Data Analysis (fcr-data-analysis)

FCR requires that data be analysed with appropriate statistics and that results be concise and address the objectives. For field-crop work that almost always means mixed models that respect the design (blocks, split-plots, environments) — not a one-way ANOVA on pooled plots. Analysis execution lives here; design decisions live in fcr-experimental-design.

When to trigger

• Building the analysis and results section from trial or modelling data
• A reviewer asked for the correct error structure, G×E modelling, or proper means separation
• Reconciling main effects with interactions across environments
• Evaluating a crop model against observations

Analysis norms FCR expects

1. Match the model to the design. Use a linear mixed model with the error structure implied by the layout: blocks, whole-plot vs. sub-plot errors (split-plot), environment as a factor, and correct random effects (e.g., environment, block, genotype-within-environment). A wrong error term inflates significance.
2. G×E done properly. Test and interpret genotype/treatment × environment; where ranking matters, use Finlay–Wilkinson, AMMI, or GGE biplot stability analysis. Report whether the treatment effect is consistent or environment-dependent.
3. Means separation the right way. Report estimated marginal (adjusted) means with SE / SED or LSD at a stated α; avoid bare means with significance stars and avoid over-using multiple-range tests on quantitative factors — fit a response curve instead (N, water, density).
4. Report uncertainty and effect size. Give the magnitude of the agronomic effect (e.g., kg ha⁻¹, % yield change) with intervals, and its agronomic meaning — not just p-values.
5. Check assumptions. Residual diagnostics, variance homogeneity across environments, and transformation/weighting where needed; consider spatial models for heterogeneous fields.
6. Meta-analysis. If synthesising published trials, use proper meta-analytic models (effect sizes, heterogeneity, weighting) — not vote-counting.

Crop-model evaluation

• Report fit statistics on independent validation data: RMSE, nRMSE, mean bias, modelling efficiency (EF), and (with care) R²; show observed-vs-simulated with the 1:1 line.
• Separate calibration from validation; state cultivar coefficients and model version.

Reproducibility while you work

• One analysis script regenerates every table and figure from the (raw or constructed) data.
• Set and report seeds for any stochastic/bootstrap/simulation step; pin software/package versions.
• Keep table/figure numbers matched to script outputs (supports the data-availability deposit — see fcr-reporting-and-data-policy).

Error-structure decision table (match the model to the layout)

The fastest way a methods reviewer rejects an analysis is a mismatch between the test and the trial's blocking structure. Read off the error terms the design implies.

Design	Fixed effects	Random / error terms
RCBD, one environment	treatment	block
Split-plot	whole-plot factor, sub-plot factor, interaction	block; whole-plot error; sub-plot (residual) error
MET (RCBD per site)	treatment	environment, environment×treatment, block-in-environment
Alpha-lattice	treatment	replicate, incomplete-block-in-replicate
Repeated measures over time	treatment, time, interaction	plot (subject); within-plot correlation

Worked analysis vignette (illustrative)

Illustrative; the inference logic matters, not the exact values. Take the split-plot MET above — a new wheat cultivar vs. a check, 5 N rates, 8 environments, 4 blocks each. A naive one-way ANOVA on the pooled plots tests cultivar against the residual and reports p < 0.001 for a 0.6 t ha⁻¹ advantage — the classic inflated result: cultivar is a sub-plot factor, but the environment×cultivar interaction is the right yardstick for a general claim. The mixed model (cultivar and N fixed; environment, block-within-environment, and environment×cultivar random) shows the advantage is ~0.9 t ha⁻¹ at 3 high-N sites but ~0.1 t ha⁻¹ (n.s.) at the 2 low-rainfall sites — a real G×E. Report adjusted means with SED per environment, fit an N response curve rather than pasting a/b/c letters on the 5 rates, and frame the conclusion conditionally. Same data, opposite paper: the second survives review because error structure and G×E are honored.

Anti-patterns

• One-way ANOVA on pooled plots, ignoring blocks/split-plot/environment structure
• Pseudoreplication: treating sub-samples within a plot as independent replicates
• Mean-separation letters (a/b/c) slapped on a quantitative dose — fit a curve
• Reporting significance without effect size, interval, or agronomic interpretation
• Pooling environments and hiding a strong G×E interaction

Output format

【Model】mixed model: fixed = ___, random = ___, error structure = ___
【G×E】tested? consistent vs. environment-dependent? stability method
【Means】adjusted means + SED/LSD at α; response curve where quantitative
【Effect size】magnitude (units) + interval + agronomic meaning
【Diagnostics】assumptions checked? spatial model if needed? [Y/N]
【Model eval (if any)】RMSE/nRMSE/EF on independent data
【Next】fcr-figures-and-tables

Supplementary resources

• ../../resources/external_tools.md — mixed-model and G×E packages (lme4, asreml, metan, SpATS) and crop-model tools
• ../../resources/official-source-map.md — appropriate-statistics and concise-results expectations