Masters-Waage et al. (2024) examine whether underrepresented minority (URM) faculty receive less favourable voting outcomes during tenure and promotion reviews. The main analyses use ordinary least squares (OLS) regression for two key continuous outcomes:
department_negative_vote_percentage): the share of department members voting against the candidatecollege_committee_negative_vote_percentage): the share of college committee members voting against the candidateBoth outcomes are proportions bounded strictly in [0, 1] and often pile up at the boundaries (unanimous votes). OLS does not respect these boundaries and can produce biased estimates when the outcome distribution is skewed or boundary-heavy. Ordered beta regression (ordbetareg) is purpose-built for such outcomes: it handles zeros and ones directly as a mixture component and models the interior with a beta distribution.
This post re-runs every lm model that uses department_negative_vote_percentage as the outcome and compares the results to ordbetareg. The question is simple: does the choice of model change the substantive conclusions?
ggplot(candidates, aes(x = department_negative_vote_percentage)) +
geom_histogram(bins = 30, fill = "#0072B2", color = "white", alpha = 0.8) +
scale_x_continuous(labels = scales::percent_format()) +
labs(
title = "Distribution of Department Negative Vote Percentage",
x = "Department negative vote %",
y = "Count"
) +
theme_minimal(base_size = 13)
lm1 <- lm(
scaled_department_negative_vote_percentage ~ candidate_urm_num,
data = candidates
)
ord1 <- ordbetareg(
department_negative_vote_percentage ~ candidate_urm_num,
data = candidates,
backend = "cmdstanr",
chains = 2,
seed = 42
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Mean chain execution time: 3.1 seconds.
Total execution time: 6.5 seconds.summary(lm1)
Call:
lm(formula = scaled_department_negative_vote_percentage ~ candidate_urm_num,
data = candidates)
Residuals:
Min 1Q Median 3Q Max
-0.5373 -0.3828 -0.3828 -0.3828 4.7822
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.01478 0.02778 -0.532 0.5947
candidate_urm_num 0.15445 0.09258 1.668 0.0955 .
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9992 on 1420 degrees of freedom
(149 observations deleted due to missingness)
Multiple R-squared: 0.001956, Adjusted R-squared: 0.001253
F-statistic: 2.783 on 1 and 1420 DF, p-value: 0.09549summary(ord1) Family: ord_beta_reg
Links: mu = identity
Formula: department_negative_vote_percentage ~ candidate_urm_num
Data: data (Number of observations: 1422)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.94 0.06 -1.05 -0.82 1.00 1840 1593
candidate_urm_num 0.30 0.13 0.03 0.54 1.00 2386 1716
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 3.57 0.26 3.07 4.10 1.00 2331 1457
cutzero 0.23 0.08 0.07 0.40 1.00 2046 1605
cutone 1.12 0.07 0.99 1.26 1.00 2212 1299
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).ms_compare(lm1, ord1, "URM → dept. negative vote % — no controls, full sample")| LM (original) | Ordbetareg (new) | |
|---|---|---|
| LM outcome: z-scored dept. negative vote %; coefficients in SD units. | ||
| Ordbetareg outcome: raw proportion [0, 1]; coefficients on log-odds scale. | ||
| Discipline and school fixed effects included but omitted from display. | ||
| 95% frequentist CI (LM) and 95% posterior CrI (Ordbetareg) shown. | ||
| URM candidate | 0.154 | |
| [-0.027, 0.336] | ||
| b_candidate_urm_num | 0.305 | |
| [0.035, 0.542] | ||
| Num.Obs. | 1422 | 1422 |
| R2 | 0.002 | 0.003 |
| R2 Adj. | 0.001 | |
| AIC | 4037.2 | |
| BIC | 4053.0 | |
| Log.Lik. | -2015.600 | |
| ELPD | -764.5 | |
| ELPD s.e. | 25.5 | |
| LOOIC | 1529.0 | |
| LOOIC s.e. | 50.9 | |
| WAIC | 1528.9 | |
| RMSE | 1.00 | 0.19 |
The full control set matches the main specification in the paper.
lm2 <- lm(
scaled_department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
candidate_discipline_stem_status_sbs_included + promotion_to_full_professor +
scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = candidates
)
ord2 <- ordbetareg(
department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
candidate_discipline_stem_status_sbs_included + promotion_to_full_professor +
scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = candidates,
backend = "cmdstanr",
chains = 2,
seed = 42
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Mean chain execution time: 27.1 seconds.
Total execution time: 56.7 seconds.summary(lm2)
Call:
lm(formula = scaled_department_negative_vote_percentage ~ candidate_urm_num +
candidate_gender_num + scaled_pi_grants + candidate_discipline_stem_status_sbs_included +
promotion_to_full_professor + scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = candidates)
Residuals:
Min 1Q Median 3Q Max
-1.3950 -0.4894 -0.2672 0.0087 4.9164
Coefficients:
Estimate Std. Error t value
(Intercept) -0.67773 0.69754 -0.972
candidate_urm_num 0.08492 0.10233 0.830
candidate_gender_num -0.05543 0.06191 -0.895
scaled_pi_grants -0.02246 0.03030 -0.741
candidate_discipline_stem_status_sbs_included 0.06232 0.16082 0.388
promotion_to_full_professor 0.16739 0.05955 2.811
scaled_years_in_present_rank 0.07755 0.02800 2.769
candidate_discipline_cip_category_as_numericAA -0.23166 1.19147 -0.194
candidate_discipline_cip_category_as_numericAB 0.58516 0.70624 0.829
candidate_discipline_cip_category_as_numericAC 0.59478 0.88799 0.670
candidate_discipline_cip_category_as_numericAD 0.40090 0.72329 0.554
candidate_discipline_cip_category_as_numericAE 0.74594 0.70737 1.055
candidate_discipline_cip_category_as_numericAF 0.41399 0.72596 0.570
candidate_discipline_cip_category_as_numericAG 0.28892 0.72944 0.396
candidate_discipline_cip_category_as_numericC 0.28556 0.78413 0.364
candidate_discipline_cip_category_as_numericD 0.30968 0.69553 0.445
candidate_discipline_cip_category_as_numericE 0.48786 0.71081 0.686
candidate_discipline_cip_category_as_numericF 0.43364 0.69904 0.620
candidate_discipline_cip_category_as_numericG 0.19531 0.71443 0.273
candidate_discipline_cip_category_as_numericH 0.29383 0.70369 0.418
candidate_discipline_cip_category_as_numericI 0.25941 0.70765 0.367
candidate_discipline_cip_category_as_numericJ 0.07596 0.72673 0.105
candidate_discipline_cip_category_as_numericL -0.16088 1.19078 -0.135
candidate_discipline_cip_category_as_numericM 0.62185 0.73835 0.842
candidate_discipline_cip_category_as_numericN 0.83905 0.71510 1.173
candidate_discipline_cip_category_as_numericO 0.37071 0.71383 0.519
candidate_discipline_cip_category_as_numericP 0.31803 0.72349 0.440
candidate_discipline_cip_category_as_numericQ 0.01440 1.19747 0.012
candidate_discipline_cip_category_as_numericR 0.48408 0.72534 0.667
candidate_discipline_cip_category_as_numericS -0.01705 0.75575 -0.023
candidate_discipline_cip_category_as_numericT 0.09067 0.71152 0.127
candidate_discipline_cip_category_as_numericU 0.34568 0.70707 0.489
candidate_discipline_cip_category_as_numericV 0.04437 0.73176 0.061
candidate_discipline_cip_category_as_numericW 0.05074 0.71733 0.071
candidate_discipline_cip_category_as_numericX 0.03997 0.75550 0.053
candidate_discipline_cip_category_as_numericY 0.32980 0.70229 0.470
candidate_discipline_cip_category_as_numericZ 0.35763 0.76127 0.470
school_code3 0.23778 0.09854 2.413
school_code4 0.24414 0.11085 2.202
school_code6 0.03145 0.14462 0.217
school_code9 0.26903 0.10903 2.468
Pr(>|t|)
(Intercept) 0.33144
candidate_urm_num 0.40676
candidate_gender_num 0.37074
scaled_pi_grants 0.45878
candidate_discipline_stem_status_sbs_included 0.69842
promotion_to_full_professor 0.00502 **
scaled_years_in_present_rank 0.00570 **
candidate_discipline_cip_category_as_numericAA 0.84587
candidate_discipline_cip_category_as_numericAB 0.40752
candidate_discipline_cip_category_as_numericAC 0.50311
candidate_discipline_cip_category_as_numericAD 0.57949
candidate_discipline_cip_category_as_numericAE 0.29185
candidate_discipline_cip_category_as_numericAF 0.56860
candidate_discipline_cip_category_as_numericAG 0.69212
candidate_discipline_cip_category_as_numericC 0.71579
candidate_discipline_cip_category_as_numericD 0.65623
candidate_discipline_cip_category_as_numericE 0.49263
candidate_discipline_cip_category_as_numericF 0.53515
candidate_discipline_cip_category_as_numericG 0.78461
candidate_discipline_cip_category_as_numericH 0.67634
candidate_discipline_cip_category_as_numericI 0.71400
candidate_discipline_cip_category_as_numericJ 0.91678
candidate_discipline_cip_category_as_numericL 0.89255
candidate_discipline_cip_category_as_numericM 0.39983
candidate_discipline_cip_category_as_numericN 0.24089
candidate_discipline_cip_category_as_numericO 0.60363
candidate_discipline_cip_category_as_numericP 0.66032
candidate_discipline_cip_category_as_numericQ 0.99041
candidate_discipline_cip_category_as_numericR 0.50466
candidate_discipline_cip_category_as_numericS 0.98200
candidate_discipline_cip_category_as_numericT 0.89862
candidate_discipline_cip_category_as_numericU 0.62500
candidate_discipline_cip_category_as_numericV 0.95165
candidate_discipline_cip_category_as_numericW 0.94362
candidate_discipline_cip_category_as_numericX 0.95782
candidate_discipline_cip_category_as_numericY 0.63872
candidate_discipline_cip_category_as_numericZ 0.63859
school_code3 0.01596 *
school_code4 0.02783 *
school_code6 0.82789
school_code9 0.01374 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9687 on 1220 degrees of freedom
(310 observations deleted due to missingness)
Multiple R-squared: 0.06089, Adjusted R-squared: 0.0301
F-statistic: 1.978 on 40 and 1220 DF, p-value: 0.0003164summary(ord2) Family: ord_beta_reg
Links: mu = identity
Formula: department_negative_vote_percentage ~ candidate_urm_num + candidate_gender_num + scaled_pi_grants + candidate_discipline_stem_status_sbs_included + promotion_to_full_professor + scaled_years_in_present_rank + candidate_discipline_cip_category_as_numeric + school_code
Data: data (Number of observations: 1261)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Regression Coefficients:
Estimate Est.Error l-95% CI
Intercept -2.60 0.90 -4.44
candidate_urm_num 0.26 0.15 -0.05
candidate_gender_num -0.06 0.10 -0.27
scaled_pi_grants -0.06 0.05 -0.15
candidate_discipline_stem_status_sbs_included -0.23 0.24 -0.68
promotion_to_full_professor 0.37 0.09 0.19
scaled_years_in_present_rank 0.09 0.04 0.01
candidate_discipline_cip_category_as_numericAA -2.80 3.44 -10.55
candidate_discipline_cip_category_as_numericAB 1.17 0.91 -0.54
candidate_discipline_cip_category_as_numericAC 1.28 1.22 -1.08
candidate_discipline_cip_category_as_numericAD 0.95 0.92 -0.85
candidate_discipline_cip_category_as_numericAE 1.40 0.91 -0.30
candidate_discipline_cip_category_as_numericAF 1.45 0.93 -0.32
candidate_discipline_cip_category_as_numericAG -1.09 1.41 -4.23
candidate_discipline_cip_category_as_numericC 0.82 0.99 -1.04
candidate_discipline_cip_category_as_numericD 0.71 0.88 -0.97
candidate_discipline_cip_category_as_numericE 1.44 0.89 -0.26
candidate_discipline_cip_category_as_numericF 0.98 0.90 -0.75
candidate_discipline_cip_category_as_numericG 0.30 1.00 -1.53
candidate_discipline_cip_category_as_numericH 0.43 0.93 -1.38
candidate_discipline_cip_category_as_numericI 0.46 0.94 -1.32
candidate_discipline_cip_category_as_numericJ 0.14 1.02 -1.92
candidate_discipline_cip_category_as_numericL -2.89 3.50 -10.46
candidate_discipline_cip_category_as_numericM 0.92 0.99 -0.99
candidate_discipline_cip_category_as_numericN 1.47 0.92 -0.30
candidate_discipline_cip_category_as_numericO 1.24 0.90 -0.47
candidate_discipline_cip_category_as_numericP 1.02 0.91 -0.66
candidate_discipline_cip_category_as_numericQ -2.63 3.66 -10.49
candidate_discipline_cip_category_as_numericR 1.42 0.94 -0.43
candidate_discipline_cip_category_as_numericS -4.39 3.01 -11.11
candidate_discipline_cip_category_as_numericT 0.71 0.90 -0.99
candidate_discipline_cip_category_as_numericU 1.06 0.90 -0.65
candidate_discipline_cip_category_as_numericV -0.70 1.19 -3.12
candidate_discipline_cip_category_as_numericW -0.62 1.17 -2.93
candidate_discipline_cip_category_as_numericX 0.73 0.99 -1.23
candidate_discipline_cip_category_as_numericY 0.88 0.90 -0.84
candidate_discipline_cip_category_as_numericZ 1.16 1.01 -0.79
school_code3 0.53 0.20 0.15
school_code4 0.50 0.22 0.09
school_code6 0.13 0.30 -0.47
school_code9 0.63 0.21 0.20
u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.87 1.02 140 336
candidate_urm_num 0.56 1.00 1904 1163
candidate_gender_num 0.14 1.00 2434 1447
scaled_pi_grants 0.02 1.00 1979 1381
candidate_discipline_stem_status_sbs_included 0.25 1.00 1601 1414
promotion_to_full_professor 0.55 1.00 1871 1550
scaled_years_in_present_rank 0.17 1.01 2035 1516
candidate_discipline_cip_category_as_numericAA 2.98 1.00 1855 1392
candidate_discipline_cip_category_as_numericAB 2.89 1.02 140 415
candidate_discipline_cip_category_as_numericAC 3.51 1.01 285 684
candidate_discipline_cip_category_as_numericAD 2.81 1.02 134 364
candidate_discipline_cip_category_as_numericAE 3.23 1.02 154 429
candidate_discipline_cip_category_as_numericAF 3.31 1.02 163 376
candidate_discipline_cip_category_as_numericAG 1.36 1.01 430 818
candidate_discipline_cip_category_as_numericC 2.86 1.02 208 482
candidate_discipline_cip_category_as_numericD 2.47 1.02 133 324
candidate_discipline_cip_category_as_numericE 3.25 1.02 134 312
candidate_discipline_cip_category_as_numericF 2.77 1.02 130 346
candidate_discipline_cip_category_as_numericG 2.26 1.02 206 439
candidate_discipline_cip_category_as_numericH 2.26 1.02 186 396
candidate_discipline_cip_category_as_numericI 2.34 1.02 145 364
candidate_discipline_cip_category_as_numericJ 2.15 1.01 186 416
candidate_discipline_cip_category_as_numericL 2.77 1.00 1928 1263
candidate_discipline_cip_category_as_numericM 2.95 1.02 181 399
candidate_discipline_cip_category_as_numericN 3.31 1.02 135 400
candidate_discipline_cip_category_as_numericO 3.04 1.02 129 333
candidate_discipline_cip_category_as_numericP 2.90 1.02 152 350
candidate_discipline_cip_category_as_numericQ 3.61 1.00 1701 1376
candidate_discipline_cip_category_as_numericR 3.29 1.02 158 393
candidate_discipline_cip_category_as_numericS 0.20 1.00 1572 1236
candidate_discipline_cip_category_as_numericT 2.54 1.02 148 298
candidate_discipline_cip_category_as_numericU 2.85 1.02 123 302
candidate_discipline_cip_category_as_numericV 1.52 1.01 250 576
candidate_discipline_cip_category_as_numericW 1.58 1.01 270 617
candidate_discipline_cip_category_as_numericX 2.65 1.02 164 519
candidate_discipline_cip_category_as_numericY 2.66 1.02 137 328
candidate_discipline_cip_category_as_numericZ 3.23 1.02 148 523
school_code3 0.93 1.00 915 990
school_code4 0.93 1.00 955 1182
school_code6 0.73 1.00 1352 1082
school_code9 1.04 1.00 927 1211
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 4.12 0.34 3.47 4.82 1.00 2214 1361
cutzero 0.09 0.09 -0.09 0.27 1.00 1592 1633
cutone 1.17 0.08 1.02 1.33 1.00 2213 1255
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).ms_compare(lm2, ord2, "URM → dept. negative vote % — full controls, full sample")| LM (original) | Ordbetareg (new) | |
|---|---|---|
| LM outcome: z-scored dept. negative vote %; coefficients in SD units. | ||
| Ordbetareg outcome: raw proportion [0, 1]; coefficients on log-odds scale. | ||
| Discipline and school fixed effects included but omitted from display. | ||
| 95% frequentist CI (LM) and 95% posterior CrI (Ordbetareg) shown. | ||
| URM candidate | 0.085 | |
| [-0.116, 0.286] | ||
| Gender (female = 1) | -0.055 | |
| [-0.177, 0.066] | ||
| PI grants (scaled) | -0.022 | |
| [-0.082, 0.037] | ||
| STEM discipline | 0.062 | |
| [-0.253, 0.378] | ||
| Promotion to full prof. | 0.167 | |
| [0.051, 0.284] | ||
| Years in present rank (scaled) | 0.078 | |
| [0.023, 0.132] | ||
| b_candidate_urm_num | 0.263 | |
| [-0.045, 0.558] | ||
| b_candidate_gender_num | -0.059 | |
| [-0.266, 0.138] | ||
| b_scaled_pi_grants | -0.061 | |
| [-0.153, 0.025] | ||
| b_candidate_discipline_stem_status_sbs_included | -0.235 | |
| [-0.683, 0.247] | ||
| b_promotion_to_full_professor | 0.370 | |
| [0.188, 0.545] | ||
| b_scaled_years_in_present_rank | 0.089 | |
| [0.007, 0.169] | ||
| Num.Obs. | 1261 | 1261 |
| R2 | 0.061 | 0.089 |
| R2 Adj. | 0.030 | |
| AIC | 3540.7 | |
| BIC | 3756.6 | |
| Log.Lik. | -1728.356 | |
| ELPD | -669.2 | |
| ELPD s.e. | 25.4 | |
| LOOIC | 1338.5 | |
| LOOIC s.e. | 50.9 | |
| WAIC | 1332.4 | |
| RMSE | 0.95 | 0.19 |
lm3 <- lm(
scaled_department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
promotion_to_full_professor + scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = candidates
)
ord3 <- ordbetareg(
department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
promotion_to_full_professor + scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = candidates,
backend = "cmdstanr",
chains = 2,
seed = 42
)Running MCMC with 2 sequential chains...
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Mean chain execution time: 22.1 seconds.
Total execution time: 44.2 seconds.summary(lm3)
Call:
lm(formula = scaled_department_negative_vote_percentage ~ candidate_urm_num +
candidate_gender_num + scaled_pi_grants + promotion_to_full_professor +
scaled_years_in_present_rank + candidate_discipline_cip_category_as_numeric +
school_code, data = candidates)
Residuals:
Min 1Q Median 3Q Max
-1.3947 -0.4876 -0.2679 0.0090 4.9156
Coefficients:
Estimate Std. Error t value
(Intercept) -0.67772 0.69730 -0.972
candidate_urm_num 0.08430 0.10228 0.824
candidate_gender_num -0.05556 0.06189 -0.898
scaled_pi_grants -0.02096 0.03004 -0.698
promotion_to_full_professor 0.16736 0.05953 2.811
scaled_years_in_present_rank 0.07768 0.02799 2.775
candidate_discipline_cip_category_as_numericAA -0.23216 1.19105 -0.195
candidate_discipline_cip_category_as_numericAB 0.58514 0.70600 0.829
candidate_discipline_cip_category_as_numericAC 0.59430 0.88768 0.670
candidate_discipline_cip_category_as_numericAD 0.40131 0.72304 0.555
candidate_discipline_cip_category_as_numericAE 0.74574 0.70712 1.055
candidate_discipline_cip_category_as_numericAF 0.47352 0.70928 0.668
candidate_discipline_cip_category_as_numericAG 0.28848 0.72919 0.396
candidate_discipline_cip_category_as_numericC 0.28554 0.78385 0.364
candidate_discipline_cip_category_as_numericD 0.31010 0.69529 0.446
candidate_discipline_cip_category_as_numericE 0.54854 0.69311 0.791
candidate_discipline_cip_category_as_numericF 0.43328 0.69880 0.620
candidate_discipline_cip_category_as_numericG 0.19532 0.71418 0.273
candidate_discipline_cip_category_as_numericH 0.29358 0.70344 0.417
candidate_discipline_cip_category_as_numericI 0.25939 0.70741 0.367
candidate_discipline_cip_category_as_numericJ 0.07522 0.72647 0.104
candidate_discipline_cip_category_as_numericL -0.16069 1.19037 -0.135
candidate_discipline_cip_category_as_numericM 0.62126 0.73809 0.842
candidate_discipline_cip_category_as_numericN 0.85910 0.71298 1.205
candidate_discipline_cip_category_as_numericO 0.43174 0.69600 0.620
candidate_discipline_cip_category_as_numericP 0.37975 0.70550 0.538
candidate_discipline_cip_category_as_numericQ 0.01606 1.19704 0.013
candidate_discipline_cip_category_as_numericR 0.52072 0.71890 0.724
candidate_discipline_cip_category_as_numericS -0.01692 0.75548 -0.022
candidate_discipline_cip_category_as_numericT 0.15185 0.69355 0.219
candidate_discipline_cip_category_as_numericU 0.38825 0.69825 0.556
candidate_discipline_cip_category_as_numericV 0.04433 0.73150 0.061
candidate_discipline_cip_category_as_numericW 0.05049 0.71708 0.070
candidate_discipline_cip_category_as_numericX 0.10089 0.73871 0.137
candidate_discipline_cip_category_as_numericY 0.33680 0.70182 0.480
candidate_discipline_cip_category_as_numericZ 0.41856 0.74460 0.562
school_code3 0.23918 0.09844 2.430
school_code4 0.24491 0.11080 2.210
school_code6 0.03262 0.14454 0.226
school_code9 0.26994 0.10897 2.477
Pr(>|t|)
(Intercept) 0.33128
candidate_urm_num 0.40999
candidate_gender_num 0.36949
scaled_pi_grants 0.48555
promotion_to_full_professor 0.00501 **
scaled_years_in_present_rank 0.00560 **
candidate_discipline_cip_category_as_numericAA 0.84548
candidate_discipline_cip_category_as_numericAB 0.40737
candidate_discipline_cip_category_as_numericAC 0.50330
candidate_discipline_cip_category_as_numericAD 0.57898
candidate_discipline_cip_category_as_numericAE 0.29181
candidate_discipline_cip_category_as_numericAF 0.50451
candidate_discipline_cip_category_as_numericAG 0.69246
candidate_discipline_cip_category_as_numericC 0.71572
candidate_discipline_cip_category_as_numericD 0.65567
candidate_discipline_cip_category_as_numericE 0.42885
candidate_discipline_cip_category_as_numericF 0.53535
candidate_discipline_cip_category_as_numericG 0.78453
candidate_discipline_cip_category_as_numericH 0.67650
candidate_discipline_cip_category_as_numericI 0.71392
candidate_discipline_cip_category_as_numericJ 0.91755
candidate_discipline_cip_category_as_numericL 0.89264
candidate_discipline_cip_category_as_numericM 0.40012
candidate_discipline_cip_category_as_numericN 0.22846
candidate_discipline_cip_category_as_numericO 0.53516
candidate_discipline_cip_category_as_numericP 0.59049
candidate_discipline_cip_category_as_numericQ 0.98930
candidate_discipline_cip_category_as_numericR 0.46901
candidate_discipline_cip_category_as_numericS 0.98213
candidate_discipline_cip_category_as_numericT 0.82673
candidate_discipline_cip_category_as_numericU 0.57829
candidate_discipline_cip_category_as_numericV 0.95169
candidate_discipline_cip_category_as_numericW 0.94388
candidate_discipline_cip_category_as_numericX 0.89139
candidate_discipline_cip_category_as_numericY 0.63138
candidate_discipline_cip_category_as_numericZ 0.57414
school_code3 0.01525 *
school_code4 0.02726 *
school_code6 0.82149
school_code9 0.01338 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.9684 on 1221 degrees of freedom
(310 observations deleted due to missingness)
Multiple R-squared: 0.06078, Adjusted R-squared: 0.03078
F-statistic: 2.026 on 39 and 1221 DF, p-value: 0.0002249summary(ord3) Family: ord_beta_reg
Links: mu = identity
Formula: department_negative_vote_percentage ~ candidate_urm_num + candidate_gender_num + scaled_pi_grants + promotion_to_full_professor + scaled_years_in_present_rank + candidate_discipline_cip_category_as_numeric + school_code
Data: data (Number of observations: 1261)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Regression Coefficients:
Estimate Est.Error l-95% CI
Intercept -2.72 0.88 -4.36
candidate_urm_num 0.25 0.15 -0.06
candidate_gender_num -0.05 0.10 -0.24
scaled_pi_grants -0.06 0.05 -0.15
promotion_to_full_professor 0.37 0.09 0.18
scaled_years_in_present_rank 0.09 0.04 0.01
candidate_discipline_cip_category_as_numericAA -2.62 3.34 -9.85
candidate_discipline_cip_category_as_numericAB 1.30 0.89 -0.39
candidate_discipline_cip_category_as_numericAC 1.40 1.16 -0.92
candidate_discipline_cip_category_as_numericAD 1.09 0.91 -0.65
candidate_discipline_cip_category_as_numericAE 1.54 0.87 -0.16
candidate_discipline_cip_category_as_numericAF 1.35 0.89 -0.34
candidate_discipline_cip_category_as_numericAG -1.00 1.48 -4.23
candidate_discipline_cip_category_as_numericC 0.94 1.00 -0.92
candidate_discipline_cip_category_as_numericD 0.84 0.87 -0.79
candidate_discipline_cip_category_as_numericE 1.34 0.86 -0.30
candidate_discipline_cip_category_as_numericF 1.11 0.88 -0.55
candidate_discipline_cip_category_as_numericG 0.42 0.96 -1.49
candidate_discipline_cip_category_as_numericH 0.55 0.93 -1.27
candidate_discipline_cip_category_as_numericI 0.60 0.90 -1.13
candidate_discipline_cip_category_as_numericJ 0.25 0.99 -1.67
candidate_discipline_cip_category_as_numericL -2.75 3.52 -10.57
candidate_discipline_cip_category_as_numericM 1.04 0.97 -0.81
candidate_discipline_cip_category_as_numericN 1.47 0.92 -0.28
candidate_discipline_cip_category_as_numericO 1.14 0.87 -0.48
candidate_discipline_cip_category_as_numericP 0.93 0.86 -0.68
candidate_discipline_cip_category_as_numericQ -2.57 3.56 -10.02
candidate_discipline_cip_category_as_numericR 1.36 0.91 -0.36
candidate_discipline_cip_category_as_numericS -4.40 3.11 -11.65
candidate_discipline_cip_category_as_numericT 0.62 0.87 -1.02
candidate_discipline_cip_category_as_numericU 1.03 0.86 -0.67
candidate_discipline_cip_category_as_numericV -0.55 1.15 -2.90
candidate_discipline_cip_category_as_numericW -0.53 1.16 -2.88
candidate_discipline_cip_category_as_numericX 0.64 0.97 -1.23
candidate_discipline_cip_category_as_numericY 0.99 0.86 -0.67
candidate_discipline_cip_category_as_numericZ 1.06 0.98 -0.84
school_code3 0.53 0.20 0.16
school_code4 0.49 0.22 0.08
school_code6 0.13 0.30 -0.47
school_code9 0.63 0.22 0.22
u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -1.06 1.01 254 446
candidate_urm_num 0.55 1.00 2619 1632
candidate_gender_num 0.14 1.00 1681 1231
scaled_pi_grants 0.02 1.00 1707 1269
promotion_to_full_professor 0.55 1.00 1878 1366
scaled_years_in_present_rank 0.17 1.00 1693 1271
candidate_discipline_cip_category_as_numericAA 3.02 1.00 1746 1188
candidate_discipline_cip_category_as_numericAB 3.01 1.01 267 428
candidate_discipline_cip_category_as_numericAC 3.62 1.01 403 689
candidate_discipline_cip_category_as_numericAD 2.90 1.01 271 454
candidate_discipline_cip_category_as_numericAE 3.23 1.01 266 416
candidate_discipline_cip_category_as_numericAF 3.06 1.01 271 437
candidate_discipline_cip_category_as_numericAG 1.65 1.00 608 725
candidate_discipline_cip_category_as_numericC 2.92 1.01 323 589
candidate_discipline_cip_category_as_numericD 2.53 1.01 255 372
candidate_discipline_cip_category_as_numericE 3.02 1.01 252 371
candidate_discipline_cip_category_as_numericF 2.81 1.01 256 386
candidate_discipline_cip_category_as_numericG 2.25 1.01 327 460
candidate_discipline_cip_category_as_numericH 2.38 1.01 281 396
candidate_discipline_cip_category_as_numericI 2.27 1.01 268 404
candidate_discipline_cip_category_as_numericJ 2.13 1.01 317 532
candidate_discipline_cip_category_as_numericL 3.14 1.00 1914 1301
candidate_discipline_cip_category_as_numericM 2.99 1.01 309 518
candidate_discipline_cip_category_as_numericN 3.23 1.01 271 423
candidate_discipline_cip_category_as_numericO 2.77 1.01 253 395
candidate_discipline_cip_category_as_numericP 2.60 1.01 260 415
candidate_discipline_cip_category_as_numericQ 3.18 1.00 1834 1353
candidate_discipline_cip_category_as_numericR 3.12 1.01 277 416
candidate_discipline_cip_category_as_numericS 0.42 1.00 1187 1163
candidate_discipline_cip_category_as_numericT 2.30 1.01 255 403
candidate_discipline_cip_category_as_numericU 2.71 1.01 259 418
candidate_discipline_cip_category_as_numericV 1.66 1.00 464 670
candidate_discipline_cip_category_as_numericW 1.59 1.01 484 810
candidate_discipline_cip_category_as_numericX 2.50 1.01 293 582
candidate_discipline_cip_category_as_numericY 2.62 1.01 268 416
candidate_discipline_cip_category_as_numericZ 2.96 1.01 303 526
school_code3 0.92 1.00 872 1134
school_code4 0.93 1.00 835 1271
school_code6 0.73 1.00 1149 1392
school_code9 1.05 1.00 893 1066
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 4.08 0.32 3.47 4.73 1.00 1622 1256
cutzero 0.09 0.09 -0.07 0.26 1.00 1816 1419
cutone 1.18 0.07 1.03 1.32 1.00 2388 1312
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).ms_compare(lm3, ord3, "URM → dept. negative vote % — controls without STEM, full sample")| LM (original) | Ordbetareg (new) | |
|---|---|---|
| LM outcome: z-scored dept. negative vote %; coefficients in SD units. | ||
| Ordbetareg outcome: raw proportion [0, 1]; coefficients on log-odds scale. | ||
| Discipline and school fixed effects included but omitted from display. | ||
| 95% frequentist CI (LM) and 95% posterior CrI (Ordbetareg) shown. | ||
| URM candidate | 0.084 | |
| [-0.116, 0.285] | ||
| Gender (female = 1) | -0.056 | |
| [-0.177, 0.066] | ||
| PI grants (scaled) | -0.021 | |
| [-0.080, 0.038] | ||
| Promotion to full prof. | 0.167 | |
| [0.051, 0.284] | ||
| Years in present rank (scaled) | 0.078 | |
| [0.023, 0.133] | ||
| b_candidate_urm_num | 0.252 | |
| [-0.063, 0.550] | ||
| b_candidate_gender_num | -0.053 | |
| [-0.243, 0.143] | ||
| b_scaled_pi_grants | -0.062 | |
| [-0.151, 0.023] | ||
| b_promotion_to_full_professor | 0.367 | |
| [0.184, 0.548] | ||
| b_scaled_years_in_present_rank | 0.090 | |
| [0.010, 0.175] | ||
| Num.Obs. | 1261 | 1261 |
| R2 | 0.061 | 0.086 |
| R2 Adj. | 0.031 | |
| AIC | 3538.9 | |
| BIC | 3749.6 | |
| Log.Lik. | -1728.434 | |
| ELPD | -667.5 | |
| ELPD s.e. | 25.4 | |
| LOOIC | 1335.0 | |
| LOOIC s.e. | 50.7 | |
| WAIC | 1331.4 | |
| RMSE | 0.95 | 0.18 |
lm4 <- lm(
scaled_department_negative_vote_percentage ~ candidate_urm_num,
data = promotion_to_associate_professor_candidates_distinct
)
ord4 <- ordbetareg(
department_negative_vote_percentage ~ candidate_urm_num,
data = promotion_to_associate_professor_candidates_distinct,
backend = "cmdstanr",
chains = 2,
seed = 42
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Mean chain execution time: 1.5 seconds.
Total execution time: 3.2 seconds.summary(lm4)
Call:
lm(formula = scaled_department_negative_vote_percentage ~ candidate_urm_num,
data = promotion_to_associate_professor_candidates_distinct)
Residuals:
Min 1Q Median 3Q Max
-0.5458 -0.2790 -0.2790 -0.2790 4.8860
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -0.11860 0.03408 -3.480 0.000531 ***
candidate_urm_num 0.26683 0.12759 2.091 0.036854 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8867 on 727 degrees of freedom
(99 observations deleted due to missingness)
Multiple R-squared: 0.005979, Adjusted R-squared: 0.004612
F-statistic: 4.373 on 1 and 727 DF, p-value: 0.03685summary(ord4) Family: ord_beta_reg
Links: mu = identity
Formula: department_negative_vote_percentage ~ candidate_urm_num
Data: data (Number of observations: 729)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -1.15 0.09 -1.33 -0.97 1.00 1505 1398
candidate_urm_num 0.56 0.21 0.13 0.96 1.00 2359 1171
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 3.65 0.41 2.92 4.52 1.00 1894 1444
cutzero 0.24 0.13 -0.01 0.50 1.00 1648 1561
cutone 1.09 0.11 0.90 1.31 1.00 2504 1172
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).ms_compare(lm4, ord4, "URM → dept. negative vote % — no controls, associate promotion subset")| LM (original) | Ordbetareg (new) | |
|---|---|---|
| LM outcome: z-scored dept. negative vote %; coefficients in SD units. | ||
| Ordbetareg outcome: raw proportion [0, 1]; coefficients on log-odds scale. | ||
| Discipline and school fixed effects included but omitted from display. | ||
| 95% frequentist CI (LM) and 95% posterior CrI (Ordbetareg) shown. | ||
| URM candidate | 0.267 | |
| [0.016, 0.517] | ||
| b_candidate_urm_num | 0.567 | |
| [0.130, 0.965] | ||
| Num.Obs. | 729 | 729 |
| R2 | 0.006 | 0.010 |
| R2 Adj. | 0.005 | |
| AIC | 1897.4 | |
| BIC | 1911.2 | |
| Log.Lik. | -945.718 | |
| F | 4.373 | |
| ELPD | -347.9 | |
| ELPD s.e. | 19.2 | |
| LOOIC | 695.8 | |
| LOOIC s.e. | 38.4 | |
| WAIC | 695.8 | |
| RMSE | 0.89 | 0.17 |
lm5 <- lm(
scaled_department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
candidate_discipline_stem_status_sbs_included + scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = promotion_to_associate_professor_candidates_distinct
)
ord5 <- ordbetareg(
department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
candidate_discipline_stem_status_sbs_included + scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = promotion_to_associate_professor_candidates_distinct,
backend = "cmdstanr",
chains = 2,
seed = 42
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Mean chain execution time: 7.7 seconds.
Total execution time: 15.6 seconds.summary(lm5)
Call:
lm(formula = scaled_department_negative_vote_percentage ~ candidate_urm_num +
candidate_gender_num + scaled_pi_grants + candidate_discipline_stem_status_sbs_included +
scaled_years_in_present_rank + candidate_discipline_cip_category_as_numeric +
school_code, data = promotion_to_associate_professor_candidates_distinct)
Residuals:
Min 1Q Median 3Q Max
-1.6398 -0.3802 -0.1961 0.0336 4.8083
Coefficients:
Estimate Std. Error t value
(Intercept) -0.44294 0.86765 -0.511
candidate_urm_num 0.17108 0.14598 1.172
candidate_gender_num -0.05410 0.07561 -0.715
scaled_pi_grants -0.07719 0.06641 -1.162
candidate_discipline_stem_status_sbs_included -0.52418 0.19702 -2.661
scaled_years_in_present_rank 0.35202 0.07963 4.421
candidate_discipline_cip_category_as_numericAB 0.36968 0.87693 0.422
candidate_discipline_cip_category_as_numericAC 0.69467 1.04920 0.662
candidate_discipline_cip_category_as_numericAD 0.18124 0.90416 0.200
candidate_discipline_cip_category_as_numericAE 0.70311 0.88494 0.795
candidate_discipline_cip_category_as_numericAF 0.71832 0.91458 0.785
candidate_discipline_cip_category_as_numericAG -0.06701 0.90799 -0.074
candidate_discipline_cip_category_as_numericC 0.26481 1.21035 0.219
candidate_discipline_cip_category_as_numericD 0.30256 0.86385 0.350
candidate_discipline_cip_category_as_numericE 0.87623 0.88426 0.991
candidate_discipline_cip_category_as_numericF 0.32968 0.86793 0.380
candidate_discipline_cip_category_as_numericG 0.06645 0.88769 0.075
candidate_discipline_cip_category_as_numericH 0.34489 0.87581 0.394
candidate_discipline_cip_category_as_numericI 0.17738 0.87769 0.202
candidate_discipline_cip_category_as_numericJ -0.05900 0.91464 -0.065
candidate_discipline_cip_category_as_numericM 0.67130 0.91438 0.734
candidate_discipline_cip_category_as_numericN 0.28863 0.88571 0.326
candidate_discipline_cip_category_as_numericO 0.86610 0.88785 0.975
candidate_discipline_cip_category_as_numericP 0.83061 0.90329 0.920
candidate_discipline_cip_category_as_numericR 0.47000 0.89964 0.522
candidate_discipline_cip_category_as_numericS -0.00291 0.94007 -0.003
candidate_discipline_cip_category_as_numericT 0.62774 0.88416 0.710
candidate_discipline_cip_category_as_numericU 0.71354 0.88022 0.811
candidate_discipline_cip_category_as_numericV -0.04044 0.90454 -0.045
candidate_discipline_cip_category_as_numericW 0.07833 0.89479 0.088
candidate_discipline_cip_category_as_numericX 0.62531 0.94549 0.661
candidate_discipline_cip_category_as_numericY 0.26487 0.87187 0.304
candidate_discipline_cip_category_as_numericZ 0.57775 0.93875 0.615
school_code3 0.20919 0.11980 1.746
school_code4 0.17095 0.13285 1.287
school_code6 0.03299 0.16655 0.198
school_code9 0.11340 0.13111 0.865
Pr(>|t|)
(Intercept) 0.60988
candidate_urm_num 0.24168
candidate_gender_num 0.47459
scaled_pi_grants 0.24553
candidate_discipline_stem_status_sbs_included 0.00801 **
scaled_years_in_present_rank 0.0000117 ***
candidate_discipline_cip_category_as_numericAB 0.67349
candidate_discipline_cip_category_as_numericAC 0.50817
candidate_discipline_cip_category_as_numericAD 0.84119
candidate_discipline_cip_category_as_numericAE 0.42720
candidate_discipline_cip_category_as_numericAF 0.43253
candidate_discipline_cip_category_as_numericAG 0.94119
candidate_discipline_cip_category_as_numericC 0.82689
candidate_discipline_cip_category_as_numericD 0.72628
candidate_discipline_cip_category_as_numericE 0.32213
candidate_discipline_cip_category_as_numericF 0.70420
candidate_discipline_cip_category_as_numericG 0.94035
candidate_discipline_cip_category_as_numericH 0.69387
candidate_discipline_cip_category_as_numericI 0.83991
candidate_discipline_cip_category_as_numericJ 0.94859
candidate_discipline_cip_category_as_numericM 0.46314
candidate_discipline_cip_category_as_numericN 0.74463
candidate_discipline_cip_category_as_numericO 0.32971
candidate_discipline_cip_category_as_numericP 0.35818
candidate_discipline_cip_category_as_numericR 0.60157
candidate_discipline_cip_category_as_numericS 0.99753
candidate_discipline_cip_category_as_numericT 0.47799
candidate_discipline_cip_category_as_numericU 0.41790
candidate_discipline_cip_category_as_numericV 0.96435
candidate_discipline_cip_category_as_numericW 0.93027
candidate_discipline_cip_category_as_numericX 0.50864
candidate_discipline_cip_category_as_numericY 0.76139
candidate_discipline_cip_category_as_numericZ 0.53850
school_code3 0.08129 .
school_code4 0.19868
school_code6 0.84307
school_code9 0.38743
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 0.8518 on 598 degrees of freedom
(193 observations deleted due to missingness)
Multiple R-squared: 0.08716, Adjusted R-squared: 0.03221
F-statistic: 1.586 on 36 and 598 DF, p-value: 0.01765summary(ord5) Family: ord_beta_reg
Links: mu = identity
Formula: department_negative_vote_percentage ~ candidate_urm_num + candidate_gender_num + scaled_pi_grants + candidate_discipline_stem_status_sbs_included + scaled_years_in_present_rank + candidate_discipline_cip_category_as_numeric + school_code
Data: data (Number of observations: 635)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Regression Coefficients:
Estimate Est.Error l-95% CI
Intercept -3.09 1.07 -5.17
candidate_urm_num 0.47 0.28 -0.06
candidate_gender_num -0.02 0.17 -0.35
scaled_pi_grants -0.24 0.15 -0.55
candidate_discipline_stem_status_sbs_included -0.81 0.44 -1.67
scaled_years_in_present_rank 0.40 0.14 0.13
candidate_discipline_cip_category_as_numericAB 1.18 1.05 -0.94
candidate_discipline_cip_category_as_numericAC 1.78 1.27 -0.73
candidate_discipline_cip_category_as_numericAD 1.09 1.11 -1.09
candidate_discipline_cip_category_as_numericAE 1.83 1.06 -0.27
candidate_discipline_cip_category_as_numericAF 1.46 1.30 -1.15
candidate_discipline_cip_category_as_numericAG -4.03 3.16 -11.25
candidate_discipline_cip_category_as_numericC 0.78 1.33 -1.96
candidate_discipline_cip_category_as_numericD 0.95 1.03 -1.06
candidate_discipline_cip_category_as_numericE 2.21 1.06 0.19
candidate_discipline_cip_category_as_numericF 1.24 1.03 -0.82
candidate_discipline_cip_category_as_numericG -4.37 3.10 -11.76
candidate_discipline_cip_category_as_numericH 0.32 1.15 -1.98
candidate_discipline_cip_category_as_numericI 0.47 1.10 -1.70
candidate_discipline_cip_category_as_numericJ -3.80 3.02 -10.69
candidate_discipline_cip_category_as_numericM 0.19 1.62 -3.30
candidate_discipline_cip_category_as_numericN 0.83 1.22 -1.50
candidate_discipline_cip_category_as_numericO 2.14 1.07 0.13
candidate_discipline_cip_category_as_numericP 2.15 1.07 0.11
candidate_discipline_cip_category_as_numericR 1.14 1.25 -1.34
candidate_discipline_cip_category_as_numericS -3.56 3.12 -10.19
candidate_discipline_cip_category_as_numericT 1.67 1.07 -0.38
candidate_discipline_cip_category_as_numericU 1.84 1.04 -0.15
candidate_discipline_cip_category_as_numericV -4.11 3.17 -11.47
candidate_discipline_cip_category_as_numericW 0.39 1.25 -2.09
candidate_discipline_cip_category_as_numericX 1.75 1.18 -0.53
candidate_discipline_cip_category_as_numericY 1.04 1.03 -0.97
candidate_discipline_cip_category_as_numericZ -3.23 3.31 -10.68
school_code3 0.76 0.35 0.09
school_code4 0.69 0.37 -0.01
school_code6 0.39 0.48 -0.58
school_code9 0.84 0.37 0.15
u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.98 1.01 359 577
candidate_urm_num 1.02 1.00 2309 1396
candidate_gender_num 0.31 1.00 2838 1187
scaled_pi_grants 0.05 1.00 3606 1379
candidate_discipline_stem_status_sbs_included 0.05 1.00 1754 1057
scaled_years_in_present_rank 0.68 1.00 3054 1461
candidate_discipline_cip_category_as_numericAB 3.28 1.01 339 505
candidate_discipline_cip_category_as_numericAC 4.29 1.01 472 704
candidate_discipline_cip_category_as_numericAD 3.25 1.01 377 614
candidate_discipline_cip_category_as_numericAE 3.89 1.01 355 545
candidate_discipline_cip_category_as_numericAF 3.97 1.01 444 748
candidate_discipline_cip_category_as_numericAG 1.00 1.00 1672 1261
candidate_discipline_cip_category_as_numericC 3.46 1.00 545 865
candidate_discipline_cip_category_as_numericD 2.95 1.01 330 482
candidate_discipline_cip_category_as_numericE 4.37 1.01 328 461
candidate_discipline_cip_category_as_numericF 3.30 1.01 342 546
candidate_discipline_cip_category_as_numericG 0.54 1.00 1606 1184
candidate_discipline_cip_category_as_numericH 2.47 1.01 396 672
candidate_discipline_cip_category_as_numericI 2.58 1.01 360 523
candidate_discipline_cip_category_as_numericJ 0.91 1.00 1871 1346
candidate_discipline_cip_category_as_numericM 3.15 1.00 805 1012
candidate_discipline_cip_category_as_numericN 3.20 1.01 461 762
candidate_discipline_cip_category_as_numericO 4.33 1.01 336 484
candidate_discipline_cip_category_as_numericP 4.34 1.01 322 525
candidate_discipline_cip_category_as_numericR 3.62 1.01 434 689
candidate_discipline_cip_category_as_numericS 1.32 1.00 1948 1544
candidate_discipline_cip_category_as_numericT 3.83 1.01 323 429
candidate_discipline_cip_category_as_numericU 4.00 1.01 315 484
candidate_discipline_cip_category_as_numericV 0.94 1.00 1595 1086
candidate_discipline_cip_category_as_numericW 2.75 1.01 475 813
candidate_discipline_cip_category_as_numericX 4.08 1.01 423 801
candidate_discipline_cip_category_as_numericY 3.06 1.01 344 496
candidate_discipline_cip_category_as_numericZ 1.91 1.00 1718 1485
school_code3 1.49 1.00 930 1192
school_code4 1.42 1.00 967 1313
school_code6 1.36 1.00 1160 1606
school_code9 1.61 1.00 936 1303
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 3.83 0.49 2.91 4.83 1.00 2318 1436
cutzero -0.00 0.15 -0.29 0.28 1.00 2274 1574
cutone 1.14 0.11 0.92 1.37 1.00 3036 1408
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).ms_compare(lm5, ord5, "URM → dept. negative vote % — controls, associate promotion subset")| LM (original) | Ordbetareg (new) | |
|---|---|---|
| LM outcome: z-scored dept. negative vote %; coefficients in SD units. | ||
| Ordbetareg outcome: raw proportion [0, 1]; coefficients on log-odds scale. | ||
| Discipline and school fixed effects included but omitted from display. | ||
| 95% frequentist CI (LM) and 95% posterior CrI (Ordbetareg) shown. | ||
| URM candidate | 0.171 | |
| [-0.116, 0.458] | ||
| Gender (female = 1) | -0.054 | |
| [-0.203, 0.094] | ||
| PI grants (scaled) | -0.077 | |
| [-0.208, 0.053] | ||
| STEM discipline | -0.524 | |
| [-0.911, -0.137] | ||
| Years in present rank (scaled) | 0.352 | |
| [0.196, 0.508] | ||
| b_candidate_urm_num | 0.468 | |
| [-0.059, 1.023] | ||
| b_candidate_gender_num | -0.016 | |
| [-0.351, 0.310] | ||
| b_scaled_pi_grants | -0.232 | |
| [-0.546, 0.046] | ||
| b_candidate_discipline_stem_status_sbs_included | -0.797 | |
| [-1.671, 0.053] | ||
| b_scaled_years_in_present_rank | 0.399 | |
| [0.127, 0.677] | ||
| Num.Obs. | 635 | 635 |
| R2 | 0.087 | 0.108 |
| R2 Adj. | 0.032 | |
| AIC | 1636.2 | |
| BIC | 1805.5 | |
| Log.Lik. | -780.114 | |
| ELPD | -308.7 | |
| ELPD s.e. | 21.4 | |
| LOOIC | 617.3 | |
| LOOIC s.e. | 42.7 | |
| WAIC | 611.1 | |
| RMSE | 0.83 | 0.16 |
lm6 <- lm(
scaled_department_negative_vote_percentage ~ candidate_urm_num,
data = promotion_to_full_professor_candidates_distinct
)
ord6 <- ordbetareg(
department_negative_vote_percentage ~ candidate_urm_num,
data = promotion_to_full_professor_candidates_distinct,
backend = "cmdstanr",
chains = 2,
seed = 42
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Mean chain execution time: 1.5 seconds.
Total execution time: 3.2 seconds.summary(lm6)
Call:
lm(formula = scaled_department_negative_vote_percentage ~ candidate_urm_num,
data = promotion_to_full_professor_candidates_distinct)
Residuals:
Min 1Q Median 3Q Max
-0.5314 -0.4967 -0.4967 -0.0994 4.6683
Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) 0.09913 0.04415 2.246 0.025 *
candidate_urm_num 0.03468 0.13331 0.260 0.795
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.097 on 691 degrees of freedom
(50 observations deleted due to missingness)
Multiple R-squared: 9.792e-05, Adjusted R-squared: -0.001349
F-statistic: 0.06767 on 1 and 691 DF, p-value: 0.7948summary(ord6) Family: ord_beta_reg
Links: mu = identity
Formula: department_negative_vote_percentage ~ candidate_urm_num
Data: data (Number of observations: 693)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Regression Coefficients:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.78 0.07 -0.92 -0.63 1.00 1820 1611
candidate_urm_num 0.07 0.17 -0.26 0.39 1.00 2268 1148
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 3.78 0.37 3.08 4.52 1.01 2055 1343
cutzero 0.18 0.11 -0.03 0.40 1.00 1939 1598
cutone 1.14 0.09 0.97 1.32 1.00 2203 1435
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).ms_compare(lm6, ord6, "URM → dept. negative vote % — no controls, full professor subset")| LM (original) | Ordbetareg (new) | |
|---|---|---|
| LM outcome: z-scored dept. negative vote %; coefficients in SD units. | ||
| Ordbetareg outcome: raw proportion [0, 1]; coefficients on log-odds scale. | ||
| Discipline and school fixed effects included but omitted from display. | ||
| 95% frequentist CI (LM) and 95% posterior CrI (Ordbetareg) shown. | ||
| URM candidate | 0.035 | |
| [-0.227, 0.296] | ||
| b_candidate_urm_num | 0.074 | |
| [-0.258, 0.393] | ||
| Num.Obs. | 693 | 693 |
| R2 | 0.000 | 0.001 |
| R2 Adj. | -0.001 | |
| AIC | 2098.4 | |
| BIC | 2112.0 | |
| Log.Lik. | -1046.200 | |
| F | 0.068 | |
| ELPD | -410.2 | |
| ELPD s.e. | 17.1 | |
| LOOIC | 820.4 | |
| LOOIC s.e. | 34.1 | |
| WAIC | 820.4 | |
| RMSE | 1.09 | 0.21 |
lm7 <- lm(
scaled_department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
candidate_discipline_stem_status_sbs_included + scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = promotion_to_full_professor_candidates_distinct
)
ord7 <- ordbetareg(
department_negative_vote_percentage ~
candidate_urm_num + candidate_gender_num + scaled_pi_grants +
candidate_discipline_stem_status_sbs_included + scaled_years_in_present_rank +
candidate_discipline_cip_category_as_numeric + school_code,
data = promotion_to_full_professor_candidates_distinct,
backend = "cmdstanr",
chains = 2,
seed = 42
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Mean chain execution time: 10.0 seconds.
Total execution time: 20.0 seconds.summary(lm7)
Call:
lm(formula = scaled_department_negative_vote_percentage ~ candidate_urm_num +
candidate_gender_num + scaled_pi_grants + candidate_discipline_stem_status_sbs_included +
scaled_years_in_present_rank + candidate_discipline_cip_category_as_numeric +
school_code, data = promotion_to_full_professor_candidates_distinct)
Residuals:
Min 1Q Median 3Q Max
-1.9903 -0.5774 -0.3305 0.0532 4.5819
Coefficients:
Estimate Std. Error t value
(Intercept) -0.614037 1.084769 -0.566
candidate_urm_num 0.009131 0.147961 0.062
candidate_gender_num -0.067993 0.101506 -0.670
scaled_pi_grants -0.007589 0.037541 -0.202
candidate_discipline_stem_status_sbs_included 0.485671 0.268293 1.810
scaled_years_in_present_rank 0.051466 0.033382 1.542
candidate_discipline_cip_category_as_numericAA -0.050231 1.515289 -0.033
candidate_discipline_cip_category_as_numericAB 0.783188 1.101313 0.711
candidate_discipline_cip_category_as_numericAC -0.061831 1.514473 -0.041
candidate_discipline_cip_category_as_numericAD 0.643818 1.120697 0.574
candidate_discipline_cip_category_as_numericAE 0.776948 1.095270 0.709
candidate_discipline_cip_category_as_numericAF 0.151352 1.125144 0.135
candidate_discipline_cip_category_as_numericAG 0.653163 1.133503 0.576
candidate_discipline_cip_category_as_numericC 0.365413 1.162953 0.314
candidate_discipline_cip_category_as_numericD 0.302765 1.083547 0.279
candidate_discipline_cip_category_as_numericE 0.247748 1.107981 0.224
candidate_discipline_cip_category_as_numericF 0.591752 1.089713 0.543
candidate_discipline_cip_category_as_numericG 0.423679 1.112765 0.381
candidate_discipline_cip_category_as_numericH 0.244955 1.094280 0.224
candidate_discipline_cip_category_as_numericI 0.274478 1.104580 0.248
candidate_discipline_cip_category_as_numericJ 0.189899 1.121597 0.169
candidate_discipline_cip_category_as_numericL -0.126567 1.514504 -0.084
candidate_discipline_cip_category_as_numericM 0.533032 1.153898 0.462
candidate_discipline_cip_category_as_numericN 1.408968 1.121020 1.257
candidate_discipline_cip_category_as_numericO 0.009664 1.112597 0.009
candidate_discipline_cip_category_as_numericP 0.019143 1.124688 0.017
candidate_discipline_cip_category_as_numericQ 0.203398 1.525837 0.133
candidate_discipline_cip_category_as_numericR 0.537580 1.134437 0.474
candidate_discipline_cip_category_as_numericS -0.010580 1.175628 -0.009
candidate_discipline_cip_category_as_numericT -0.283631 1.110208 -0.255
candidate_discipline_cip_category_as_numericU 0.098679 1.100112 0.090
candidate_discipline_cip_category_as_numericV 0.066468 1.146941 0.058
candidate_discipline_cip_category_as_numericW 0.067558 1.113487 0.061
candidate_discipline_cip_category_as_numericX -0.368260 1.171541 -0.314
candidate_discipline_cip_category_as_numericY 0.474899 1.095924 0.433
candidate_discipline_cip_category_as_numericZ 0.477462 1.202223 0.397
school_code3 0.200911 0.162398 1.237
school_code4 0.267947 0.183424 1.461
school_code6 -0.032357 0.250655 -0.129
school_code9 0.376165 0.184027 2.044
Pr(>|t|)
(Intercept) 0.5716
candidate_urm_num 0.9508
candidate_gender_num 0.5032
scaled_pi_grants 0.8399
candidate_discipline_stem_status_sbs_included 0.0708 .
scaled_years_in_present_rank 0.1237
candidate_discipline_cip_category_as_numericAA 0.9736
candidate_discipline_cip_category_as_numericAB 0.4773
candidate_discipline_cip_category_as_numericAC 0.9674
candidate_discipline_cip_category_as_numericAD 0.5659
candidate_discipline_cip_category_as_numericAE 0.4784
candidate_discipline_cip_category_as_numericAF 0.8930
candidate_discipline_cip_category_as_numericAG 0.5647
candidate_discipline_cip_category_as_numericC 0.7535
candidate_discipline_cip_category_as_numericD 0.7800
candidate_discipline_cip_category_as_numericE 0.8231
candidate_discipline_cip_category_as_numericF 0.5873
candidate_discipline_cip_category_as_numericG 0.7035
candidate_discipline_cip_category_as_numericH 0.8230
candidate_discipline_cip_category_as_numericI 0.8038
candidate_discipline_cip_category_as_numericJ 0.8656
candidate_discipline_cip_category_as_numericL 0.9334
candidate_discipline_cip_category_as_numericM 0.6443
candidate_discipline_cip_category_as_numericN 0.2093
candidate_discipline_cip_category_as_numericO 0.9931
candidate_discipline_cip_category_as_numericP 0.9864
candidate_discipline_cip_category_as_numericQ 0.8940
candidate_discipline_cip_category_as_numericR 0.6358
candidate_discipline_cip_category_as_numericS 0.9928
candidate_discipline_cip_category_as_numericT 0.7984
candidate_discipline_cip_category_as_numericU 0.9286
candidate_discipline_cip_category_as_numericV 0.9538
candidate_discipline_cip_category_as_numericW 0.9516
candidate_discipline_cip_category_as_numericX 0.7534
candidate_discipline_cip_category_as_numericY 0.6649
candidate_discipline_cip_category_as_numericZ 0.6914
school_code3 0.2165
school_code4 0.1446
school_code6 0.8973
school_code9 0.0414 *
---
Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
Residual standard error: 1.065 on 586 degrees of freedom
(117 observations deleted due to missingness)
Multiple R-squared: 0.08866, Adjusted R-squared: 0.028
F-statistic: 1.462 on 39 and 586 DF, p-value: 0.0374summary(ord7) Family: ord_beta_reg
Links: mu = identity
Formula: department_negative_vote_percentage ~ candidate_urm_num + candidate_gender_num + scaled_pi_grants + candidate_discipline_stem_status_sbs_included + scaled_years_in_present_rank + candidate_discipline_cip_category_as_numeric + school_code
Data: data (Number of observations: 626)
Draws: 2 chains, each with iter = 2000; warmup = 1000; thin = 1;
total post-warmup draws = 2000
Regression Coefficients:
Estimate Est.Error l-95% CI
Intercept -2.40 0.89 -4.15
candidate_urm_num 0.10 0.19 -0.28
candidate_gender_num -0.09 0.14 -0.38
scaled_pi_grants -0.03 0.05 -0.13
candidate_discipline_stem_status_sbs_included 0.01 0.35 -0.66
scaled_years_in_present_rank 0.04 0.05 -0.05
candidate_discipline_cip_category_as_numericAA -2.97 3.77 -11.18
candidate_discipline_cip_category_as_numericAB 1.45 0.95 -0.34
candidate_discipline_cip_category_as_numericAC -2.78 3.55 -10.59
candidate_discipline_cip_category_as_numericAD 1.39 0.95 -0.40
candidate_discipline_cip_category_as_numericAE 1.41 0.93 -0.38
candidate_discipline_cip_category_as_numericAF 1.53 0.91 -0.23
candidate_discipline_cip_category_as_numericAG -0.16 1.51 -3.25
candidate_discipline_cip_category_as_numericC 1.12 1.15 -1.08
candidate_discipline_cip_category_as_numericD 0.96 0.91 -0.72
candidate_discipline_cip_category_as_numericE 1.52 0.89 -0.16
candidate_discipline_cip_category_as_numericF 1.23 0.91 -0.44
candidate_discipline_cip_category_as_numericG 1.18 0.99 -0.64
candidate_discipline_cip_category_as_numericH 0.70 0.95 -1.04
candidate_discipline_cip_category_as_numericI 0.64 1.00 -1.33
candidate_discipline_cip_category_as_numericJ 0.68 1.03 -1.34
candidate_discipline_cip_category_as_numericL -2.83 3.61 -10.54
candidate_discipline_cip_category_as_numericM 1.33 1.04 -0.67
candidate_discipline_cip_category_as_numericN 1.99 0.96 0.15
candidate_discipline_cip_category_as_numericO 1.19 0.90 -0.58
candidate_discipline_cip_category_as_numericP 0.99 0.91 -0.78
candidate_discipline_cip_category_as_numericQ -2.54 3.67 -10.70
candidate_discipline_cip_category_as_numericR 1.66 0.94 -0.22
candidate_discipline_cip_category_as_numericS -3.91 3.22 -11.13
candidate_discipline_cip_category_as_numericT 0.70 0.90 -1.07
candidate_discipline_cip_category_as_numericU 1.13 0.89 -0.58
candidate_discipline_cip_category_as_numericV 0.07 1.21 -2.34
candidate_discipline_cip_category_as_numericW -4.32 3.05 -11.51
candidate_discipline_cip_category_as_numericX 0.28 1.18 -2.01
candidate_discipline_cip_category_as_numericY 1.26 0.94 -0.51
candidate_discipline_cip_category_as_numericZ 1.75 1.03 -0.34
school_code3 0.37 0.25 -0.11
school_code4 0.40 0.27 -0.11
school_code6 -0.20 0.42 -1.02
school_code9 0.61 0.28 0.04
u-95% CI Rhat Bulk_ESS Tail_ESS
Intercept -0.69 1.00 229 584
candidate_urm_num 0.47 1.00 2724 1462
candidate_gender_num 0.18 1.00 2755 1544
scaled_pi_grants 0.07 1.00 3091 1390
candidate_discipline_stem_status_sbs_included 0.74 1.00 1772 1300
scaled_years_in_present_rank 0.13 1.01 3289 1553
candidate_discipline_cip_category_as_numericAA 3.17 1.00 1668 882
candidate_discipline_cip_category_as_numericAB 3.34 1.00 245 568
candidate_discipline_cip_category_as_numericAC 3.13 1.00 1880 1428
candidate_discipline_cip_category_as_numericAD 3.22 1.01 267 684
candidate_discipline_cip_category_as_numericAE 3.28 1.00 245 542
candidate_discipline_cip_category_as_numericAF 3.36 1.00 235 628
candidate_discipline_cip_category_as_numericAG 2.65 1.00 527 1081
candidate_discipline_cip_category_as_numericC 3.46 1.00 365 779
candidate_discipline_cip_category_as_numericD 2.76 1.00 233 503
candidate_discipline_cip_category_as_numericE 3.31 1.01 219 570
candidate_discipline_cip_category_as_numericF 3.09 1.00 232 517
candidate_discipline_cip_category_as_numericG 3.16 1.00 270 705
candidate_discipline_cip_category_as_numericH 2.57 1.00 251 608
candidate_discipline_cip_category_as_numericI 2.53 1.00 285 706
candidate_discipline_cip_category_as_numericJ 2.69 1.00 285 599
candidate_discipline_cip_category_as_numericL 3.08 1.00 2100 1330
candidate_discipline_cip_category_as_numericM 3.41 1.01 294 719
candidate_discipline_cip_category_as_numericN 3.85 1.00 249 590
candidate_discipline_cip_category_as_numericO 3.02 1.01 219 491
candidate_discipline_cip_category_as_numericP 2.82 1.01 222 580
candidate_discipline_cip_category_as_numericQ 3.39 1.00 1584 1225
candidate_discipline_cip_category_as_numericR 3.49 1.00 244 502
candidate_discipline_cip_category_as_numericS 1.11 1.00 1640 1216
candidate_discipline_cip_category_as_numericT 2.50 1.00 227 608
candidate_discipline_cip_category_as_numericU 2.94 1.01 217 553
candidate_discipline_cip_category_as_numericV 2.39 1.00 347 838
candidate_discipline_cip_category_as_numericW 0.31 1.00 1053 1272
candidate_discipline_cip_category_as_numericX 2.57 1.00 312 782
candidate_discipline_cip_category_as_numericY 3.14 1.00 230 681
candidate_discipline_cip_category_as_numericZ 3.81 1.01 296 659
school_code3 0.86 1.00 1188 1186
school_code4 0.95 1.00 1115 1248
school_code6 0.60 1.00 1662 1608
school_code9 1.15 1.00 1273 1274
Further Distributional Parameters:
Estimate Est.Error l-95% CI u-95% CI Rhat Bulk_ESS Tail_ESS
phi 4.34 0.47 3.44 5.28 1.00 2418 1241
cutzero 0.03 0.12 -0.21 0.28 1.00 1828 1250
cutone 1.20 0.09 1.01 1.38 1.00 2541 1071
Draws were sampled using sample(hmc). For each parameter, Bulk_ESS
and Tail_ESS are effective sample size measures, and Rhat is the potential
scale reduction factor on split chains (at convergence, Rhat = 1).ms_compare(lm7, ord7, "URM → dept. negative vote % — controls, full professor subset")| LM (original) | Ordbetareg (new) | |
|---|---|---|
| LM outcome: z-scored dept. negative vote %; coefficients in SD units. | ||
| Ordbetareg outcome: raw proportion [0, 1]; coefficients on log-odds scale. | ||
| Discipline and school fixed effects included but omitted from display. | ||
| 95% frequentist CI (LM) and 95% posterior CrI (Ordbetareg) shown. | ||
| URM candidate | 0.009 | |
| [-0.281, 0.300] | ||
| Gender (female = 1) | -0.068 | |
| [-0.267, 0.131] | ||
| PI grants (scaled) | -0.008 | |
| [-0.081, 0.066] | ||
| STEM discipline | 0.486 | |
| [-0.041, 1.013] | ||
| Years in present rank (scaled) | 0.051 | |
| [-0.014, 0.117] | ||
| b_candidate_urm_num | 0.097 | |
| [-0.282, 0.467] | ||
| b_candidate_gender_num | -0.089 | |
| [-0.378, 0.178] | ||
| b_scaled_pi_grants | -0.031 | |
| [-0.128, 0.066] | ||
| b_candidate_discipline_stem_status_sbs_included | 0.011 | |
| [-0.660, 0.742] | ||
| b_scaled_years_in_present_rank | 0.044 | |
| [-0.046, 0.130] | ||
| Num.Obs. | 626 | 626 |
| R2 | 0.089 | 0.109 |
| R2 Adj. | 0.028 | |
| AIC | 1896.0 | |
| BIC | 2078.0 | |
| Log.Lik. | -907.002 | |
| ELPD | -382.0 | |
| ELPD s.e. | 17.8 | |
| LOOIC | 764.0 | |
| LOOIC s.e. | 35.6 | |
| WAIC | 757.9 | |
| RMSE | 1.03 | 0.20 |
This section assembles the URM coefficient from every model pair into a single table and forest plot.
How to read the scales:
# ── Helper: extract URM row from each model pair ───────────────────────────
extract_urm <- function(lm_mod, ord_mod, label, lm_sd) {
# LM
lm_row <- coef(summary(lm_mod))["candidate_urm_num", ]
# Ordbetareg: fixef() from brms returns Estimate, Est.Error, Q2.5, Q97.5
ord_row <- fixef(ord_mod)["candidate_urm_num", ]
# Posterior probability > 0
draws <- as_draws_df(ord_mod)[["b_candidate_urm_num"]]
pp_pos <- round(mean(draws > 0), 3)
tibble(
analysis = label,
lm_est = lm_row[["Estimate"]],
lm_lo = lm_row[["Estimate"]] - 1.96 * lm_row[["Std. Error"]],
lm_hi = lm_row[["Estimate"]] + 1.96 * lm_row[["Std. Error"]],
lm_p = lm_row[["Pr(>|t|)"]],
lm_est_prop = lm_row[["Estimate"]] * lm_sd,
lm_lo_prop = (lm_row[["Estimate"]] - 1.96 * lm_row[["Std. Error"]]) * lm_sd,
lm_hi_prop = (lm_row[["Estimate"]] + 1.96 * lm_row[["Std. Error"]]) * lm_sd,
ord_est = ord_row[["Estimate"]],
ord_lo = ord_row[["Q2.5"]],
ord_hi = ord_row[["Q97.5"]],
pp_pos = pp_pos,
same_sign = sign(lm_row[["Estimate"]]) == sign(ord_row[["Estimate"]]),
lm_sig = lm_row[["Pr(>|t|)"]] < 0.05,
ord_sig = ord_row[["Q2.5"]] > 0 | ord_row[["Q97.5"]] < 0
)
}
results <- bind_rows(
extract_urm(lm1, ord1, "1. Full sample – no controls", dnvp_sd),
extract_urm(lm2, ord2, "2. Full sample – full controls", dnvp_sd),
extract_urm(lm3, ord3, "3. Full sample – no STEM control", dnvp_sd),
extract_urm(lm4, ord4, "4. Associate subset – no controls", dnvp_sd_assoc),
extract_urm(lm5, ord5, "5. Associate subset – with controls", dnvp_sd_assoc),
extract_urm(lm6, ord6, "6. Full-prof subset – no controls", dnvp_sd_full),
extract_urm(lm7, ord7, "7. Full-prof subset – with controls", dnvp_sd_full)
)
# ── Forest plot ────────────────────────────────────────────────────────────────
lm_plot <- results |>
mutate(analysis = fct_rev(factor(analysis))) |>
ggplot(aes(x = lm_est_prop, xmin = lm_lo_prop, xmax = lm_hi_prop,
y = analysis, color = lm_sig)) +
geom_pointrange(size = 0.7, linewidth = 1) +
geom_vline(xintercept = 0, linetype = "dashed", color = "grey40") +
scale_color_manual(
values = c("TRUE" = "#D55E00", "FALSE" = "#0072B2"),
labels = c("TRUE" = "p < .05", "FALSE" = "p ≥ .05")
) +
labs(
title = "LM",
x = "Effect on dept. negative vote\n(proportion units, LM coef × SD)",
y = NULL,
color = NULL
) +
theme_minimal(base_size = 12) +
theme(legend.position = "bottom", panel.grid.minor = element_blank())
ord_plot <- results |>
mutate(analysis = fct_rev(factor(analysis))) |>
ggplot(aes(x = ord_est, xmin = ord_lo, xmax = ord_hi,
y = analysis, color = ord_sig)) +
geom_pointrange(size = 0.7, linewidth = 1) +
geom_vline(xintercept = 0, linetype = "dashed", color = "grey40") +
scale_color_manual(
values = c("TRUE" = "#D55E00", "FALSE" = "#0072B2"),
labels = c("TRUE" = "95% CrI ≠ 0", "FALSE" = "95% CrI spans 0")
) +
labs(
title = "Ordbetareg",
x = "Effect on dept. negative vote\n(log-odds, latent beta scale)",
y = NULL,
color = NULL
) +
theme_minimal(base_size = 12) +
theme(
legend.position = "bottom",
panel.grid.minor = element_blank(),
axis.text.y = element_blank(),
axis.ticks.y = element_blank()
)
lm_plot + ord_plot +
plot_annotation(
title = "URM Effect on Department Negative Vote Percentage",
subtitle = "Orange = statistically/credibly different from zero; Blue = interval spans zero",
theme = theme(plot.title = element_text(size = 14, face = "bold"))
)
results |>
transmute(
Analysis = analysis,
`LM β (SD)` = round(lm_est, 3),
`LM 95% CI` = paste0("[", round(lm_lo, 3), ", ", round(lm_hi, 3), "]"),
`LM p` = round(lm_p, 3),
`Ord median` = round(ord_est, 3),
`Ord 95% CrI` = paste0("[", round(ord_lo, 3), ", ", round(ord_hi, 3), "]"),
`P(β > 0)` = pp_pos,
`Same direction` = if_else(same_sign, "✓", "✗"),
`Both sig.` = case_when(
lm_sig & ord_sig ~ "Both",
lm_sig & !ord_sig ~ "LM only",
!lm_sig & ord_sig ~ "Ord only",
TRUE ~ "Neither"
)
) |>
gt() |>
tab_header(
title = "URM Effect on Dept. Negative Vote %: LM vs. Ordbetareg",
subtitle = "Key predictor: candidate_urm_num"
) |>
tab_spanner(label = "Linear Model", columns = c(`LM β (SD)`, `LM 95% CI`, `LM p`)) |>
tab_spanner(label = "Ordbetareg", columns = c(`Ord median`, `Ord 95% CrI`, `P(β > 0)`)) |>
tab_style(
style = cell_fill(color = "#fff3cd"),
locations = cells_body(rows = `Same direction` == "✗")
) |>
tab_style(
style = cell_fill(color = "#d4edda"),
locations = cells_body(rows = `Both sig.` == "Both")
) |>
tab_footnote(
footnote = "LM β in SD units (z-scored outcome). Ordbetareg median on log-odds (latent beta) scale. P(β > 0): posterior probability URM coefficient is positive (more negative votes). Yellow = models disagree on direction; green = both find a credible/significant effect.",
locations = cells_title()
)| URM Effect on Dept. Negative Vote %: LM vs. Ordbetareg1 | ||||||||
| Key predictor: candidate_urm_num1 | ||||||||
| Analysis |
Linear Model
|
Ordbetareg
|
Same direction | Both sig. | ||||
|---|---|---|---|---|---|---|---|---|
| LM β (SD) | LM 95% CI | LM p | Ord median | Ord 95% CrI | P(β > 0) | |||
| 1. Full sample – no controls | 0.154 | [-0.027, 0.336] | 0.095 | 0.301 | [0.035, 0.542] | 0.988 | ✓ | Ord only |
| 2. Full sample – full controls | 0.085 | [-0.116, 0.285] | 0.407 | 0.257 | [-0.045, 0.558] | 0.951 | ✓ | Neither |
| 3. Full sample – no STEM control | 0.084 | [-0.116, 0.285] | 0.410 | 0.251 | [-0.063, 0.55] | 0.948 | ✓ | Neither |
| 4. Associate subset – no controls | 0.267 | [0.017, 0.517] | 0.037 | 0.563 | [0.13, 0.965] | 0.993 | ✓ | Both |
| 5. Associate subset – with controls | 0.171 | [-0.115, 0.457] | 0.242 | 0.469 | [-0.059, 1.023] | 0.954 | ✓ | Neither |
| 6. Full-prof subset – no controls | 0.035 | [-0.227, 0.296] | 0.795 | 0.074 | [-0.258, 0.393] | 0.686 | ✓ | Neither |
| 7. Full-prof subset – with controls | 0.009 | [-0.281, 0.299] | 0.951 | 0.098 | [-0.282, 0.467] | 0.688 | ✓ | Neither |
| 1 LM β in SD units (z-scored outcome). Ordbetareg median on log-odds (latent beta) scale. P(β > 0): posterior probability URM coefficient is positive (more negative votes). Yellow = models disagree on direction; green = both find a credible/significant effect. | ||||||||
sessionInfo()R version 4.4.2 (2024-10-31)
Platform: aarch64-apple-darwin20
Running under: macOS Sequoia 15.7.3
Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/4.4-arm64/Resources/lib/libRlapack.dylib; LAPACK version 3.12.0
locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8
time zone: America/New_York
tzcode source: internal
attached base packages:
[1] stats graphics grDevices utils datasets methods base
other attached packages:
[1] rstan_2.32.6 StanHeaders_2.32.10 fmsb_0.7.6
[4] gt_0.11.1 broom.mixed_0.2.9.6 patchwork_1.3.0
[7] marginaleffects_0.32.0 modelsummary_2.6.0 posterior_1.6.0
[10] ordbetareg_0.8 brms_2.23.1 Rcpp_1.1.1
[13] lubridate_1.9.5 forcats_1.0.0 stringr_1.6.0
[16] dplyr_1.2.0 purrr_1.2.1 readr_2.1.5
[19] tidyr_1.3.1 tibble_3.3.1 ggplot2_4.0.2
[22] tidyverse_2.0.0
loaded via a namespace (and not attached):
[1] gridExtra_2.3 inline_0.3.19 sandwich_3.1-1
[4] rlang_1.1.7 magrittr_2.0.4 multcomp_1.4-26
[7] furrr_0.3.1 otel_0.2.0 matrixStats_1.4.1
[10] compiler_4.4.2 loo_2.8.0 reshape2_1.4.4
[13] vctrs_0.7.2 pkgconfig_2.0.3 crayon_1.5.3
[16] fastmap_1.2.0 backports_1.5.0 labeling_0.4.3
[19] effectsize_1.0.1 cmdstanr_0.9.0 rmarkdown_2.30
[22] tzdb_0.4.0 ps_1.9.1 bit_4.6.0
[25] xfun_0.57 litedown_0.9 jsonlite_2.0.0
[28] tinytable_0.16.0 broom_1.0.6 parallel_4.4.2
[31] R6_2.6.1 tables_0.9.31 stringi_1.8.7
[34] RColorBrewer_1.1-3 parallelly_1.46.1 lmtest_0.9-40
[37] estimability_1.5.1 knitr_1.51 future.apply_1.11.2
[40] zoo_1.8-12 parameters_0.28.3 bayesplot_1.11.1
[43] Matrix_1.7-1 splines_4.4.2 timechange_0.4.0
[46] tidyselect_1.2.1 abind_1.4-8 yaml_2.3.12
[49] codetools_0.2-20 curl_7.0.0 processx_3.8.6
[52] listenv_0.9.1 pkgbuild_1.4.4 plyr_1.8.9
[55] lattice_0.22-6 bayestestR_0.17.0 withr_3.0.2
[58] bridgesampling_1.1-2 S7_0.2.1 coda_0.19-4.1
[61] evaluate_1.0.5 archive_1.1.9 future_1.70.0
[64] survival_3.7-0 RcppParallel_5.1.9 xml2_1.5.2
[67] pillar_1.11.1 tensorA_0.36.2.1 checkmate_2.3.2
[70] stats4_4.4.2 insight_1.4.6 distributional_0.5.0
[73] generics_0.1.4 vroom_1.6.5 hms_1.1.4
[76] rstantools_2.4.0 scales_1.4.0 globals_0.19.1
[79] xtable_1.8-8 glue_1.8.0 emmeans_1.10.5
[82] tools_4.4.2 data.table_1.18.2.1 mvtnorm_1.3-1
[85] grid_4.4.2 QuickJSR_1.3.1 datawizard_1.3.0
[88] colorspace_2.1-1 nlme_3.1-166 performance_0.16.0
[91] cli_3.6.5 Brobdingnag_1.2-9 V8_8.0.1
[94] gtable_0.3.6 sass_0.4.10 digest_0.6.39
[97] TH.data_1.1-2 htmlwidgets_1.6.4 farver_2.1.2
[100] htmltools_0.5.9 lifecycle_1.0.5 bit64_4.6.0-1
[103] MASS_7.3-61