a population

18. General Model • Two instruments measure the same characteristic when there exist a monotone continuous function ψ such that • The function ψ is referred to as the calibration model • The instruments measure the same thing but in different units • Typical forms for ψ are linear or log linear functions • Note that ψ is given by • The calibration model implies a form of measurement invariance (Meredith, 1993) in the sense that

19. General Model • Assume for now that the function ψ is known • From the observations we can only calculate ψ(Y1,i) • This leads to a joint distribution for (Y2,i , ψ(Y1,i)): • If Y1,i and Y2,i are independent conditioned on the latent variables • The joint distribution H is a measure for how well Y2,i can be predicted from Y1,i • The joint distribution H equals the marginal distribution of ψ(Y1,i) or Y2,i when Y2,i = ψ(Y1,i) : perfect prediction • A simpler predictability measure is the correlation coefficient between Y2,i and ψ(Y1,i)

20. General Model • In a few studies separate calibration models ψ1, ψ2, ….., ψkmay be estimable from the data • The invariance principle would require that all calibration models are identical ψ1=ψ2=…..=ψk • This form of invariance does not imply that the latent variable distribution is identical across studies: • With subject i(k1) from study k1 and subject i(k2) from study k2 • Factorial invariance in measurement reliability (Meredith & Teresi, 2006) does require equality • If two subjects have the same ability they have the same probability of correctly answering the items

21. Analysis Motivating Example Study HUI RAVLT Free BCRP Cued BCRP CCHS 5 15 NA NA CSHA NA 15 12 12 NuAge NA NA 16 16 Statistical Model • Let Zi be a latent variable for memory on subject i • Consider covariates sex (x1), age (x2), and education (x3) • Assume binomial distribution with logit link function

22. Analysis Motivating Example Statistical Model • Assume a linear calibration model • This calibration model leads to • The invariance principle within a study implies that a and b are independent of subject i, which leads to gh,0 = g0 • Another aspect of invariance is that the calibration model is independent of studies

23. Analysis Motivating Example Baseline Characteristics • Education is measured in “years of education” • Substantial difference between studies

24. Analysis Motivating Example

25. Analysis Motivating Example Estimates Binomial latent variable model

26. Analysis Motivating Example • Rey is more difficult than the free recall of Buschke • Free recall of Buschke is easier than the Cued recall • Older age reduces memory: consistent in all studies • There is an effect of gender on memory performance in the CCHS and NuAge, but not in CSHA • The Buschke test discriminates between gender in the CSHA study but not in the CCHS and NuAge • The gain in memory in the latent variable from the free Buschke to the cued Buschke is different between studies • Thus the Buschke test is not consistent across studies violating the invariance principle

27. Analysis Motivating Example • Harmonization of Rey and Free Buschke seems more realistic, although we can not test for invariance • A subject-specific ICC was calculated for Rey and Free Buschke • Thus predictability of the memory scores for Rey and Free Buschke is quite good

28. Harmonization

29. References • Astin, A.V., Karo, H.A., and Mueller, F.H. (1959), Refinement of values for the yard and the pound, US Federal Register. • Bauer DJ, Hussong AM, Psychometric Approaches for Developing Commensurate Measures across Independent Studies: Traditional and New Models, Psychological Methods, 2009, 14(2), 101-125 • Meridith, W. (1993), "Measurement invariance, factor analysis and factorial invariance", Psychometrika, 58(4), 525-543. • Meredith, W., and Teresi, J.A. (2006), "An essay on measurment and factorial invariance", Medical Care, 44, S69-S77. • Van Buuren S, Eyres S, Tennant A, Hopman-Rock M, Improving comparability of Existing Data by Response Conversion, Journal of Official Statistics, 2005, 21(1), 53-72