Nested events: MDE and severe MDE

Self-paced bonus activity · what happens when one event is a subset of another

Welcome to a self-paced bonus activity. In the M06 lab you worked with two overlapping events (MDE × SUD) on a 2×2 table. This bonus works through one more cross-tab — structurally different from anything in the lab — and points out a subtle pattern in the probability rules that becomes obvious once you see it. It’s a great one to revisit when you’re reviewing the probability rules for exams.

Everything happens in the sandbox chunks on this page — nothing here goes into your lab notebook. The 2019 NSDUH subset (nsduh_2019) is already loaded, and tidyverse and gtsummary are attached.

NSDUH has a second MDE-related variable — mde_pastyear_severe — that flags whether the past-year MDE was classified as severe (roughly, a high-interference score on at least one of the four severity domains). Here’s the key structural difference from MDE × SUD:

Severe MDE can only happen if MDE has already happened. A person who does not have a past-year MDE cannot, by definition, have a severe past-year MDE.

This is called a subset relationship (or a nested events relationship). Think of it as one circle sitting entirely inside another:

Contrast this with the MDE × SUD pair from the lab: those two circles overlap, but neither one sits entirely inside the other. Here, the two circles are nested.

Build the cross-tabs

Build two cross-tabs of mde_pastyear × mde_pastyear_severe for all 2019 adolescents (both sexes — don’t filter by sex): one with raw counts, then one with percent = "cell". This is the same structural pattern as the lab’s first cross-tab, minus the sex filter and with different variables.

A note about drop_na() on this pair: mde_pastyear_severe is a full-sample indicator — respondents who did not have past-year MDE are coded "Negative" on it (not missing), so its "Negative" blends “no MDE at all” with “MDE, but below the severity threshold.” drop_na() here removes only the small number of respondents with genuinely missing values. Keeping the non-MDE respondents in the table is exactly what makes the peculiarity below pop out — it’s the nested structure of the two events, not any sample restriction, that produces it.

Counts

Select mde_pastyear and mde_pastyear_severe, then tbl_cross() with mde_pastyear on the row and mde_pastyear_severe on the column.

Look at the interior of the table. One of the four cells should have a count of zero — this is the “structural zero” that the nested relationship forces.

Cell percentages

Same call as the counts version, adding percent = "cell" so joints and marginals print as percentages.

The structural zero is now 0 (0%). That’s the algebraic signature of the nested-events relationship — it’s what forces the peculiarity you’ll spot in a moment.

Row and column percentages

The two conditionals below — (d) and (f) — are read straight off these, exactly as in the lab’s Step 3: percent = "row" divides by the row total, percent = "column" by the column total.

percent = "row" gives you P(MDE Severe | MDE) in the MDE-positive row. percent = "column" gives you P(MDE | MDE Severe) in the severe-positive column — and that one is the surprise.

Look at the column-percent table before you go on. Because severe MDE is a subset of MDE, the severe-positive column is 100% MDE-positive — the nesting makes that conditional certain, and (f) below asks you to say why.

Work through the core probabilities

Read the probabilities below off the four tables you just built, the same way you did in the lab’s Step 3 — the counts table for (c), cell percentages for (a), (b) and (e), row percentages for (d), and column percentages for (f). Work each one out on paper or in your head — record a one-sentence interpretation for each.

Work through each of these

(a) Marginal probability of MDE: P(MDE). What share of this 2019 sample had past-year MDE?

(b) Marginal probability of severe MDE: P(MDE Severe). What share of this 2019 sample had a severe past-year MDE?

(c) Joint probability: P(MDE ∩ MDE Severe). What share had both MDE and severe MDE in the past year?

(d) Conditional probability: P(MDE Severe | MDE). Given that someone had past-year MDE, what share were severe? This is the clinically useful quantity: “of the adolescents who experienced an MDE, how many of them had an episode bad enough to cross the severity threshold?”

(e) Union of the two events: P(MDE ∪ MDE Severe). What share had MDE, severe MDE, or both? Use the addition rule from the lab.

(f) The reverse conditional: P(MDE | MDE Severe). Given that someone had a severe past-year MDE, what share had past-year MDE at all? Because severe MDE is a subset of MDE, this one is a certainty — it equals 1. It’s the most memorable number in a nested pair: conditioning on the smaller event guarantees the larger one.

Stop and notice something strange

Look carefully at what you computed in (b), (c), (d), and (e). Before you read further, see if you can spot the peculiarity on your own.

The nested-events peculiarity

Two of your numbers should look suspicious — they match numbers that, at first glance, you wouldn’t expect them to match.

Peculiarity 1 — the joint (and) equals one of the marginals.

Formal notation:

\[P(\text{MDE} \cap \text{MDE Severe}) = P(\text{MDE Severe})\]

Plain English:

\[P(\text{MDE} \;\text{and}\; \text{MDE Severe}) = P(\text{MDE Severe})\]

Why? The set of people with “both MDE and severe MDE” is exactly the same set as the people with severe MDE. You cannot have severe MDE without first having MDE — so every person in the “severe MDE” bucket is automatically in the “both” bucket. The intersection (the and) of the two events is the smaller of the two events.

Peculiarity 2 — the union (or) equals the other marginal.

Formal notation:

\[P(\text{MDE} \cup \text{MDE Severe}) = P(\text{MDE})\]

Plain English:

\[P(\text{MDE} \;\text{or}\; \text{MDE Severe}) = P(\text{MDE})\]

Why? The event “had MDE OR had severe MDE” is the same set of people as “had MDE” — because anyone who had severe MDE is already in the MDE circle. There’s no one in the severe-MDE circle who isn’t also in the MDE circle, so the union can’t be any larger than the MDE circle itself.

You can verify this algebraically using the addition rule:

\[P(\text{MDE} \cup \text{MDE Severe}) = P(\text{MDE}) + P(\text{MDE Severe}) - P(\text{MDE} \cap \text{MDE Severe})\]

But because \(P(\text{MDE} \cap \text{MDE Severe}) = P(\text{MDE Severe})\) (Peculiarity 1), the last two terms cancel and you’re left with \(P(\text{MDE})\).

This is the general pattern whenever one event is a subset of another: the joint (and) collapses to the smaller event, and the union (or) collapses to the larger event. The addition rule still works — it just simplifies dramatically.

Contrast with the lab’s MDE × SUD pair

Jot a short paragraph (3–5 sentences) — on paper or just in your head — contrasting the MDE × SUD pair from the lab with the MDE × severe-MDE pair you just built. Some prompts:

  • For MDE × SUD, the joint probability is smaller than either marginal, and the union is larger than either marginal. Why?
  • For MDE × severe MDE, the joint equals the smaller marginal and the union equals the larger marginal. Why?
  • Draw (in words, or literally sketch it on paper and describe it) the two-circle Venn diagram for each pair. What’s different about how the circles relate?

The goal here is to understand how the algebraic rules of probability play out in real data, and how the structure of the relationship between two events (overlapping vs. nested) creates different patterns in the probabilities.

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