Appendix 3F: Interpretation in y* Metric

The binary logit model can be interpreted in three ways:
- y* metric
- Odds ratios
  - Log odds
- Predicted probabilities
The binary logit or probit model can be interpreted in terms of y*
- This is uncommon in typical applications of the models
  - It is not the method I recommend/prefer
- But useful to understand and used sometimes (e.g., in structural equation modeling [SEM])
- Also, useful for ordinal logit/probit models (more later)
The coefficients can be directly interpreted in terms of y*. Remember the model is linear in the y* metric

For a unit change in x_k, y* is expected to change by β_k units, holding all other variables constant

use "https://tdmize.github.io/data/data/cda_anes", clear

(American National Election Study 2020 - Pre-election | CDA | 2021-09-26)

drop if missing(age, hhincome, woman, lgb, race4, ///
    edyrs, relimport, wrongtrack)

(948 observations deleted)

logit vote_trump hhincome edyrs i.woman i.lgb i.race4 ///
    i.relimport i.wrongtrack c.age##c.age

Iteration 0:  Log likelihood = -4658.8753  
Iteration 1:  Log likelihood =  -2888.204  
Iteration 2:  Log likelihood = -2862.3173  
Iteration 3:  Log likelihood = -2862.0466  
Iteration 4:  Log likelihood = -2862.0463  

Logistic regression                                    Number of obs =   6,857
                                                       LR chi2(14)   = 3593.66
                                                       Prob > chi2   =  0.0000
Log likelihood = -2862.0463                            Pseudo R2     =  0.3857

-------------------------------------------------------------------------------
   vote_trump | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
     hhincome |     -0.000      0.001   -0.897   0.370       -0.002       0.001
        edyrs |     -0.140      0.015   -9.539   0.000       -0.169      -0.111
              |
        woman |
       Woman  |     -0.355      0.067   -5.289   0.000       -0.486      -0.223
              |
          lgb |
         LGB  |     -1.164      0.176   -6.607   0.000       -1.510      -0.819
              |
        race4 |
       Black  |     -2.928      0.188  -15.543   0.000       -3.297      -2.559
    Hispanic  |     -1.185      0.126   -9.382   0.000       -1.433      -0.938
       Other  |     -0.648      0.120   -5.419   0.000       -0.883      -0.414
              |
    relimport |
2. Very im..  |     -0.598      0.097   -6.170   0.000       -0.788      -0.408
3. Moderat..  |     -0.950      0.097   -9.817   0.000       -1.140      -0.760
4. A littl..  |     -1.200      0.111  -10.841   0.000       -1.417      -0.983
5. Not imp..  |     -1.949      0.107  -18.279   0.000       -2.158      -1.740
              |
   wrongtrack |
 Wrong track  |     -3.308      0.098  -33.781   0.000       -3.500      -3.116
          age |      0.027      0.013    2.129   0.033        0.002       0.053
              |
  c.age#c.age |     -0.000      0.000   -2.509   0.012       -0.001      -0.000
              |
        _cons |      5.268      0.385   13.692   0.000        4.514       6.023
-------------------------------------------------------------------------------

est store logitmod

library(haven)           # import Stata data and labels
library(dplyr)           # data manipulation

anes <- read_dta("https://tdmize.github.io/data/data/cda_anes.dta")

# Drop any missing data on the model IVs
anes <- anes[complete.cases(anes[c("age", "hhincome", "woman", "lgb", "race4",
  "edyrs", "relimport", "wrongtrack")]), ]

# Treat binary/nominal variables as factors using their value labels
anes <- anes |>
  mutate(woman = as_factor(woman), lgb = as_factor(lgb), race4 = as_factor(race4),
    relimport = as_factor(relimport), wrongtrack = as_factor(wrongtrack)) |>
  droplevels()

logitmod <- glm(vote_trump ~ hhincome + edyrs + woman + lgb + race4 + relimport +
  wrongtrack + age + I(age^2), family = binomial(link = "logit"), data = anes)
summary(logitmod)


Call:
glm(formula = vote_trump ~ hhincome + edyrs + woman + lgb + race4 + 
    relimport + wrongtrack + age + I(age^2), family = binomial(link = "logit"), 
    data = anes)

Coefficients:
                                   Estimate Std. Error z value Pr(>|z|)    
(Intercept)                       5.2684570  0.3847783  13.692  < 2e-16 ***
hhincome                         -0.0004911  0.0005477  -0.897   0.3698    
edyrs                            -0.1401283  0.0146899  -9.539  < 2e-16 ***
womanWoman                       -0.3547981  0.0670774  -5.289 1.23e-07 ***
lgbLGB                           -1.1642765  0.1762055  -6.607 3.91e-11 ***
race4Black                       -2.9277400  0.1883485 -15.544  < 2e-16 ***
race4Hispanic                    -1.1850771  0.1263068  -9.383  < 2e-16 ***
race4Other                       -0.6484143  0.1196631  -5.419 6.00e-08 ***
relimport2. Very important       -0.5977685  0.0968775  -6.170 6.81e-10 ***
relimport3. Moderately important -0.9499901  0.0967647  -9.818  < 2e-16 ***
relimport4. A little important   -1.2000537  0.1106925 -10.841  < 2e-16 ***
relimport5. Not important at all -1.9488242  0.1066167 -18.279  < 2e-16 ***
wrongtrackWrong track            -3.3081474  0.0979287 -33.781  < 2e-16 ***
age                               0.0274182  0.0128806   2.129   0.0333 *  
I(age^2)                         -0.0003113  0.0001241  -2.509   0.0121 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 9317.8  on 6856  degrees of freedom
Residual deviance: 5724.1  on 6842  degrees of freedom
  (475 observations deleted due to missingness)
AIC: 5754.1

Number of Fisher Scoring iterations: 5

Each additional year of education leads to a 0.140 decrease in y* (propensity to vote for Trump)

Women are about 0.35 lower on y* (propensity to vote for Trump) than men

Example y* interpretations

“The problem with this interpretation is that the variance of y* is unknown, so the meaning of a change of β_k in y* is unclear.” (Long 1997 p.70)

Results from a binary probit model illustrate Long’s point:

probit vote_trump hhincome edyrs i.woman i.lgb i.race4 ///
    i.relimport i.wrongtrack c.age##c.age

Iteration 0:  Log likelihood = -4658.8753  
Iteration 1:  Log likelihood =  -2884.606  
Iteration 2:  Log likelihood = -2872.7891  
Iteration 3:  Log likelihood = -2872.7698  
Iteration 4:  Log likelihood = -2872.7698  

Probit regression                                      Number of obs =   6,857
                                                       LR chi2(14)   = 3572.21
                                                       Prob > chi2   =  0.0000
Log likelihood = -2872.7698                            Pseudo R2     =  0.3834

-------------------------------------------------------------------------------
   vote_trump | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
--------------+----------------------------------------------------------------
     hhincome |     -0.000      0.000   -1.085   0.278       -0.001       0.000
        edyrs |     -0.078      0.008   -9.444   0.000       -0.095      -0.062
              |
        woman |
       Woman  |     -0.197      0.038   -5.170   0.000       -0.272      -0.122
              |
          lgb |
         LGB  |     -0.623      0.092   -6.798   0.000       -0.802      -0.443
              |
        race4 |
       Black  |     -1.531      0.089  -17.253   0.000       -1.704      -1.357
    Hispanic  |     -0.642      0.069   -9.285   0.000       -0.778      -0.507
       Other  |     -0.348      0.067   -5.225   0.000       -0.479      -0.218
              |
    relimport |
2. Very im..  |     -0.346      0.056   -6.152   0.000       -0.456      -0.236
3. Moderat..  |     -0.543      0.056   -9.713   0.000       -0.653      -0.433
4. A littl..  |     -0.687      0.063  -10.831   0.000       -0.811      -0.563
5. Not imp..  |     -1.091      0.059  -18.475   0.000       -1.206      -0.975
              |
   wrongtrack |
 Wrong track  |     -1.897      0.050  -37.652   0.000       -1.995      -1.798
          age |      0.015      0.007    2.096   0.036        0.001       0.030
              |
  c.age#c.age |     -0.000      0.000   -2.399   0.016       -0.000      -0.000
              |
        _cons |      2.952      0.213   13.854   0.000        2.534       3.370
-------------------------------------------------------------------------------

est store probitmod

probitmod <- glm(vote_trump ~ hhincome + edyrs + woman + lgb + race4 + relimport +
  wrongtrack + age + I(age^2), family = binomial(link = "probit"), data = anes)
summary(probitmod)


Call:
glm(formula = vote_trump ~ hhincome + edyrs + woman + lgb + race4 + 
    relimport + wrongtrack + age + I(age^2), family = binomial(link = "probit"), 
    data = anes)

Coefficients:
                                   Estimate Std. Error z value Pr(>|z|)    
(Intercept)                       2.952e+00  2.148e-01  13.744  < 2e-16 ***
hhincome                         -3.339e-04  3.083e-04  -1.083   0.2787    
edyrs                            -7.832e-02  8.283e-03  -9.456  < 2e-16 ***
womanWoman                       -1.971e-01  3.806e-02  -5.179 2.23e-07 ***
lgbLGB                           -6.225e-01  9.161e-02  -6.796 1.08e-11 ***
race4Black                       -1.531e+00  9.049e-02 -16.913  < 2e-16 ***
race4Hispanic                    -6.424e-01  6.966e-02  -9.222  < 2e-16 ***
race4Other                       -3.483e-01  6.718e-02  -5.185 2.16e-07 ***
relimport2. Very important       -3.461e-01  5.618e-02  -6.161 7.22e-10 ***
relimport3. Moderately important -5.431e-01  5.586e-02  -9.722  < 2e-16 ***
relimport4. A little important   -6.869e-01  6.343e-02 -10.830  < 2e-16 ***
relimport5. Not important at all -1.091e+00  5.910e-02 -18.453  < 2e-16 ***
wrongtrackWrong track            -1.897e+00  5.042e-02 -37.618  < 2e-16 ***
age                               1.524e-02  7.296e-03   2.089   0.0367 *  
I(age^2)                         -1.688e-04  7.045e-05  -2.396   0.0166 *  
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

(Dispersion parameter for binomial family taken to be 1)

    Null deviance: 9317.8  on 6856  degrees of freedom
Residual deviance: 5745.5  on 6842  degrees of freedom
  (475 observations deleted due to missingness)
AIC: 5775.5

Number of Fisher Scoring iterations: 5

y* standardized interpretations

One solution is to y* standardize the coefficient; i.e. put it in standard deviation of y* units
First, need to calculate the variance/standard deviation of y*

\[\widehat{Var}(y^{*}) = \widehat{\boldsymbol{\beta}}' Var(\mathbf{x}) \widehat{\boldsymbol{\beta}} + Var(\varepsilon)\]

Where the variance of the error is assumed (1 in probit; π²/3 in logit)
Then, put coefficient in y* standard deviation units:

\[\beta_{k}^{Sy*} = \frac{\beta_{k}}{\sigma_{y*}}\]

For a unit increase in x_k, y* is expected to increase by \(\boldsymbol{\beta_{k}^{Sy*}}\) standard deviations, holding all other variables constant

y* standardized coefficients

est restore logitmod

(results logitmod are active now)

listcoef, std

logit (N=6857): Unstandardized and standardized estimates 

  Observed SD:  0.4932
    Latent SD:  2.7871

-------------------------------------------------------------------------------
             |         b        z    P>|z|    bStdX    bStdY   bStdXY     SDofX
-------------+-----------------------------------------------------------------
    hhincome |   -0.0005   -0.897    0.370   -0.033   -0.000   -0.012    67.851
       edyrs |   -0.1401   -9.539    0.000   -0.349   -0.050   -0.125     2.493
             |
       woman |
      Woman  |   -0.3548   -5.289    0.000   -0.177   -0.127   -0.063     0.499
             |
         lgb |
        LGB  |   -1.1643   -6.607    0.000   -0.290   -0.418   -0.104     0.249
             |
       race4 |
      Black  |   -2.9277  -15.543    0.000   -0.839   -1.050   -0.301     0.287
   Hispanic  |   -1.1851   -9.382    0.000   -0.335   -0.425   -0.120     0.283
      Other  |   -0.6484   -5.419    0.000   -0.182   -0.233   -0.065     0.281
             |
   relimport |
2. Very i..  |   -0.5978   -6.170    0.000   -0.235   -0.214   -0.084     0.393
3. Modera..  |   -0.9500   -9.817    0.000   -0.375   -0.341   -0.135     0.395
4. A litt..  |   -1.2001  -10.841    0.000   -0.406   -0.431   -0.146     0.338
5. Not im..  |   -1.9488  -18.279    0.000   -0.802   -0.699   -0.288     0.412
             |
  wrongtrack |
Wrong track  |   -3.3081  -33.781    0.000   -1.448   -1.187   -0.520     0.438
         age |    0.0274    2.129    0.033    0.468    0.010    0.168    17.075
             |
 c.age#c.age |   -0.0003   -2.509    0.012   -0.548   -0.000   -0.197  1761.316
             |
    constant |    5.2685   13.692    0.000        .        .        .         .
-------------------------------------------------------------------------------

est restore probitmod

(results probitmod are active now)

listcoef, std

probit (N=6857): Unstandardized and standardized estimates 

  Observed SD:  0.4932
    Latent SD:  1.5524

-------------------------------------------------------------------------------
             |         b        z    P>|z|    bStdX    bStdY   bStdXY     SDofX
-------------+-----------------------------------------------------------------
    hhincome |   -0.0003   -1.085    0.278   -0.023   -0.000   -0.015    67.851
       edyrs |   -0.0783   -9.444    0.000   -0.195   -0.050   -0.126     2.493
             |
       woman |
      Woman  |   -0.1971   -5.170    0.000   -0.098   -0.127   -0.063     0.499
             |
         lgb |
        LGB  |   -0.6225   -6.798    0.000   -0.155   -0.401   -0.100     0.249
             |
       race4 |
      Black  |   -1.5305  -17.253    0.000   -0.439   -0.986   -0.283     0.287
   Hispanic  |   -0.6424   -9.285    0.000   -0.182   -0.414   -0.117     0.283
      Other  |   -0.3483   -5.225    0.000   -0.098   -0.224   -0.063     0.281
             |
   relimport |
2. Very i..  |   -0.3461   -6.152    0.000   -0.136   -0.223   -0.088     0.393
3. Modera..  |   -0.5431   -9.713    0.000   -0.214   -0.350   -0.138     0.395
4. A litt..  |   -0.6869  -10.831    0.000   -0.232   -0.442   -0.150     0.338
5. Not im..  |   -1.0906  -18.475    0.000   -0.449   -0.702   -0.289     0.412
             |
  wrongtrack |
Wrong track  |   -1.8966  -37.652    0.000   -0.830   -1.222   -0.535     0.438
         age |    0.0152    2.096    0.036    0.260    0.010    0.168    17.075
             |
 c.age#c.age |   -0.0002   -2.399    0.016   -0.297   -0.000   -0.192  1761.316
             |
    constant |    2.9520   13.854    0.000        .        .        .         .
-------------------------------------------------------------------------------

# Latent SD: variance of the linear predictor + variance of the error
sd_ystar_logit <- sqrt(var(predict(logitmod, type = "link")) + pi^2 / 3)
sd_ystar_probit <- sqrt(var(predict(probitmod, type = "link")) + 1)
c(logit = sd_ystar_logit, probit = sd_ystar_probit)

   logit   probit 
2.787081 1.552388

# y* standardized coefficients
data.frame(b_logit = coef(logitmod), bStdY_logit = coef(logitmod) / sd_ystar_logit,
  b_probit = coef(probitmod), bStdY_probit = coef(probitmod) / sd_ystar_probit)[-1, ] |>
  round(4)

                                 b_logit bStdY_logit b_probit bStdY_probit
hhincome                         -0.0005     -0.0002  -0.0003      -0.0002
edyrs                            -0.1401     -0.0503  -0.0783      -0.0505
womanWoman                       -0.3548     -0.1273  -0.1971      -0.1270
lgbLGB                           -1.1643     -0.4177  -0.6225      -0.4010
race4Black                       -2.9277     -1.0505  -1.5305      -0.9859
race4Hispanic                    -1.1851     -0.4252  -0.6424      -0.4138
race4Other                       -0.6484     -0.2326  -0.3483      -0.2244
relimport2. Very important       -0.5978     -0.2145  -0.3461      -0.2230
relimport3. Moderately important -0.9500     -0.3409  -0.5431      -0.3498
relimport4. A little important   -1.2001     -0.4306  -0.6869      -0.4425
relimport5. Not important at all -1.9488     -0.6992  -1.0906      -0.7025
wrongtrackWrong track            -3.3081     -1.1870  -1.8966      -1.2217
age                               0.0274      0.0098   0.0152       0.0098
I(age^2)                         -0.0003     -0.0001  -0.0002      -0.0001

y* coefficients differ noticeably; y* standardized coefficients are virtually identical in both logit and probit model

Each additional year of education is associated with a 0.05 standard deviation decrease in the propensity to vote for Trump
Compared to men, women are 0.127 standard deviations lower on propensity to vote for Trump

Advice on y* interpretations

y* interpretations only really make sense if you think an underlying latent continuum exists and you can name and talk about it cogently
Standardized coefficients are useful to facilitate comparisons across models: all are on the same standard deviation scale
- If you are going to use y* interpretations, I recommend y* standardized coefficients
Ultimately, the interpretations are not very satisfying as the substantive magnitude of an effect is unclear—even with y* standardized coefficients
- I find marginal effects a better default interpretation strategy (see 3.6 Marginal Effects)