Table 2.2 summarises the studies that contributed data to this chapter, as well as the number of participants tested for each study. Data were collected from five different studies and are combined for the analysis. The initial piloting study was
Table 2.2: Overview of the studies that contributed data to this chapter, grouped by age and task. The values in the cells are the number of participants contributed by the study.
Experiment Age Group FDR BDR Corsi OOO
Pilot 6-7 y.o. 56 55 57 55
Pilot 7-8 y.o. 57 53 57 57
Pilot 8-9 y.o. 53 53 53 53
Pilot 9-10 y.o. 60 58 59 60
Ch. 3, Exp 1 7-8 y.o. 29 29 28 28
Ch. 3, Exp 1 8-9 y.o. 26 26 24 25
Ch. 3, Exp 1 9-10 y.o. 32 32 32 32
Ch. 3, Exp 2 7-8 y.o. 29 28 29 28
Ch. 3, Exp 2 8-9 y.o. 29 29 29 29
Ch. 3, Exp 2 9-10 y.o. 29 28 28 28
Ch. 3, Exp 3 7-8 y.o. 29 29 29 29
Ch. 3, Exp 3 8-9 y.o. 28 28 28 28
Ch. 3, Exp 3 9-10 y.o. 29 29 27 26
Ch. 4, Exp 1 8-9 y.o. 41 41 41 42
Ch. 4, Exp 1 9-10 y.o. 58 58 57 58
1 FDR = forward digit recall; BDR = backward digit recall;
OOO = odd-one-out
designed to collect data from the measures to ensure they ‘worked’, where ‘working’
involves exhibiting a number of properties expected for successful WM measure, such as sequence length effects, and minimal ceiling and floor effects. The piloting stage also provided an opportunity to update the measures if, for example, they appeared to be too easy or difficult. In practice the piloting was very successful, and minimal changes were made to the measures. Thus, the data from the pilot study are combined with the data from four subsequent experimental studies for this chapter. The experimental studies for which the measures were used are described in greater detail in Chapters 3 and 4.
Table 2.3: Sample sizes for each age group combining across experiments. The values in the Ns column show the minimum and maximum number of participants tested in each age group, as the sample sizes were not identical between tasks.
Age group Ns Mean age SD age 6-7 y.o. 55 - 57 6.95 0.302 7-8 y.o. 139 - 144 7.98 0.325 8-9 y.o. 175 - 177 9.01 0.31 9-10 y.o. 203 - 208 10 0.312
2.3.1 Participants
Participants were recruited from primary schools in Bradford, UK. The majority of the children tested were from low SES British-Pakistani neighbourhoods. Table 2.3 summarises the number of participants that were tested in each age group. Six to seven year-olds were only tested for the pilot study, explaining the low sample size in Table 2.3.
Exclusions. The sample sizes reported in Tables 2.2 and 2.3 are prior to any exclusions being made. Participants were excluded from the subsequent analyses for three reasons: (i) they had special educational needs; (ii) notes from the experimenters indicated that they were distracted/not trying during testing; or (iii) they scored at or below chance. Table 2.4 summarises the exclusions made for each task. The counts are presented such that SEN children are not included in the ‘Distracted’ column even if they were also distracted during testing. In addition, only those children scoring below chance who were not distracted or SEN are counted in the ‘Chance’ column.
SEN children were excluded for a number of reasons. The inclusion of SEN children could potentially either inflate or suppress the observed relationship between tasks. If SEN children have a specific difficulty with, say, verbal material, then that may inflate the relationship between verbal measures; they would perform poorly on both measures for reasons unrelated to the structure of WM.
Table 2.4: Overview of the number of participants excluded for each task.
Task Total N Distracted SEN Chance Final N
BDR 576 9 94 3 470
Corsi 578 9 97 0 472
FDR 585 6 101 2 476
OOO 578 18 94 35 431
1 SEN = special educational needs; Chance = performing at chance
The relationship between tasks could also be suppressed if SEN children had an equivalent global difficulty with all tasks. This could then mask performance differences and specific relationships between tasks. We also did not have information on the particular reason that individual children were categorised as SEN. Consequently, interpretation of any outcomes for SEN children would be fundamentally incomplete. In practice, the majority of SEN children were either distracted, scoring below chance, or had incomplete data.
2.3.2 Materials
All the tasks were written using PsychoPy (Peirce, 2007) and presented on touchscreen tablets (screen size: 25.7 x 14.4cm; resolution: 1920 x 1080). The same pseudorandom stimulus sequences were use for all participants constrained such that, for example, the same stimuli were not presented twice within a trial (see below for further details).
2.3.3 Procedure
Participants were always tested one-on-one with an experimenter to ensure they remained engaged with the tasks. To ensure efficient testing of large numbers of children, teams of experimenters worked within the same space, each testing a single child at a time. Testing sessions ranged from 20 to 40 minutes, with the
measures being completed alongside experimental tasks for the studies described in Chapter 3 and 4.
2.3.4 Analysis plan
For all the tasks the outcome measure was accuracy at the item level. To respect the fact that accuracy data is not normally distributed binomial logistic regression was used (Jaeger, 2008). Bayesian binomial logistic regression models were estimated in R (R Core Team, 2017) using rstanarm (Gabry and Goodrich, 2017). Visualisation of the posterior estimates will be used to evaluate the absence or presence of different effects. This will be supplemented by posterior odds and posterior contrasts where appropriate. With Bayesian estimation a single estimate of some parameter is not made, as with maximum likelihood techniques. This is because Bayes Rule cannot be solved analytically for anything but the simplest models. Instead, sampling techniques are used to determine the most likely parameter values given the data. A sampling procedure is constructed to explore the posterior distribution of probability(parameter|data) such that the frequency with which different values appear in the set of samples approximates the analytic posterior distribution. This provides the distribution of likely parameter values given the data. All the models were run for 20,000 iterations, split across 4 chains. An additional 20,000 iterations were used to
‘warm-up’ the sampling algorithm, prior the 20,000 draws from the posterior used to make the inferences below. The default, very weak, priors in rstanarm (Gabry and Goodrich, 2017) were used for this model. These are shown in Figure 2.2.
Serial position, sequence length and age. A single model is used to estimate effects of serial position, sequence length, and age group, though they will be discussed separately. For the sake of computational ease interactions between the predictors were not included. In addition, preliminary analysis of the data did not support interactions between these factors. The effect of these
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Figure 2.2: Default priors for rstanarm Version 2.15.3 for the models used here.
The figures show the priors for the intercept, coefficients, and variance parameters, respectively.
factors on accuracy was investigated with a Bayesian binomial logistic regression.
Preliminary analyses of the response times for the tasks indicated that there were not enough trials to make precise estimates of the effects of serial position or sequence length. This preliminary analysis revealed a small reduction in response times with age, but for the sake of brevity an analysis of reaction times is not included here.
Relationship between tasks. The relationship between the tasks is investigated by predicting each measure from the other measures, again using a Bayesian logistic regression.
Relationship to additional outcomes. School-based measures of academic attainment were obtained for a subset of the sample. The relationship between the measures described and academic attainment is investigated using Bayesian linear regression.