SOCI 620: Quantitative methods 2

Agenda

Parsimony
& overfitting

  1. Logistic with predictors
  2. Interpreting coefficients
  3. Hands on:
    Prior predictive simulation in R

Slides are licensed under CC BY-NC-SA 4.0

Logistic with predictors

The 'trolley problem' meme template: a crude, black-and-white line drawing of a trolley on tracks. The tracks split into two paths: one going straight towards a row of people tied to the tracks, the other turning toward a single person tied to the tracks. Another person stands by the tracks with their hand on a lever and a frown on their face.

Cocaine use among adolescents

Prior predictive plots

Prior predictive plots

Model estimates (posterior)

Mean 95% C.I.

-3.52 (-3.68, -3.46)

0.16 (0.08, 0.25)

Interpreting logistic regression coefficients

A person examines a small leaf using a foldable loupe.

Interpreting coefficients

Mean 95% C.I.

-3.52 (-3.68, -3.46)

0.16 (0.08, 0.25)

Intercept (α)

  • The expected probability of having tried cocaine for a student in grade 9 (Gi = 0) is:

    logit-1(-3.52) = 0.029 = 2.9%

  • The probability that a student in grade 9 has tried cocaine is most likely between 2.5% and 3.4% (95% CI).

Coefficient (β)

  • Coefficients are not so simple

  • Some options:

    Odds ratio
    Selected cases
    Average marginal effect (AME)
    Posterior visualization

Interpreting coefficients

Odds ratio

The odds ratio is the proportional change in the odds of the outcome associated with a one-unit change in the predictor.

“For every unit increase in the covariate, the expected odds of the outcome is multiplied by exp(β)”

“For every grade a student completes, the expected odds of having tried cocaine is multiplied by 13% (multiplied by 1.13)”

Odds ratios are unintuitive

The probability scale of the odds ratio depends on "initial" probability

Interpreting without odds ratios

Cases

An average 7th grade student has about a 2.1% (1.6%, 2.8%) chance of having tried cocaine, while for an average 12th grader, that probability is about 4.7% (3.7%, 5.7%)

Posterior visualization

Average marginal effects

Among the students in our sample, completing a grade is predicted to increase the probability of having tried cocaine by about 17.3% (17.1%, 17.4%) on average
Multiplicative on posterior probability

Adding covariates

Gi: Grade in school (centered at grade 9)

Di: Delinquency (standardized)

Wi: White (indicator)

Mean 95% C.I.

-4.48 (-4.81, -4.16)

0.22 (0.13, 0.32)

0.89 (0.78, 1.00)

0.62 (0.32, 0.94)

An increase by x in odds for an unlikely event is much more impactful than the same increase for a likely event. E.g. odds going from 0 to 1 -> prob goes from 0 to 0.5 odds going from 1 to 2 -> prob goes from 0.5 to 0.67

--- # Adding covariates <div class=gridded style="grid-template: auto 1fr / 1.3fr 1fr"> <div style="margin-top:10px"> $$ \begin{aligned} C_i &\sim\mathrm{Norm}(\mu_i,\sigma)\\ \mu_i &= \alpha + \beta_G G_i + \beta_D D_i + \beta_W W_i\\ \\ \alpha &\sim\mathrm{Norm}(0,1.7)\\ \beta_G,\beta_D,\beta_W &\sim \mathrm{Norm}(0,0.4) \end{aligned} $$ </div> <div class=regtable style="font-size:70%;margin:auto;margin-top:10px"> <table style="margin:auto"> <tr> <th></th> <th>Mean</th> <th>95% C.I.</th> </tr> <tr> <td> $\alpha$</td> <td>-4.48</td> <td>(-4.81, -4.16)</td> </tr> <tr> <td> $\beta_G$</td> <td>0.22</td> <td>(0.13, 0.32)</td> </tr> <tr> <td> $\beta_D$</td> <td>0.89</td> <td>(0.78, 1.00)</td> </tr> <tr> <td> $\beta_W$</td> <td>0.62</td> <td>(0.32, 0.94)</td> </tr> </table> </div> <div style="grid-area:2/1/3/3; display:grid; grid-template: repeat(3, max-content) / 30% 1fr 1fr"> <div style="grid-area:1/1/4/2; border-right:5px solid var(--dark);padding:0;padding-right:10px;margin-right:10px;text-align:right; font-size:80%"> ## Interpreting results using _odds ratios_ Each coefficient has an easy-to-calculate (but hard-to-interpret) meaning </div> <div> **exp(_α_)** = </div> </div>