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Language 92.3, September 2016 s1 SNAP JUDGMENTS: A SMALL N ACCEPTABILITY PARADIGM (SNAP) FOR LINGUISTIC ACCEPTABILITY JUDGMENTS: ONLINE APPENDICES KYLE MAHOWALD PETER GRAFF Massachusetts Institute of Technology Intel Corporation JEREMY HARTMAN EDWARD GIBSON University of Massachusetts Amherst Massachusetts Institute of Technology APPENDIX A: RATING STUDY RESULTS ‘z-bad’ is the average z-score for the hypothesized ‘bad’ option. ‘z-good’ is the average z-score for the hypothesized good option. ‘Z.diff’ is the difference between z-good and z-bad and is the effect size. Beta is the estimate from the linear mixed-effects model, which has a standard error ‘SE’ and a t-value ‘t’. ‘χ2 ’ is the chi-squared value comparing the full model to an intercept-only model, and ‘χ2 p’ is the p-value obtained by that comparison. Simple ‘p’ is just the p-value calculated using the t-value. Pred is TRUE if the effect goes in the significant direction. Sig is TRUE if there is a significant effect. Rows in yellow are rows in which the effect goes in the predicted direction but is not significant. EXPERIMENT z-BAD z-GOOD Z.DIFF BETA SE t χ2 χ2 p p PRED SIG 35.3.Hazout:36–36 −0.05 −0.04 0.01 0.00 0.06 0.08 0.01 0.9350 0.9360 TRUE FALSE 34.4.Lasnik:24a–24b 0.20 0.21 0.01 0.01 0.08 0.12 0.02 0.9010 0.9040 TRUE FALSE 34.1.Basilico:11a–12a −0.46 −0.44 0.03 0.02 0.10 0.24 0.06 0.8130 0.8100 TRUE FALSE 34.4.Lasnik:22a–22b 0.03 0.06 0.03 0.03 0.06 0.48 0.23 0.6290 0.6310 TRUE FALSE 35.3.Hazout:73b–73b −0.29 −0.17 0.11 0.11 0.07 1.72 2.89 0.0890 0.0850 TRUE FALSE 33.1.Fox:47c–48b −0.39 −0.26 0.12 0.12 0.07 1.69 2.68 0.1020 0.0910 TRUE FALSE 32.2.Nunes:fn 35iia– fn35iib −0.89 −0.78 0.12 0.12 0.06 2.02 4.06 0.0440 0.0430 TRUE TRUE 35.2.Hazout:1b–1b −0.35 −0.18 0.17 0.17 0.07 2.59 6.57 0.0100 0.0100 TRUE TRUE 32.4.Lopez:9c–10c −0.56 −0.36 0.20 0.20 0.05 3.59 11.01 0.0010 < 0.0001 TRUE TRUE 32.3.Culicover:37a–37a −0.37 −0.15 0.22 0.21 0.07 3.21 9.95 0.0020 0.0010 TRUE TRUE 33.4.Neeleman:97a–98 −0.33 −0.09 0.24 0.24 0.13 1.80 2.93 0.0870 0.0720 TRUE FALSE 40.1.Heck:51–52 −0.63 −0.39 0.24 0.24 0.09 2.69 5.87 0.0150 0.0070 TRUE TRUE 34.3.Landau:7c–7c 0.88 1.13 0.25 0.25 0.12 2.05 3.74 0.0530 0.0400 TRUE FALSE 41.3.Landau:11a–11a 0.28 0.54 0.26 0.26 0.06 4.08 15.21 < 0.0001 < 0.0001 TRUE TRUE 35.2.Larson:44b–44b 0.54 0.80 0.27 0.27 0.08 3.50 10.76 0.0010 < 0.0001 TRUE TRUE 34.1.Phillips:59c–60c −0.74 −0.45 0.29 0.29 0.11 2.58 5.52 0.0190 0.0100 TRUE TRUE 33.2.Bowers:49c–49c −0.74 −0.46 0.29 0.29 0.09 3.35 8.65 0.0030 0.0010 TRUE TRUE 34.2.Caponigro:11b–11c –0.50 0.82 0.32 0.32 0.06 5.50 20.80 < 0.0001 < 0.0001 TRUE TRUE 32.3.Fanselow:61a–61b −0.96 −0.63 0.33 0.33 0.06 5.24 22.00 < 0.0001 < 0.0001 TRUE TRUE 34.1.Phillips:23a–25a −0.02 0.39 0.40 0.40 0.09 4.50 12.78 < 0.0001 < 0.0001 TRUE TRUE 35.1.Bhatt:93a–b −0.69 −0.29 0.41 0.40 0.08 5.02 14.82 < 0.0001 < 0.0001 TRUE TRUE 32.3.Culicover:46a–48a −0.20 0.21 0.41 0.41 0.07 6.30 22.52 < 0.0001 < 0.0001 TRUE TRUE 34.3.Landau:38a–38c −0.28 0.14 0.42 0.42 0.07 5.68 18.67 < 0.0001 < 0.0001 TRUE TRUE 39.1.Sobin:8b–8f −0.36 0.06 0.42 0.42 0.07 6.32 25.00 < 0.0001 < 0.0001 TRUE TRUE 34.4.Boskovic:fn6iie– fn6iid −0.34 0.12 0.46 0.46 0.10 4.54 13.78 < 0.0001 < 0.0001 TRUE TRUE 35.3.Embick:62b–62b.Cf 0.40 0.86 0.46 0.46 0.08 5.54 17.72 < 0.0001 < 0.0001 TRUE TRUE 34.4.Haegeman:2a–2b −0.15 0.30 0.46 0.46 0.06 7.27 25.89 < 0.0001 < 0.0001 TRUE TRUE 34.3.Landau:fn12i–fn12ii −0.88 −0.42 0.46 0.46 0.06 7.73 27.48 < 0.0001 < 0.0001 TRUE TRUE s2 34.1.Basilico:37a–37b −0.37 0.10 0.47 0.47 0.09 5.31 16.56 < 0.0001 < 0.0001 TRUE TRUE 39.1.Sobin:8c–8f −0.32 0.15 0.47 0.47 0.06 8.02 35.62 < 0.0001 < 0.0001 TRUE TRUE 35.2.Hazout:1a–1a −0.86 −0.34 0.52 0.52 0.06 8.77 34.21 < 0.0001 < 0.0001 TRUE TRUE 33.2.Bowers:7d–7d 0.56 1.12 0.56 0.56 0.10 5.48 17.11 < 0.0001 < 0.0001 TRUE TRUE 34.1.Phillips:61a–61b −1.00 −0.41 0.59 0.59 0.07 7.86 25.39 < 0.0001 < 0.0001 TRUE TRUE 39.1.Sobin:20a–21a −0.18 0.41 0.60 0.60 0.08 7.41 23.31 < 0.0001 < 0.0001 TRUE TRUE 35.3.Embick:7a–7b −0.47 0.14 0.62 0.62 0.09 6.98 22.55 < 0.0001 < 0.0001 TRUE TRUE 34.1.Fox:37a–37b −0.17 0.45 0.62 0.62 0.08 8.26 27.10 < 0.0001 < 0.0001 TRUE TRUE 35.1.Bhatt:fn25ia–fn25ib −1.08 −0.43 0.65 0.65 0.08 7.78 26.37 < 0.0001 < 0.0001 TRUE TRUE 34.3.Takano:2b–d −0.52 0.14 0.66 0.66 0.12 5.71 16.37 < 0.0001 < 0.0001 TRUE TRUE 41.3.Landau:32a–32b −0.60 0.06 0.66 0.66 0.07 8.98 29.15 < 0.0001 < 0.0001 TRUE TRUE 34.1.Phillips:23a–24a −0.41 0.30 0.71 0.71 0.13 5.28 14.52 < 0.0001 < 0.0001 TRUE TRUE 33.2.Bowers:20a–20b −0.53 0.18 0.71 0.71 0.12 5.84 16.21 < 0.0001 < 0.0001 TRUE TRUE 34.4.Haegeman:2c–2b −0.73 0.00 0.73 0.73 0.08 8.80 28.07 < 0.0001 < 0.0001 TRUE TRUE 32.3.Culicover:25c–25d. WithOneself −0.25 0.52 0.77 0.77 0.10 7.76 22.87 < 0.0001 < 0.0001 TRUE TRUE 41.4.Bruening:61b–62b. StarredVariantIn61 −0.24 0.54 0.78 0.78 0.16 4.89 13.65 < 0.0001 < 0.0001 TRUE TRUE 35.1.Bhatt:fn5ia–fn5ia 0.03 0.80 0.78 0.78 0.09 8.89 25.75 < 0.0001 < 0.0001 TRUE TRUE 32.1.Martin:50b–51b −0.49 0.29 0.78 0.78 0.07 11.65 42.71 < 0.0001 < 0.0001 TRUE TRUE 32.3.Culicover:46b–46b −0.32 0.49 0.81 0.81 0.08 9.89 30.41 < 0.0001 < 0.0001 TRUE TRUE 40.4.Hicks:2a–2b 0.21 1.04 0.82 0.83 0.12 6.65 19.30 < 0.0001 < 0.0001 TRUE TRUE 34.2.Panagiotidis:12a–b 0.07 0.92 0.84 0.84 0.13 6.50 19.02 < 0.0001 < 0.0001 TRUE TRUE 38.2.Hornstein:fn2.iii–iii −0.24 0.60 0.84 0.84 0.10 8.68 28.84 < 0.0001 < 0.0001 TRUE TRUE 32.3.Culicover:34c–34e −0.28 0.56 0.84 0.84 0.08 10.76 34.15 < 0.0001 < 0.0001 TRUE TRUE 32.1.Martin:50a–51a −0.21 0.65 0.87 0.87 0.09 9.20 27.00 < 0.0001 < 0.0001 TRUE TRUE 35.2.Hazout:5a–5c −0.67 0.21 0.89 0.89 0.06 15.54 42.33 < 0.0001 < 0.0001 TRUE TRUE 40.4.Hicks:10a–10b −0.80 0.11 0.91 0.91 0.08 11.63 36.63 < 0.0001 < 0.0001 TRUE TRUE 32.3.Culicover:23c–23d. SentenceDP 0.20 1.12 0.92 0.93 0.15 6.11 16.76 < 0.0001 < 0.0001 TRUE TRUE 34.3.Takano:2a–c −0.36 0.58 0.94 0.93 0.09 10.10 27.12 < 0.0001 < 0.0001 TRUE TRUE 34.1.Fox:4–4 −1.01 −0.09 0.93 0.93 0.08 11.24 36.45 < 0.0001 < 0.0001 TRUE TRUE 41.4.Bruening:62a–87a. StarredVariantIn87 −0.31 0.65 0.95 0.95 0.12 8.01 23.20 < 0.0001 < 0.0001 TRUE TRUE 34.3.Heycock:93a–93b 0.04 1.01 0.98 0.97 0.07 14.00 38.46 < 0.0001 < 0.0001 TRUE TRUE 38.3.Landau:62a–62b −0.12 0.87 0.98 0.98 0.10 10.32 29.29 < 0.0001 < 0.0001 TRUE TRUE 32.3.Culicover:44a–45a −0.61 0.40 1.01 1.01 0.06 16.01 44.47 < 0.0001 < 0.0001 TRUE TRUE 40.2.Johnson:78–79 −0.57 0.48 1.05 1.04 0.08 12.30 31.13 < 0.0001 < 0.0001 TRUE TRUE 41.3.Constantini:1b– 1b.BothVsBothBoth −0.14 0.91 1.04 1.04 0.08 13.69 37.01 < 0.0001 < 0.0001 TRUE TRUE 34.2.Caponigro:fn6ia– fn6ib.EagerlyIn2ndPos −0.06 0.98 1.03 1.04 0.07 15.02 37.98 < 0.0001 < 0.0001 TRUE TRUE 34.1.Basilico:29b–30b −0.97 0.07 1.05 1.05 0.10 10.71 30.49 < 0.0001 < 0.0001 TRUE TRUE 34.3.Takano:11a–11b −0.92 0.12 1.05 1.05 0.09 11.40 31.77 < 0.0001 < 0.0001 TRUE TRUE 34.1.Fox:1–1 −1.01 0.08 1.09 1.09 0.07 14.94 46.32 < 0.0001 < 0.0001 TRUE TRUE 37.4.Nakajima:fn1ia– fn1iiia −0.91 0.21 1.12 1.11 0.13 8.69 23.29 < 0.0001 < 0.0001 TRUE TRUE 35.1.Bhatt:5a–5c −0.40 0.72 1.12 1.12 0.07 15.57 43.81 < 0.0001 < 0.0001 TRUE TRUE 33.2.Bowers:56c–56d −0.37 0.77 1.14 1.14 0.16 7.09 20.34 < 0.0001 < 0.0001 TRUE TRUE 32.2.Alexiadou:fn11iib– fn11iic −0.56 0.58 1.14 1.14 0.10 11.71 32.22 < 0.0001 < 0.0001 TRUE TRUE 33.1.denDikken:56a–58a −0.51 0.66 1.17 1.17 0.09 13.22 35.62 < 0.0001 < 0.0001 TRUE TRUE 32.3.Culicover:fn6ia–fn6ib −0.77 0.41 1.18 1.18 0.07 17.10 48.58 < 0.0001 < 0.0001 TRUE TRUE 36.4.denDikken:35a–35b −0.26 0.95 1.21 1.21 0.09 13.25 37.76 < 0.0001 < 0.0001 TRUE TRUE 35.1.Bhatt:1b–1b −0.52 0.69 1.21 1.21 0.07 17.76 50.67 < 0.0001 < 0.0001 TRUE TRUE s3 34.3.Landau:fn13ii–fn13ii −0.49 0.73 1.22 1.23 0.10 12.21 34.15 < 0.0001 < 0.0001 TRUE TRUE 41.3.Vicente:6b–8b −0.98 0.26 1.24 1.24 0.08 15.94 42.74 < 0.0001 < 0.0001 TRUE TRUE 33.2.Bowers:7a–7a. PerfectlyIn2ndPos3rdPos −0.42 0.82 1.25 1.25 0.07 17.18 40.23 < 0.0001 < 0.0001 TRUE TRUE 41.1.Muller:28a–28b −0.86 0.42 1.28 1.28 0.11 11.30 31.48 < 0.0001 < 0.0001 TRUE TRUE 35.1.McGinnis:63a–63b −0.35 0.94 1.28 1.28 0.09 14.49 36.90 < 0.0001 < 0.0001 TRUE TRUE 38.2.Hornstein:2b–2c −0.12 1.24 1.35 1.35 0.09 14.74 45.23 < 0.0001 < 0.0001 TRUE TRUE 33.2.Bowers:19a–19b −0.35 1.02 1.37 1.37 0.10 13.18 39.73 < 0.0001 < 0.0001 TRUE TRUE 35.3.Embick:72a–72b −0.37 1.05 1.41 1.41 0.13 11.17 30.27 < 0.0001 < 0.0001 TRUE TRUE 32.1.Martin:15a–15b −0.33 1.12 1.45 1.45 0.13 11.35 29.79 < 0.0001 < 0.0001 TRUE TRUE 32.4.Lopez:16a–16b −0.44 1.03 1.48 1.48 0.07 20.04 55.90 < 0.0001 < 0.0001 TRUE TRUE 34.1.Basilico:50–51 −0.81 0.68 1.49 1.49 0.14 10.89 29.51 < 0.0001 < 0.0001 TRUE TRUE 33.1.denDikken:57a–57b −0.65 0.87 1.51 1.52 0.10 15.16 39.21 < 0.0001 < 0.0001 TRUE TRUE 34.1.Basilico:7a–7b −0.46 1.06 1.52 1.52 0.09 16.29 40.28 < 0.0001 < 0.0001 TRUE TRUE 32.1.Martin:48a–48b −0.93 0.67 1.60 1.60 0.12 13.14 33.95 < 0.0001 < 0.0001 TRUE TRUE 32.3.Fanselow:59a–59b −0.49 1.12 1.61 1.61 0.08 20.91 48.82 < 0.0001 < 0.0001 TRUE TRUE 35.3.Hazout:30a–30a −0.67 0.98 1.64 1.64 0.15 10.76 27.84 < 0.0001 < 0.0001 TRUE TRUE 37.2.deVries:70a–70b −0.68 0.97 1.65 1.65 0.07 22.04 50.10 < 0.0001 < 0.0001 TRUE TRUE 35.2.Larson:61a–61b −0.81 0.85 1.66 1.66 0.09 17.71 43.62 < 0.0001 < 0.0001 TRUE TRUE 38.4.Boskovic:74–75 −0.80 0.87 1.67 1.67 0.08 20.91 44.93 < 0.0001 < 0.0001 TRUE TRUE 35.3.Hazout:65a–65b −0.99 0.73 1.72 1.72 0.12 13.94 35.00 < 0.0001 < 0.0001 TRUE TRUE 33.2.Bowers:13b–13b −0.81 0.98 1.79 1.79 0.11 16.49 40.15 < 0.0001 < 0.0001 TRUE TRUE 35.1.Bhatt:13a–13a −1.03 0.79 1.82 1.82 0.07 27.59 61.45 < 0.0001 < 0.0001 TRUE TRUE 34.1.Basilico:4b–4c −0.81 1.03 1.84 1.84 0.06 28.80 64.27 < 0.0001 < 0.0001 TRUE TRUE 36.4.denDikken:38b–38b −1.02 0.89 1.91 1.91 0.08 24.38 58.24 < 0.0001 < 0.0001 TRUE TRUE 37.2.Sigurdsson:3c–3e −0.92 1.08 2.00 2.00 0.07 29.93 58.39 < 0.0001 < 0.0001 TRUE TRUE APPENDIX B: FORCED-CHOICE RESULTS ‘Gramm’ is the proportion of people who choose the hypothesized acceptable sentence. ‘Beta’ is the model estimate of the effect size, which has a standard error of SE and a z-value (distance from 0 in units of standard error) of z. The p-value is calculated directly from the z-value. Pred is TRUE if the effect goes in the significant direction, FALSE otherwise. Sig is TRUE if there is a significant effect. Rows in red represent contrasts where the effect is significant in the opposite direction of that predicted . Rows in pink show effects in the opposite direction of what was predicted but are not significant. Rows in yellow are rows in which the effect goes in the predicted direction but is not significant. EXPERIMENT GRAMM BETA z SE p PRED SIG 35.3.Hazout:36–36 0.39 −0.79 −4.03 0.20 < 0.0001 FALSE TRUE 34.4.Lasnik:24a–24b 0.35 −0.73 −3.43 0.21 0.0010 FALSE TRUE 32.2.Nunes:fn35iia–fn35iib 0.44 −0.25 −1.40 0.18 0.1620 FALSE FALSE 32.4.Lopez:9c–10c 0.46 −0.19 −1.20 0.15 0.2300 FALSE FALSE 39.1.Sobin:8b–8f 0.46 −0.19 −1.09 0.17 0.2760 FALSE FALSE 34.4.Lasnik:22a–22b 0.47 −0.17 −0.70 0.25 0.4840 FALSE FALSE 34.1.Basilico:11a–12a 0.51 0.04 0.17 0.26 0.8650 TRUE FALSE 34.4.Haegeman:2a–2b 0.58 0.51 2.20 0.23 0.0280 TRUE TRUE 34.1.Phillips:23a–25a 0.62 0.65 2.45 0.26 0.0140 TRUE TRUE 33.4.Neeleman:97a–98 0.62 0.70 1.93 0.37 0.0540 TRUE FALSE 40.1.Heck:51–52 0.69 0.94 3.88 0.24 < 0.0001 TRUE TRUE 39.1.Sobin:8c–8f 0.70 1.02 4.60 0.22 < 0.0001 TRUE TRUE 34.3.Landau:fn12i–fn12ii 0.71 1.03 5.47 0.19 < 0.0001 TRUE TRUE 34.1.Basilico:37a–37b 0.72 1.04 4.59 0.23 < 0.0001 TRUE TRUE s4 33.1.Fox:47c–48b 0.71 1.05 4.06 0.26 < 0.0001 TRUE TRUE 35.2.Larson:44b–44b 0.69 1.11 3.85 0.29 < 0.0001 TRUE TRUE 34.3.Landau:7c–7c 0.71 1.18 6.07 0.19 < 0.0001 TRUE TRUE 34.1.Phillips:61a–61b 0.77 1.30 10.07 0.13 < 0.0001 TRUE TRUE 34.2.Panagiotidis:12a–b 0.75 1.45 4.10 0.35 < 0.0001 TRUE TRUE 34.4.Boskovic:fn6iie–fn6iid 0.73 1.54 4.56 0.34 < 0.0001 TRUE TRUE 32.3.Fanselow:61a–61b 0.78 1.55 8.99 0.17 < 0.0001 TRUE TRUE 32.3.Culicover:37a–37a 0.79 1.57 9.36 0.17 < 0.0001 TRUE TRUE 41.3.Constantini:1b– 1b.BothVsBothBoth 0.79 1.58 7.19 0.22 < 0.0001 TRUE TRUE 41.3.Landau:11a–11a 0.80 1.64 7.28 0.22 < 0.0001 TRUE TRUE 32.3.Culicover:25c–25d.WithOneself 0.80 1.77 5.75 0.31 < 0.0001 TRUE TRUE 41.4.Bruening:61b– 62b.StarredVariantIn61 0.74 1.84 3.77 0.49 < 0.0001 TRUE TRUE 34.2.Caponigro:11b–11c 0.83 2.01 8.24 0.24 < 0.0001 TRUE TRUE 35.3.Embick:7a–7b 0.83 2.03 6.12 0.33 < 0.0001 TRUE TRUE 34.1.Phillips:23a–24a 0.83 2.06 5.95 0.35 < 0.0001 TRUE TRUE 35.1.Bhatt:93a–b 0.85 2.14 9.59 0.22 < 0.0001 TRUE TRUE 39.1.Sobin:20a–21a 0.83 2.18 6.41 0.34 < 0.0001 TRUE TRUE 40.4.Hicks:2a–2b 0.86 2.24 6.22 0.36 < 0.0001 TRUE TRUE 33.1.denDikken:57a–57b 0.87 2.24 8.85 0.25 < 0.0001 TRUE TRUE 33.2.Bowers:49c–49c 0.85 2.26 8.48 0.27 < 0.0001 TRUE TRUE 35.1.Bhatt:fn25ia–fn25ib 0.89 2.31 8.81 0.26 < 0.0001 TRUE TRUE 34.3.Landau:38a–38c 0.88 2.41 7.65 0.32 < 0.0001 TRUE TRUE 35.1.Bhatt:fn5ia–fn5ia 0.88 2.49 6.74 0.37 < 0.0001 TRUE TRUE 32.3.Culicover:46b–46b 0.89 2.51 7.03 0.36 < 0.0001 TRUE TRUE 34.1.Phillips:59c–60c 0.86 2.59 7.84 0.33 < 0.0001 TRUE TRUE 41.4.Bruening:62a– 87a.StarredVariantIn87 0.83 2.60 4.57 0.57 < 0.0001 TRUE TRUE 34.3.Takano:2b–d 0.86 2.65 5.06 0.52 < 0.0001 TRUE TRUE 32.1.Martin:48a–48b 0.89 2.75 7.90 0.35 < 0.0001 TRUE TRUE 32.1.Martin:50a–51a 0.91 2.79 7.20 0.39 < 0.0001 TRUE TRUE 34.3.Takano:2a–c 0.90 2.80 7.52 0.37 < 0.0001 TRUE TRUE 33.2.Bowers:20a–20b 0.88 2.82 8.01 0.35 < 0.0001 TRUE TRUE 35.2.Hazout:1b–1b 0.86 2.91 3811.11 0.00 < 0.0001 TRUE TRUE 35.3.Embick:62b–62b.Cf 0.87 2.93 7.25 0.40 < 0.0001 TRUE TRUE 33.2.Bowers:7d–7d 0.90 2.95 7.00 0.42 < 0.0001 TRUE TRUE 35.3.Hazout:73b–73b 0.90 2.98 8.21 0.36 < 0.0001 TRUE TRUE 38.3.Landau:62a–62b 0.92 3.00 7.66 0.39 < 0.0001 TRUE TRUE 33.2.Bowers:56c–56d 0.89 3.04 5.60 0.54 < 0.0001 TRUE TRUE 38.2.Hornstein:fn2.iii–iii 0.95 3.06 13.01 0.23 < 0.0001 TRUE TRUE 32.3.Culicover:34c–34e 0.91 3.32 6.97 0.48 < 0.0001 TRUE TRUE 35.3.Hazout:65a–65b 0.97 3.38 11.94 0.28 < 0.0001 TRUE TRUE 32.3.Fanselow:59a–59b 0.97 3.38 12.16 0.28 < 0.0001 TRUE TRUE 35.2.Hazout:1a–1a 0.90 3.41 5.78 0.59 < 0.0001 TRUE TRUE 32.1.Martin:50b–51b 0.91 3.42 6.34 0.54 < 0.0001 TRUE TRUE 33.2.Bowers:7a– 7a.PerfectlyIn2ndPos3rdPos 0.97 3.54 14.39 0.25 < 0.0001 TRUE TRUE 32.3.Culicover:46a–48a 0.90 3.57 5.13 0.70 < 0.0001 TRUE TRUE 32.3.Culicover:23c–23d.SentenceDP 0.94 3.60 5.91 0.61 < 0.0001 TRUE TRUE 34.1.Fox:4–4 0.91 3.63 5.55 0.65 < 0.0001 TRUE TRUE 34.3.Takano:11a–11b 0.92 3.65 5.10 0.71 < 0.0001 TRUE TRUE 34.1.Fox:1–1 0.93 3.77 5.43 0.69 < 0.0001 TRUE TRUE 35.3.Hazout:30a–30a 0.97 3.79 9.45 0.40 < 0.0001 TRUE TRUE 37.4.Nakajima:fn1ia–fn1iiia 0.93 3.84 5.63 0.68 < 0.0001 TRUE TRUE s5 34.1.Basilico:29b–30b 0.94 3.93 5.71 0.69 < 0.0001 TRUE TRUE 34.1.Basilico:4b–4c 0.98 3.99 13.10 0.30 < 0.0001 TRUE TRUE 35.2.Hazout:5a–5c 0.93 4.03 4.92 0.82 < 0.0001 TRUE TRUE 41.3.Landau:32a–32b 0.92 4.17 3.88 1.07 < 0.0001 TRUE TRUE 33.2.Bowers:13b–13b 0.99 4.44 11.69 0.38 < 0.0001 TRUE TRUE 37.2.Sigurdsson:3c–3e 0.99 4.44 11.69 0.38 < 0.0001 TRUE TRUE 40.4.Hicks:10a–10b 0.92 4.79 3.11 1.54 0.0020 TRUE TRUE 34.3.Landau:fn13ii–fn13ii 0.92 5.52 4.18 1.32 < 0.0001 TRUE TRUE 34.1.Fox:37a–37b 0.92 6.06 5.86 1.04 < 0.0001 TRUE TRUE 40.2.Johnson:78–79 0.94 6.70 5.87 1.14 < 0.0001 TRUE TRUE 35.2.Larson:61a–61b 0.95 6.76 4.62 1.46 < 0.0001 TRUE TRUE 32.2.Alexiadou:fn11iib–fn11iic 0.95 7.45 7.53 0.99 < 0.0001 TRUE TRUE 34.4.Haegeman:2c–2b 0.93 7.50 7.21 1.04 < 0.0001 TRUE TRUE 34.1.Basilico:7a–7b 0.97 7.59 6.80 1.12 < 0.0001 TRUE TRUE 32.1.Martin:15a–15b 0.97 7.88 7.75 1.02 < 0.0001 TRUE TRUE 38.4.Boskovic:74–75 0.97 7.95 1318.99 0.01 < 0.0001 TRUE TRUE 32.3.Culicover:fn6ia–fn6ib 0.96 7.96 7.70 1.03 < 0.0001 TRUE TRUE 33.2.Bowers:19a–19b 0.95 8.04 6.54 1.23 < 0.0001 TRUE TRUE 34.2.Caponigro:fn6ia– fn6ib.EagerlyIn2ndPos 0.96 8.17 7.60 1.08 < 0.0001 TRUE TRUE 37.2.deVries:70a–70b 0.97 8.26 7.25 1.14 < 0.0001 TRUE TRUE 41.1.Muller:28a–28b 0.96 8.40 8.05 1.04 < 0.0001 TRUE TRUE 38.2.Hornstein:2b–2c 0.97 8.49 7.31 1.16 < 0.0001 TRUE TRUE 41.3.Vicente:6b–8b 0.97 8.49 7.31 1.16 < 0.0001 TRUE TRUE 35.1.Bhatt:1b–1b 0.98 8.51 1004.54 0.01 < 0.0001 TRUE TRUE 34.3.Heycock:93a–93b 0.96 8.61 6.13 1.40 < 0.0001 TRUE TRUE 35.1.Bhatt:13a–13a 0.98 8.69 1834.83 0.00 < 0.0001 TRUE TRUE 36.4.denDikken:38b–38b 0.96 8.76 8.92 0.98 < 0.0001 TRUE TRUE 35.1.McGinnis:63a–63b 0.93 8.78 6.95 1.26 < 0.0001 TRUE TRUE 34.1.Basilico:50–51 0.96 8.82 7.08 1.25 < 0.0001 TRUE TRUE 32.4.Lopez:16a–16b 0.98 9.10 6.59 1.38 < 0.0001 TRUE TRUE 35.3.Embick:72a–72b 0.98 9.10 7.07 1.29 < 0.0001 TRUE TRUE 32.3.Culicover:44a–45a 0.96 9.33 5.31 1.76 < 0.0001 TRUE TRUE 35.1.Bhatt:5a–5c 0.96 9.74 7.12 1.37 < 0.0001 TRUE TRUE 33.1.denDikken:56a–58a 0.95 10.35 6.80 1.52 < 0.0001 TRUE TRUE 36.4.denDikken:35a–35b 0.96 10.61 6.56 1.62 < 0.0001 TRUE TRUE s6 APPENDIX C: REFERENCES FOR LINGUISTIC INQUIRY PAPERS See full set of materials in the Materials folder at the Open Science Foundation, http://osf.io/5wm2a. ALEXIADOU, ARTEMIS, and ELENA ANAGNOSTOPOULOU. 2001. The subject-in-situ generalization and the role of case in driving computations. Linguistic Inquiry 32.193–231. DOI: 10.1162/00243890152001753. BASILICO, DAVID. 2003. The topic of small clauses. Linguistic Inquiry 34.1–35. DOI: 10.1162/00243890376 3255913. BECKER, MISHA. 2006. There began to be a learnability puzzle. Linguistic Inquiry 37.441–56. DOI: 10.1162/ ling.2006.37.3.441. BECK, SIGRID, and KYLE JOHNSON. 2004. Double objects again. Linguistic Inquiry 35.97–123. DOI: 10.1162/ 002438904322793356. BHATT, RAJESH, and ROUMYANA PANCHEVA. 2004. Late merger of degree clauses. Linguistic Inquiry 35.1– 45. DOI: 10.1162/002438904322793338. BOECKX, CEDRIC, and SANDRA STJEPANOVIĆ. 2001. Head-ing toward PF. Linguistic Inquiry 32.345–55. DOI: 10.1162/00243890152001799. BOŠKOVIĆ, ŽELJKO. 2002. On multiple wh-fronting. Linguistic Inquiry 33.351–83. DOI: 10.1162/002438902 760168536. BOŠKOVIĆ, ŽELJKO. 2007. On the locality and motivation of Move and Agree: An even more minimal theory. Linguistic Inquiry 38.589–644. DOI: 10.1162/ling.2007.38.4.589. BOŠKOVIĆ, ŽELJKO, and HOWARD LASNIK. 2003. On the distribution of null complementizers. Linguistic Inquiry 34.527–46. DOI: 10.1162/002438903322520142. BOWERS, JOHN. 2002. Transitivity. Linguistic Inquiry 33.183–224. DOI: 10.1162/002438902317406696. BRUENING, BENJAMIN. 2010a. Double object constructions disguised as prepositional datives. Linguistic Inquiry 41.287–305. DOI: 10.1162/ling.2010.41.2.287. BRUENING, BENJAMIN. 2010b. Ditransitive asymmetries and a theory of idiom formation. Linguistic Inquiry 41.519–62. DOI: 10.1162/LING_a_00012. CAPONIGRO, IVANO, and LISA PEARL. 2009. The nominal nature of where, when, and how: Evidence from free relatives. Linguistic Inquiry 40.155–64. DOI: 10.1162/ling.2009.40.1.155. CAPONIGRO, IVANO, and CARSON T. SCHÜTZE. 2003. Parameterizing passive participle movement. Linguistic Inquiry 34.293–307. DOI: 10.1162/002438903321663415. COSTANTINI, FRANCESCO. 2010. On infinitives and floating quantification. Linguistic Inquiry 41.487–96. DOI: 10.1162/LING_a_00006. CULICOVER, PETER W., and RAY JACKENDOFF. 2001. Control is not movement. Linguistic Inquiry 32.493– 512. DOI: 10.1162/002438901750372531. DEN DIKKEN, MARCEL. 2005. Comparative correlatives comparatively. Linguistic Inquiry 36.497–532. DOI: 10.1162/002438905774464377. DEN DIKKEN, MARCEL, and ANASTASIA GIANNAKIDOU. 2002. From hell to polarity: ‘Aggressively non-Dlinked ’ wh-phrases as polarity items. Linguistic Inquiry 33.31–61. DOI: 10.1162/002438902317382170. DE VRIES, MARK. 2006. The syntax of appositive relativization: On specifying coordination, false free relatives , and promotion. Linguistic Inquiry 37.229–70. DOI: 10.1162/ling.2006.37.2.229. EMBICK, DAVID. 2004. On the structure of resultative participles in English. Linguistic Inquiry 35.355–92. DOI: 10.1162/0024389041402634. FANSELOW, GISBERT. 2001. Features, θ-roles, and free constituent order. Linguistic Inquiry 32.405–37. DOI: 10.1162/002438901750372513. FOX, DANNY. 2002. Antecedent-contained deletion and the copy theory of movement. Linguistic Inquiry 33.63–96. DOI: 10.1162/002438902317382189. FOX, DANNY, and HOWARD LASNIK. 2003. Successive-cyclic movement and island repair: The difference between sluicing and VP-ellipsis. Linguistic Inquiry 34.143–54. DOI: 10.1162/002438903763255959. s7 HADDICAN, BILL. 2007. The structural deficiency of verbal pro-forms. Linguistic Inquiry 38.539–47. DOI: 10.1162/ling.2007.38.3.539. HAEGEMAN, LILIANE. 2003. Notes on long adverbial fronting in English and the left periphery. Linguistic Inquiry 34.640–49. DOI: 10.1162/ling.2003.34.4.640. HAEGEMAN, LILIANE. 2010. The movement derivation of conditional clauses. Linguistic Inquiry 41.595–621. DOI: 10.1162/LING_a_00014. HAZOUT, ILAN. 2004a. Long-distance agreement and the syntax of for-to infinitives. Linguistic Inquiry 35. 338–43. DOI: 10.1162/ling.2004.35.2.338. HAZOUT, ILAN. 2004b. The syntax of existential constructions. Linguistic Inquiry 35.393–430. DOI: 10.1162/ 0024389041402616. HECK, FABIAN. 2009. On certain properties of pied-piping. Linguistic Inquiry 40.75–111. DOI: 10.1162/ ling.2009.40.1.75. HEYCOCK, CAROLINE, and ROBERTO ZAMPARELLI. 2003. Coordinated bare definites. Linguistic Inquiry 34. 443–69. DOI: 10.1162/002438903322247551. HICKS, GLYN. 2009. Tough-constructions and their derivation. Linguistic Inquiry 40.535–66. DOI: 10.1162/ ling.2009.40.4.535. HIROSE, TOMIO. 2007. Nominal paths and the head parameter. Linguistic Inquiry 38.548–53. DOI: 10.1162/ ling.2007.38.3.548. HORNSTEIN, NORBERT. 2007. A very short note on existential constructions. Linguistic Inquiry 38.410–11. DOI: 10.1162/ling.2007.38.2.410. JOHNSON, KYLE. 2009. Gapping is not (VP-) ellipsis. Linguistic Inquiry 40.289–328. DOI: 10.1162/ling .2009.40.2.289. KALLULLI, DALINA. 2007. Rethinking the passive/anticausative distinction. Linguistic Inquiry 38.770–80. DOI: 10.1162/ling.2007.38.4.770. LANDAU, IDAN. 2003. Movement out of control. Linguistic Inquiry 34.471–98. DOI: 10.1162/0024389033 22247560. LANDAU, IDAN. 2007. EPP extensions. Linguistic Inquiry 38.485–523. DOI: 10.1162/ling.2007.38.3.485. LANDAU, IDAN. 2010. The explicit syntax of implicit arguments. Linguistic Inquiry 41.357–88. DOI: 10.1162/ LING_a_00001. LARSON, RICHARD K., and FRANC MARUŠIČ. 2004. On indefinite pronoun structures with APs: Reply to Kishimoto . Linguistic Inquiry 35.268–87. DOI: 10.1162/002438904323019075. LASNIK, HOWARD, and MYUNG-KWAN PARK. 2003. The EPP and the subject condition under sluicing. Linguistic Inquiry 34.649–60. DOI: 10.1162/ling.2003.34.4.649. LÓPEZ, LUIS. 2001. On the (non)complementarity of θ-theory and checking theory. Linguistic Inquiry 32.694– 716. DOI: 10.1162/002438901753373050. MARTIN, ROGER. 2001. Null case and the distribution of PRO. Linguistic Inquiry 32.141–66. DOI: 10.1162/ 002438901554612. MCGINNIS, MARTHA. 2004. Lethal ambiguity. Linguistic Inquiry 35.47–95. DOI: 10.1162/00243890432 2793347. MÜLLER, GEREON. 2010. On deriving CED effects from the PIC. Linguistic Inquiry 41.35–82. DOI: 10.1162/ ling.2010.41.1.35. NAKAJIMA, HEIZO. 2006. Adverbial cognate objects. Linguistic Inquiry 37.674–84. DOI: 10.1162/ling.2006 .37.4.674. NEELEMAN, AD, and HANS VAN DE KOOT. 2002. The configurational matrix. Linguistic Inquiry 33.529–74. DOI: 10.1162/002438902762731763. NUNES, JAIRO. 2001. Sideward movement. Linguistic Inquiry 32.303–44. DOI: 10.1162/00243890152001780. s8 PANAGIOTIDIS, PHOEVOS. 2003. One, empty nouns, and θ-assignment. Linguistic Inquiry 34.281–92. DOI: 10.1162/ling.2003.34.2.281. PHILLIPS, COLIN. 2003. Linear order and constituency. Linguistic Inquiry 34.37–90. DOI: 10.1162/0024389 03763255922. REZAC, MILAN. 2010. ϕ-Agree versus ϕ-feature movement: Evidence from floating quantifiers. Linguistic Inquiry 41.496–508. DOI: 10.1162/LING_a_00007. RICHARDS, NORVIN. 2004. Against bans on lowering. Linguistic Inquiry 35.453–63. DOI: 10.1162/00243890 41402643. SIGURÐSSON, HALLDÓR ÁRMANN. 2006. The nominative puzzle and the low nominative hypothesis. Linguistic Inquiry 37.289–308. DOI: 10.1162/ling.2006.37.2.289. SOBIN, NICHOLAS. 2004. Expletive constructions are not ‘lower right corner’ movement constructions. Linguistic Inquiry 35.503–8. DOI: 10.1162/ling.2004.35.3.503. SOBIN, NICHOLAS. 2008. Do so and VP. Linguistic Inquiry 39.147–60. DOI: 10.1162/ling.2008.39.1.147. STEPANOV, ARTHUR; and PENKA STATEVA. 2009. When QR disobeys superiority. Linguistic Inquiry 40.176– 85. DOI: 10.1162/ling.2009.40.1.176. STROIK, THOMAS. 2001. On the light verb hypothesis. Linguistic Inquiry 32.362–69. DOI: 10.1162/ling.2001 .32.2.362. TAKANO, YUJI. 2003. How antisymmetric is syntax? Linguistic Inquiry 34.516–26. DOI: 10.1162/ling.2003 .34.3.516. VICENTE, LUIS. 2010. A note on the movement analysis of gapping. Linguistic Inquiry 41.509–17. DOI: 10.1162/LING_a_00008. APPENDIX D: DISCUSSION OF ITEMS THAT DO NOT SHOW CLEAR RESULTS IN THE PREDICTED DIRECTION 35.3.Hazout:36 (#) There seem/*seems to have appeared [some new candidates] in the course of the presidential campaign. The rating study revealed no significant difference between the two variants (β = 0), and the starred variant was significantly preferred in the forced-choice experiment. This judgment seems to reflect a trend in colloquial English to use the singular There seems in these ‘verbal existential sentences’, even when the agreeing phrase is plural. At the very least, there may be individual variation in sentences like this. 34.4.Lasnik:24a–24b a. *?The detective asserted two students to have been at the demonstration during each other’s hearings. b. ?*The detective asserted that two students were at the demonstration during each other’s hearings. Example (b) is proposed to be unacceptable only when the final PP modifies the matrix clause and not the embedded clause. Our items were written to ensure that this is the only plausible interpretation, but participants still preferred (b) by a significant margin in the forced-choice experiment. 34.4.Lasnik:22a–22b a. John proved three chapters to have been plagiarized with one convincing example each. b. ?*John proved that three chapters were plagiarized with one convincing example each. s9 This example showed a nonsignificant trend in favor of (a) in the rating study and a nonsignificant trend toward (b) in the forced-choice study. Again, we took care to ensure that the final PP modifies the matrix verb across all our items. 32.4.Lopez:9c–10c a. We proved Smith to the authorities to be the thief. b. *We proved to the authorities Smith to be the thief. People significantly preferred (a) in the rating study, but the opposite trend emerged in the forced-choice study, which suggests that this is not a clear contrast. In fact, Hartman (2011) has argued that sentences like (a) are degraded on independent grounds, which might explain why most subjects did not prefer them over (b). 39.1.Sobin:8b–8f a. Bill devoured a ham, and Mary did a similar thing with a chicken. b. *Bill devoured a ham, and Mary did so with a chicken. In this contrast, we found a significant predicted effect in the rating study but a trend in the opposite direction in the forced-choice experiment. It is possible, in this case, that the did so construction in (b) is semantically unclear out of context, but clearer (and more natural sounding) when presented with the more semantically transparent (a). This would explain the difference between the rating study and the forced-choice study.1 APPENDIX E: MATH BEHIND SNAP JUDGMENTS Formally, we can think of our experiment as a draw from a binomial distribution, where p is the underlying population parameter for how likely someone is to choose sentence A over sentence B, n is the total number of trials, and k is the number of trials on which someone chose sentence A over sentence B. 𝑃(𝑘|𝑛, 𝑝) = � 𝑛 𝑘 � 𝑝𝑘 (1 − 𝑝)𝑛−𝑘 To obtain a confidence interval from a binomial distribution where the sample is unanimous while also taking advantage of our prior knowledge about how MOST experiments turn out, we will use a Bayesian credible interval—which is the Bayesian version of a confidence interval and can be thought of as the probability that a given parameter falls within some interval—on the posterior distribution. We get the posterior distribution by combining our binomial likelihood with a beta prior distribution (Gelman et al. 2004) on the parameter p, which gives a distribution of possible values for our parameter p. This prior distribution is the distribution over the value of p BEFORE we have collected any data. In other words, before we flip the coin, we do not know its weight p. We might think that it is very likely that the coin is fair and that p is near 0.50. Or maybe we think that p is close to 1. The shape of the distribution is controlled by the shape parameters α and β. Formally, the beta distribution is: 𝑃(𝑝|α, β) = 𝑝𝛼−1(1−𝑝)β−1 𝐵(α,β) , 1 These results also demonstrate that different experimental tasks can sometimes give different results. Specifically , it seems that (b)’s unacceptability is largely context-dependent. s10 where B is the beta function. We could, in principle, use any distribution with support on [0,1], but we use the beta distribution because it is the conjugate prior for the binomial and thus lets us obtain a closed-form solution. Informally, we can think of the job of the prior as being to add in our prior belief about the underlying distribution. We can literally think of this as adding the results of imaginary trials that we have not actually conducted. For instance, if we suspect that the coin is fair, we might use a beta prior of Beta(5, 5)—meaning α and β are both 5. Then, we present five people with sentence A and sentence B and ask which is better. In this case, p is the underlying probability of choosing A. We get the following result: A A A A A Without the prior, our best guess for the underlying parameter p is 1 since 5/5 is 1. If we use the Beta(5, 5) prior, however, we can think of this as adding five a priori As and five a priori Bs to our five experimentally obtained As such that we imagine we have ten As and five Bs, as in the following (where the italicized values come from the prior): A A A A A B B B B B A A A A A In this case, our best estimate of the underlying parameter p is (5 As + 5 As) / (15 trials) = 0.66. If we were very confident that the sentences are equally acceptable (i.e. the coin is fair; p ~ 0.5), we could use a Beta(100, 100) prior. With a prior like that, we would have to conduct many more trials in order to move our estimate substantially away from 0.50. After getting five As, we would still have an estimate of 51%. If we thought it was very likely that one of the sentences was better, but we did not know which, we might instead use a beta prior of Beta(.1, .1). This would mean that, after asking five people who all choose A, our new estimate for how likely a random person is to choose A would be: 5.1/(5.1 + .1) = 98%. Figure 3 shows the shape of the beta distribution for two possible settings of the shape parameters. If the shape parameters are unequal, then the distribution is skewed. When the two shape parameters are equal, the distribution is symmetric. Formally, we can multiply the beta prior and the binomial likelihood together to get the posterior probability. 𝑃(𝑘|𝑛, 𝑝) ∗ 𝑃(𝑝|α, β) = 𝑃(𝑘|𝑛, α, β) = � 𝑛 𝑘 � 𝐵(𝑘 + α, 𝑛 − 𝑘 + β) 𝐵(α, β) s11 FIGURE 3. The histograms represent a density map of a draw from a beta distribution with the shape parameters indicated . The red line is the probability density of the beta distribution at each value for p between 0 and 1. The plot on the left conforms to an instance in which, most of the time, the probability p is extreme (toward 0 or 1), as in the experiments we tested here. The plot on the right corresponds to a situation in which we have a strong prior belief that the probability p is near 0.5. In our case, we want to know what our prior expectations about p should be. Should our prior look more like Figure 3a or Figure 3b? Because we have formal results for 100 contrasts, we can use these empirical results to set our prior.2 In other words, when we have a new contrast for which we do not have much data but which we believe likely to produce a unanimous result, we can imagine that the contrast has an underlying parameter p (where p is once again the probability of choosing sentence A) and that p is drawn from the same distribution of judgments that gave rise to the 100 contrasts we observed. If we do not believe that the contrast is likely to produce a unanimous result, the assumption that the parameter p is drawn from the same distribution as the 100 contrasts we tested experimentally is potentially invalid since, in general, the effects that we tested were hypothesized to be very strong. 2 The prior that is obtained by our experimental results ends up very similar to what is obtained from the results from Sprouse, Schütze, and Almeida’s (2013) data (available on Jon Sprouse’s webpage). s12 FIGURE 4. This plot corresponds to a smoothed histogram (averaged over many trials) of the data from our forced-choice experiment where, for each contrast, one variant is randomly assigned to be sentence A and one to be sentence B. Most of the time, there is a strong preference for one sentence or the other. The best fit for the beta distribution is Beta(5.9, 1.1)—which is shown by the red line. In order to determine the prior empirically, for each contrast in our experiment, we assume that the hypothesized ‘good’ sentence is sentence A. We then draw a histogram of the effect sizes and fit the beta distribution to the histogram (as seen in Fig. 4). Averaging over 100 samples, the best fit is Beta(5.9, 1.1), with standard error .12 and .01, respectively. Rounding to the nearest whole number, we can think of this as having seen six As and one B BEFORE we run our experiment. Thus, if we run an experiment and get three As and zero Bs, we can act as if we have nine As and one B. We can use this prior to construct 95% Bayesian credible intervals for the underlying probability in the population of someone preferring sentence A over sentence B. Specifically, the Bayesian credible interval gives us a continuous interval, for which there is a 95% probability that the true underlying probability falls in that region. We also checked to see if the recommendations here were robust to other reasonable choices of prior. There is some theoretical question as to whether it makes sense to use the full available information in order to set the prior or if we should instead ‘forget’ which sentence is hypothesized to be good and assume that it is equally likely that the good sentence is A or B. The logic here is that including information as to which sentence is supposed to be good would be equivalent to doing an experiment where a researcher wants to test the efficacy of a medicine and then includes her prior belief that the medicine will probably work as evidence in the experiment. While she might be very confident in the medicine’s efficacy , she cannot include that prior belief as part of her analysis or else she could end up concluding that data that are consistent with pure noise are actually a result in favor of the hypothesis. But, because the s13 whole point of the SNAP Judgment paradigm is to use the existing information, we do not believe those concerns are particularly relevant here.3 To check how robust the paradigm is to choice of prior, we tried this approach where A and B are equally likely to be the ‘good’ sentence. To do that, we randomly assign one sentence in each contrast as A and one as B. Using this approach, we find a Beta(.6, .6) prior. For five unanimous participants, this gives us a mean of .90 with a 95% CI of [.67, 1]. So the CI’s lower bound is only slightly lower than when we include all the information. To get the lower bound to .75 when we use this prior, we would need to include seven participants in the experiment (as compared to five in our main analysis). We would also arrive at similar conclusions if we used the Jeffreys uninformative prior Beta(.5, .5)—a prior that is standardly used in many applications since it is locally uniform. Hence, the outcome is similar under other plausible alternative priors. We use the asymmetric, full-information prior in our main analysis, but we recognize that there may be good theoretical reasons to instead use the symmetric prior. APPENDIX F: STATISTICAL POWER The idea of computing statistical power is to ask, if there is an underlying ‘true effect’ size D that is being looked for in the experiment, what is the likelihood that the experiment correctly detects a significant effect? (Note that, in reality, we can never know the ‘true effect size’ because that would require infinite data. We can only sample.) If D = .8 for a sentence in the forced-choice experiment, that would mean the true underlying effect was .80. If the statistical power of our experiment is .95 (based on the sample size and design), that would mean that 95% of the time we would find a significant effect given the underlying effect size of .80. (Power would be lower if the effect size were smaller.) To compute statistical power and possible error rates using linear mixed-effects models, we repeated the following procedure 100 times for each contrast, took the mean of those 100 iterations, and then averaged across contrasts. a) Fit a linear mixed-effects model to the real data as described in the main text. b) Use the random-effects structure and residual variance from the model fit to the actual data in a). For the fixed-effect estimate, use D, which we systematically vary and report for several values in the table below. In effect, this lets us use the actual variance in the world (by subject, by item, and residual variance) to estimate the noise we should expect in an experiment. c) Use the parameters from b) to simulate a new set of data equivalent in sample size to the original experiment and with the same subject and item breakdown as the original experiment. d) Fit a new linear mixed-effects model to the simulated data in c) and test for effect size and significance. e) Use the effect sizes and significance levels found in d) to calculate power, type S, and type M error. We used the simulated effect size and significance measures to calculate statistical power given varying underlying effect sizes as well as two measures recommended: type S (sign) error and type M (magnitude ) error (Gelman & Carlin 2014). Power here refers to the proportion of the time a ‘true effect’ would be detected in the experiment given true effect size D. Type S error refers to the proportion of the time a significant effect is found in the OPPOSITE direction of the true effect. That is, if the type S error rate is .05, that means that 5% of the time, we should expect to find a significant effect in the opposite direction of the true effect. Type M error refers to the expected absolute overestimation rate given that a significant 3 See Cox & Mayo 2011 and Gelman 2012 for more discussion of how to use prior information responsibly in scientific inference. s14 effect is found (that is, when significant, the absolute value of the estimated effect size divided by the true effect size). This means that, conditioned on finding a significant effect, we should expect it to be M times more extreme than the underlying true effect. The tables below report power and estimated error rates for various true effect sizes. Note that, in the rating study, a true effect size less than .4 is quite small (only 19% of our estimated effect sizes are this small) and possibly not large enough for robust acceptability generalizations. For the forced-choice study, an effect size less than .70 is quite small, and only 11% of our data fits that description. D (TRUE EFFECT SIZE) STATISTICAL POWER TYPE S ERROR RATE TYPE M ERROR RATE .2 0.63 0.0 1.29 .4 0.96 0.0 1.01 .6 1.00 0.0 1.00 TABLE F1. Ratings study (all values where significance is defined by p < 0.05). D (TRUE EFFECT SIZE) STATISTICAL POWER TYPE S ERROR RATE TYPE M ERROR RATE .6 .48 .04 1.71 .7 .80 0.0 1.17 .8 .93 0.0 1.06 TABLE F2. Forced-choice study (all values where significance is defined by p < 0.05). * Note that for the forced-choice study, the type M error rate refers to the overestimation rate of the difference between the effect size D (defined as the proportion choosing the good sentence) and .5 (50% baseline in which neither sentence is better than another). So a 1.17 type M error rate for D = .7 means that, on average, if the contrast is significant at p < 0.05, the difference between the estimated d and .5 is 1.17 higher than it should be (where what it ‘should be’ is .7 − .5 = .2). REFERENCES COX, SIR DAVID, and DEBORAH MAYO. 2011. Statistical scientist meets a philosopher of science: A conversation. Rationality, Markets and Morals 2.103–14. Online: http://www.rmm-journal.de/ downloads/Article_Cox_Mayo.pdf. GELMAN, ANDREW. 2012. Ethics and statistics: Ethics and the statistical use of prior information. CHANCE 25.52–54. DOI: 10.1080/09332480.2012.752294. GELMAN, ANDREW, and JOHN CARLIN. 2014. Beyond power calculations: Assessing type S (sign) and type M (magnitude) errors. Perspectives on Psychological Science 9.641–51. DOI: 10.1177/174569161455 1642. GELMAN, ANDREW; JOHN B. CARLIN; HAL S. STERN; and DONALD B. RUBIN. 2004. Bayesian data analysis. Boca Raton, FL: CRC Press. SPROUSE, JON; CARSON T. SCHÜTZE; and DIOGO ALMEIDA. 2013. A comparison of informal and formal acceptability judgments using a random sample from Linguistic Inquiry 2001–2010. Lingua 134.219– 48. DOI: 10.1016/j.lingua.2013.07.002. [kmahowald@gmail.com] [graffmail@gmail.com] [hartman@linguist.umass.edu] [egibson@mit.edu] ...

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ISSN
1535-0665
Print ISSN
0097-8507
Pages
pp. s1-s14
Launched on MUSE
2016-09-09
Open Access
No
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