Word embedding association test

Overview

The test asks whether two sets of target words sit closer to one attribute set than to the other. Each word gets an association score with the two attribute sets. A split of the target words scores the summed association of the first set minus the same sum for the second 1903.03862 §4. The p-value comes from a permutation test: reshuffle the pooled target words into two equal halves and count how often a reshuffled split scores as high as the observed one 1903.03862 §3.

The two papers below use the same kind of word-set test in different ways. Gonen and Goldberg run it as a permutation test on debiased embeddings, with word sets of 8 words per side 1903.03862 §4. No before-and-after comparison is reported for this test, unlike the other checks in that paper. Wan et al. report the score, called WEAT, as an effect size on word sets drawn from LLM-generated recommendation letters, and report no p-value for it 2310.09219 §4.3.5 2310.09219 tab. 3.

Variants

The variants differ in how the p-value is computed, or whether one is computed at all. The first two come from the same paper and share one score function.

Exact enumeration (the paper's setting)

Enumerate every way to split the pooled target words into two equal halves. With 8 + 8 words that is 12,870 splits 1903.03862 §4. The p-value is the share of splits that score at least as high as the observed one. Reported p-values for the debiased spaces are 0 and 7.7e-5 (names against family and career words), 0.00016 and 0.00031 (names against arts and maths words), and 0.0467 and 0.0064 (names against arts and science words), for Hard-Debiased and GN-GloVe respectively 1903.03862 §4.

Sampled permutations (authors' code only)

The authors' notebook refuses word sets larger than 10 and points to a sampled version, which caps the number of samples at 1,000,000. The paper text does not describe this version, so treat it as an implementation choice rather than a tested method 1903.03862 §4.

Effect size on generated text (Wan et al.)

Wan et al. report WEAT(MF), with male and female names, and WEAT(CF), with career and family words, for ChatGPT and Alpaca letters, split into nouns and adjectives 2310.09219 §4.3.5 2310.09219 tab. 3. Each word list has 8 entries 2310.09219 app. F.4. The values for ChatGPT are 0.393 and 0.901 for nouns, and 0.493 and 0.535 for adjectives. The values for Alpaca are 0.579 and 0.419 for nouns, and 1.009 and 0.419 for adjectives 2310.09219 tab. 3. The card flags that both Alpaca CF values are 0.419, so check the source before reusing them. Card: Wan et al., 2023.

The paper says WEAT measures embedding distance but does not name the embedding model 2310.09219 §4.3.5. It gives no p-value, no permutation count and no formula for these values. Do not assume they equal the split score above.

The sign conventions differ in wording. Wan et al. say a positive WEAT(MF) means female words lean toward female-stereotyped traits [arxiv:2310.09219 §4.3.5]. Gonen and Goldberg define the score as the first set minus the second 1903.03862 §4. Check which set is "first" before comparing signs across papers.

When to use which

Implementation sketch

The score and the exact p-value share one core. Only the enumeration changes between the two variants. The effect-size variant needs only the observed score.

# Score of one split of target words  [arxiv:1903.03862 §4]
def split_score(first, second, attrs_a, attrs_b, space):
    # Summed association for the first set...
    total = 0
    for w in first:
        total += association(w, attrs_a, attrs_b, space)
    # ...minus the same sum for the second set
    for w in second:
        total -= association(w, attrs_a, attrs_b, space)
    return total

# Exact permutation p-value  [arxiv:1903.03862 §3] [arxiv:1903.03862 §4]
def exact_p_value(xs, ys, attrs_a, attrs_b, space):
    observed = split_score(xs, ys, attrs_a, attrs_b, space)
    pooled = xs + ys
    hits = 0
    total = 0
    # Each choice of first half is one split; 8 + 8 gives 12,870
    for half in combinations(pooled, len(xs)):
        total += 1
        rest = [w for w in pooled if w not in half]
        # The authors' code counts strictly greater; the text says at least
        if split_score(half, rest, attrs_a, attrs_b, space) > observed:
            hits += 1
    return hits / total

Contradictions

The paper's text and its code disagree on the permutation count. The text describes counting splits that score equal or greater than the observed one 1903.03862 §3. The authors' code counts only splits scoring strictly greater, which the paper text does not state. The two rules differ only when a split ties with the observed score, so the gap is small in practice, but an implementer should pick one and state it.

Papers

• Gonen and Goldberg, 2019: defines the test over Caliskan et al.'s word sets and runs three experiments on debiased spaces, with exact p-values reported.
• Wan et al., 2023: applies WEAT(MF) and WEAT(CF) to ChatGPT and Alpaca recommendation letters, reporting effect sizes only.