Longitudinal Inference Without Longitudinal Data

Responding to Measurement-Theory Questions about TAMP, Copulas, and Growth

Damian W. Betebenner

National Center for the Improvement of Educational Assessment

Henry I. Braun

Lynch School of Education, Boston College

Betebenner-Braun Working Update · May 14, 2026

Purpose of This Update

Henry’s comments are not asking for another explanation of copulas.

They are asking whether the current framework has found the right measurement-theoretic object:

What operations are admissible on IRT-produced test scores?
What is the difference between an interval-scale claim and a vertical-scale claim?
Does TAMP already exhaust what cross-sectional surveys can tell us?
If copulas add information, what kind of information do they add?
Should the basic reporting target be mean SGPc, median SGPc, or both?
How would the gain become visible in TIMSS or NAEP terms?

The response is: TAMP identifies the right invariant comparison language; copulas identify the missing dependence object. TAMPc keeps Braun’s operator, but moves it to fitted growth-regime space.

The Short Answer

What TAMP gets exactly right

Work in percentile space, not raw score space
Compare distributions through CDF/quantile composition
Avoid unsupported interval-scale and vertical-scale gain claims
Preserve conclusions under strictly increasing transformations

What TAMP leaves implicit

The coupling between the two marginal distributions
The assumption of exact percentile preservation
The absence of a stochastic growth regime
The uncertainty attached to any pseudo-longitudinal bridge

TAMP is not wrong. It is a boundary case. The copula framework makes the boundary explicit and then asks whether real educational data live near it.

Henry’s Questions as an Agenda

Mean or median SGPc?
Which summary best balances measurement defensibility and statistical precision?
Admissible operations.
What remains meaningful under monotone transformations of IRT score scales?
Interval versus vertical scale skepticism.
What does each claim buy, and which parts of growth inference require them?
TAMP optimality.
In the absence of true linkage, does TAMP exhaust the comparative growth information?

Linkage premium.
How much does “some species of longitudinal data” buy us once we restrict ourselves to invariant claims?
TIMSS demonstration.
What policymaker-facing contrast would show the gain from TAMP to TAMPc?
References.
Add Luce-Tukey conjoint measurement; verify the Martineau vertical-scaling citation; keep Michell central.

The deck follows those questions in order.

Claim 1: The Mean/Median Trade-off

The clean measurement-theory answer points toward median SGPc.

It is a quantile of a percentile-scale object
It is robust to tail behavior and regime skew
It aligns with the ordinal/monotone-invariant logic of the paper
It has an immediate interpretation: the typical conditional percentile in the subgroup’s growth regime

But the empirical error results point strongly toward mean SGPc.

It has better statistical behavior across the validation work
Its error is consistently smaller
Its uncertainty bands are narrower
It is more attractive for decision making

Summary	Strength	Vulnerability
Mean SGPc	Precision, error, decision stability	Easier to attack as less purely ordinal
Median SGPc	Typical growth, ordinal defensibility	Larger error, lower precision
Regime CDF	Full distributional comparison	Harder to summarize
Regime Q-Q map	Braun-style dominance claims	Requires careful uncertainty bands

Recommendation: Use mean SGPc as the primary decision statistic, but explicitly present it as a pragmatic precision choice. Pair it with median SGPc and regime-level diagnostics as measurement-theory safeguards.

Claim 2: Mean SGPc Is a Pragmatic Compromise

The choice resembles a bias-variance trade-off.

Median SGPc protects the measurement argument

It is a rank/quantile summary of a rank-based object
It resists leverage from tails and skew
It is easier to defend under admissible-transformation scrutiny
It matches the language of “typical” growth

Mean SGPc protects the decision problem

It aggregates information more efficiently
It has lower sampling error in the current results
It produces more stable comparisons
It behaves better when the task is classification, monitoring, or policy triage

The honest position is not “mean is measurement-pure.” It is: mean SGPc buys precision at the cost of admitting a little less measurement-preserving purity.

Claim 3: Means Are Not Enough

A mean gap can be real and still fail to answer Braun’s original comparative question.

Two groups may differ in mean latent SGPc because:

one regime uniformly dominates the other;
the regimes cross;
one group is more polarized;
the centers match but dispersion differs;
differences appear only in the lower or upper tail.

Three-panel schematic comparing two fitted Beta growth regimes through densities, cumulative distribution functions, and a regime Q-Q map against the diagonal. — A TAMPc comparison built from two illustrative fitted Beta regimes. The density, CDF, and Q-Q views show why distributional comparison is richer than a mean gap.

Braun’s TAMP was never merely about one number. TAMPc should inherit that distributional ambition.

Claim 4: The Right Invariance Group

The measurement issue can be stated cleanly.

Let the admissible transformations be coordinatewise strictly increasing functions:

$X^\star = a(X), \qquad Y^\star = b(Y).$

These transformations can warp score units, stretch tails, and change apparent gains.

But they do not change:

$U = F_X(X), \qquad V = F_Y(Y),$

or the copula linking $U$ and $V$ .

Equivalence classes/orbits: all score systems related by monotone transformations belong to the same measurement orbit. The copula is the dependence object that survives across the orbit.

Three Different Scale Questions

The presentation should keep these distinct.

Question	Transformation group	What it buys
Ordinal scale	Strictly increasing recodings	Order, ranks, percentiles, copula dependence
Interval scale	Positive affine recodings: $x \mapsto ax+b$	Differences, means, slopes, additive change
Vertical scale	One affine meaning across grades/construct regions	Cross-grade developmental differences on a common unit

Interval scale and vertical scale are not synonyms. Interval is a claim about meaningful differences on a scale; vertical is the stronger claim that the same difference meaning holds across heterogeneous grade/content regions.

Claim 5: Copulas Are the Maximal Invariant of Dependence

Sklar’s theorem separates the joint distribution into:

$H(x,y) = C\{F_X(x), F_Y(y)\}.$

Under ordinally defensible transformations, the margins may change expression, but the copula remains fixed.

That makes the copula more than a modeling convenience.

The copula is the measurement-invariant dependence structure.

A t-copula CDF surface over the unit square in percentile coordinates, illustrating dependence after both score margins have been transformed to ranks.

What Interval and Vertical Scales Do Not Buy

Interval and vertical scaling concern the margins, not the dependence.

What interval/vertical scaling can add

Meaningful score differences
Means and average gains on reported scores
Slopes and linear growth parameters
A stronger interpretation of scale-score change

What they do not add

More ordinal dependence information
A better copula
More information about rank mobility
A substitute for student-level linkage

Dependence lives in $C$ . Measurement choices live in the marginals.

A Literature Repair

Henry’s bibliographic point is important because it sharpens the measurement argument.

Already central or present

Braun (1988): TAMP as ordinally safe CDF/quantile comparison
Stevens (1946): permissible transformations
Michell (1997): critique of quantitative assumptions in psychology
Luce & Suppes (2002): representational measurement theory
Narens (2002): meaningfulness and transformation groups
Schulz & Nicewander (1997): IRT representations of growth

Should be added or verified

Luce & Tukey (1964): simultaneous conjoint measurement
Martineau vertical-scaling critique/dissertation: exact citation to verify
Additional vertical-scale references only after separating them from the interval-scale argument

The paper’s argument becomes stronger if it is presented as a contribution to the “meaningfulness under permissible transformations” tradition, not only as a new growth estimator.

TAMP: The Original Ordinal Move

Braun’s TAMP operator is:

$e(x) = F_1^{-1}\{F_0(x)\}.$

It maps:

score to percentile;
percentile to the later distribution;
comparison to distributional position rather than raw gain.

This is why TAMP is robust to monotone rescaling.

Reproduction of Braun's 1988 TAMP figure showing equipercentile mapping across two score distributions through an identity line in percentile space. — TAMP equipercentile mapping for two groups, reproduced with enhancements from Braun (1988).

TAMP’s enduring contribution is not an interval-scale or vertical-scale assumption. It is a way to ask a growth-like comparative question using only order and percentile position.

The Hidden Assumption in TAMP

Once TAMP is written in percentile coordinates,

$U = F_0(X), \qquad V = F_1(Y),$

the equipercentile rule becomes:

$V = U.$

That is exact rank preservation.

In copula language, this is the Fréchet-Hoeffding upper bound:

$C^+(u,v) = \min(u,v).$

TAMP implicitly supplies the missing longitudinal bridge by choosing the comonotonic copula. That is mathematically elegant, ordinally safe, and very strong.

Does TAMP Exhaust the Information?

Yes, in one precise sense.

If we insist on:

no external longitudinal information;
no dependence model;
only observed marginals;
monotone-invariant comparisons;
deterministic percentile transport;

then TAMP is the natural endpoint.

It is the monotone rearrangement coupling.

No, in the broader scientific sense.

If we allow:

a dependence template learned from real longitudinal data;
explicit bridge uncertainty;
a stochastic transition kernel;
validation against known longitudinal truth;

then TAMP does not exhaust the information.

It supplies the boundary case.

The likely theorem is not “TAMP is universally optimal.” It is closer to: TAMP is the unique deterministic, monotone-invariant, rank-preserving transport given only two marginals.

From TAMP to TAMPc

TAMP

$\text{Braun operator} + \text{implicit deterministic coupling}$

The object compared is an observed marginal distribution.

The bridge is fixed:

$V = U.$

TAMPc

$\text{Braun operator on fitted regimes} + \text{explicit stochastic coupling}$

The object compared is a latent growth-regime distribution:

$P_S \sim H_S.$

The operator survives. The object being compared changes.

What the Copula Adds

The copula supplies a transition kernel:

$F_0(v \mid u) = \frac{\partial C_0(u,v)}{\partial u}.$

The subgroup supplies the observed prior and current marginals.

The estimator solves for the latent growth regime $H_S$ whose forward image best reproduces the observed current marginal.

Observed prior-year and current-year marginal densities in pseudo-observation space, with no student-level pairings available.

Heatmap of candidate Beta growth-regime parameters, with color indicating Wasserstein distance between observed and predicted current-year CDFs.

Forward CDF check comparing the observed current-year CDF with the CDF predicted by the estimated growth regime and canonical copula kernel.

LIwLD in three steps: observe unlinked marginals, estimate a growth regime, and verify the predicted current marginal.

The added information is not interval-scale gain and not vertical-scale gain. It is a modeled, validated dependence bridge that turns two marginals into an inferable subgroup growth regime.

The Identification Boundary

TAMPc must be stated with care.

Identified or estimated

subgroup growth-regime CDF $H_S$
mean SGPc
median SGPc
dispersion and tail/bucket probabilities
regime CDF and Q-Q comparisons

Not identified without linkage

individual realized growth percentiles
student-by-student trajectories
unrestricted nonparametric dependence
causal effects of schooling without additional assumptions
interval-scale gain units
vertical-scale developmental units across grades

This boundary is a strength, not a weakness. The method is explicit about what the data plus bridge can and cannot support.

How Much Does Linkage Buy?

Longitudinal linkage buys information, but not everything people often imagine.

If the target remains invariant under monotone transformations, linkage primarily buys:

direct observation of the dependence structure;
individual conditional percentiles;
smaller uncertainty bands;
validation of the canonical bridge.

It does not license interval-scale gain interpretations by itself, and it certainly does not establish a vertical scale across grades.

For subgroups of $N \approx 2{,}500$ :

Metric	Paired	Unlinked	Ratio
SE, mean SGPc	0.60	1.48	2.5x
CI width, mean	2.16	5.65	2.6x
SE, median SGPc	0.79	1.96	2.5x
CI width, median	2.91	7.48	2.6x

Mean SGPc is not only more precise in the paired case; it remains more precise after linkage is broken.

The linkage premium is real and quantifiable. The mean/median gap is also real: mean SGPc buys additional precision even inside the rank-based framework.

Accuracy of the Bridge

Validation asks:

If we deliberately forget the links, can the cross-sectional method recover the linked-data truth?

Across 239 subgroups:

$r = 0.995$ between inferred and true mean SGPc
MAE = 0.40 SGPc points
Points cluster within a narrow +/- 2 SGPc envelope

The next reporting step is to present the comparison directly: mean SGPc as the low-error decision statistic, median SGPc as the cleaner measurement-theory companion.

Scatter plot of inferred mean SGPc versus true mean SGPc across 239 subgroups, with points tightly clustered around the identity line. — Cross-condition validation: inferred versus true mean SGPc across 239 subgroups.

The TIMSS Demonstration We Need

The policy-facing illustration should compare growth regimes, not just score distributions.

For two countries, jurisdictions, or subgroups $A$ and $B$ :

Estimate $H_A$ and $H_B$
Report mean SGPc for each as the decision statistic
Report median SGPc as the typical-growth companion
Plot the two regime CDFs
Plot the regime Q-Q map:

$Q_{B \leftarrow A}(p) = H_B^{-1}\{H_A(p)\}$

Attach replicate or bootstrap uncertainty bands

The interpretation:

Entire Q-Q curve below diagonal: $A$ stochastically dominates $B$
Curve crosses diagonal: advantage depends on which part of the growth regime we inspect
Similar medians but different spread: same typical growth, different inequality of growth
Tail differences: different mobility for initially low or high performers

This is where TAMPc becomes useful to policymakers: it separates the decision signal from the evidentiary support around it.

Median SGP Functions Across Prior Scores

Henry’s proposed comparison can be framed as a conditional median function:

$m_S(u) = Q_{V \mid U=u, S}(0.5).$

$m_A(u) > m_B(u) \qquad \forall u \in (0,1),$

then group $A$ shows higher conditional median current rank at every prior rank.

This complements TAMP because it asks a rank-conditional question: at each starting percentile, which group sits higher in the predicted current distribution?

But it is weaker than full conditional stochastic dominance.

Dominance Claims: A Ladder of Strength

Claim	What it supports
Mean SGPc $_A$ > Mean SGPc $_B$	Efficient decision signal, especially with uncertainty bands
Median SGPc $_A$ > Median SGPc $_B$	Typical growth advantage
$m_A(u) > m_B(u)$ for all $u$	Conditional median advantage across the prior range
$H_A(t) \le H_B(t)$ for all $t$	Regime-level stochastic dominance
$F_{V\|U=u,A}(v) \le F_{V\|U=u,B}(v)$ for all $u,v$	Uniform conditional stochastic dominance

The strongest claim implies the weaker ones.

The converse is generally false.

This gives us a disciplined reporting language: do not say “uniformly accelerated progress” unless the dominance evidence supports that level of claim.

A Candidate Optimality Statement

The mathematically honest statement may be:

Candidate Theorem

Given only two continuous marginal distributions and the requirement that comparisons be invariant under strictly increasing transformations of each score scale, TAMP is the canonical deterministic rank-preserving transport. It is the comonotone coupling and therefore the upper Fréchet-Hoeffding boundary case.

What TAMPc adds is not a contradiction of that result.

It adds an external, empirically validated dependence prior:

$C_0 \quad \Rightarrow \quad F_0(v \mid u) \quad \Rightarrow \quad \widehat{H}_S.$

So the research question becomes: when is the extra bridge information strong enough to justify moving beyond TAMP?

What Would Convince a Skeptic?

Measurement skeptic

No raw gain claims
Explicit transformation group
Copula as invariant dependence object
Mean SGPc defended as a precision-driven decision choice, not as pure ordinal meaning
Interval-scale and vertical-scale claims separated and justified independently

Policy skeptic

Accuracy validated by hiding real links
Linkage premium quantified
TIMSS/NAEP sample-size uncertainty bands
Comparison statements graded by strength
Clear distinction between typical, distributional, and uniform-growth claims

The method should be sold as disciplined humility: it extracts more from cross-sectional surveys, but only by saying exactly what has been assumed.

Proposed Revision to the Paper Argument

Current arc

Longitudinal inference seems impossible without longitudinal data
Copulas separate marginals from dependence
Educational dependence is stable enough to model
Cross-sectional growth regimes can be recovered

Stronger arc after Henry’s comments

Growth claims must respect admissible transformations
TAMP solved this by moving to percentile space
Interval and vertical scaling add metric marginal claims, not dependence
TAMP’s hidden bridge is comonotonic dependence
Copulas make the bridge explicit and testable
TAMPc estimates growth-regime distributions, reported through mean SGPc for decisions, median SGPc for typical growth, and dominance maps for distributional claims

The new framing makes Henry’s measurement concerns the foundation of the paper, not an appendix.

Questions for Us

Should mean SGPc be the default decision headline, with median SGPc always reported as the measurement-theory companion?
Do we want to state a formal TAMP optimality theorem, or present it as a conjecture/interpretive proposition?
Which exact Martineau vertical-scaling reference should anchor the stronger cross-grade scale skepticism section?
Should the TIMSS example prioritize:
- country-to-country comparisons,
- subgroup comparisons within countries,
- or both?
What is the minimum uncertainty display needed before we use the phrase uniformly accelerated progress?

My recommendation: settle these five questions before the next public version of the presentation or manuscript.

Working Summary

Mean SGPc should become the primary decision summary. It is consistently superior on error and precision, while median SGPc remains the cleaner typical-growth and invariance companion.
TAMP is the deterministic invariant benchmark. It likely has an optimality result under strict “two marginals only” conditions.
TAMPc is the stochastic extension. It does not replace Braun’s operator; it relocates the operator to fitted latent growth regimes.
Interval and vertical scales are separate commitments. Interval scaling licenses additive score differences; vertical scaling asks that this same unit meaning hold across grades or construct regions. Neither adds dependence information.
TIMSS should be reported through decisions plus safeguards. Mean SGPc provides the decision signal; median SGPc, regime CDFs, Q-Q maps, and uncertainty bands provide the measurement-theory guardrails.

The compact thesis: TAMP tells us what can be compared safely; copulas tell us how much more can be inferred once dependence is made explicit.

Selected References

Betebenner, D. W. (2009). Norm- and criterion-referenced student growth. Educational Measurement: Issues and Practice, 28(4), 42–51.

Braun, H. I. (1988). A new approach to avoiding problems of scale in interpreting trends in mental measurement data. Journal of Educational Measurement, 25(3), 171–191.

Luce, R. D., & Suppes, P. (2002). Representational measurement theory. In H. Pashler & J. Wixted (Eds.), Stevens’ Handbook of Experimental Psychology (3rd ed., Vol. 4, pp. 1–41). Wiley.

Luce, R. D., & Tukey, J. W. (1964). Simultaneous conjoint measurement: A new type of fundamental measurement. Journal of Mathematical Psychology, 1(1), 1–27.

Michell, J. (1997). Quantitative science and the definition of measurement in psychology. British Journal of Psychology, 88(3), 355–383.

Narens, L. (2002). Theories of Meaningfulness. Lawrence Erlbaum Associates.

Nelsen, R. B. (2006). An Introduction to Copulas (2nd ed.). Springer.

Schweizer, B., & Wolff, E. (1981). On nonparametric measures of dependence for random variables. Annals of Statistics, 9(4), 879–885.

Schulz, E. M., & Nicewander, W. A. (1997). Grade Equivalent and IRT Representations of Growth (ACT Research Report No. 97-2). ACT.

Sklar, A. (1959). Fonctions de repartition a n dimensions et leurs marges. Publications de l’Institut de Statistique de l’Universite de Paris, 8, 229–231.

Longitudinal Inference Without Longitudinal Data

Responding to Measurement-Theory Questions about TAMP, Copulas, and Growth

Damian W. Betebenner
National Center for the Improvement of Educational Assessment
dbetebenner@nciea.org

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