Analysis of Correlated Data. Patrick J. Heagerty PhD Department of Biostatistics University of Washington


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1 Analysis of Correlated Data Patrick J Heagerty PhD Department of Biostatistics University of Washington Heagerty, 6
2 Course Outline Examples of longitudinal data Correlation and weighting Exploratory data analysis between and withinperson variation correlation / covariance Regression analysis Models for the mean Models for the correlation Linear mixed model GEE Heagerty, 6
3 Course Outline Missing data Binary and Count Data Models for the mean Models for the correlation Generalized linear mixed model GEE Transition models Timedependent covariates Multilevel models Design considerations 3 Heagerty, 6
4 Bio & Notes Patrick J Heagerty Professor, University of Washington Collaborative roles = RWJ, VA ERIC, NIAMS MCRC, K Books: Diggle, Heagerty, Liang & Zeger Analysis of Longitudinal Data Oxford, van Belle, Fisher, Heagerty & Lumley Biostatistics Wiley, 4 (introductory chapter on LDA) 4 Heagerty, 6
5 Course Notes & Slides UW Biostat 57 = PhD applied core sequence Winter 999,,, UM Epi 766 = Longitudinal Data Analysis / Epi Summer (Summer 4 with VA/UW Biostat/Epi) Second Seattle Symposium (with S Zeger) Fall RAND short course Fall NICHD short course Fall 3 5 Heagerty, 6
6 Longitudinal Data Analysis INTRODUCTION to EXAMPLES AND ISSUES 6 Heagerty, 6
7 Outline Examples of longitudinal data between and withinperson variation correlation / covariance Scientific motivation Opportunities Issues Time scales Crosssectional contrasts Longitudinal contrasts Correlation and weighting Impact of correlation Weighted estimation 7 Heagerty, 6
8 Continuous Longitudinal Data Example : Cystic Fibrosis and Lung Function There is a large registry of cystic fibrosis patient data Annual measurements include standard pulmonary function measures: FVC, FEV primary outcome: FEV percent predicted covariates: age, gender, genotype Q: Does change in lung function differ by gender and/or genotype? 8 Heagerty, 6
9 8 Heagerty, fev age
10 Continuous Longitudinal Data Example : Betacarotene and vitamin E a phase II study to ascertain the pharmacokinetics of betacarotene supplementation and the subsequent impact on vitamin E levels primary outcome: plasma measures taken monthly for 3 months prior to, 9 months during, and 3 months after supplementation covariates: dose (, 5, 3, 45, 6 mg/day) and time Q: What is the time course? Doseresponse? Relationship between betacarotene and vitamin E? 9 Heagerty, 6
11 9 Heagerty, 6 Dose = 45 Subject = 9 Subject = Subject = Subject = 3 betacarotene (solid) betacarotene (solid) betacarotene (solid) betacarotene (solid) months since randomization months since randomization months since randomization months since randomization Subject = 3 Subject = 3 Subject = 46 Subject = 47 betacarotene (solid) betacarotene (solid) betacarotene (solid) betacarotene (solid) Vitamin E (dashed) months since randomization months since randomization months since randomization months since randomization
12 9 Heagerty, 6 Dose = 45 Subject = 9 Subject = Subject = Subject = 3 log(betacarotene) (solid) log(betacarotene) (solid) log(betacarotene) (solid) log(betacarotene) (solid) months since randomization months since randomization months since randomization months since randomization Subject = 3 Subject = 3 Subject = 46 Subject = 47 log(betacarotene) (solid) log(betacarotene) (solid) log(betacarotene) (solid) log(betacarotene) (solid) Vitamin E (dashed) months since randomization months since randomization months since randomization months since randomization
13 Categorical Longitudinal Data Example 3: Maternal Stress and Child Morbidity daily indicators of stress (maternal), and illness (child) primary outcome: illness, utilization covariates: employment, stress Q: association between employment, stress and morbidity? Heagerty, 6
14  Heagerty, 6
15 Illness Percent Employed= Employed= Day Stress Percent Employed= Employed= Day  Heagerty, 6
16 3 Heagerty, 6 subject = 4 subject = subject = 56 Y illness N Y stress N Day Y illness N Y stress N Day Y illness N Y stress N Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day Y illness N Y stress N subject = Day
17 Continuous Longitudinal Data Example 4: PANSS Data PANSS is a standard symptom assessment for schizophrenic patients This study compares different doses of a new agent to a standard agent and to placebo primary outcome: PANSS covariates: treatment, time Q: What s the best treatment? Heagerty, 6
18  Heagerty, 6 Last Visit = 8 Last Visit = 6 Last Visit = 4 PANSS Score PANSS Score PANSS Score Time Time Time Last Visit = Last Visit = Last Visit = PANSS Score PANSS Score PANSS Score Time Time Time
19 Longitudinal Data In longitudinal studies of health, we typically observe two distinct kinds of outcomes Times of clinical or other key events Repeated values of markers of the health status of participants In general terms, the scientific question is how explanatory variables affect times to clinical events and markers of the level or change in health status over time The relationship between the event times and markers can also be of interest Use markers as predictors of, or surrogates for, the clinical event time Heagerty, 6
20 Longitudinal Studies Benefits of longitudinal studies: Incident events are recorded Measure the new occurrence of disease Timing of disease onset can be correlated with recent changes in patient exposure and/or with chronic exposure Prospective ascertainment of exposure Participants can have their exposure status recorded at multiple followup visits This can alleviate recall bias Temporal order of exposures and outcomes is observed 3 Heagerty, 6
21 3 Measurement of individual change in outcomes A key strength of a longitudinal study is the ability to measure change in outcomes and/or exposure at the individual level Longitudinal studies provide the opportunity to observe individual patterns of change 4 Separation of time effects: Cohort, Period, Age When studying change over time there are many time scales to consider cohort scale is the time of birth such as 945 or 963 period is the current time such as 4 age is (period  cohort) A longitudinal study with times t, t, t n can characterize multiple time scales such as age and cohort effects using covariates derived from the calendar time and birth year: age of subject i at time t j is age ij = (t j birth i ); and cohort is cohort ij = birth i 4 Heagerty, 6
22 5 Control for cohort effects In a crosssectional study the comparison of subgroups of different ages combines the effects of aging and the effects of different cohorts That is, comparison of outcomes measured in 3 among 58 year old subjects and among 4 year old subjects reflects both the fact that the groups differ by 8 years (aging) and the fact that the subjects were born in different eras In a longitudinal study the cohort under study is fixed and thus changes in time are not confounded by cohort differences An nice overview of LDA opportunities in respiratory epidemiology is presented in Weiss and Ware (996) Lebowitz (996) discusses age, period, and cohort effects 5 Heagerty, 6
23 Longitudinal Studies The benefits of a longitudinal design are not without cost There are several challenges posed: Challenges of longitudinal studies: Participant followup Risk of bias due to incomplete followup, or dropout of study participants If subjects that are followed to the planned end of study differ from subjects who discontinue followup then a naive analysis may provide summaries that are not representative of the original target population 6 Heagerty, 6
24 Analysis of correlated data Statistical analysis of longitudinal data requires methods that can properly account for the intrasubject correlation of response measurements If such correlation is ignored then inferences such as statistical tests or confidence intervals can be grossly invalid 7 Heagerty, 6
25 3 Timevarying covariates Although longitudinal designs offer the opportunity to associate changes in exposure with changes in the outcome of interest, the direction of causality can be complicated by feedback between the outcome and the exposure Example = MSCM with stresss and illness Although scientific interest generally lies in the effect of exposure on health, reciprocal influence between exposure and outcome poses analytical difficulty when trying to separate the effect of exposure on health from the effect of health on exposure How to choose exposure lag? eg Is it the air pollution today, yesterday, or last week that is the important predictor of morbidity today? 8 Heagerty, 6
26 Longitudinal Studies The Scientific Opportunity Observe individual changes over time Characterize the timecourse of disease Outcome Measures A single outcome at a fixed followup time The time until an event occurs Repeated measures taken over time 9 Heagerty, 6
27 Motivation Cystic Fibrosis and Pulmonary Function Several specific aspects are of interest: What is the rate of decline in FEV? Is the time course different for males and females? 3 Is the time course different for F58 homozygous subjects? Reference: Davis PB (997) Journal of Pediatrics Heagerty, 6
28 Data ID = patient id FEV = percentpredicted forced expiratory volume in second AGE = age (years) GENDER = sex (=male, =female) PSEUDOA = infection with Pseudomonas Aeruginosa (=no, 3=yes) F58 = genotype (=homozygous, =heterozygous, 3=none) PANCREAT = pancreatic enzyme supplmentation (,=no, =yes) Heagerty, 6
29  Heagerty, 6 EDA: Numerical Summaries Total number of subjects = Number of observations (number of subjects with ni): Distribution of males / females male female 98 Number of mutations of f
30  Heagerty, 6 EDA: Numerical Summaries Age at entry N = Median = 9655 Quartiles = 7758, 5335 Decimal point is at the colon 5 : : : : : : 3349 : : : : : : : : : 4 : 3778 : 5577 : : 8
31 ID = 57 ID = 5796 ID = 577 ID = 774 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 7 ID = 6345 ID = 884 ID = 749 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 3564 ID = 587 ID = 439 ID = 7755 FEV 5 5 FEV 5 5 FEV PA PA PA PA PC PC PC PC Heagerty, FEV 5 5
32 ID = 743 ID = 977 ID = 876 ID = 83 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 3399 ID = 979 ID = 8645 ID = 7 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 679 ID = 5 ID = 35 ID = 8 FEV 5 5 FEV 5 5 FEV PA PA PA PA PC PC PC PC Heagerty, FEV 5 5
33 ID = 9847 ID = 67 ID = 97 ID = 597 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 736 ID = 4367 ID = 63 ID = 945 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 6699 ID = 394 ID = 54 ID = 7 FEV 5 5 FEV 5 5 FEV PA PA PA PA PC PC PC PC Heagerty, FEV 5 5
34 ID = 837 ID = 74 ID = 76 ID = 74 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 7483 ID = 4864 ID = 786 ID = 754 FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC FEV 5 5 PA PC Age Age Age Age ID = 66 ID = 8579 ID = 383 ID = 9469 FEV 5 5 FEV 5 5 FEV PA PA PA PA PC PC PC PC Heagerty, FEV 5 5
35 7 Heagerty, 6 FEV versus Age fev 5 5 slope = age
36 FEV versus AgeatEntry FEV 5 5 slope = Age at Entry FEV change FEV Change versus AgesinceEntry slope = Age since Entry 8 Heagerty, 6
37 Distinguishing Crosssectional and Longitudinal Associations Crosssectional data Y i = β C x i + ɛ i, i =,, m () β C represents the difference in average Y across two subpopulations which differ by one unit in x Longitudinal data Y ij = β C x i + β L (x ij x i ) + ɛ ij, j =,, n i i =,, m () Heagerty, 6
38 When j =, the two equations are the same; β C has the same crosssectional interpretation Subtract equations above to obtain (Y ij Y i ) = β L (x ij x i ) + (ɛ ij ɛ i ) β L represents the expected change in Y per unit change in x 3 Heagerty, 6
39 3 Heagerty, 6 A  CrossSectional Data Reading Score o o o o o o o o o o o o Age
40 Crosssectional Longitudinal Response 5 5 Response Months Months Both Response Months 3 Heagerty, 6
41 Age change in FEV males females FEV by Male/Female Age change in FEV f58 f58 f58 FEV by f Heagerty, 6
42 EDA Summary Observations Systematic trends: time, gender, F58 Random variation: individual, observation Questions Two time scales? Estimation / testing for rates of decline? Other? 4 Heagerty, 6
43 Longitudinal Data Analysis INTRODUCTION to CORRELATION and WEIGHTING 5 Heagerty, 6
44 Longitudinal Data The basic statistical problem is that variables from a given individual are correlated over time (generic) Q: So what? () ignoring dependence can lead to invalid inference () often limited information regarding dependence (+) can observe change for individuals over time (+) variety of statistical approaches that are available 6 Heagerty, 6
45 Longitudinal Data need to account for the dependence (generic) Q: How? Choice of Model Choice of Estimator 3 Choice of Summaries 7 Heagerty, 6
46 Dependent Data and Proper Variance Estimates Let X ij = denote placebo assignment and X ij = denote active treatment () Consider (Y i, Y i ) with (X i, X i ) = (, ) for i = : n and (X i, X i ) = (, ) for i = (n + ) : n ˆµ = n ˆµ = n n i= j= n Y ij i=n+ j= var(ˆµ ˆµ ) = n {σ ( + ρ)} 8 Heagerty, 6 Y ij
47 8 Heagerty, 6 Scenario subject control treatment time time time time ID = Y, Y, ID = Y, Y, ID = 3 Y 3, Y 3, ID = 4 Y 4, Y 4, ID = 5 Y 5, Y 5, ID = 6 Y 6, Y 6,
48 Dependent Data and Proper Variance Estimates () Consider (Y i, Y i ) with (X i, X i ) = (, ) for i = : n and (X i, X i ) = (, ) for i = (n + ) : n ˆµ = n ˆµ = n { n Y i + i= { n Y i + i= n i=n+ n i=n+ Y i } Y i } var(ˆµ ˆµ ) = n {σ ( ρ)} 9 Heagerty, 6
49 9 Heagerty, 6 Scenario subject control treatment time time time time ID = Y, Y, ID = Y, Y, ID = 3 Y 3, Y 3, ID = 4 Y 4, Y 4, ID = 5 Y 5, Y 5, ID = 6 Y 6, Y 6,
50 Dependent Data and Proper Variance Estimates If we simply had n independent observations on treatment (X = ) and n independent observations on control then we d obtain var(ˆµ ˆµ ) = σ n + σ n = n σ Q: What is the impact of dependence relative to the situation where all (n + n) observations are independent? () positive dependence, ρ >, results in a loss of precision () positive dependence, ρ >, results in an improvement in precision! 3 Heagerty, 6
51 Therefore: Dependent data impacts proper statements of precision Dependent data may increase or decrease standard errors depending on the design 3 Heagerty, 6
52 Weighted Estimation Consider the situation where subjects report both the number of attempts and the number of successes: (Y i, N i ) Examples: live born (Y i ) in a litter (N i ) condoms used (Y i ) in sexual encounters (N i ) SAEs (Y i ) among total surgeries (N i ) Q: How to combine these data from i = : m subjects to estimate a common rate (proportion) of successes? 3 Heagerty, 6
53 Weighted Estimation Proposal : ˆp = i Y i / i N i Proposal : Simple Example: ˆp = m Y i /N i i Data : (, ) (, ) ˆp = ( + )/() = 3 ˆp = {/ + /} = 5 33 Heagerty, 6
54 Weighted Estimation Note: Each of these estimators, ˆp, and ˆp, can be viewed as weighted estimators of the form: ˆp w = { i w i Y i N i } / i We obtain ˆp by letting w i = N i, corresponding to equal weight given each to binary outcome, Y ij, Y i = N i j= Y ij We obtain ˆp by letting w i =, corresponding to equal weight given to each subject w i Q: What s optimal? 34 Heagerty, 6
55 Weighted Estimation A: Whatever weights are closest to /variance of Y i /N i (stat theory called GaussMarkov ) If subjects are perfectly homogeneous then and ˆp is best V (Y i ) = N i p( p) If subjects are heterogeneous then, for example V (Y i ) = N i p( p){ + (N i )ρ} and an estimator closer to ˆp is best 35 Heagerty, 6
56 Summary Longitudinal (dependent) data are common (and interesting!) Inference must account for the dependence Consideration as to the choice of weighting will depend on the variance/covariance of the response variables 36 Heagerty, 6
57 Summary Methodological Issues in the Study of Cognitive Decline Morris et al AJE (999) In modeling associations between risk factors and change in cognitive function, the usual analytic issues are complicated by the need to consider time in the analysis The potential for both fruitful investigation and misinterpretation of the data is increased Analyses that make the most effective use of the data usually require the most advanced methods of longitudinal analysis 37 Heagerty, 6
58 Summary Methodological Issues in the Study of Cognitive Decline Morris et al AJE (999) Use of the longitudinal designs and advanced statistical models described here is well worth the extra effort required Meeting the special challenges of measuring change in cognitive function should improve our ability to identify risk factors for cognitive decline and other diseases related to aging Many of these issues apply to any study of disease processes entailing change with age 38 Heagerty, 6
59 Some References: Books Diggle PJ, Heagerty PJ, Liang KY, Zeger SL () Analysis of Longitudinal Data, Second Edition, Oxford University Press Fitzmaurice GM, Laird NM, Ware JM (4) Applied Longitudinal Analysis, Wiley Verbeke G, Molenberghs G () Linear Mixed Models for Longitudinal Data, Springer Singer JD, Willett JB (3) Applied Longitudinal Data Analysis, Oxford University Press 39 Heagerty, 6
60 Longitudinal Data Analysis INTRODUCTION to REGRESSION APPROACHES 4 Heagerty, 6
61 Linear Mixed Model Regression model: mean response as a function of covariates systematic variation Random effects: variation from subjecttosubject in trajectory random betweensubject variation Withinsubject variation: variation of individual observations over time random withinsubject variation 4 Heagerty, 6
62 Scientific Questions as Regression Questions concerning the rate of decline refer to the time slope for FEV: E[ FEV X = age, gender, f58] = β (X) + β (X) time Time Scales Let age = ageatentry, age i Let agel = timesinceentry, age ij age i 4 Heagerty, 6
63 CF Regression Model Model: E[FEV X i ] = β +β age + β agel +β 3 female +β 4 f58 = + β 5 f58 = +β 6 female agel +β 7 f58 = agel + β 8 f58 = agel = β (X i ) + β (X i ) agel 43 Heagerty, 6
64 43 Heagerty, 6 Intercept f58= f58= f58= male β + β age β + β age β + β age +β 4 +β 5 female β + β age β + β age β + β age +β 3 +β 3 + β 4 +β 3 + β 5
65 43 Heagerty, 6 Slope f58= f58= f58= male β β + β 7 β + β 8 female β β + β 7 β + β 8 +β 6 +β 6 +β 6
66 433 Heagerty, 6 Gender Groups (f58==) FEV Male Female Age (years)
67 434 Heagerty, 6 Genotype Groups (male) FEV f58= f58= f58= Age (years)
68 Define Y ij = FEV for subject i at time t ij X i = (X ij,, X ini ) X ij = (X ij,, X ij,,, X ij,p ) age, agel, gender, genotype Issue: Response variables measured on the same subject are correlated cov(y ij, Y ik ) 44 Heagerty, 6
69 Some Notation It is useful to have some notation that can be used to discuss the stack of data that correspond to each subject Let n i denote the number of observations for subject i Define: Y i = Y i Y i Y ini If the subjects are observed at a common set of times t, t,, t m then E(Y ij ) = µ j denotes the mean of the population at time t j 45 Heagerty, 6
70 Dependence and Correlation Recall that observations are termed independent when deviation in one variable does not predict deviation in the other variable Given two subjects with the same age and gender, then the blood pressure for patient ID= is not predictive of the blood pressure for patient ID=334 Observations are called dependent or correlated when one variable does predict the value of another variable The LDL cholesterol of patient ID= at age 57 is predictive of the LDL cholesterol of patient ID= at age 6 46 Heagerty, 6
71 Dependence and Correlation Recall: The variance of a variable, Y ij (fix time t j for now) is defined as: σ j = E [ (Y ij µ j ) ] = E [(Y ij µ j )(Y ij µ j )] The variance measures the average distance that an observation falls away from the mean 47 Heagerty, 6
72 Dependence and Correlation Define: The covariance of two variables, Y ij, and Y ik (fix t j and t k ) is defined as: σ jk = E [(Y ij µ j )(Y ik µ k )] The covariance measures whether, on average, departures in one variable, Y ij µ j, go together with departures in a second variable, Y ik µ k In simple linear regression of Y ij on Y ik the regression coefficient β in E(Y ij Y ik ) = β + β Y ik is the covariance divided by the variance of Y ik : β = σ jk σ k 48 Heagerty, 6
73 Dependence and Correlation Define: The correlation of two variables, Y ij, and Y ik (fix t j and t k ) is defined as: ρ jk = E [(Y ij µ j )(Y ik µ k )] σ j σ k The correlation is a measure of dependence that takes values between  and + Recall that a correlation of implies that the two measures are unrelated (linearly) Recall that a correlation of implies that the two measures fall perfectly on a line one exactly predicts the other! 49 Heagerty, 6
74 Why interest in covariance and/or correlation? Recall that on pages 8 and 9 our standard error for the sample mean difference µ µ depends on ρ In general a statistical model for the outcomes Y i = (Y i, Y i,, Y ini ) requires the following: Means: µ j Variances: σ j Covariances: σ jk, or correlations ρ jk Therefore, one approach to making inferences based on longitudinal data is to construct a model for each of these three components 5 Heagerty, 6
75 Something new to model cov(y i ) = var(y i ) cov(y i, Y i ) cov(y i, Y ini ) cov(y i, Y i ) var(y i ) cov(y i, Y ini ) cov(y ini, Y i ) cov(y ini, Y i ) var(y ini ) = σ σ σ ρ σ σ ni ρ ni σ σ ρ σ σ σ ni ρ ni σ ni σ ρ ni σ ni σ ρ ni σ n i 5 Heagerty, 6
76 Mean and Covariance Models for FEV Models: E(Y ij X i ) = µ ij (regression) cov(y i X i ) = Σ i = Z i DZ T i } {{ } + R }{{} i between subjects within subjects Q: What are appropriate covariance models for the FEV data? 5 Heagerty, 6
77 5 Heagerty, 6 Wide Data Data array for the residuals (first rows of rmat ) age8 age age age4 age6 age8 age age age4 [,] NA NA NA NA NA [,] NA NA NA NA NA [3,] NA NA NA NA NA [4,] NA NA NA NA NA NA NA 46 [5,] NA NA NA NA NA [6,] NA NA NA NA [7,] NA NA NA NA NA [8,] NA NA NA NA NA [9,] NA NA NA NA NA NA [,] NA NA NA NA
78 age age age age 4 age age 8 age age *^ 99975*^ 6*^ age 4 5 Heagerty, 6
79 53 Heagerty, 6 Empirical Covariance Matrix: [,] [,] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,] NA NA NA [,] NA NA NA [3,] NA [4,] [5,] [6,] [7,] [8,] [9,] 687 Empirical Correlation Matrix: [,] [,] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,] NA NA NA [,] NA NA NA [3,] NA [4,] [5,] [6,] [7,] [8,] 78 [9,]
80 54 Heagerty, 6 Number of observations (pairs): [,] [,] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,] [,] [3,] [4,] [5,] [6,] [7,] [8,] 66 4 [9,] 48
81 How to build models for correlation? Mixed models random effects withinsubject similarity due to sharing trajectory Serial correlation close in time implies strong similarity correlation decreases as time separation increases 53 Heagerty, 6
82 Linear Mixed Model Regression model: mean response as a function of covariates systematic variation Random effects: variation from subjecttosubject in trajectory random betweensubject variation Withinsubject variation: variation of individual observations over time random withinsubject variation 54 Heagerty, 6
83 54 Heagerty, 6 Two Subjects Response Months
84 Levels of Analysis We first consider the distribution of measurements within subjects: Y ij = β,i + β,i t ij + e ij e ij N (, σ ) E[Y i X i, β i ] = β,i + β,i t ij = [, time ij ] β,i β,i = X i β i 55 Heagerty, 6
85 Levels of Analysis We can equivalently separate the subjectspecific regression coefficients into the average coefficient and the specific departure for subject i: β,i = β + b,i β,i = β + b,i This allows another perspective: Y ij = β,i + β,i t ij + e ij = (β + β t ij ) + (b,i + b,i t ij ) + e ij E[Y i X i, β i ] = X i β } {{ } + X i b } {{ } i mean model betweensubject 56 Heagerty, 6
86 56 Heagerty, 6 Sample of Lines FEV Age (years)
87 56 Heagerty, 6 Intercepts and Slopes Slope Deviation (b) Intercept Deviation (b)
88 Levels of Analysis Next we consider the distribution of patterns (parameters) among subjects: equivalently β i N (β, D) b i N (, D) Y i = X i β } {{ } + X i b } {{ } i + e }{{} i mean model betweensubject withinsubject 57 Heagerty, 6
89 57 Heagerty, 6 Fixed intercept, Fixed slope Random intercept, Fixed slope beta + beta * time beta + beta * time time Fixed intercept, Random slope time Random intercept, Random slope beta + beta * time beta + beta * time time time
90 Betweensubject Variation We can use the idea of random effects to allow different types of betweensubject heterogeneity: The magnitude of heterogeneity is characterized by D: b i = var(b i ) = b,i b,i D D D D 58 Heagerty, 6
91 Betweensubject Variation The components of D can be interpreted as: D the typical subjecttosubject deviation in the overall level of the response D the typical subjecttosubject deviation in the change (time slope) of the response D the covariance between individual intercepts and slopes If positive then subjects with high levels also have high rates of change If negative then subjects with high levels have low rates of change 59 Heagerty, 6
92 59 Heagerty, 6 Fixed intercept, Fixed slope Random intercept, Fixed slope beta + beta * time beta + beta * time time Fixed intercept, Random slope time Random intercept, Random slope beta + beta * time beta + beta * time time time
93 Betweensubject Variation: Examples No random effects: Y ij = β + β t ij + e ij = [, time ij ]β + e ij Random intercepts: Y ij = (β + β t ij ) + b,i + e ij = [, time ij ]β + [ ]b,i + e ij Random intercepts and slopes: Y ij = (β + β t ij ) + b,i + b,i t ij + e ij = [, time ij ]β + [, time ij ]b i + e ij 6 Heagerty, 6
94 6 Heagerty, 6 Mixed Models and Covariances/Correlation Q: What is the correlation between outcomes Y ij and Y ik under these random effects models? Random Intercept Model Y ij = β + β t ij + b,i + e ij Y ik = β + β t ik + b,i + e ik var(y ij ) = var(b,i ) + var(e ij ) = D + σ cov(y ij, Y ik ) = cov(b,i + e ij, b,i + e ik ) = D
95 6 Heagerty, 6 Mixed Models and Covariances/Correlation Random Intercept Model corr(y ij, Y ik ) = D D + σ D + σ = D D + σ = between var between var + within var Therefore, any two outcomes have the same correlation Doesn t depend on the specific times, nor on the distance between the measurements Exchangeable correlation model Assuming: var(e ij ) = σ, and cov(e ij, e ik ) =
96 63 Heagerty, 6 Mixed Models and Covariances/Correlation Random Intercept and Slope Model Y ij = (β + β t ij ) + (b,i + b,i t ij ) + e ij Y ik = (β + β t ik ) + (b,i + b,i t ik ) + e ik var(y ij ) = var(b,i + b,i t ij ) + var(e ij ) = D + D t ij + D t ij + σ cov(y ij, Y ik ) = cov[(b,i + b,i t ij + e ij ), (b,i + b,i t ik + e ik )] = D + D (t ij + t ik ) + D t ij t ik
97 64 Heagerty, 6 Mixed Models and Covariances/Correlation Random Intercept and Slope Model ρ ijk = corr(y ij, Y ik ) = D + D (t ij + t ik ) + D t ij t ik D + D t ij + D t ij + σ D + D t ik + D t ik + σ Therefore, two outcomes may not have the same correlation Correlation depends on the specific times for the observations, and does not have a simple form Assuming: var(e ij ) = σ, and cov(e ij, e ik ) =
98 Linear Mixed Model Regression model: mean response as a function of covariates systematic variation Random effects: variation from subjecttosubject in trajectory random betweensubject variation Withinsubject variation: variation of individual observations over time random withinsubject variation 6 Heagerty, 6
99 Covariance Models Serial Models Linear mixed models assume that each subject follows his/her own line In some situations the dependence is more local meaning that observations close in time are more similar than those far apart in time 6 Heagerty, 6
100 Covariance Models Define e ij = ρ e ij + ɛ ij ɛ ij N (, σ ( ρ ) ) ɛ i N (, σ ) This leads to autocorrelated errors: cov(e ij, e ik ) = σ ρ j k 63 Heagerty, 6
101 ID=, rho=5 ID=, rho=5 ID=3, rho=5 ID=4, rho=5 y y y y months months months months ID=5, rho=5 ID=6, rho=5 ID=7, rho=5 ID=8, rho=5 y y y y months months months months ID=9, rho=5 ID=, rho=5 ID=, rho=5 ID=, rho=5 y y y y months months months months ID=3, rho=5 ID=4, rho=5 ID=5, rho=5 ID=6, rho=5 y y y y months months months months ID=7, rho=5 ID=8, rho=5 ID=9, rho=5 ID=, rho=5 y y y y months months months months 63 Heagerty, 6
102 ID=, rho=9 ID=, rho=9 ID=3, rho=9 ID=4, rho=9 y y y y months months months months ID=5, rho=9 ID=6, rho=9 ID=7, rho=9 ID=8, rho=9 y y y y months months months months ID=9, rho=9 ID=, rho=9 ID=, rho=9 ID=, rho=9 y y y y months months months months ID=3, rho=9 ID=4, rho=9 ID=5, rho=9 ID=6, rho=9 y y y y months months months months ID=7, rho=9 ID=8, rho=9 ID=9, rho=9 ID=, rho=9 y y y y months months months months 63 Heagerty, 6
103 633 Heagerty, 6 Two Subjects Response line for subject average line line for subject subject dropout Months
104 Linear Mixed Model Regression model: mean response as a function of covariates systematic variation Random effects: variation from subjecttosubject in trajectory random betweensubject variation Withinsubject variation: variation of individual observations over time random withinsubject variation 64 Heagerty, 6
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