Lecture 11 Notes

Author

Alec L. Robitaille

Published

April 4, 2024

Ordered categorical

Ordered categorical variables are discrete types with ordered relationships.

not a count variable, not continuous
bounded between min, max (eg. Trolley 1-7)
eg. good, bad, excellent - not “cat, dog, chicken”
distances between categories is not known or the same
often there are “anchor points” and not everyone shares the same anchor point

Modeling an ordered categorical requires converting the orders to cumulative frequency. Instead of the probability of 5 out of 10, we have the probability of 5 or less out of 10.

Recall, cumulative log-odds is the log of the probability of x occurring divided by the probability of x not occurring.

Now we are estimating cutpoints. The number of cutpoints we estimate is the number of categories - 1, since the cumulative probability of the last category is the total cumulative probability.

Then estimating the probability of a specific category is the probability of all categories up to that category subtracting the probability of all categories up to the preceding category.

To make it a function of the variables, either by stratifying the cutpoints or by offsetting each cutpoint by the value of a linear model \(\phi\).

\(\phi_{i} = \beta x_{i}\)

\(log \frac{Pr(R_{i} <= k)}{1-Pr(R_{i} <= k)} = \alpha_{k} + \phi_{i}\)

\(R_{i} \sim OrderedLogit(\phi_{i}, \alpha)\)

Example: Trolley problem

Runaway trolley, actor standing next to a switch
If you don’t pull the switch, it will strike 5 people on the track
If you do pull the switch, it will strike 1 person on the track

Scenarios that contain three principles

Action: taking an action is less morally permissible than not taking an action
Intention: eg. pulling the lever to strike the 1 person and this stopping the trolley - this is necessary and intentional to save the 5 people
Contact: eg. pushing a large person on the track

Data:

331 individuals
action, intention, contact
30 different trolley problems
voluntary participation

Response: how appropriate from 1-7

Estimand

How do action, intention and contact influence response to a trolley story?

coords <- data.frame(
    name = c('X', 'R', 'S', 'E', 'Y', 'G',  'P'),
    x =    c(1,    2,   3,   1,    2,   3,  2),
    y =    c(0,    0,   0,   -2,  -1,  -2,  -3)
)

dagify(
    R ~ X + S + E + Y + G,
    E ~ Y + G,
  coords = coords
) |> ggdag(seed = 2, layout = 'auto') + theme_dag()

R: response to the trolley story
X: action, intention, contact
S: story
E: education
Y: age
G: gender

Initial model

\(R_{i} \sim OrderedLogit(\phi_{i}, \alpha)\)

\(\phi_{i} = \beta_{A}A_{i} + \beta_{C}C_{i} + \beta_{I}I_{i}\)

\(\beta \sim Normal(0, 0.5)\)

\(\alpha_{j} \sim Normal(0, 1)\)

Total causal effect of gender

\(R_{i} \sim OrderedLogit(\phi_{i}, \alpha)\)

\(\phi_{i} = \beta_{A}A_{i, G_{i}} + \beta_{C}C_{i, G_{i}} + \beta_{I}I_{i, G_{i}}\)

\(\beta \sim Normal(0, 0.5)\)

\(\alpha_{j} \sim Normal(0, 1)\)

dagify(
    R ~ X + S + E + Y + G,
    E ~ Y + G,
    exposure = 'G',
  coords = coords
) |> ggdag_status(seed = 2, layout = 'auto') + theme_dag() + guides(color = 'none')

Endogenous selection

Note - this is a voluntary sample and participation is influenced by age, education and gender. There is sample selection from this participation meaning the data is already stratified by the features of the participating population.

In addition, participation is a collider and conditioning on it makes education, gender and age co-vary in the sample. Therefore, we cannot estimate the total causal effect of gender.

dagify(
    R ~ X + S + E + Y + G,
    E ~ Y + G,
    P ~ E + Y + G,
    exposure = 'P',
  coords = coords
) |> ggdag_status(seed = 2, layout = 'auto') + 
    theme_dag() + 
    guides(color = 'none')

To get the direct effect of gender, we need to stratify by age and education. We’ll represent education as an ordered monotonic predictors.

Ordered monotonic predictors

Education is an ordered category, but it is unlikely that each level has the same effect for an equivalent unit change. We want a parameter for each level but we need to enforce ordering so there is always a larger or smaller effect than the previous level.

Each category is the sum of the previous and the last category is equal to the maximum effect of the variable. Then the deltas are all summed up (0-1) and multiplied by the maximum effect. The delta parameters form a simplex, a vector that sums to 1.

We rescale the \(\phi\) variables to sum to 1 (\(\beta_{E}\)).

\[\phi_{i} = \beta_{E} \sum_{j=0}^{E_{i}-1} \delta_{j}\]

We model a simplex using a Dirichlet distribution. Dirichlet distributions take a vector \(a\) (concentration parameters) that defines the differences among the categories. Larger values in \(a\) indicate that there are smaller differences between categories.

plot_dirichlet <- function(n, a) {
    diri <- data.table(rdirichlet(n, a))
    setnames(diri, as.character(seq_along(a)))
    diri[, i := .I]
    melt_diri <- melt(diri, id.vars = 'i')
    ggplot(melt_diri) +
        geom_line(aes(variable, value, group = i), alpha = 0.2) +
        labs(x = 'category',
                 y = 'probability',
                 title = paste('alpha =', paste0(a, collapse = ', '))) +
        ylim(0, 1)
}
n <- 100
n_cat <- 7
plot_dirichlet(n, rep(2, n_cat))

plot_dirichlet(n, rep(7, n_cat))

plot_dirichlet(n, seq(1, n_cat))

Example: Trolley problem

Recall

R: response to the trolley story
X: action, intention, contact
S: story
E: education
Y: age
G: gender

bA, bI, bC are action, intention, contact

There are only direct effects for:

gender (eg. bA[1], bA[2]) because we stratified for education given the participation effect (a collider)
education (bE[1], bE[2]) again because of participation
age (bY[1], bY[2]) again because of participation and education

This is a great example of complex causal effects. Instead of using the current sample which is contaminated by the participation bias, we can post-stratify (marginalize) to a new target that is representative of your estimand. Post-stratification simulation is the same as all previous examples of generative simulations using posterior samples.

Bonus: non-representative samples and post-stratification

Post-stratification or transport is a transparent, principled method for extrapolating from sample to population. It requires causal models of reasons the sample differs from population. All problems, whether they are descriptive, predictive or causal, have to consider the differences between the sample and the target population. Post stratification requires a causal model of the reasons that the sample differs from the population. Commonly used in surveys and elections (see multi-level regression and post-stratification “MRP”).

The quality of data is more important than quantity of data. Bigger samples amplify biases.

Example: larger surveys of COVID vaccination rate largely overestimated the CDC benchmark, while the smaller survey better represented the population

Non-representative samples can outperform representative samples if we know how the sample differs from our target population.

Example: Wang et al 2014, Xbox survey of election results for Obama vs Romney, post-stratified to target population

Example of four age groups. Just because the sample differs from the population doesn’t tell us what we need to do statistically.

Selection nodes indicate reasons why the sample differs from the population. Eg. selection by age.

Some selection is recoverable, eg. where age stratification in the sample that influences X. Influence on the outcome variable, eg. where anarchists don’t answer the phone, are not recoverable because the results you need are not in the data.

Many questions are really post-stratification questions: - justified descriptions require causal information and post-stratification - causal effects require causal information and post-stratification - time trends should account for changes in measurement/population - comparisons are post-stratification between populations