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Let $W_{aj\ell }$ be recall for age level $a$, processing level $j$, and replicate $\ell =1,\dots ,10$. Write $u_a=1$ for Younger and $0$ for Older. With Older and A as the reference levels in a regression factor, the additive two-factor normal linear model under treatment coding iswhere the errors are independent $N(0,{\sigma }^2)$, with common positive variance. The design matrix is fixed; subjects supply independent observations; the specified conditional mean is correct; and the absence of an interaction term means the age difference is constant across processing levels. These are model assumptions, not consequences of random assignment. The treatment allocation supports independence between processing assignment and background characteristics, but age itself was not randomized.

An older subject in A has $u_a=0$ and ${\gamma }_A=0$, so the estimated mean is $11.35$ words. Its standard error is the intercept's $0.7632$. There are $100-6=94$ residual degrees of freedom, giving the 95% confidence interval for this mean:This is a mean-response confidence interval, not a prediction interval for a new individual's recall; the latter also includes the new observation's error variance.

The second two-factor normal linear model adds age-by-processing interaction terms:Its ten free mean parameters are equivalent to one mean for each of the ten cells. To compare the fits, the null hypothesis is ${\delta }_B={\delta }_C={\delta }_D={\delta }_E=0$. The full residual sum of squares is $722.30$ on $90$ degrees of freedom. Adding interaction reduces the additive model's residual sum of squares by $190.30$, so the reduced value is $912.60$. The nested-model F-test statistic isIts p-value is $0.0002793$. Reject additivity and retain the interaction model. The significant interaction means that a single age effect averaged over all processing methods does not adequately summarize the data.

The regression intercept $11.0$ is the older-A mean. The age coefficient $3.8$ is the younger-minus-older contrast specifically in A. The processing coefficients $(-4.0,2.4,1.0,-4.1)$ compare B, C, D and E with A specifically among older subjects. The interaction coefficients $(-4.3,0.4,3.5,-3.1)$ are differences between the age contrast in each of those methods and its value in A. Adding the relevant coefficients givesThus A, C and especially D favour younger subjects in their estimated means, whereas B and E have little estimated age difference. In the younger group, D has the largest estimated recall, followed by C and A; B and E are much lower. Among older subjects C is largest, D and A are intermediate, and B and E are lowest. Comparing these orders is a description of estimates, not a claim that every pairwise difference is significant.

Each coefficient's printed standard error estimates its uncertainty; its $t$ statistic divides the estimate by that error, and its two-sided p-value uses $t_{90}$ under the relevant zero-contrast null hypothesis. For example, the older B-versus-A and E-versus-A contrasts have $p=0.00217$ and $0.00170$. The C-versus-A and D-versus-A older contrasts have $p=0.06139$ and $0.43201$. The B interaction has $p=0.01846$, while the D interaction is borderline at $0.05387$ and the other two individual interaction tests are not significant at $5\%$. These individual tests do not override the joint interaction F-test, and multiple comparisons require care. A simple younger-versus-older contrast outside A combines coefficients and must use their estimated covariance, rather than adding their marginal errors.

The residual standard error $2.833$ estimates $\sigma$. The coefficient of determination $0.7293$ is the fraction of centered sample recall variation explained by the ten-cell model. The overall $F=26.93$ with null law $F_{9,90}$ tests all nine non-intercept coefficients jointly against zero, giving extremely strong evidence that the cell means are not all equal. The sequential analysis of variance separates age, process, and their interaction; the balanced design makes the main-effect sums orthogonal, whereas the coefficient table uses the specified reference-level contrasts.

To assess pooling into three processing types, retain age-by-type interaction because the previous test rejects additivity. The reduced model $W\sim \mathrm{Age}\ast \mathrm{Type}$ has six cell means. Its null hypothesis says A and C have equal means within each age, and B and E have equal means within each age: four restrictions. Compare it with the ten-cell model usingIn fact the supplied fitted cell means and equal cell sizes determine the increase without raw observations. Pooling two ten-person cells with means $m_1,m_2$ adds $5(m_1-m_2{)}^2$ to the residual sum of squares. Here the four differences are $2.4,2.8,-0.1,1.1$, soTherefore $F_{\mathrm{pool}}=2.3083$ and $p\simeq 0.0640$. At $5\%$, the data do not reject the three-type reduction, though the result is borderline and is not proof that the pooled means are identical. This factor-level pooling test keeps the previously supported interaction structure; reducing the categories and removing all interaction simultaneously would test a different set of restrictions.

The intercept-only model has $49$ residual degrees of freedom, so the sample size is $50$. The one-covariate model and full model therefore have respectively $48$ and $47$ residual degrees of freedom. The completed sequential analysis of variance table hasThus the missing sums of squares are $133.617$ and $0.014$, and the corresponding F-statistics are approximately $203.8$ and $0.0214$. The first p-value is below $2\times 10^{-16}$; the second is approximately $0.8845$. The tiny second increment is consistent with the reported t-test because, for one added coefficient, the partial F-statistic equals the squared t-statistic up to displayed rounding.

For a one-factor analysis of variance, let $n_j,{\bar{x}}_j,s_j$ be the four seasonal sizes, means and standard deviations. The weighted grand arithmetic mean isThis agrees with the rounded annual mean $5.07$. The between-season and within-season sums of squares areThere are $4-1=3$ between-season statistical degrees of freedom, because the four fitted seasonal means are compared subject to their weighted grand-mean constraint. There are $N-4=6882$ within-season statistical degrees of freedom, because one mean was estimated in each group. The total has $N-1=6885$, equal to their sum. HenceUnder $H_0:{\mu }_W={\mu }_{Sp}={\mu }_{Su}={\mu }_F$, compare with the F-distribution on $(3,6882)$ statistical degrees of freedom. The $5\%$ critical value is about $2.60620$, with p-value approximately $8.9\times 10^{-63}$ from the rounded summaries. Reject equality of all four means very strongly under the stated model. The omnibus hypothesis test establishes that at least one mean differs; it does not identify every pairwise difference.

The summer variance is substantially lower than the other three, so the exact common-variance ANOVA assumptions deserve checking, as does temporal independence. Raw observations would permit diagnostics and a heteroscedastic or time-dependent analysis. As a cross-check on arithmetic, $S{S}_B+S{S}_W=64499.774183$ implies annual sample standard deviation $3.06074$, matching the table's $3.06$. The numerical significance must not be mistaken for verification of the unobserved Attachment C or of the sampling model.

Peter Rostron is a co-editor, with Michael H. Ramsey and Stephen L. R. Ellison, of the 2019 guide to measurement uncertainty arising from analytical sampling. Its editorial page identifies him with the University of Sussex. The guide connects laboratory analysis with the earlier act of collecting material.^[ref]

An illuminating experimental design question is where to place replication. Repeating an analysis of one collected sample reveals a different source of variation from collecting another sample from the target material. A nested design can distinguish the corresponding variance components. The guide also describes an unbalanced sampling-uncertainty design with two analyses of one field sample and one analysis of its field duplicate.^[ref]

An original counting example makes the design concrete. At each of ten targets, two collected samples each analysed twice require $10\times 2\times 2=40$ analytical results. Analysing the first sample twice and its field duplicate once instead requires $10\times (2+1)=30$. The design retains a within-sample analytical contrast at each target and a contrast between separately collected samples, with fewer analytical results. The information content and the appropriate analysis of variance must nevertheless follow the actual replication pattern; thirty results do not automatically have the same precision as forty.

This distinction matters when deciding whether to spend effort on another instrument reading or on another collection. Rostron's verified contribution here is editorial participation in the shared guidance, rather than sole authorship of every method it discusses. Repeated collection can reveal variability without, by itself, proving the absence of a common collection bias.

An analysis of variance for a design containing repeated observations on the same experimental units, with an appropriate within-unit error structure. It differs from treating every recorded response as independent. The allocation and covariance assumptions, not just the factor count, determine the applicable tests.

For nonsingular within-group scatter $E$ and between-group scatter $H$, the statistic is ${\prod }_j(1+{\theta }_j{)}^{-1}$, where ${\theta }_j$ solve the generalized eigenvalue problem $Hv=\theta Ev$. It is small when the group means differ strongly relative to within-group variation. Maximizing a common-covariance normal likelihood under equal and unrestricted means gives the likelihood ratio ${\Lambda }^{N/2}$. Its determinant factors cancel under any nonsingular change of response coordinates. For one response it is equivalent to the analysis of variance F-test.