Latent Variable Modeling - PUBH 7435 Improved Name: Latent variable measurement models and path analysis Slide 9:45 - :00 Tuesday and Thursday Fall 2006 Melanie M. Wall Division of Biostatistics School of Public Health University of Minnesota Office: A426 Mayo email: melanie@biostat.umn.edu What is a latent variable? A variable that is not observable or is not directly measurable. A variable that is measured with error or can only be measured with error. A latent variable can be used to represent a true variable which is measured with error, OR a single conceptual variable, OR a construct which is a summarization of a complex concept. Slide 2 Examples of true variables that are measured with error: calcium intake measured by a Food Frequency, physical activity measured by self-report, self-reported weight, sleep latency recorded with sleep diaries, lung capacity measured by FEV (Forced Expiratory Volume in second) Examples of conceptual variables and constructs that are desirable to measure: liberalism, quality of life, stress, self-esteem, social economic status, coping, unhealthy dieting, math ability, parenting skill, satisfaction, social support, speech difficulties, asthma severity, self-restraint problems, etc.
Latent variable measurement models Our goal is to use statistical models to measure latent variables by relating them with things that can be observed (e.g. questionnaire items, test results, any observable tool) Slide 3 Philosophical debates about the fundamental existence or non-existence of conceptual latent variables will be avoided and a more pragmatic point of view will be taken focused on obtaining useful information and explanations for relationships from data using statistical modeling. This course will focus more on latent variables representing concepts or constructus. Latent variables as mathematical convenience Latent variables are also used in different statistical modeling techniques as a mathematical convenience where they often are not of primary interest, i.e. the goal is not to measure them per se: Unobserved heterogeneity (e.g. frailties in survival analysis, random effects in longitudinal data or clustered data) Slide 4 Missing data Counterfactuals or potential outcomes
Need for measuring latent variables in Health Sciences Traditionally latent variables and methods for measuring them have been dealt with in the realm of psychology, sociology and education. Need in clinical research Need in assessment/decisions in health services Need in behavioral public health Slide 5 Need in clinical research Slide 6 In the past 20 years or so, the situation in clinical research has become more complex. The effects of new drugs or surgical procedures on quantity of life is likely to be marginal. Conversely, there is increased awareness of the impact of health and health care on the quality of human life. Therapeutic efforts in many disciplines of medicine- psychiatry, respirology, rheumatology, oncology-other health professionsnursing, physiotherapy, occupational therapy- are directed equally if not primarily to the improvement of quality, not quantity of life. If the efforts of these disciplines are to be placed on a sound scientific basis, methods must be devised to measure what was previously thought to be unmeasurable, and assess in a reproducible and valid fashion those subjective states which cannot be converted into the position of a needle on a dial. From: Streiner, D.L. and Norman, G.R. (200) Health Measurement scales: A practical guide to their development and use 3rd ed. Oxford Medical Publications. Examples: quality of life, stress, pain worsening, compliance with treatment, perimenopause, stages of Alzheimers, diagnostic for myocardial infarction
Slide 7 Need in assessment/decisions in health services Example: Your interest is in developing programs of interventions to aid individuals who are considering undergoing genetic testing for inheritable cancer. In particular, you are interested in identifying decision-making concerns among women who are at familial risk for breast cancer. By identifying these concerns, you will be better able to devise counseling programs that are specific to this vulnerable group. Need a standardized scale that will provide you and your colleagues with a reliable, valid, and easy-to-use assessment of the genetic testing and cancer outcome concerns of this population. From: Pett, M.A., Lackey, N.R., Sullivan, J.J. (2003) Making sense of factor analysis: The use of factor analysis for instrument development in health care research Sage Publications. Examples: proneness to falls, risk for developing perineal dermatitis Need in behavioral public health So many health issues are directly related to our behavior. Much theoretical work done in sociology building theories to describe different aspects of personal and social/familial networks that influence our behaviors. Many of these phenomena are not things easily measured. Examples: Slide 8 peer relationship strains family connectedness intimidation/bully-ing self efficacy for screening practices taste preferences body satisfaction motivation of alcoholism treatment post traumatic stress disorder dieting and other weight control behavior physical activity alcohol consumption violence in the workplace
Latent Variable Modeling Course Two general areas Measurement of latent variables - Statistical models are used to describe the way that observed variables are related to the latent variables. Slide 9 Path Analysis - Statistical models are used to evaluate the presumed causal relations (direct and indirect) among several variables (possibly latent). Slide 0 Latent Variable Modeling Course - Syllabus Measurement Models (7 weeks) Fundamental ideas of measurement Measuring continuous latent variables Factor Analysis (and using Principal component analysis) Factor Analysis for binary and ordered categorical variables Measuring categorical latent variables Latent class analysis. Hidden Markov Models Path Analysis (7 weeks) Intro to Causality Path analysis Structural equation modeling. Latent class regression.
Measurement Models NAMES OF MODELS: exploratory and confirmatory factor analysis, latent trait models or item response theory models or Rasch models, latent class models or latent mixture models or hidden Markov models Measures the latent variables reduces the dimensionality of the data Slide find patterns of correlations among several observed variables that are measuring the same thing observed variables are just a reflection of some underlying phenomena (i.e. latent variable) goal is to lose as little information as possible when reducing the dimensionality goal is to quantify how well each observed variables actually measure the latent variable Measurement Models Like observed variables, latent variables can be (hypothesized to be) continuous or categorical and if they are categorical they can be ordinal (ordered) or nominal (unordered). Depending upon what is assumed about the distribution of the latent variable and upon what kind of observed variables are used to measure them (i.e. continuous or categorical), the method for estimating the measurement model will change. Slide 2 latent latent continuous categorical factor class observed continuous factor analysis latent mixture model observed categorical latent trait model latent class model
Factor analysis for continuous observed variables - EFA and CFA Exploratory Factor Analysis (EFA) - Method originated by Spearman in 904 EFA general purposes: To determine how many underlying factors are necessary to explain most of the correlations and variance in the data. Slide 3 To determine the relationship via rotation between each of these underlying factors with each of the observed variables in a meaningful way so that the factors can be interpreted and named. To weed out observed variables that do not tend to measure well the underlying factors shared by the other variables. To propose blocks of variables that may be subsequently be used to create a simple sum scale. To propose a CFA model Factor analysis for continuous observed variables - EFA and CFA Confirmatory Factor Analysis (CFA) CFA general purposes: Slide 4 To define a measurement model for the relationship between multivariate observations and underlying factors To test the statistical significance of factor loadings and correlations. May be interested in testing whether rotated factor loadings from an EFA that look close to zero are, in fact, significantly different from zero or not. To test whether the measurement model for one group is the same as the measurement model for some other group As a precursor to a Structural equation model
x x2 e e2 x x2 e e2 f x x2 e e2 x3 e3 x3 e3 x3 e3 x4 e4 f x4 e4 x4 e4 f x5 x6 e5 e6 x5 x6 e5 e6 f2 x5 x6 e5 e6 x7 e7 f2 x7 e7 x7 e7 Slide 5 x8 x9 e8 e9 x8 x9 e8 e9 f3 x8 x9 e8 e9 x0 e0 x0 e0 x0 e0 x = µ + Λf + ǫ. x : p-dimensional vector of continuous observed variables f : q-dimensional vector of underlying latent factors. Often called common factors. Assume f is random such that E(f) = 0 and V ar(f) = Φ ǫ: p-dimensional vector of random error. Often called unique factors or specific factors. V ar(ǫ) = Ψ x e f x2 e2 x3 e3 f2 x4 x5 x6 e4 e5 e6 x7 e7 Slide 6 x8 e8 f3 x9 e9 x0 e0 In CFA usually several elements in Λ are fixed to zero and it is possible to consider correlated ǫ which means that Ψ is not necessarily diagonal. Furthermore, it is usually assumed that the factors are correlated so that no restriction is placed on Φ.
EFA for developing scales from questionnaires Example: Concerns about decisions to get genetic testing (from Pett, Lackey, Sullivan 2003) Slide 7 Slide 8
Slide 9 Slide 20 Factor analysis for ordered categorical observed variables - Latent Trait Models Originally methods comes from Education testing, (latent variable are labeled as traits), Item Response Theory (IRT), large literature related to IRT Answer (0,) to a series of p questions, thus there are 2 p possible response patterns (dichotomous data). Answer (,2,...c) to a series of p questions, thus there are c p possible response patterns (polytomous data). Questions to answer:. How much of the differences in these responses can be explained by supposing all items depend on one or more continuous latent variables? 2. How many underlying variables are there? 3. Which observed variables help discriminate individuals the best? 4. What is the best way to combine the observed variables in order to create a scale or score for each individual?
Factor analysis with ordered categorical observed variables This has been called Item response theory Factor analysis for categorical data Latent trait modeling Slide 2 There are basically two general approaches which can be considered if we want to take into account the categorical nature of the observed variables Underlying response variable approach Response function approach (item response theory approach) The difference in these two approaches boils down to where the categorical nature of the observed data is taken into account. Can be Exploratory or Confirmatory Latent Class Analysis Measuring categorical latent variables Credit usually given to Paul Lazarsfeld as being the originator of LCA, Foundation book is Lazarsfeld, P.F. and Henry, N.W. (968) Latent Structure Analysis. Houghton Mifflin. Slide 22 Latent class analysis is a statistical method for finding subtypes of related cases from multivariate categorical data. Questions to answer:. How many underlying classes are there? 2. What is the prevalence in each of the latent classes? 3. What is the relationship between the observed responses and the latent classes 4. What is the probability that a particular individual will be in a particular class?
Latent Class Analysis Measuring categorical latent variables EXAMPLE: Measuring unhealthy weight control behavior. Hypothetically a categorical latent variable. Have you done the following in the last year in order to lose weight or maintain your weight: (yes, no) Slide 23 marginal 2-class 3-class 4-class To control weight 2 2 3 2 3 4 fasted 7.9 38.8 2.8 58.5 32.6 2.6 24.9 7.4 29.2 2.6 ate little 44. 92.5 9.0 94.2 89.9 7.9 74. 00.0 87.5 6.9 diet pills 6.3 3.6. 40.4 6.5.2 49.4 3.0 6. 0.8 vomit 6.3 5.0 0. 45.0 7. 0. 33.2 43.6 5.6 0. laxatives.6 3.5 0.2 7. 0. 0.2 20.7.7 0.0 0.2 diuretics.4 3.3 0. 6. 0. 0. 29.4 7.9 0.0 0. food substitutes 9.3 9.2 2. 4.6 3. 2. 54.4 34.5 2.0.7 skipped meals 44.4 89.7. 85.7 89.7 9.5 43.5 00.0 87.4 8.5 smoked more cigs 9.3 8.6 2.5 39. 3. 2.5 9.8 47.6.2 2.4 % in each class 00 42.0 58.0 8.4 35.2 56.4 2.6 8.0 34.7 54.7 Estimated θ jk (probability of saying yes to the variable j given that the individual is in latent class k) under latent class models with different K Latent Mixture Models categorical latent variables distinguishing longitudinal profiles Recently there has been a lot of statistical methods work on more complex models for longitudinal data. Slide 24
Path Analysis Generally, path analysis is the combination of assumed causal theory with empirical evidence. allows researcher to translate idea about how causes are related to effects into a model total effect of one variable on another can be broken down into direct and indirect effects Slide 25 mediation and moderation Can be used with or without latent variables, that is variables of interest can be observed directly (i.e. no need for a measurement model). When there are no latent variables or when latent variables are treated as if they can be observed, it is often called path analysis, when there are latent variables and the measurement error in them is taken into account statistically by incorporating a measurement model, it is often called structural equation modeling. The general ideas are introduced using models without measurement error models included, then the measurement models will be added in later. Slide 26 Path Analysis What path analysis CANNOT do for you... Take non-experimental data and prove whether one variable actually causes another. Take non-experimental data and prove the direction of causal order between variables Take non-experimental data and distinguish between models that results in identical correlation patterns. What path analysis CAN do for you... Provide a graphical way to represent your assumed theory Provide a way to empirically estimate the relationships in your assumed theory, in particular to estimate whether the relationships are positive, negative, and importantly to test whether the relationship is zero and hence not supported by the data. Provide a way to estimate the assumed causal effect that one variable has on another through its assumed causal effect on other variables. Take experimental data (e.g. interventions) and prove whether the experimentally changed variable actually causes an outcome.
Mediation - Total and Partial Generally when we talk about mediation we are asking whether the causal relationship between two variables e X tau Z Slide 27 can be broken down into a series of intermediate causal paths. e Y alpha beta e2 X tau' Z From MacKinnon DP, Taborga MP, Morgan-Lopez AA (2002) Mediation designs for tobacco prevention research Drug and Alcohol Dependence, 68, S69- S83. Slide 28 Fig. 2. One independent variable, six-mediator model illustrating the incorporation of action theory and conceptual theory for tobacco use prevention.
Path Analysis Slide 29 Friend Level Personal Level Family Level Parent-Youth Strain Illness Strains and Worries Seeking Support Peer Relationship Strain Parent Support Risk Taking Behaviors Friend Support Self Esteem Self-Care Motivation Self-Care Discouragement Non Adherence.5-Year FEV Mean.5-Year AI Mean Figure. Conceptual model showing relationship between strains, resources, nonadherence feelings/behaviors and health outcomes for youth with CF Path Analysis Slide 30 Friend Level Personal Level Family Level Parent-Youth Strain Illness Strains and Worries Seeking Support Peer Relationship Strain.30*** -.28* Parent Support Risk Taking Behaviors.25* Friend Support Self Esteem.39***.34***.3***.27*.22** Self-Care Motivation.26*** Self-Care Discouragement.23* Girls.36*** Non Adherence * p <.0 ** p <.05 *** p <.0
Path Analysis Slide 3 Friend Level Personal Level Family Level Parent-Youth Strain Illness Strains and Worries Peer Relationship Strain.32***.26** -.3** Risk Taking Behaviors Self Esteem.32**.44*** Self-Care Motivation Self-Care Discouragement Boys.25*** Non Adherence * p <.0 ** p <.05 *** p <.0 Structural Equation Modeling Combines measurement models with path analysis Takes the measurement error into account Slide 32 Rather than taking scales with less than perfect reliability and using them as if they are perfect measurements of the latent variable, SEM models incorporates the measurement error and thus adjusts the correlations and path coefficients appropriately. Assuming the model specification is correct (as usual).
Use full SEM - Incorporate CFA into the Path Analysis Slide 33 Final results of SEM Slide 34 Figure 4. Final model testing among adolescent girls: Correlates of unhealthy weight-control behaviors. BMI body mass index. * p.0. Figure 5. Final model testing among adolescent boys: Correlates of unhealthy weight-control behaviors. BMI body mass index. * p.0.