Modelling executive function test performance in children

Transcription

1 Modelling executive function test performance in children Ivonne Solis-Trapala NCRM Lancaster-Warwick Node ESRC Research Festival 2008

2 Collaborative work Peter J. Diggle, Department of medicine Charlie Lewis, Department of psychology funded by the UK Economic and Social Research Council

3 Background What is executive function? Components: inhibitory control, attentional flexibility, working memory, planning Competing models Experiments conducted by Shimmon (2004)

4 Executive tests used Component Measure Version 1 Version 2 Inhibitory Stroop Day/night Abstract pattern control Attentional Card-sort Face-down Face-up flexibility (DCCS) Working Boxes tasks Scrambled Stationary memory Digit-span Backward Forward (Times 2 & 3) Planning Tower of London Subgoal No-subgoal Tower/Mixed Tower/Mixed

5 A longitudinal study 115 participants were randomised to one of two groups: Group 1 (58 participants) harder tasks precede easier tasks Group 2 (57 participants) easier tasks precede harder tasks single testing session (hard tasks) single testing session, (easy tasks) a week later hard tasks, 6 months later easy tasks, a week later hard tasks, 6 months later easy tasks, a week later Time 1 Time 2 Time 3 single testing session (easy tasks) single testing session, (hard tasks) a week later easy tasks, 6 months later hard tasks, a week later easy tasks, 6 months later hard tasks, a week later

6 Aims of the study 1. Methodological questions concerning each executive function. For example, - identify patterns on the dynamics of test performance, within single sessions and over time periods; - evaluate the influence of one test upon another. 2. Relationships between executive functions

7 Inhibitory control abstract pattern (control) 16 trials at each session a week apart 3 sessions 6 months apart

8 Methodological questions Analyse key changes in the dynamics of test performance. Compare performance between abstract pattern and day/night tests. Evaluate the influence of one test upon another. Identify factors that influence performance e.g. age.

9 Modelling approach of IC data We assume the existence of an unobservable underlying ability, for each child. We represent such unobservable ability by a subject specific effect. Conditional on the subject specific effect we specify a dynamic model (Aalen et al, 2004) for each series of dependent outcomes. We extend the model to include the effect of time between test sessions.

10 Model specification: Part I conditioning on the past and subject specific effects Y ijk i = 1,...,32 j = 1,...,115 k = 1,2,3 π ijk Pr { Y ijk = 1 Y i 1,j,k,S ijk,x jk,z ijk,u j ;φ } logit ( π ijk ) = log ( πijk ) 1 + π ijk = X jkβ + Z ijkδ + γ 1 Y i 1,j,k + γ 2 S ijk + U j We assume the U j s to be an independent random sample from a normal distribution.

11 Model specification. Part II Two ways of looking at longitudinal change (i) specify different sets of regression parameters at each time period logit ( π ijk ) = X jkβ k + Z ijkδ k + γ 1k Y i 1,j,k + γ 2k S ijk + U jk (ii) consider common regression parameters at three time periods and a period effect logit ( π ijk ) = X jkβ + Z ijkδ + γ 1 Y i 1,j,k + γ 2 S ijk + η k + U j

12 Likelihood factorisation Notation We omit the index k without loss of generality. Let φ = (β,δ,γ 1,γ 2,η) and W ij = (X j,z ij,y i 1,j,S ij ). Thus η ij = logit(π ij ) = W ijφ + U j Vector φ contains the parameters of primary interest, and U j s are regarded as nuisance parameters.

13 Likelihood factorisation (cont.) The likelihood function is proportional to j exp ( i y ij η ij + U j t j ) i {1 + exp(η ij + U j )}, where t j = i y ij, and can be expressed as: exp { } l y lj η ij + U j t j j L i {1 + exp(η ij + U j )} j = j exp { i y ij η ij } L exp { l y lj η ij } L M (φ,u j ;t j ) L C (φ;y ij t j ) j

14 Statistical inference Statistical inference for φ based on j L C (φ;y ij t j ) above is suitable because it does not make distributional assumptions about the subject-specific effects; however regression coefficients of covariates that do not change within cluster are non-identifiable. Therefore we adopt a random effects model, but we compare our results for the identifiable parameters.

15 Results Different sets of regression parameters at each time period Table: MLE of parameters from random effects model (i) Para- Time 1 Time 2 Time 3 meters Est. SE Est. SE Est. SE Age β Test δ Gp δ T gp δ Pr. ob. η S. ord. η LogL:

16 Estimated posterior modes of random effects Time Time 2 Time

17 Separating between and within effects of covariates Neuhaus (2006) suggests to separate the effects of covariates in generalised linear mixed effects models in order to avoid a potential model misspecification. Note that separation of covariates into within- (W ij W j ) and between- ( W j ) components in the conditional likelihood L C yields: exp { i y ij (W ij W j ) η ij } L exp { l y lj (W ij W j ) η ij } Thus the conditional approach only estimates withincomponents of covariates effects.

18 Results: Separating between and within cluster age effect Common regression parameters at three time periods Table: MLE of parameters from a random effects model (ii) Parameters Model 1 Model 2 Estimate SE Estimate SE Age β Age mean β B Age dif. β W Test (DN vs. AP) δ Group (2 vs. 1) δ Prev. obs. η Serial order η Time (2 vs. 1) γ Time (3 vs. 1) γ Test group δ Log-likelihood:

19 Graphical representation of results age mean, x j age diff., ( x xj) jk DN/AP, z jk1 previous succ., y i - 1,j,k serial order, s ijk succ/fail, y ijk group, z jk2 child's IC ability, U j time point, t jk

20 Results in words Maximum likelihood estimates of regression coefficients suggest: 1. A fatigue effect in the performance of a given child, as indicated by the negative effect associated to trial index (ˆη 2 = 0.043, se(ˆη 2 ) = 0.008). In contrast, 2. a success in the previous trial increases the chances of success in subsequent trials (ˆη 1 = 2.05, se(ˆη 2 ) = 0.075). 3. Children perform better at the AP task than at the DN task (ˆδ 1 = 1.082, se(ˆδ 1 ) = 0.098). 4. Children who took AP before DN task performed better than those who took the test in the reverse order ˆδ 2 = 0.38, se(ˆδ 2 ) = 0.19),

21 Working memory stationary vs. scramble boxes sequences of succ/fail until retrieving 6 sweets 3 sessions 6 months apart

22 Boxes tasks data Let Z jk = (z ijk,...,z nj jk) fail/succ to retrieve a sweet in n j trials at time period k Let S ijk = 5 i l=1 z ljk No. of sweets that remain to be retrieved at trial i th and time period k. We model P ijk = Pr(z ijk = 1 s ijk = s), for s = 1,...,5 as logit(p ijk ) = α s + X ijkβ k + γ k + U j

23 Statistical inference The parameters of primary interest are the regression parameters and the subject-specific effects are regarded as nuisance parameters. Recall that S ijk = 5 i l=1 z ljk. The likelihood function is: L(α s,β;z jk ) = jk {[ s 1 failures ( 1 Pijk ) ] P ijk } f (U j ;θ)du j, where f (U j ;θ) is the density function of the latent variable U j

24 Statistical inference (cont.) As with the inhibitory control data we adopt a random effects model, but we also compare results with a conditional likelihood approach. Similarly we investigate for within and between effects of age.

25 Table: MLE of parameters for boxes tasks data from a random effects model Parameters Estimate SE Age mean β B Age dif. β W Test (Scr vs. Sta) δ Group (2 vs. 1) δ Time (2 vs. 1) γ Time (3 vs. 1) γ Test*group δ Time (2 vs. 1)*Test γ 2 δ Time (3 vs. 1)*Test γ 3 δ Log-likelihood:-2170

26 Recall... Test order at each time point Week 1 Group 1: harder tasks Group 2: easier tasks Week 2 Group 1: easier tasks Group 2: harder tasks

27 Plots of overall logodds for boxes tasks data Logodds Boxes stationary Boxes scrambled Logodds Group 1 Group Time Time Logodds Boxes stationary, Gp 1 Boxes stationary, Gp 2 Logodds Boxes scrambled, Gp 1 Boxes scrambled, Gp Time Time

28 Impurity of boxes tests strong effect of order (of a different nature to that of IC tests) children who took the easy test version first, performed better at the stationary but not at scramble version

29 Concluding remarks We investigated how succ/fail in previous trials affect future performance, aggregates of succ/fail will loose information on the dynamics of the sequence. There is value in separating practice effects from age effects. Finally, we emphasize that testing order should not be ignored.

30 References Aalen, O., Fosen J., Weedon-Fekjær, H., Borgan, Ø., and Husebye E. (2004) Dynamic analysis of multivariate failure time data. Biometrics 60, Neuhaus, J.M. and McCulloch, C.E. (2006) Separating between and within cluster covariate effects by using conditional and partitioning methods. JRSS B Shimmon, K. L. (2004).The development of executive control in young children and its relationship with mental-state understanding: a longitudinal study. Ph.D. Thesis, Lancaster University.