Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 35 / 2

The experimental unit is a volunteer on a particular afternoon, since that volunteer-session receives a program assignment. There are 36 such experimental units; a volunteer is a repeated block, rather than a unit that receives only one program throughout the experiment.

The observational unit is the individual completed sketch, equivalently the session's single recorded count of correctly represented features. The separate map features contribute to that response variable; they are not independently randomized replicates. Nor are repeated sessions on one volunteer independent subjects.

Use two crossed sets of blocks in experimental design: volunteers as rows and afternoons as columns. Volunteer blocks control persistent ability, drawing style and prior experience; afternoon blocks control conditions shared that day, including common task difficulty and general practice over time. Neither partition is nested in the other.

Give every volunteer each program twice, and use each program twice each afternoon. The row-column design is then balanced for both block systems. In the additive modelthe program contrasts are orthogonal to both centered block spaces. This controls additive volunteer and afternoon effects; it does not automatically eliminate individual learning differences or carryover effects.

Start with six program sequences: $ABCABC$, $ACBACB$, $BACBAC$, $BCABCA$, $CABCAB$ and $CBACBA$. This gives two appearances of each program per row and per column. It also balances transitions: each ordered pair of distinct programs occurs five times across all adjacent sessions.

Apply restricted randomization by randomly assigning the six sequences to the six volunteers and independently permuting the three program labels. Using the supplied numbers, rank the first six from smallest to largest: the sequence indices are $(4,1,5,2,3,6)$. Assign these, in order, to volunteers 1 through 6. Rank the next three numbers attached to symbols $(A,B,C)$: their order is $(B,C,A)$. Assign actual program labels $(A,B,C)$ to these ordered symbols, so base $A\mapsto C$, base $B\mapsto A$, base $C\mapsto B$. The final ready-to-use randomized row-column design is:

Programs retain their real identities after this label permutation. The researcher should follow the table across chronological afternoons; arbitrary column permutations are deliberately excluded so that transition balance survives. Each program receives twelve sessions and occurs twice in every block. Under the additive block model, the analysis of variance has row, column, program and residual statistical degrees of freedom $5,5,2,23$, respectively, besides the grand mean. Transition balance is useful against differential carryover effects, but is not a proof that they are absent.

Consider learning and memory of the maps. Repeating an identical map can improve later scores through recall rather than the assigned program, and a skill learned using one program can transfer to another. Provide comparable familiarization before recording responses and use new, difficulty-matched maps on later afternoons. If all volunteers face the same map on a given afternoon, common map difficulty is absorbed by the afternoon block.

The balanced transitions in the proposed row-column design help with first-order carryover effects, but general practice, higher-order memory and program-dependent learning may still require an explicit interaction term or carryover effect in the analysis. Merely applying randomization to treatment labels cannot make these effects disappear.