WHY MATHEMATICS?

Why Mathematics? | CellPhoneDB v5 Communication Specificity: Multimeric Complexes, Geometric Means, Label Permutations and Interaction Scores

How CellPhoneDB v5 turns curated ligand-receptor knowledge and grouped single-cell expression into ordered communication candidates while keeping scores, empirical evidence and biological claims separate.

Why mathematics is important becomes easier to see when a real decision depends on it. This guide follows the mechanism from definitions and dimensions to a worked calculation, validation, practical learning and responsible use. It keeps factual evidence separate from interpretation and treats every illustrative scenario as an example rather than a promise of results.

Choose a reading route

Represent ligands, receptors, cofactors and required complex subunits
Summarize expression for ordered sender-receiver group pairs
Test cell-type specificity with an auditable permutation null
Rank interactions without calling scores binding probabilities

Chapter map

1. Curated interaction universe
2. Ordered cell-group pairs
3. Expression prevalence gate
4. Group mean expression
5. Multimeric complex rule
6. Geometric mean
7. Observed interaction mean
8. Label permutation
9. Empirical tail probability
10. Multiple testing family
11. Cross-cell-type scaling
12. Interaction product score
13. Microenvironment restriction
14. Receptor-TF prioritization
15. Visualization and sharing
16. Governance and learning

Visit the How Mathematics Works hub for the wider series and its connections to education, reasoning, science, computing, design and careers.

1. Curated interaction universe

The central mathematical idea is concrete. A versioned database defines eligible protein and complex relationships.

A concrete situation

One ligand may bind two receptor complexes.

What the mathematics does not license

Missing or context-mismatched edges limit inference. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Record database version. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

2. Ordered cell-group pairs

Here the geometry becomes operational. Sender and receiver labels create directed comparisons.

A concrete situation

Myeloid-to-T differs from T-to-myeloid.

What the mathematics does not license

Labels may hide within-group heterogeneity. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Repeat at plausible resolutions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

3. Expression prevalence gate

This step deserves a careful pause. A partner must be detected in a declared fraction of cells.

A concrete situation

The threshold is ten percent.

What the mathematics does not license

Dropout and group size affect passage. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Publish counts and thresholds. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

4. Group mean expression

The useful question is not only how to calculate, but what the calculation assumes. Expression is averaged within each annotated group.

A concrete situation

Ligand L has mean 1.8 in senders.

What the mathematics does not license

A mean can hide bimodality. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Show distributions. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

Did You Know? In this chapter, the most important number is useful only together with its construction rule. Changing scaling, sample membership or validation design can change the numerical answer without changing the arithmetic.

5. Multimeric complex rule

A small example makes the mechanism visible. Required subunit values are combined before interaction scoring.

A concrete situation

Three subunits yield one geometric mean.

What the mathematics does not license

One absent subunit can invalidate the complex. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Inspect every subunit. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

6. Geometric mean

This is where modelling choices begin to affect evidence. The nth root of the subunit product balances scale and joint presence.

A concrete situation

Values 1, 4 and 9 form a complex summary.

What the mathematics does not license

Zeros and transformations matter. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

State zero handling. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

7. Observed interaction mean

The central mathematical idea is concrete. Partner summaries produce a statistic for each ordered group pair.

A concrete situation

L in A and R in B form one value.

What the mathematics does not license

The statistic is method-specific. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Preserve raw components. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

8. Label permutation

Here the geometry becomes operational. Cell labels are shuffled while expression values remain.

A concrete situation

One thousand reshuffles form a null.

What the mathematics does not license

Donor or condition confounding breaks exchangeability. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Use blocked designs where possible. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

9. Empirical tail probability

This step deserves a careful pause. The observed statistic is ranked against permuted values.

A concrete situation

Twelve null values exceed the observation.

What the mathematics does not license

Finite permutations limit resolution. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Report numerator and denominator. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

10. Multiple testing family

The useful question is not only how to calculate, but what the calculation assumes. Thousands of interactions and group pairs are examined.

A concrete situation

One table contains fifty thousand tests.

What the mathematics does not license

Unadjusted tails create many false positives. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Report correction and family. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

11. Cross-cell-type scaling

A small example makes the mechanism visible. Partner means are rescaled across cell types for scoring.

A concrete situation

The largest mean maps near ten.

What the mathematics does not license

Adding a cell type can change every scale. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Freeze the comparison set. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

12. Interaction product score

This is where modelling choices begin to affect evidence. Scaled partner summaries are multiplied.

A concrete situation

Scores favor both partners being specific.

What the mathematics does not license

Product scale is not biological probability. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Use rank language. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

13. Microenvironment restriction

The central mathematical idea is concrete. Spatial compartments can limit eligible cell pairs.

A concrete situation

Only co-located groups are compared.

What the mathematics does not license

A predicted niche can be circular evidence. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Use independent boundaries. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

14. Receptor-TF prioritization

Here the geometry becomes operational. Active transcription factors add downstream consistency.

A concrete situation

A receptor links to an active TF.

What the mathematics does not license

Prior links and TF scores have uncertainty. A trustworthy workflow records the choice before seeing the final test result.

Try it, check it, transfer it

Keep layers separate. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

15. Visualization and sharing

This step deserves a careful pause. Matrices, dots and networks summarize many tests.

A concrete situation

Dot size and color encode distinct quantities.

What the mathematics does not license

Visual salience can merge score and significance. Evidence becomes stronger when the limitation is tested instead of hidden.

Try it, check it, transfer it

Label every channel. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

16. Governance and learning

The useful question is not only how to calculate, but what the calculation assumes. Version annotations, thresholds, complexes, permutations and prioritizers.

A concrete situation

A student computes a three-subunit mean and empirical tail.

What the mathematics does not license

A candidate remains a hypothesis. Keeping this boundary clear makes the result easier to audit and transfer.

Try it, check it, transfer it

Write a CellPhoneDB evidence card. Then explain the result to someone who has not seen the formula. If the explanation needs a hidden assumption, add that assumption to the model record.

A seven-day practice lab

Day 1: define the decision, observation unit, X variables and Y or reference condition.
Day 2: reproduce the smallest worked example and check every dimension, unit and sign.
Day 3: build a transparent baseline and write what it assumes.
Day 4: fit the main method inside a training-only pipeline; record every parameter.
Day 5: challenge one assumption with a deliberately difficult dataset or operating regime.
Day 6: compare validation metrics with the practical workload created by errors.
Day 7: present one supported conclusion, one unresolved uncertainty and one safe next experiment.

Questions for a student, parent or mentor

Which part of the model is a mathematical definition, and which part is a choice?
What small calculation would catch a sign, scale or dimension error?
What evidence comes from training, validation and the untouched test?
What benign condition could look like a fault or prediction failure?
How would the answer change for another response, asset, school project or time period?
What should a human review before acting on the score?

A method-specific audit before deployment

CellPhoneDB begins with a curated interaction catalogue in which proteins can be ligands, receptors, cofactors or parts of multimeric complexes. The complex matters mathematically because every required subunit must be represented. Cell-level expression is summarized within annotated cell groups, so the unit of inference becomes an ordered sender-group and receiver-group pair rather than one observed molecular encounter.

The statistical method asks whether an interaction looks unusually specific to a particular pair of cell groups. It repeatedly permutes cell labels, recomputes partner means and compares the observed interaction mean with the empirical null distribution. This is a conditional randomization argument. Donor, batch, condition and tissue structure determine whether unrestricted label exchange is defensible; a small empirical p value does not measure the probability that communication occurred.

CellPhoneDB v5 also supplies a scoring module. Genes expressed in too few cells are filtered, subunits of a heteromer are combined with a geometric mean, values are scaled across cell types and the two interacting partners are multiplied. This creates a relevance score whose scale depends on preprocessing, thresholds and the set of cell types. The score is useful for ranking but is not a binding affinity or a normalized probability.

The newest workflow can prioritize results with microenvironment restrictions, receptor-linked transcription-factor activity and user-specified interaction subsets. These additions strengthen a declared biological question only when the auxiliary evidence is independent and correctly aligned. Mathematics keeps the layers honest: expression, specificity, permutation evidence, prioritization filters and biological interpretation should be stored as separate columns rather than collapsed into one confident arrow.

Frequently asked questions

What is CellPhoneDB v5?

CellPhoneDB v5 is a curated interaction resource and analysis toolkit for prioritizing potential communication between annotated cell groups from single-cell and complementary data.

How are multimeric complexes handled?

Required subunits are represented explicitly; in the v5 scoring module their group means are combined with a geometric mean before interaction partners are scored.

What does the statistical method test?

It compares an observed interaction mean for a cell-type pair with a null distribution made by permuting cell labels and recomputing the statistic.

Is its p value the probability that cells communicated?

No. It is an empirical tail measure under the label-exchange null, conditional on annotation, preprocessing, database and permutation design.

What does the v5 interaction score mean?

After expression filtering and cross-cell-type scaling, partner values are multiplied to rank expression-supported specificity. It is not a binding affinity or normalized probability.

What evidence strengthens a candidate?

Spatial proximity, receptor-linked transcription-factor activity, protein measurements, perturbation, timing and replication can support different parts of the mechanism.

Primary source and fact-check note

Kevin Troulé, Robert Petryszak, Batuhan Çakır et al., “CellPhoneDB v5: inferring cell-cell communication from single-cell multiomics data,” Nature Protocols 20, 3412–3440 (2025), DOI 10.1038/s41596-024-01137-1. Definitions and core claims in this article were checked against this primary publication. Calculations shown here were independently recomputed. Examples are educational illustrations; they are not reported industrial performance results.

Useful next reading

Keep the options open. Mathematics education supports clearer questions, transferable problem-solving skills and informed choices across science, engineering, computing, AI, statistics, design and many other careers. It does not guarantee intelligence, income, admission or employment; those outcomes depend on far more than one subject or model.

Extended laboratory: topic-specific validation and responsible use

Database audit

Version interactions, complexes and cofactors.

Annotation audit

Repeat plausible cell-group resolutions.

Complex audit

Inspect every required subunit and zero.

Permutation audit

Respect donors, conditions and batches.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Multiplicity audit

Name the complete family of tests.

Score audit

Preserve prevalence, means, scales and products.

Validation audit

Use spatial, protein, TF and perturbation evidence.

Final claim

An empirical tail or score is not communication probability.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Mechanism workshop: calculate, challenge and transfer

Workshop 1: Curated interaction universe

Begin with this mechanism: A versioned database defines eligible protein and complex relationships. Recreate the concrete case: One ligand may bind two receptor complexes. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Missing or context-mismatched edges limit inference. Complete the transfer step: Record database version. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 2: Ordered cell-group pairs

Begin with this mechanism: Sender and receiver labels create directed comparisons. Recreate the concrete case: Myeloid-to-T differs from T-to-myeloid. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Labels may hide within-group heterogeneity. Complete the transfer step: Repeat at plausible resolutions. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 3: Expression prevalence gate

Begin with this mechanism: A partner must be detected in a declared fraction of cells. Recreate the concrete case: The threshold is ten percent. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Dropout and group size affect passage. Complete the transfer step: Publish counts and thresholds. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 4: Group mean expression

Begin with this mechanism: Expression is averaged within each annotated group. Recreate the concrete case: Ligand L has mean 1.8 in senders. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: A mean can hide bimodality. Complete the transfer step: Show distributions. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 5: Multimeric complex rule

Begin with this mechanism: Required subunit values are combined before interaction scoring. Recreate the concrete case: Three subunits yield one geometric mean. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: One absent subunit can invalidate the complex. Complete the transfer step: Inspect every subunit. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 6: Geometric mean

Begin with this mechanism: The nth root of the subunit product balances scale and joint presence. Recreate the concrete case: Values 1, 4 and 9 form a complex summary. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Zeros and transformations matter. Complete the transfer step: State zero handling. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 7: Observed interaction mean

Begin with this mechanism: Partner summaries produce a statistic for each ordered group pair. Recreate the concrete case: L in A and R in B form one value. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: The statistic is method-specific. Complete the transfer step: Preserve raw components. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 8: Label permutation

Begin with this mechanism: Cell labels are shuffled while expression values remain. Recreate the concrete case: One thousand reshuffles form a null. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Donor or condition confounding breaks exchangeability. Complete the transfer step: Use blocked designs where possible. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 9: Empirical tail probability

Begin with this mechanism: The observed statistic is ranked against permuted values. Recreate the concrete case: Twelve null values exceed the observation. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Finite permutations limit resolution. Complete the transfer step: Report numerator and denominator. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 10: Multiple testing family

Begin with this mechanism: Thousands of interactions and group pairs are examined. Recreate the concrete case: One table contains fifty thousand tests. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Unadjusted tails create many false positives. Complete the transfer step: Report correction and family. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 11: Cross-cell-type scaling

Begin with this mechanism: Partner means are rescaled across cell types for scoring. Recreate the concrete case: The largest mean maps near ten. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Adding a cell type can change every scale. Complete the transfer step: Freeze the comparison set. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 12: Interaction product score

Begin with this mechanism: Scaled partner summaries are multiplied. Recreate the concrete case: Scores favor both partners being specific. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Product scale is not biological probability. Complete the transfer step: Use rank language. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Did You Know? A sophisticated adjustment can be algebraically correct and still answer the wrong scientific question when samples, labels, batches or validation boundaries are wrong.

Workshop 13: Microenvironment restriction

Begin with this mechanism: Spatial compartments can limit eligible cell pairs. Recreate the concrete case: Only co-located groups are compared. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: A predicted niche can be circular evidence. Complete the transfer step: Use independent boundaries. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 14: Receptor-TF prioritization

Begin with this mechanism: Active transcription factors add downstream consistency. Recreate the concrete case: A receptor links to an active TF. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Prior links and TF scores have uncertainty. Complete the transfer step: Keep layers separate. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 15: Visualization and sharing

Begin with this mechanism: Matrices, dots and networks summarize many tests. Recreate the concrete case: Dot size and color encode distinct quantities. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: Visual salience can merge score and significance. Complete the transfer step: Label every channel. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.

Workshop 16: Governance and learning

Begin with this mechanism: Version annotations, thresholds, complexes, permutations and prioritizers. Recreate the concrete case: A student computes a three-subunit mean and empirical tail. Write group sizes, prevalence fractions, subunit means, geometric means, observed statistics, permutation ranks, empirical tails and interaction products with dimensions, direction and normalization. Calculate the smallest non-trivial example by hand, then reproduce it in code. Challenge this limitation: A candidate remains a hypothesis. Complete the transfer step: Write a CellPhoneDB evidence card. Change exactly one annotation, threshold, subunit, permutation block, resource, score direction, candidate universe, specificity denominator, propagation setting, prize-cost scale or forest penalty. Explain which mathematical object changes, which conclusion survives and which claim must weaken. Keep measurement, model output, null evidence, interpretation and decision on separate notebook lines. Reserve complete donors, tissues, conditions or interaction families outside tuning when testing transfer.