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Section B: The pupil samples

B.1 Background

The SSA, like the AAP before it, is principally intended to produce attainment estimates for the population of pupils at a stage, whether taught in the maintained or the independent sector, however large or small their schools, and wherever they might be located. The only pupils deliberately excluded in the 2005 survey were those being taught through the medium of Gaelic, and those who were at the time in special schools. Pupils with special educational needs who were being taught in mainstream schools were not excluded, although schools could use their discretion and withdraw such pupils from their samples, before or during testing, should they consider the experience potentially or actually distressing for them.

The SSA aim to produce attainment estimates for reading, writing and numeracy by local authority as well as nationally demanded a large increase in the scale of the new survey programme compared with the AAP. This is because the proportionate sampling scheme used for decades in the AAP would not produce sufficient numbers of pupils in most of the 32 authorities to support separate reporting with acceptable margins of error.

For example, in the 2004 AAP Mathematics survey the number of schools providing P5 pupils for assessment ranged from none in Eilean Siar (one of the smallest authorities) to a maximum of 27 in Glasgow City (the largest authority), with the number of pupils therefore ranging from none to a maximum of 540 at an intended 20 pupils per school (small schools obviously would provide fewer than 20 pupils each). At S2, the number of schools ranged from none in the Shetland Islands to 21 in Glasgow City, with the number of pupils therefore ranging from zero to 630, at an intended 30 pupils per school. Under proportionate sampling, only Glasgow City would be represented by at least 500 pupils at each stage, allowing attainment reporting with margins of error of at best four percentage points or so if all 500 pupils were assessed in the same subject area. Well over a third of authorities would be represented by fewer than 150 pupils at S2, and almost two-thirds would be in this position for P5.

The SSA sample would clearly need to be much larger than a regular AAP sample to achieve its wider objectives. In order to lower the new survey burden on schools nationally, a policy decision was made to report only half the 32 authorities in the first SSA in 2005, with the other half being reported in 2006. The 16 authorities to have separate attainment reporting in this first survey (see Table B1) were not randomly selected, but were identified by HMIE on the basis of their preparedness to make best use for their own system evaluation purposes of the attainment data that would be produced for their pupils. Although not necessarily selected to be representative of all 32 Scottish authorities, the set of 16 reporting authorities does nevertheless include authorities from across the country, large and small, urban and rural, socially deprived and socially advantaged.

The general brief for producing the SSA sample for 2005 was to boost the samples that would typically represent the 'reporting' authorities in an AAP sample, whilst changing as little as possible in other respects in order to minimise the impact of the transition from AAP to SSA for the maximum number of participating schools.

But what should be the extent of the sample boosting? When producing estimated population proportions on the basis of simple random samples, a sample size of 1,000 pupils would produce an estimate with a maximum associated margin of error of around three percentage points ³. So, we might say that the estimated proportion of P3 pupils deemed to be working at Level B in numeracy is 57% plus or minus 3%, having assessed 1,000 P3 pupils nationally. With a sample size of 500 pupils the margin of error would increase to more than four percentage points. With 250 pupils the margin of error would be around six percentage points.

A decision was made to aim for sample sizes of around 400 pupils in each reporting authority ⁴, to give authority attainment estimates with margins of error of around five percentage points ⁵ (the 400 was in practice raised to 445 to allow for an estimated 10% or so pupil loss through absence). It was further decided to select a total of 1,600 pupils at each stage to represent the group of 16 non-reporting authorities, that is an average of 100 pupils per authority, the actual number per authority reflecting that authority's population size. The group of independent schools would be represented by 100 pupils at each stage.

Table B1
Division of the 32 local education authorities into 'reporting' and 'non-reporting' for the 2005 SSA

Wherever possible, and following recent practice in the AAP, primary schools would be asked to provide 20 pupils at a single stage for assessment (P3 or P5 or P7), and secondary schools would be asked to provide 30 pupils (S2). Another constraint was that around 110 schools that had been invited to participate in the pre-testing of assessment material for the survey should not be selected for survey involvement, unless absolutely unavoidable.

While both constraints could be met in the non-reporting authorities, the independent sector, and four of the reporting authorities, a different sampling strategy had to be adopted in 12 of the reporting authorities (see below), where school numbers are very small, and schools involved in task pre-testing had also to be included in the survey samples.

B.2 Sampling in non-reporting authorities and in the independent sector

The 16 non-reporting authorities were treated as a single group for sampling purposes, with the independent sector forming a separate group. In the non-reporting authority group, the usual ' AAP' two-stage proportionate sampling scheme was applied to produce the 1,600 pupils needed at each stage, with separate school samples drawn without replacement for the three primary stages.

Before sampling began, maintained schools in the non-reporting authority group were first classified by authority (16 of these) and by size (two size groups: less than 20 pupils and 20+ pupils in the relevant stage in the primary sector; less than 30 pupils and 30+ pupils in the secondary sector). The intention behind the size stratification was that any selected small schools would, for their own convenience, be asked to provide all their relevant pupils for assessment. Because every relevant pupil in selected small schools would therefore have the same, 100%, chance of selection, the schools, too, were selected with equal selection probabilities. This strategy gives every pupil in every school in a 'small school' stratum the same probability of selection, and therefore in principle produces an unbiased sample of 'small school' pupils.

In the group of larger schools at each stage in each authority, schools were drawn with probabilities of selection proportional to stage size, and each selected school was asked to provide the same number of randomly selected pupils for assessment at the stage concerned (20 in primaries and 30 in secondaries). Again, this strategy, the usual AAP strategy, gave every pupil in a 'large school' stratum an equal chance of selection, producing an unbiased pupil sample.

The numbers of pupils at each stage selected to represent the various strata (school size groups in 16 authorities) reflected respective pupil population sizes. In other words, the sampling was proportionate.

No school stratification was imposed in the independent sector before sampling was carried out. Rather, the requirement for around 100 pupils at a stage, with 20 in each primary school and 30 in each secondary school, determined the number of schools needed, and these were selected with probability of selection proportional to size.

B.3 Sampling in reporting authorities

In each reporting authority, the intention was to draw a sample of around 445 pupils, to allow for a typical 10% loss of pupils through absence on assessment days.

Four of the 16 reporting authorities are sufficiently large in terms of school and pupil numbers that it was possible to use the usual AAP sampling strategy to produce their pupil samples. These authorities are Edinburgh City, Highland, North Lanarkshire and South Lanarkshire. Within each of these authorities maintained schools were stratified by size (as above) prior to sampling, and the same two sampling strategies described above for the non-reporting authorities were applied, with separate pupil samples being drawn at each of the four stages (with no schools selected at more than one stage).

In the remaining 12 reporting authorities, a different sampling strategy was needed, because in most of these authorities there were simply too few schools available in either sector for the usual restriction on pupil numbers per school to be possible, viz. 20 pupils maximum at a single primary stage and 30 pupils maximum in a school at S2. In these cases all the schools in the authority in both sectors were selected by default for survey participation, and primary schools had to provide pupils at all three stages for assessment.

Given their unavoidable inclusion in the authority samples, every school in each of the 12 smaller authorities therefore had a 100% chance of survey selection. In order, therefore, to produce unbiased pupil samples for each of the 12 reporting authorities, all the schools in each authority had to provide the same proportion of their pupils for assessment rather than a fixed number as in the AAP. The proportion concerned - the sampling fraction - was given by the proportion of the authority's population size at a stage that the required 445 pupils represented: the sampling fraction varied from authority to authority and from stage to stage (e.g. 45.9% at P3 in Stirling, 41.5% at S2 in Inverclyde, 21.2% at S2 in West Lothian) ⁶.

The results of this sampling were adjusted in two ways, to make survey involvement slightly less burdensome for the schools. Firstly, in primary schools with fewer than 10 pupils at the three stages combined, all the pupils were automatically included in the sample. Secondly, where primary school pupil samples exceeded 60, as they did in a handful of cases in two or three authorities, the number was capped at 60.

B.4 Summary of sampling strategies

Non-reporting authorities

(Aberdeenshire, Argyll & Bute, Clackmannanshire, Dumfries & Galloway, Dundee City, East Lothian, Eilean Siar, Falkirk, Fife, Glasgow City, Midlothian, Moray, Orkney Islands, Scottish Borders, Shetland Islands, West Dunbartonshire)

• Around 1,600 pupils were selected at random at each stage from maintained schools to represent the whole group of 16 authorities, through 2-stage proportionate stratified sampling
• The school population was stratified by authority and school size (stage size: <20 and 20+ for primaries, <30 and 30+ for secondaries) prior to sampling
• Separate school samples were drawn for each stage, with no overlap in the primary samples
• In the small school-size strata, schools were selected by simple random sampling (equal probabilities of selection), with all pupils at the relevant stage automatically selected for assessment
• In the large school-size strata, schools were selected by pps sampling (probability of selection proportional to size of stage) and then 20 (primary stages) or 30 (S2) pupils were selected at random from within each school for assessment.

Largest reporting authorities

(Edinburgh City, Highland, North Lanarkshire, South Lanarkshire)

• Around 445 pupils were selected at random at each stage from maintained schools to represent the individual authority, through 2-stage proportionate stratified sampling
• The authority's school population was stratified by school size (stage size: <20 and 20+ for primaries, <30 and 30+ for secondaries) prior to sampling
• Separate school samples were drawn for each stage, with no overlap in the primary samples
• In the small school-size strata, schools were selected by simple random sampling (equal probabilities of selection), with all pupils at the relevant stage automatically selected for assessment
• In the large school-size strata, schools were selected by pps sampling (probability of selection proportional to size of stage) and then 20 (primary stages) or 30 (S2) pupils were selected at random from within each school for assessment.

Other reporting authorities

(Aberdeen City, Angus, East Ayrshire, East Dunbartonshire, East Renfrewshire, Inverclyde, North Ayrshire, Perth & Kinross, Renfrewshire, South Ayrshire, Stirling, West Lothian)

• Around 445 pupils were selected at random at each stage from maintained schools to represent the individual authority, through proportionate sampling
• No school sampling was involved in either sector, since every school needed to participate
• Primary schools provided pupils at all three stages (P3, P5 and P7)
• A fixed proportion of pupils was randomly selected at each relevant stage from within each school, the proportion being given by the 445 pupils needed divided by the number available in the authority's pupil population at the stage concerned
• In primary schools with fewer than 10 pupils in total across the three stages, all the pupils were selected for assessment, while in primary schools whose pupil sample would normally exceed 60 pupils, a 60-pupil maximum was applied.

Independent schools

(Nationwide)

• At each stage around 100 pupils were randomly selected for assessment, using 2-stage sampling
• Separate school samples were drawn for each stage
• Schools were selected by pps sampling, and a fixed number of pupils then selected from within each selected school for assessment: 20 pupils at the relevant stage in primaries and 30 S2 pupils in secondaries
• Schools with fewer than 20 (primary stages) or 30 (S2) pupils were to provide all their pupils for assessment.

The result of this complex sampling was an intended national pupil sample at each stage of around 9,000 pupils, or around 15% of the pupil population. The stage samples were by design disproportionate, with reporting authorities over-represented within them and non-reporting authorities under-represented. It follows that the intended SSA pupil samples were not self-weighting, and that during data analysis, when the estimated national attainment proportions were being calculated, appropriate adjustment (date weighting) would be required to compensate for the deliberate bias in authority representation. The adjustment process would also compensate for sample imbalances due to school refusals to participate and pupil absences.

The advantage of the different sampling schemes used in the different reporting authorities is that the pupil sample at every stage in an authority was designed to be fully representative of its population. This means that if all the selected schools in an authority participated in the survey and all the selected pupils were assessed, then no data weighting would be needed during authority-level analysis. The pupil samples could be treated as though they were simple random samples, with all the analysis advantages this offers.

B.5 The sampling strategy for the practical assessments

Practically-based assessment is more costly and more logistically challenging than is pencil and paper assessment, and for this reason it was decided that practical assessments would be undertaken in a subsample, rather than all of, the survey schools. The results of the practical assessments would also be reported at national level only.

Following practice in recent AAP surveys, and working on the basis of recruitment feasibility and cost, it was planned to recruit 164 practising teachers to work as itinerant field officers for the purpose of the practical assessment. These individuals would work in pairs, each pair spending a day in each of five assigned schools, organising and supervising pupil assessments, and sometimes making attainment judgments themselves. Clearly, 164 field officers in total, working in pairs, each pair visiting five schools, suggests 410 school visits in total, or just over a quarter of the schools in the main survey, with up to 8,000 pupils assessed in total over the four stages.

The practical pupil samples were to be nationally representative, i.e. there would be no over-representation of reporting authorities. Thus, if x% of the pupils in the country were in Authority X, then x% of the pupils in the nationally representative practical sample should also have been in Authority X.

There were two constraints on school sampling for the practical assessment, however. To maximise use of the field officers' time, it was decided (as in the AAP) to select for practical assessments schools that were within easy travelling of the field officers' homes, and that had at least 20 pupils at a stage in their main survey sample - or at least 20 sample pupils in P3 and P5 combined. Given the location constraint, it would not be possible to finalise the subsample of schools that would be asked to participate in the practical component of the survey until the final list of field officers was known. But a provisional sample of schools was drawn well before the survey took place, by randomly selecting schools with 20 or more pupils at one stage in their written survey sample, in appropriate numbers from each authority. The selected subsample was larger than needed, since it was expected that for one reason or another not every school in the list would be able to be visited, either because insufficient numbers of field officers would be available or because the school's location would prohibit a field officer visit.

All 32 local education authorities were invited to nominate practising teachers to serve as field officers. The numbers of field officers requested from each authority reflected the authority's relative size, in terms of pupil population. This is because assessing the x% of sample pupils from Authority X in the sample would require x% of the recruited field officers to be from Authority X, since, for efficiency reasons, field officers would generally be visiting schools in their own authorities.

In the event, just 132 teachers were nominated by their authorities to serve as field officers, and released from their schools for the required seven days each (one day of pre-survey task orientation, five days of schools visits, and one day post-survey de-briefing). As a result, working as 66 pairs, the maximum number of schools that could be visited was 330, and the maximum possible number of pupils that could be tested at each stage was something just over 1,500.

B.6 The intended pupil samples

Just over 36,000 pupils across the four stages were selected for 'written' assessment, that is approximately 15% of each stage population. The pupils were drawn from almost 1,500 different schools throughout the country: 1,246 primary schools and 1,238 secondary schools. Table B2 provides a detailed sample breakdown ⁷.

As the table shows, in the reporting authorities over 900 primary schools and just over 170 secondary schools were selected, in principle contributing a total of just under 26,500 pupils for assessment (just over 7,100 at each stage). The total number of pupils selected in each school varied from one pupil to the capped 60 pupils in the primary sector (P3, P5 and P7 combined), and from 11 pupils to 97 pupils in the secondary sector (S2). The authority target sample size of 445 pupils at each stage varied from 438 to 452 (variation caused by the fact that the sampling fraction to be applied to each school's stage roll within an authority to identify the appropriate number of pupils to select - see section B.3 - had to be dynamically adjusted in order to produce integer numbers of pupils).

In the group of 'non-reporting' authorities, 317 primary schools and 61 secondary schools were selected for survey participation, and within them a total of just over 7100 pupils were randomly selected for assessment: roughly 1,700-1,800 per stage. In all but one of the 'non-reporting' authorities, the secondary schools were all large enough to provide 30 pupils each for assessment. In primary schools, while most schools were able to supply 20 pupils at the assigned stage there were again schools that were so small that they had only one or two pupils at the stage concerned.

In the independent sector, 18 schools were randomly selected to provide pupils for assessment at the primary stages (six schools per stage) and five schools were randomly selected to provide pupils for assessment at S2. Between them, these schools were to provide for assessment 442 randomly selected pupils across the four stages.

Of all the pupils in the intended pupil sample, 55% were to be assessed for reading and 45% for numeracy, and for a random 33% of the pupils schools would be asked to submit a piece of recent writing of a given genre.

Table B2
The intended pupil samples for written assessment in the 2005 SSA
(Numbers of schools and pupils selected for survey participation)

B.7 School participation rates

Table B3 presents the statistics on school participation in the main survey (reading and numeracy).

Table B3
School participation statistics: reading/numeracy*

A very small number of schools were withdrawn by their authorities from the initial sample lists, principally because of staffing problems or school amalgamation/closure. Rather more schools declined the invitation to participate in the survey or failed to respond to the invitation by the due date. Where reasons were offered by schools for declining the invitation to participate, the most frequent reason given was 'staffing' (13 schools), followed by HMIE inspections (six schools), NFER testing (four schools), accommodation problems (four schools), and mergers (three schools). Among those schools that did agree to participate, a number failed to return completed test booklets; four secondary schools returned completed booklets for reading but not for numeracy, perhaps suggesting confusion over the status of numeracy assessment in this 'English language' survey.

The participation rate among the originally selected schools was 90% among primary schools and 82% among secondary schools. Interestingly there was no evidence of any tendency to decline to participate or to fail to return booklets the larger the pupil sample requested.

B.8 The achieved pupil samples

The numbers of pupils originally selected for participation in the assessment of reading or numeracy at the four stages were immediately reduced when schools were withdrawn from the survey sample by their authorities, and were reduced further as schools declined the invitation to participate. In addition, some schools that had agreed to participate did not in the event do so (completed tests were not returned), and this resulted in further losses in the pupil samples. Finally, in the schools that did undertake the assessments a very small number of pupils could not be assessed, because they had left the school since the sample was drawn, because they were withdrawn from the sample by the schools (a tiny number of special needs pupils), or because they were absent during the assessment period. Absence was the major contributor to pupil loss at this stage. The result of these losses is that in the primary stages around 80% of the pupils originally selected for written assessment were actually assessed, compared with just over 70% at S2 (see Table B4).

Table B4
Pupil participation statistics: reading/numeracy*

Table B5 provides details of the final analysis samples for the reporting authorities for reading and numeracy, along with the corresponding participation rates as percentages of all pupils originally selected for assessment in the authorities concerned. As the table shows, the achieved pupil sample sizes vary from 203 to 416, representing participation rates of 46% to 93%.

Table B5
Achieved pupil samples by reporting authority: reading/numeracy
(The target sample size per stage was 445; bracketed figures show the achieved samples as percentages of the intended samples)

Table B6 records the achieved authority sample sizes for reading and numeracy separately.

Table B6
Achieved pupil samples for reading and numeracy, by reporting authority
(The 445-pupil samples were divided into reading and numeracy in the ratio 55:45, i.e. 245 pupils for reading and 200 for numeracy)

The intended authority samples were selected to be fully representative of the pupil populations in the authorities. High pupil losses from the intended samples obviously threaten population representativeness. The achieved samples were therefore explored for evidence of possible bias, in terms of the only two variables available for this purpose, viz. gender and 'deprivation'.

Table B7 compares the achieved samples for reading and numeracy with the authority populations at each stage, in terms of the gender mix (shown as proportions of boys). In most cases the gender mix in the samples was close to that in the stage population, but there are also a number of instances where the samples appear unbalanced (though in only five cases is the imbalance large enough to reach statistical significance).

Table B8 provides comparative information for the deprivation variable. The table shows the proportions of all pupils in the achieved reading samples and in the achieved numeracy samples that were living in geographical areas classified as among the 20% most deprived areas in the country ⁸, and compares these proportions with the population figures. On the evidence presented in Table B8, the large majority of achieved authority samples would appear to have been representative of their populations in deprivation terms.

Gender and deprivation imbalances were redressed during attainment estimation, through appropriate data weighting.

Table B7
Gender make-up of authority populations and achieved samples
(Proportions of boys)

Table B8
Deprivation make-up of authority populations and achieved samples
(Proportions of pupils living in areas among the 20% most deprived areas in Scotland*)

B.9 Data weighting procedures

First, a note on notation. Since there are many variables involved in the computation of weights for this survey, use of conventional subscript notation would result routinely in expressions involving six or seven subscripts, which could be very difficult to read. In this section, therefore, square brackets are used rather than reduced-font subscripts. Thus the expression p _iskgdv/b will normally appear here as p[i,s,k,g,d,v/b].

The variables involved in the computation of weights for individual pupil results are as follows:

• School, designated s, ranging over all Scottish schools.
• Stages, designated k, drawn from the set {P3,P5,P7,S2}.
• Pupils within schools, designated i.
• Gender, designated g, drawn from the set {G,B,N}, standing for Girl, Boy and Not specified, respectively.
• Deprivation index d = 1 if a pupil lies within deprivation decile 1 or 2,
  = 2 if a pupil lies within deprivation deciles 3-10,
  = 0 otherwise (typically unspecified)
• Level, designated v, drawn from the set {A,B,C,D,E,F}
• Authority band, designated b. There are two categories of authority: the 16 reporting authorities, and the 16 non-reporting authorities. Reporting authorities were treated separately, each as a single band. Non-reporting authorities were considered together in a single band. Independent schools were also grouped together, regardless of their location, in a single band. Schools are, of course, completely nested in bands.

Summation over a particular subscript is indicated by a dot. Thus p[.,s,k,.,.,v/b] denotes the total number of pupils in school s at stage k tested at level v. For the special case of level, the dot represents aggregation over pupils tested at one or more levels; an asterisk is used here as a special notation to denote aggregation over all pupils, whether tested or not. Thus p[.,s,k,.,.,./b] denotes the total number of pupils tested at stage k in school s, while p[.,s,k,.,.,*/b] stands for the total pupil roll size for stage k in school s, including pupils not tested. Similarly, p[.,.,k,.,.,*/.] denotes the total size of the pupil population in Scotland at stage k.

As a convenient shorthand, a pupil at stage k with gender g and deprivation index d is referred to as belonging to the group kgd. This shorthand can also be extended to cover aggregates, so that, for example, the group k.. contains all pupils at stage k.

The quantity r[i,s,k,g,d,v/b] is of interest not so much in itself but for its contribution to the aggregate r[.,s,k,g,d,v/b], which is equal to the roll size of group kgd in school s, provided school s contributed to the kgd sample at level v, and zero otherwise.

Under certain circumstances, it can happen that the actual number of pupils sampled at a given stage in a particular school, p[.,s,k,g,d,v/b], turns out to be greater than r[.,s,k,g,d,v/b], the reported group roll size. In order to avoid such paradoxes, in practice for computing weightings this composite value is used:

{1}

Each pupil in school s, tested at level v, with gender g and deprivation index d, has weighting:

{2}

The first part of {2} is the ratio of the total roll of group kgd pupils in school s to the total number of group kgd pupils in the same school s tested at level v. It represents the weight associated with school s in group kgd at level v.

The second part of {2} is the weight associated with the whole of authority band b, computed as the ratio of the total group kgd roll in authority band b to the total group roll size considering only schools in that authority which contributed to the kgd sample at level v.

Summing {2} over pupils and schools, we should obtain

{3}

In other words, the sum of weights of all sampled pupils at level v in group kgd within an authority band should equal the total population roll size for that group within the band.

It is often convenient to normalise the basic weighting by dividing by the total roll size and multiplying by 100:

By substituting the total population roll size at stage k, p[.,.,k,.,.,*/.], for the divisor in {4}, we obtain the normalised weight for a pupil within the country, rather than within the authority alone.

To restrict attention to a particular group, simply do not aggregate over the group. For example, the expression for the weight for pupils in a given school, restricted to deprived girls, considered within the authority band, would be:

{5}

The corresponding normalised weighting would be:

{6}

Now define 0 = f[i,s,k,g,d,v/b] = 1 as the proportion of correct marks scored by pupil i from school s in the level v assessment. f[i,s,k,g,d,v/b] is undefined for p[i,s,k,g,d,v/b] = 0.

f[i,s,k,g,d,v/b] and p[i,s,k,g,d,v/b] can be abbreviated to f[i,s,v] and p[i,s,v], respectively, where there is no ambiguity. Similarly, we can usually abbreviate w[i,s,k,g,d,v/b] to w[i,s,v/b], when there is no risk of ambiguity.

Now write f ^p(i,s,v) = 1 when f[i,s,v] = p, 0 otherwise. Then f ^0.5(i,s,v) = 1 characterises a "good start" at level v in the subject, a pupil showing f ^0.65(i,s,v) = 1 is deemed to have "well-established" skills at level v, and pupils such that f ^0.8(i,s,v) = 1 are said to have "very good" attainment at level v. f ^p(i,s,v) can be written in full as f ^p(i,s,k,g,d,v/b), when necessary to avoid ambiguity.

If, now, for each sampled pupil in the group of interest, we multiply f ^p(i,s,v) by w[i,s,v] and sum over all pupils in the group, we obtain an estimate of the number of pupils achieving p relative to the corresponding group in the population.

For example

{7}

estimates the number of pupils at stage k in authority band b, of gender g and deprivation index d, achieving a "well-established" result at level v.

To express the same quantity as a percentage of all pupils at stage k in band b, of gender g and deprivation index d relative to level v, replace w[i,s,k,g,d,v/b] with the normalised w', as in

{8}

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