Varians- og regressionsanalyse

Transcription

1 Faculty of Health Sciences Varians- og regressionsanalyse Variance component models Lene Theil Skovgaard Department of Biostatistics

2 Variance component models Definitions and motivation One-way anova with random variation estimation interpretations Two-way anova with random variation Crossed random effects (interactions) Special features and pitfalls Homepages: 2 / 75

3 Traditional assumption so far Independence One observation per individual (unit) No twins/siblings... Why is this important? Otherwise some observations look alike (correlated) Variation between these become unrealistically small Number of observations become misleadingly high 3 / 75

4 Mixed models Two basic types of generalizations: Anova Variance component models Regression Random regression SAS: Proc Mixed Mix of systematic and random effects 4 / 75

5 Neglectance of correlation will lead to errors Typical errors: Wrong standard errors (too small or too big) Wrong confidence intervals (too narrow or too wide) Wrong conclusions (type I or TYpe II errors).. will be further explained 5 / 75

6 Terminology for correlated measurements Cluster/multilevel design: Same outcome (response) measured on all individuals in a number of families/villages/school classes Repeated measurements: Same outcome (response) measured in different situations (or at different spots) for the same individual. Longitudinal measurements: Same outcome (response) measured consecutively over time for each individual. Multivariate outcome: Several outcomes (responses) for each individual, e.g. a number of hormone measurements that we want to study simultaneously. 6 / 75

7 Variance component models Generalizations of ANOVA-type models, involving several sources of random variation (variance components) geographical/environmental variation between regions, hospitals or countries biological variation variation between individuals, families or animals within-individual variation variation between arms, teeth, injection sites, days variation due to uncontrollable circumstances time of day, temperature, observer measurement error 7 / 75

8 Cluster designs Typical studies involve data from: a number of family members from a sample of households pupils from a sample of school classes measurements on several spots of each individual Alternative name (for some of them): Multilevel models variation on each level (variance component) possibly systematic effects (covariates) on each level 8 / 75

9 Examples of hierarchies level 1 level 2 level 3 subjects twin pairs countries subjects families regions students classes schools spots rabbits fields sections rats visits subjects centres Measurements belonging together in the same cluster look alike (are correlated) On all levels, we may have random variation (variance components), as well as covariates 9 / 75

10 Merits of cluster designs Certain effects may be estimated more precisely, 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 be of help in deciding the number of repetitions needed at each level 10 / 75

11 Example: Number of infections Number of positive swabs in 5 family members from each of 18 families 11 / 75

12 Drawbacks of cluster designs When making inference (estimation and testing), it is important to take all sources of variation into account, and effects have to be evaluated using the relevant variation! Bias may result, if one or more sources of variation are disregarded possible bias in the mean value structure low efficiency (type 2 error) for evaluation of level 1 covariates (within-cluster effects) too small standard errors (type 1 error) for estimates of level 2 effects (between-cluster effects) 12 / 75

13 Examples Rabbits: 6 rabbits, 6 measurements on each Rats: Sections from pancreas, subdivided into fields Dogs: 2 groups, 4 treatments for each dog Eyes: Two eyes on 7 patients, 4 lenses All examples involve small data sets some of them too small to allow for trustworthy interpretations illustrative precisely because of their limited size 13 / 75

14 Example: Evaluate vaccine More precisely: We seek an estimate of the swelling due to the vaccine. Experiment: 6 rabbits, each vaccinated in 6 spots on the back Outcome y rs : swelling in cm 2, where r = 1,, R = 6 denotes the rabbit, s = 1,, S = 6 denotes the spot We have observed a total of 36 swelling areas, but we must expect swelling to be specific to the individual rabbit. 14 / 75

15 Scatter plot x-axis: Arbitrary numbering of rabbits 15 / 75

16 One-way anova Each rabbit has its own level, i.e. rabbit is a factor proc glm data=rabbit; class rabbit; model swelling=rabbit / solution; run; Model: swelling = grand mean + rabbit deviation + variation y rs = µ + α r + ε rs, ε rs N (0, σ 2 ), The variation (σ) can be regarded either as within-rabbit variation or measurement error (probably a combination of the two). 16 / 75

17 Output from anova model Source DF Type III SS Mean Square F Value Pr > F rabbit Standard Parameter Estimate Error t Value Pr > t Intercept B <.0001 rabbit B rabbit B rabbit B rabbit B rabbit B rabbit B... But: Do we get any useful information from this? We are not interested in these particular 6 rabbits, only in rabbits in general, as a species! We assume these 6 rabbits to have been randomly selected from the species. 17 / 75

18 We choose to model rabbit variation instead of rabbit levels: swelling = grand mean + between-rabbit variation + within-rabbit variation y rs = µ + a r + ε rs, where the a r s and the ε rs s are assumed independent, normally distributed with Var(a r )=ω 2 B, Var(ε rs)=σ 2 W The variation between rabbits is now a random factor ωb 2 and σ2 W are variance components, and the model is also called a two-level model 18 / 75

19 Fixed vs. random effects? Fixed: Random: all values of the factor present (typically only a few, e.g. treatment) allows inference for these particular factor values only must include a reasonable number of observations for each factor value a representative sample of values of the factor is present allows inference to be extended beyond the values in the experiment (e.g. geographical areas, classes, rabbits) is necessary when we have a covariate for this level, e.g. groups of individuals. 19 / 75

20 Formulation in terms of correlation All observations have common mean and variance: y rs N (µ, ω 2 B + σ 2 W ) But: Measurements made on the same rabbit are correlated with the intra-class correlation Corr(y r1, y r2 ) = ρ = ω 2 B ω 2 B + σ2 W Measurements made on the same rabbit tend to look more alike than measurements made on different rabbits. All measurements on the same rabbit look equally much alike. This correlation structure is called compound symmetry (CS) or exchangeability. 20 / 75

21 Estimation of variance components In balanced situations: E(MS B ) = Rω 2 B + σ2 W (R: the number of rabbits, here 6) E(MS W ) = σ 2 W and from this we get the estimates σ 2 W = MS W ω 2 B = MS B MS W R 21 / 75

22 Variances are positive! But note: It may happen that σ B 2 becomes negative! by a coincidence as a result of competition between units belonging together, e.g. when measuring yield for plants grown in the same pot In such a case, it will be reported as a zero 22 / 75

23 Reading in data in SAS data rabbit_orig; input spot $ y1-y6; datalines; a b c d e f ; run; data rabbit; set rabbit_orig; rabbit=1; swelling=y1; output; rabbit=2; swelling=y2; output; rabbit=3; swelling=y3; output; rabbit=4; swelling=y4; output; rabbit=5; swelling=y5; output; rabbit=6; swelling=y6; output; run; 23 / 75

24 Estimation in SAS proc mixed data=rabbit; class rabbit; model swelling = / s; random rabbit; run; Covariance Parameter Estimates Cov Parm Estimate rabbit Residual Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept < / 75

25 Interpretation of variance components Proportion of Variation Variance component Estimate variation Between ωb % Within σw % Total ωb 2 + σ2 W % Typical differences (95% Prediction Intervals): for spots on the same rabbit ± = ±2.16 cm 2 for spots on different rabbits ± = ±2.70 cm 2 25 / 75

26 Interpretation of variance components, cont d Approx. 2 3 of the variation in the measurements comes from the variation within rabbits. Could there be a systematic difference between the injection sites? Two-way anova: Source DF Type III SS Mean Square F Value Pr > F rabbit spot but this does not seem to be the case (P=0.26). 26 / 75

27 Design considerations, R rabbits and S spots For R=#rabbits, varying from 3 to 20: For S=#spots, varying from 1 to 10: Var(ȳ) = ω2 B R + σ2 W RS 27 / 75

28 Effective sample size If we had only one observation for each of k rabbits, how many rabbits would we then need to obtain the same precision? k = R S 1 + ρ(s 1) We have here ρ = ω2 B ω 2 B +σ2 W = = k = 12.8 Effectively, we have only approximately two independent observations from each rabbit! 28 / 75

29 Quantification of overall swelling Forget rabbit: Pool all 36 measurements, wrongly assuming independence. ˆµ = 7.367(0.155) Fixed rabbit: Estimate the mean swelling of exactly these 6 rabbits (using only within-rabbit variation). ˆµ = 7.367(0.127) Rabbit averages: Start out by taking averages for each rabbit. ˆµ = 7.367(0.267) Random rabbit: Estimate the mean swelling of rabbits as a species (in general the correct approach). ˆµ = 7.367(0.267) 29 / 75

30 Estimation of individual rabbit means Two different approaches: Traditional averages ȳ r. BLUP s (best linear unbiased predictor) rely on the assumption that individuals come from the same population, and become weighted averages: ω 2 B ω 2 B + σ2 W S ȳ r. + σ 2 W S ω 2 B + σ2 W S ȳ.. which have been shrinked towards the overall mean, ȳ.. 30 / 75

31 BLUPs vs. averages, shrinkage 31 / 75

32 Quantification for reduced dataset When the 3 smallest measurements from rabbit 2 (largest level) are omitted, the results become: Forget rabbit: We have omitted some of the largest observation. ˆµ = 7.367(0.155) ˆµ = 7.291(0.163) Fixed rabbit: rabbit 2 has a lower weight in the average due to only 3 observations ˆµ = 7.367(0.127) ˆµ = 7.291(0.136) 32 / 75

33 Quantification for reduced dataset, cont d Weighted rabbit averages: rabbit 2 has a lower weight in the average due to only 3 observations ˆµ = 7.367(0.267) ˆµ = 7.291(0.265) Unweighted rabbit averages: average for rabbit 2 has increased ˆµ = 7.367(0.267) ˆµ = 7.436(0.333) Random rabbit: rabbit 2 has a lower weight in the average due to a larger standard error ˆµ = 7.367(0.267) ˆµ = 7.390(0.298) 33 / 75

34 BLUPs for the reduced data set Larger shrinkage than before, for rabbit no / 75

35 Confidence limits for the variance components...just as a warning Intra-individual variation σ 2 W : Inter-individual variation ω 2 B : < σ 2 W < < ω 2 B < 2.48 So, we should take care not to over-interpret / 75

36 Now imagine, that rabbits are grouped in two (grp=1,2) proc mixed data=rabbit; class grp rabbit; model swelling = grp / s; random rabbit(grp); run; Cov Parm Estimate rabbit(grp) < this changes Residual < this stays the same Solution for Fixed Effects Standard Effect grp Estimate Error DF t Value Pr > t Intercept <.0001 grp grp / 75

37 Such a comparison can not be performed in the usual way (ignoring the rabbits), since we then perform the comparison/test against a wrong variation. Type I error will occur! proc glm data=rabbit; class grp; model swelling=grp / solution; run; T for H0: Pr > T Std Error of Parameter Estimate Parameter=0 Estimate Intercept grp / 75

38 Two-level model level Unit Variation Covariates 1 rabbit*spot within rabbit spot 2 rabbit between rabbits group overall mean Errors if the random rabbit variation is ignored: low efficiency (type 2 error) for evaluation of level 1 covariates (spot) too small standard errors (type 1 error) for estimates of level 2 effects (group, overall mean) 38 / 75

39 Factor diagrams In the traditional one-way anova: [I] = [R*S] [R] 0 In case of grouping: [I] = [R*S] [R] G 0 We have here used the notation arrows indicating simplifications / groupings [ ] for the random effects, corresponding to variance components on the various levels. 39 / 75

40 Example of 3-level model Number of nuclei per cell in the rat pancreas, used for the evaluation of cytostatica 4 rats (R) 3 sections for each rat (S) 5 randomly chosen fields from each section (F) Henrik Winther Nielsen, Inst. Med. Anat. Hierarchy: fields sections rats σ 2 τ 2 ω 2 Factor diagram: [I] = [R*S*F] [R*S] [R] 0 40 / 75

41 Scatter plot, with jitter Symbols indicate sections 41 / 75

42 3-level model in SAS proc mixed data=nuclei; class rat section; model nuclei= / outpm=pm s; random rat section(rat); run; Covariance Parameter Estimates Cov Parm Estimate rat section(rat) Residual Solution for Fixed Effects Standard Effect Estimate Error DF t Value Pr > t Intercept / 75

43 Estimates of variance components Proportion of Variation Variance component Estimate variation Rats ω % Sections τ % Fields σ % Total ω 2 + τ 2 + σ % Almost all variation is on the lowest level! 43 / 75

44 Typical differences between two measurements: for different fields on the same section ± = ±1.255 for different sections on the same rat ±2 2 ( ) = ±1.264 for sections on different rats ±2 2 ( ) = ± / 75

45 Correlations vary, depending on Measurements on the same section: Corr(y rs1, y rs2 ) = ω 2 + τ 2 ω 2 + τ 2 + σ 2 = Measurements on different sections of the same rat: Corr(y r11, y r22 ) = ω 2 ω 2 + τ 2 + σ 2 = Measurements from different rats are independent 45 / 75

46 Example: Compare osmolality in 2 groups 2 groups of dogs (5 resp. 6 dogs). Each treated in 4 different ways 46 / 75

47 Spaghettiplot is more illustrative Do we have repetitions? Dogs with a high level also vary more than dogs with a low level. 47 / 75

48 Residual plot, for untransformed observations We see a trumpet shape Solution: Make a logarithmic transformation 48 / 75

49 On logarithmic scale and residual plot 49 / 75

50 Two-level model level unit variation covariates 1 dog*treat within dogs grp*treat treat 2 dog between dogs group overall mean proc mixed data=dogs; class grp treat dog; model losmol=grp treat grp*treat / outpm=fit1 ddfm=satterth; random dog(grp); run; 50 / 75

51 Class Level Information Class Levels Values grp treat 4 cont00 peep10 peep15 recv dog Covariance Parameter Estimates Cov Parm Estimate dog(grp) Residual P=0.08 for test of interaction, It can be omitted. Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F grp treat <.0001 grp*treat / 75

52 Additive model, no interaction proc mixed covtest data=dogs; class grp treat dog; model losmol=grp treat / outpm=fit2 ddfm=satterth s; random dog(grp); run; Covariance Parameter Estimates Standard Z Cov Parm Estimate Error Value Pr Z dog(grp) Residual < / 75

53 Output from additive model, cont d Solution for Fixed Effects Standard Effect treat grp Estimate Error DF t Value Pr > t Intercept grp grp treat cont treat peep treat peep <.0001 treat recv Type 3 Tests of Fixed Effects Num Den Effect DF DF F Value Pr > F grp treat < / 75

54 Incorrect analysis ignoring correlation If we forget the random dog-effect, we have a traditional two-way anova: proc glm data=dogs; class grp treat; model losmol=grp treat / solution; run; The GLM Procedure Dependent Variable: losmol Source DF Type III SS Mean Square F Value Pr > F grp treat / 75

55 Incorrect analysis, cont d Standard Parameter Estimate Error t Value Pr > t Intercept B <.0001 grp B grp B... treat cont B treat peep B treat peep B treat recv B / 75 Type 2 error for effect of time (level 1 covariate) time is evaluated in an unpaired fashion Type 1 error for effect of grp (level 2 covariate) we think we have more information than we actually have (we disregard the correlation)

56 Factor diagram [Dog] Grp [I ] [Dog Treat] Grp Treat Treat We note the following: The effect of grp*treat is evaluated against dog*treat If grp*treat is not considered significant, we thereafter evaluate treat against dog*treat grp against dog, also called og(grp) 56 / 75

57 Interpretation of group effect The estimated group difference is (0.166), with a 95% confidence interval of (-0.053,0.611). But this is on a logarithmic scale! Back transformation with the exponential function yields the factor exp(0.279)=1.321, which means that group 1 lies 32.1% above group 2. The 95% confidence interval is (exp(-0.053),exp(0.611))=(0.948,1.842) 57 / 75

58 Example: Visual acuity Individuals are looking at a screen, where a light flash appears. Outcome: Example of a non-hierarchical model: time in msec. from the stimulus to the response 7 individuals (patient), 2 eyes for each individual (eye) 4 lens magnifications (power) for each eye Crowder & Hand (1990) 58 / 75

59 59 / 75

60 Covariates for visual acuity Main effects: Systematic (mean value): eye, power Random: patient Interactions: Systematic (mean value): eye*power Random: patient*eye, patient*power Example of crossed random factors: The model is non-hierarchical 60 / 75

61 Factor diagram [I ] = [Pa Ey Po] [Pa Ey] Ey Po [Pa Po] Ey [Pa] Po 0 61 / 75

62 Not quite a multilevel model, but.. Level Unit Covariates 1 single measurements Ey*Po 2 interactions 2A [Pa*Ey] Ey 2B [Pa*Po] Po 3 individuals, [Pa] overall level 62 / 75

63 proc mixed data=visual; class patient eye power; model acuity=eye power eye*power / ddfm=satterth; random patient patient*eye patient*power; run; Cov Parm Estimate patient patient*eye patient*power Residual Type 3 Tests of Fixed Effects Effect Num DF Den DF F Value Pr > F eye power eye*power / 75

64 Ex: Peak expiratory flow rate 17 subjects, 2 measurement devices Each measured twice (Bland and Altman, 1986). subject Wright mini Wright nr. X 1p1 X 1p2 X 2p1 X 2p X SD / 75

65 Illustration of all data 65 / 75

66 Variance component model p = 1,..., 17, m = 1, 2, j = 1, 2 X pmj = µ + β m + A p + C pm + ε pmj where A p N (0, ω 2 ) C pm N (0, τ 2 ) ε pmj N (0, σ 2 ) 66 / 75

67 SAS-programming proc mixed data=wright; class method id; model wr=method / ddfm=satterth s; random intercept method / subject=id; run; 67 / 75

68 Output Class Level Information Class Levels Values method 2 mini wright id Covariance Parameter Estimates Cov Parm Subject Estimate Intercept id method id Residual Solution for Fixed Effects Standard Effect method Estimate Error DF t Value Pr > t Intercept <.0001 method mini method wright / 75

69 Estimates Variance components: ω 2 = τ 2 = σ 2 = Systematic difference between measuring devices: ˆβ 1 ˆβ 2 = 6.03(8.05), P = 0.46 How can we use these?? 69 / 75

70 Precision of each method Difference between double measurements: D pm = X pmj1 X pmj2 = ε p1j1 ε p2j2 N (0, 2σ 2 ) Limits-of-agreement: ±2 2σ 2 = ± / 75

71 Agreement between the two methods Difference between single measurements by the two methods: D p = X p1j1 X p2j2 = β 1 β 2 + C p1 C p2 + ε p1j1 ε p2j2 N (β 1 β 2, 2τ 2 + 2σ 2 ) Limits-of-agreement: ±2 2(τ 2 + σ 2 ) = ±75.31 (where we have ignored the nonsignificant systematic difference between the two...) 71 / 75

72 Difference in precision?? New model: X pmj = µ + β m + A p + C pm + ε pmj A p N (0, ω 2 ) C pm N (0, τ 2 ) ε pmj N (0, σ m 2 ) proc mixed data=wright; class method id; model wr100=method / ddfm=satterth s; random intercept method / subject=id; repeated / group=method type=simple subject=id*method; run; 72 / 75

73 Strange looking results Covariance Parameter Estimates Cov Parm Subject Group Estimate Intercept id method id 0 Residual method*id method mini Residual method*id method wright We have to scale wr100=wr/100 hmmm / 75

74 Output Covariance Parameter Estimates Cov Parm Subject Group Estimate Intercept id method id Residual method*id method mini Residual method*id method wright Solution for Fixed Effects Standard Effect method Estimate Error DF t Value Pr > t Intercept <.0001 method mini method wright / 75

75 Results Precisions: Wright: σ 2 1 = mini Wright: σ 2 2 = Conclusion: Wright is better than mini Wright, but is it significantly better? No... F = σ2 2 σ 2 1 = = 1.67 F(17, 17) P = / 75