The fashionable theory of PIE laryngeals offers plausible explanations for many phenomena, but plausibility is not proof. Are any implications of the postulated laryngeals amenable to statistical test? This would be the standard of proof in hard disciplines.
What I have in mind is this: One could try to compare cognates from two branches of the IE family to find a statistically significant correlation between some contrasting features allegedly determined by an ancient laryngeal. I would call this the method of parallel watersheds.
I have tried to apply this strategy to distinctive reflexes of “heavy” CṚ:/CḶ:(C) roots. These are expected to give rise to ŪR and ĪR in Sanskrit, as in pūrņa, stīrņa, tīrņa, gīrņa, jīrņa, ūrņā, dīrgha, bhūrja; ΡΗ/ΡΩ and vowel-liquid-vowel (VLV) sequences in Greek, as in πληθος, βρωτος, γηραω, ληνος, στρωτος/ στορε-, δολιχος; and RĀ and (arguably) RACC sequences in Latin, as in plēnus, strātus, trātus, vorātus, lāna, longus, fraxinus. “Light” CṚ/CḶ(C) roots, by contrast, are expected to give rise to LV/VL sequences with a short vowel to one side only.
The key step in the correlation analysis is to pigeonhole paired cognates into heavy/heavy, heavy/light, light/heavy, and light/light boxes. Loosely speaking, matches should preponderate over mismatches, but the technical definition of correlation is a bit more complicated.
Applying a χ² test to limited data, I found correlations “significant at the 5% level”. This bit of jargon means that the odds against finding a spurious correlation due to sampling error, when none would actually be present in abundant data, are better than 20:1. This would be considered adequate in most disciplines, and allowances might be made for the limited nature of the fossil record, because they don’t make more classical languages while you sleep.
The degree of concordance is not as impressive. There are a fair number of heavy/light mismatches without obvious explanation, e.g., sūrkšati/στεργεται, pṛthus/πλατυς/plānus. (I tried not to cherry-pick the data, but I may have applied unconscious bias.)
I leave you two questions: Has a statistical strategy been tried, and what did it conclude? Are there any other implications of laryngeal theory that can be so tested?
I would think that some, maybe even most, supposed implications of laryngeals defy testing by this method. For example: The leading vowels on Greek words such as ακουω, ανηρ, αστηρ, ελευθερος, ελαχυς, εννεϝα, ερεβος, ερυθρος, ομιχω, οριγω, οφρυς are commonly attributed to laryngeal prefixes, but no other IE branch (Hittite aside) displays them. The number of possible exceptions is too small to support a statistical test, and ονομα, ομφαλος, ονυξ are easily explained in terms of syllabic nasals.
EDIT: The point of such tests is not to prove the existence of PIE laryngeals, but rather to assess the strength of evidence (in a mathematically rigorous fashion) for their supposed manifestations in daughter languages by ruling out the possibility of random transformations. The evidence for some phenomena is stronger than for others.