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

Transcription

1 Analysis of Longitudinal Data Patrick J Heagerty PhD Department of Biostatistics University of Washington Auckland 8

2 Session One Outline Examples of longitudinal data Scientific motivation Opportunities Issues Time scales Cross-sectional contrasts Longitudinal contrasts Exploratory data analysis between- and within-person variation correlation / covariance Auckland 8

3 Longitudinal Data Analysis ITRODUCTIO to EXAMPLES AD ISSUES 3 Auckland 8

4 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? 4 Auckland 8

5 4- Auckland fev age

6 Continuous Longitudinal Data Example : Beta-carotene and vitamin E a phase II study to ascertain the pharmacokinetics of beta-carotene 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? Dose-response? Relationship between beta-carotene and vitamin E? 5 Auckland 8

7 5- Auckland 8 Dose = 45 Subject = 9 Subject = Subject = Subject = 3 beta-carotene (solid) beta-carotene (solid) beta-carotene (solid) beta-carotene (solid) months since randomization months since randomization months since randomization months since randomization Subject = 3 Subject = 3 Subject = 46 Subject = 47 beta-carotene (solid) beta-carotene (solid) beta-carotene (solid) beta-carotene (solid) Vitamin E (dashed) months since randomization months since randomization months since randomization months since randomization

8 5- Auckland 8 Dose = 45 Subject = 9 Subject = Subject = Subject = 3 log(beta-carotene) (solid) log(beta-carotene) (solid) log(beta-carotene) (solid) log(beta-carotene) (solid) months since randomization months since randomization months since randomization months since randomization Subject = 3 Subject = 3 Subject = 46 Subject = 47 log(beta-carotene) (solid) log(beta-carotene) (solid) log(beta-carotene) (solid) log(beta-carotene) (solid) Vitamin E (dashed) months since randomization months since randomization months since randomization months since randomization

9 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? 6 Auckland 8

10 6- Auckland 8

11 Illness Percent Employed= Employed= Day Stress Percent Employed= Employed= Day 6- Auckland 8

12 6-3 Auckland 8 subject = 4 subject = subject = 56 illness stress Day illness stress Day illness stress Day illness stress subject = Day illness stress subject = Day illness stress subject = Day illness stress subject = Day illness stress subject = Day illness stress subject = Day illness stress subject = Day illness stress subject = Day illness stress subject = Day

13 Continuous Longitudinal Data Example 4: SS Data SS 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: SS covariates: treatment, time Q: What s the best treatment? 7 Auckland 8

14 Schizophrenia Treatment Trial Reasons for dropout: Abnormal lab result 4 Adverse experience 6 Inadequate response 83 Inter-current illness 3 Lost to follow-up 3 Uncooperative 5 Withdrew consent 9 Other 7 This combines the 6 treatment arms 8 Auckland 8

15 8- Auckland 8 Last Visit = 8 Last Visit = 6 Last Visit = 4 SS Score SS Score SS Score Time Time Time Last Visit = Last Visit = Last Visit = SS Score SS Score SS Score Time Time Time

16 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 9 Auckland 8

17 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 follow-up visits This can alleviate recall bias Temporal order of exposures and outcomes is observed Auckland 8

18 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, 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 Auckland 8

19 5 Control for cohort effects In a cross-sectional 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 Auckland 8

20 Longitudinal Studies The benefits of a longitudinal design are not without cost There are several challenges posed: Challenges of longitudinal studies: Participant follow-up Risk of bias due to incomplete follow-up, or drop-out of study participants If subjects that are followed to the planned end of study differ from subjects who discontinue follow-up then a naive analysis may provide summaries that are not representative of the original target population 3 Auckland 8

21 Analysis of correlated data Statistical analysis of longitudinal data requires methods that can properly account for the intra-subject correlation of response measurements If such correlation is ignored then inferences such as statistical tests or confidence intervals can be grossly invalid 4 Auckland 8

22 3 Time-varying 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? 5 Auckland 8

23 5- Auckland 8 USRDS Dialysis Data ID = EpoDose month Hematocrit month

24 5- Auckland 8 USRDS Dialysis Data ID = 765 EpoDose month Hematocrit month

25 Longitudinal Studies The Scientific Opportunity Observe individual changes over time Characterize the time-course of disease Outcome Measures A single outcome at a fixed follow-up time The time until an event occurs Today s focus: Repeated measures taken over time 6 Auckland 8

26 Motivation Cystic Fibrosis and Pulmonary Function Several specific aspects are of interest: What is the rate of decline in FEV? Change Is the time course different for males and females? Group Differences 3 Is the time course different for F58 homozygous subjects? Group Differences Reference: Davis PB (997) Journal of Pediatrics ext: some exploratory data analysis (EDA) 7 Auckland 8

27 Data ID = patient id FEV = percent-predicted forced expiratory volume in second AGE = age (years) GEDER = sex (=male, =female) PSEUDOA = infection with Pseudomonas Aeruginosa (=no, 3=yes) F58 = genotype (=homozygous, =heterozygous, 3=none) CREAT = pancreatic enzyme supplmentation (,=no, =yes) Auckland 8

28 8- Auckland 8 EDA: umerical Summaries Total number of subjects = umber of observations (number of subjects with ni): Distribution of males / females male female 98 umber of mutations of f

29 8- Auckland 8 EDA: umerical Summaries at entry = Median = 9655 Quartiles = 7758, 5335 Decimal point is at the colon 5 : : : : : : 3349 : : : : : : : : : 4 : 3778 : 5577 : : 8

30 ID = 57 ID = 5796 ID = 577 ID = 774 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 7 ID = 6345 ID = 884 ID = 749 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 3564 ID = 587 ID = 439 ID = 7755 FEV 5 5 FEV 5 5 FEV Auckland FEV 5 5

31 ID = 743 ID = 977 ID = 876 ID = 83 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 3399 ID = 979 ID = 8645 ID = 7 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 679 ID = 5 ID = 35 ID = 8 FEV 5 5 FEV 5 5 FEV Auckland FEV 5 5

32 ID = 9847 ID = 67 ID = 97 ID = 597 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 736 ID = 4367 ID = 63 ID = 945 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 6699 ID = 394 ID = 54 ID = 7 FEV 5 5 FEV 5 5 FEV Auckland FEV 5 5

33 ID = 837 ID = 74 ID = 76 ID = 74 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 7483 ID = 4864 ID = 786 ID = 754 FEV 5 5 FEV 5 5 FEV 5 5 FEV ID = 66 ID = 8579 ID = 383 ID = 9469 FEV 5 5 FEV 5 5 FEV Auckland FEV 5 5

34 Choosing Time Scale(s) : use AGE ij as the time variable Assumes: decline from age to age experienced is the same as that from from age to age experienced (eg no period effects) -since-entry: use AGE ij AGE i as the time variable Here: this is the same as the calendar year (for most subjects) Assumes: decline experienced from is the same for children that aged from to years old, and children that aged from to years old (eg no cohort effects) 9 Auckland 8

35 Choosing Time Scale(s) -at-entry: use AGE i as the time variable Here: this is the same as CHILD AGE in the year data collection started (eg 986) Assumes: children may be different at entry to study, but do not change further during follow-up (eg no aging effects) Auckland 8

36 - Auckland 8 FEV versus fev 5 5 slope = age

37 FEV versus -at-entry FEV 5 5 slope = at Entry FEV Change versus -since-entry FEV change slope = since Entry - Auckland 8

38 Distinguishing Cross-sectional and Longitudinal Associations Cross-sectional data i = β C X i + ɛ i, i =,, m () β C represents the difference in average across two sub-populations which differ by one unit in X EDA: plot i versus X i Auckland 8

39 Distinguishing Cross-sectional and Longitudinal Associations Longitudinal data ij = β C X i + β L (X ij X i ) + ɛ ij, j =,, n i i =,, m () When j =, the two equations are the same; β C has the same cross-sectional interpretation Subtract equations above to obtain ( ij i ) = β L (X ij X i ) + (ɛ ij ɛ i ) β L represents the expected change in per unit change in X EDA: plot ij i versus X ij X i Auckland 8

40 Cross-sectional Longitudinal Response 5 5 Response Months Months Both Response Months - Auckland 8

41 change in FEV males females FEV by Male/Female change in FEV f58 f58 f58 FEV by f58 - Auckland 8

42 EDA Summary Observations Systematic trends: time, gender, F58 Random variation: individual, observation Questions Two time scales? Estimation / testing for rates of decline? Models for analysis? 3 Auckland 8

43 Some References: Books Diggle PJ, Heagerty PJ, Liang K-, Zeger SL () Analysis of Longitudinal Data, Second Edition, Oxford University Press Fitzmaurice GM, Laird M, Ware JM (4) Applied Longitudinal Analysis, Wiley Singer JD, Willett JB (3) Applied Longitudinal Data Analysis, Oxford University Press Verbeke G, Molenberghs G () Linear Mixed Models for Longitudinal Data, Springer 4 Auckland 8