Variance component models part I

Transcription

1 Faculty of Health Sciences Variance component models part I Analysis of repeated measurements, 30th November 2012 Julie Lyng Forman & Lene Theil Skovgaard Department of Biostatistics, University of Copenhagen

2 Plan for this lecture Variance component models (aka random effects models) for repeated measurements or clustered data. Two- and three-level random effects models (Linear models for repeated measurements.) Advantages/necessity of modeling random effects. Analysis with SAS Strategies for (mis)handling repeated measurements. Ecological analysis. 2 / 60

3 university of copenhagen d e pa rt m e n t o f b i o s tat i s t i c s What are repeated measurents? Repeated measurements refer to data where the same outcome has been measured in different situations (or at different spots) on the same individuals. I 3 / 60 Special case: longitudinal means repeatedly over time.

4 university of copenhagen d e pa rt m e n t o f b i o s tat i s t i c s What is clustered data? Repeated measurements are termed clustered data when the same outcome is measured on groups of individuals from the same families/workplaces/school classes/villages/etc. 4 / 60

5 Outline The two-level model (random effects ANOVA) SAS proc mixed Variance components in general Example: A three-level model Strategies for (mis)handling repeated measurements Ecological analyses 5 / 60

6 Case study: Retinopathy of prematurity 34 premature children were assesed for retinopathy of prematurity on one eye by four leading experts. Each eye was ranked on a scale from 0 to 5 by all four experts. Outcome: Severity of desease as measured by the score. Aims: Describe desease severity in the populatioin. Assess the difference in use of the scale among the experts. 6 / 60

7 Case study: Retinopathy of prematurity Eye Expert Mean no no 1 no 2 no 3 no 4 score We expect that severity of desease varies from child to child. 7 / 60

8 Case study: Retinopathy of prematurity 8 / 60

9 Random effects ANOVA Comparison of data from k groups (individuals/clusters): The groups are of no individual interest and it is of no relevance to test whether they have identical means (most likely they don t). The groups may be thought of as representatives from a population, that we want to describe. We don t have the same amount of information as if all observations were sampled independently. Our statistical analysis must reflect this! 9 / 60

10 The two-level model We model the deviations (variances) instead of the levels (means): score y ij = µ + a i + ε ij, i = 1,..., 34 j = 1,..., 4 = population mean + between-eyes deviation + within-eye deviation Assuming the deviations (the a i s and ε ij s) are independent and normally distributed: a i N (0, ω 2 B), ε ij N (0, σ 2 W ) The deviations between and within eyes are considered random and their variances, are the so-called variance components. 10 / 60

11 Case study: Retinopathy of prematurity 11 / 60

12 Interpretaion of model parameters All observations follow the same normal distribution y ij N (µ, ω 2 B + σ 2 W ) µ is the population mean (the grand mean). ω 2 B + σ2 W is the population variance (the total variation). But: Measurements made on the same eye are correlated with the so-called intra-class correlation Corr(y i1, y i2 ) = ρ = ω 2 B ω 2 B + σ2 W 0.82 I.e. scores of the same eye tend to look more alike than scores from different eyes. 12 / 60

13 Case study: Retinopathy of prematurity Parameter estimates: Variance components (random effects): Variation Variance component Estimate %of variation Between ω 2 B % Within σ 2 W % Total ω 2 B + σ2 W % Covariates (fixed effects): Effect Parameter Estimate 95%CI Population mean µ 1.90 (1.43, 2.37) 13 / 60

14 Outline The two-level model (random effects ANOVA) SAS proc mixed Variance components in general Example: A three-level model Strategies for (mis)handling repeated measurements Ecological analyses 14 / 60

15 Case study: retinopathy of prematurity PROC MIXED DATA=eyedata COVTEST; CLASS eye; MODEL score = / SOLUTION CL; RANDOM intercept / subject=eye; RUN; Syntax is similar to PROC GLM with a MODEL-statement specifying the (linear) relationship between outcome and the covariates. Note: fixed effects only!. Random effects are specified in a separate RANDOM-statement. Cathegorical variable (fixed as well as random) must be declared with CLASS. 15 / 60

16 SAS: proc mixed output Model Information Data Set Dependent Variable Covariance Structure Estimation Method Residual Variance Method Fixed Effects SE Method Degrees of Freedom Method WORK.EYEDATA score Variance Components REML Profile Model-Based Containment Class Level Information Class Levels Values eye Dimensions Covariance Parameters 2 Columns in X 1 Columns in Z Per Subject 1 Subjects 34 Max Obs Per Subject 4 16 / 60

17 SAS: proc mixed output Number of Observations Number of Observations Read 136 Number of Observations Used 133 Number of Observations Not Used 3 Iteration History Iteration Evaluations -2 Res Log Like Criterion Convergence criteria met. Always check that the numerical optimisation has converged. 17 / 60

18 SAS: proc mixed output Covariance Parameter Estimates Cov Parm Estimate Std.Error Value Pr > Z eye <.0001 Residual <.0001 Fit Statistics -2 Res Log Likelihood AIC (smaller is better) AICC (smaller is better) BIC (smaller is better) Solution for Fixed Effects Effect Estimate Std.Error DF t Value Pr > t Alpha Lower Upper Intercept < At last: Parameter estimates, tests, goodnes of fit criteria. 18 / 60

19 Case study: Retinopathy of prematurity Testing fixed effects level variation covariates 1 within eye expert 2 between eyes group Part of the variation within eyes could be explained by systematic differences between the experts. Part of the variation between eyes could be explained by systematic differences between the groups (e.g. treatment). 19 / 60

20 Case study: Retinopathy of prematurity Are there differences among the experts? Eye Expert Mean no no 1 no 2 no 3 no 4 score Mean / 60

21 Case study: Retinopathy of prematurity PROC MIXED DATA=eyedata; CLASS eye expert; MODEL score = expert / SOLUTION CL; RANDOM intercept / subject=eye; RUN; Solution for Fixed Effects Effect Estimate Std.Error DF t Value Pr > t Alpha Lower Upper Intercept < expert < expert expert expert Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F expert <.0001 Conclusion: Highly significant differences between experts. 21 / 60

22 Case study: Retinopathy of prematurity PROC MIXED DATA=eyedata; CLASS eye expert; MODEL score = expert / SOLUTION CL; RANDOM intercept / subject=eye; RUN; Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr > Z eye <.0001 Residual <.0001 Note: Smaller estimate of residual variance because expert differences explain some of the within eye variation. 22 / 60

23 Case study: Retinopathy of prematurity PROC MIXED DATA=eyedata; CLASS eye grp; MODEL score = grp / SOLUTION CL; RANDOM intecept / subject=eye; RUN; Solution for Fixed Effects Effect Estimate Std.Error DF t Value Pr > t Alpha Lower Upper Intercept < grp grp Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F grp Conclusion: No significant difference between the two groups. 23 / 60

24 Case study: Retinopathy of prematurity Differences between scores of the same eye for expert 1 and 2: y i1 y i2 N (µ 1 µ 2, 2ωW 2 ) Typical differences (95% normal region)? ( ) ± = 0.88 ± 1.32 Differences between scores of the same eye for expert 3 and 4: y i3 y i4 N (µ 3 µ 4, 2ωW 2 ) Typical differences? ± = 0.12 ± 1.32 Don t compute normal regions unless your data is normal! 24 / 60

25 Outline The two-level model (random effects ANOVA) SAS proc mixed Variance components in general Example: A three-level model Strategies for (mis)handling repeated measurements Ecological analyses 25 / 60

26 General variance component models Generalisations of ANOVA and GLM models involving several sources of random variation, so-called variance components. Examples of sources of random variation: Environmental variation. Between regions, hospitals or countries. Biological variation. Between individuals, families or animals. Within-individual variation. Between arms, teeth, days. Variation due to uncontrollable circumstances. E.g. time of day, temperature, observer. Measurement error. 26 / 60

27 Multilevel models Variance component models are also called multilevel models. We have variation (i.e. a variance component) on each level. And possibly systematic effects (covariates) on each level. individual context/cluster context/cluster level 1 level 2 level 3 students classes schools patient clinic regions asessment eye child On all levels, we have random variation (random effect or a variance component) and possibly covariates (fixed effects). 27 / 60

28 Fixed or random effect? How do we decide whether a factor should be modeled as fixed or random? Fixed All values of the factor present (typically few, e.g. treatment). Allows inference for these particular values only. Demands a decent number of observations for each value. Random A representative sample of values of the factor is present. Allows inference to be extended beyond the values in the experiment and to the population they were sampled from. 28 / 60

29 Merits of multilevel models We get a better understanding of the various sources of variation. Certain effects may be estimated more precisely (higher power), since some sources of variation are eliminated, e.g. by making comparisons within a family. This is analogous to the paired comparison situation. When planning subsequent investigations, the knowledge of the relative sizes of the variance components will (in principle) be of help in deciding the number of repetitions needed at each level. 29 / 60

30 Drawbacks of multilevel models Their statistical analysis is more difficult. When making inference (estimation and testing), it is important to take all sources of variation into account, and effects have to be evaluated against the relevant variation. Results may be biased if one or more sources of variation are disregarded! 30 / 60

31 Necessity of multilevel models Measurements belonging together in the same cluster tend to look alike they are correlated. If we fail to take this correlation into account, we will experience: Possible bias in the mean value estimates. Too small standard errors (type 1 error) for estimates of level 2 covariates (between-cluster effects). Too large standard errors (type 2 error) for estimates of level 1 covariates (within-cluster effects) 31 / 60

32 Case study: Retinopathy of prematurity When the repeated measurements are ignored: PROC GLM DATA=eyedata; CLASS expert; MODEL score=expert / SOLUTION; RUN; Source DF Type III SS Mean Square F Value Pr > F expert Parameter Estimate Std.Error t Value Pr > t 95% Confidence Limits Intercept < expert expert expert expert Too large standard errors and type 2 error for level 1 effect! 32 / 60

33 Case study: Retinopathy of prematurity When the repeated measurements are ignored: PROC GLM DATA=eyedata; CLASS grp; MODEL score=grp / SOLUTION; RUN; Source DF Type III SS Mean Square F Value Pr > F grp Parameter Estimate Std.Error t Value Pr > t 95% Confidence Limits Intercept < grp grp Too small standard errors and type 1 error for level 2 effect! 33 / 60

34 Outline The two-level model (random effects ANOVA) SAS proc mixed Variance components in general Example: A three-level model Strategies for (mis)handling repeated measurements Ecological analyses 34 / 60

35 Case study: Prisme Outcome: Covariates: Kortisol koncentration in blood samples taken morning and evening in workers in Aarhus amt and kommune in 2007 (3536 participants) with similar follow-up in 2009 (2408 participants) Age, gender, BMI, level of education, stressors: lifeevents, Effort Reward Index. level variation covariates 3 between persons gender 2 within person: between days age, bmi, stressors 1 within person: within days time of day 35 / 60

36 Case study: Prisme Sample data from 8 men: NOTE: koncentration on logarithmic scale. 36 / 60

37 Case study: Prisme PROC MIXED DATA=prisme COVTEST; WHERE kvinde EQ 0; CLASS idnr us_aar tid; MODEL logkonc = tid / SOLUTION CL; RANDOM idnr idnr*us_aar; RUN; Covariance Parameter Estimates Cov Parm Estimate Std.Error Z Value Pr > Z idnr <.0001 idnr*us_aar 0... Residual <.0001 Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F tid <.0001 One of the variance component estimates is a zero. 37 / 60

38 Negative variance components In case on of the variance component estimates becomes negative, SAS repports a zero. What does it mean? The zero-estimate may be a chance finding due to statistical uncertainty. Or it might be the result of truly negative correlation within clusters - e.g. from competition. Correlation between days (within persons) might be negative -?? What can we do about it? Re-fit the model without the problematic random effect Use a marginal model which allows for negative correlation 38 / 60 Include exact time of day as a covariate

39 Case study: Prisme Estimated variance components Level Variation Estimate 3 between persons (ω 2 ) (10.0%) 2 between days (τ 2 ) (0.0%) 1 within days (σ 2 ) (90.0%) Total (100%) Level 2 covariates (stressors) can only have very little impact on individual cortisol koncentrations! Correlation between two measurements made on the same person: ω 2 + τ 2 ω 2 + τ 2 + σ 2 = ω 2 ω 2 + τ 2 + σ 2 = 0.10 regardless of whether they are measured on the same day or on different days! 39 / 60

40 Case study: Prisme Solution for Fixed Effects Standard Effect tid Estimate Error DF t Value Pr > t Alpha Lower Upper Intercept < tid < tid Effect of time of day is measured on log-scale. Backtransformation exp(2.0137) 7.49 yields that median levels of kortisol is an estimated 7.5 times higher in the morning than in the evening. 40 / 60

41 Outline The two-level model (random effects ANOVA) SAS proc mixed Variance components in general Example: A three-level model Strategies for (mis)handling repeated measurements Ecological analyses 41 / 60

42 Comparison of four modeling strategies Quantifying overall severity of desease in the ROP-data Four strategies for estimating the population mean method estimate (s.e.) 1: ignore repeated (0.125) 2: ordinary ANOVA (0.125) 3: average scores (0.230) 4: two-level model (0.230) 1. We (wrongfully) assume independence of all measurements 2. We estimate the mean score of these particular eyes by ordinary one-way ANOVA 3. We reduce the data to a sample of averages for the individual eyes and use one-sample formulae. 4. We estimate the population mean from the randomly sampled eyes using the two-level model. 42 / 60

43 Comments on the strategies: 1. Ignoring the repeated measurements is wrong! leads to systematic underestimation of the standard error. 2. In the ordinary one-way ANOVA the grand mean has a different meaning. leads to systematic underestimation of the standard error. 3. Taking averages is often OK. At least if the number of replicates is the same for all individuals. But we loose information on within subject variation. The expert-effect cannot be tested! 43 / 60

44 Outline The two-level model (random effects ANOVA) SAS proc mixed Variance components in general Example: A three-level model Strategies for (mis)handling repeated measurements Ecological analyses 44 / 60

45 Ecological analyses An easy way of dealing with repeated measurements: Compute summary statistics for each cluster/individual. Perform a traditional analysis on the sample of summary statistics rightfully assuming that these are independent. Summary statistics could be: Sample mean or standard deviation. AUC (area under the curve). Intercept and slope of regression line. Beware of: Loss of relevant information. Inhomogeneous variances in case of unbalanced designs. 45 / 60

46 Ecological vs two-level analysis Blood pressure and social inequity: women in 17 regions of Malmø. Covariates: Individual (level 1): low educational achievement (x) (less than 9 years of school) age group Regional (level 2): rate of people with low educational achievement (z) from the Skåne Council Statistics Office 46 / 60

47 Diastolic blood pressure and area of residence... From: 47 / 60 J. Merlo etal, J. Epidemiol Community Health 55, 2001

48 Ecological analysis Average blood pressure in region vs rate of people with low educational achievement. Size of circle indicates size of investigation. Estimate of regression coefficient: with SE Seems an important explanatory variable?!? 48 / 60

49 Results of the two-level analysis Estimates are: variances within and between regions, explained variation, and 49 / 60 intra-region correlation

50 Estimates from variance component model Covariates Individual variation Rate of variation in low education between low education between model individuals regions x σw 2 z ωb 2 none age x, age (0.170) z, age (1.345) x, z, age (0.167) (1.250) / 60

51 Ecological analysis vs the two-level model Region as a random effect could only account for 0.36% of the variation in blood pressures. Thus, regional variables such as rate of low-income will have very little impact on individual blood presures! The ecological analysis sums up the individual and the regional effects, but is not able to distinguish between the two. It overestimates the level 2 effect. It cannot be interpreted as a level 1 effect. 51 / 60

52 Individual and regional blood presure 52 / 60

53 Covariate effects on level 1 and 2 can be quite different Example: Reading ability, as a function of age and cohort: / 60

54 y Simulated data: Random effect of individual p Level Variation standard deviation (SD) 1 within individuals ˆσ W = between individuals ˆω B = / 60

55 y level 1 (individual) covariate, e.g. time/age x Level Variation SD regression coefficient 1 within individuals ˆσ W = 0.41 β x = 1.028(0.046) 2 between individuals ˆω B = / 60

56 y Add a level 2 covariate, e.g. baseline age x Level Variation SD regression coefficient 1 within individuals ˆσ W = 0.41 ˆβx = 1.033(0.046) 2 between individuals ˆω B = 1.14 ˆβz = 1.316(0.206) 56 / 60

57 Comparison of estimates within individual between individual Model ˆβx sd, ˆσ W ˆβz sd, ˆω B x (0.046) z (0.201) 1.03 x, z (0.046) (0.206) / 60

58 Effects of model misspecification Misspecification missing random effect missing z missing x Result - type 2 error for x (unpaired) - type 1 error for z (too many df s, wrong variation) - estimate of ωb 2 too big - estimate of σw 2 perhaps too big (in unbalanced designs) - estimate of ωb 2 too big or too small - estimate of σw 2 too big 58 / 60

59 Example: suicide and religion Ecological analysis of regions: Percent of suicides increases with percent of protestants. Are protestants more likely to commit suicide? Two-level model: level unit variation covariates 1 individuals within region, σ 2 W religion, x 2 regions between regions, ω 2 B % protestants, z Finding: Interaction between individual effect (x) and region covariate (z) 59 / 60

60 Example: suicide and religion More suicides among catholics in regions with many protestants. 60 / 60