With an intraclass correlation of 0.3 and 20 pupils per classroom, what is the effective sample size?
answer
- share of variance between clusters
- correlation between two classmates
- design effect uses cluster size minus one
- divide n by the design effect
- 1 + 19 times 0.3
basics
~10 sThe design effect is 1 + (20 - 1) * 0.3 = 6.7, so divide the pupil count by 6.7. An 800-pupil study in 40 classrooms carries the information of about 119 independent pupils.
solid answer
~50 sAn intraclass correlation of 0.3 says that two pupils in the same classroom correlate 0.3 on the outcome, equivalently that 30% of the total variance sits between classrooms rather than within them. For equal cluster sizes the variance of a mean is inflated by the design effect `DEFF = 1 + (m - 1) * ICC`, which here is `1 + 19 * 0.3 = 6.7`. Effective sample size is the actual count divided by that factor: 800 pupils across 40 classrooms behave like about 119 independent pupils, and standard errors are `sqrt(6.7)` — roughly 2.6 times — wider than an analysis assuming independence would report. The practical lesson is that with a substantial intraclass correlation, extra pupils per classroom buy very little; effective sample size tends to a ceiling of the number of classrooms divided by the intraclass correlation, so power comes from adding classrooms.
go deeper
Know what the intraclass correlation measures — how alike two members of the same group are — and that a positive value means your real sample is smaller than your row count suggests.
Be able to compute it on the spot: design effect 1 + (m - 1) * ICC, effective size n divided by that, standard errors up by its square root. Getting the m minus one right is the whole point of the question.
Show you use it for design, not just diagnosis: pick the intraclass correlation from a pilot or history, plan around the ceiling of clusters divided by ICC, and flag unequal cluster sizes as making the simple formula optimistic.
Frame it as a budget argument. Be ready to argue for spending on more sites, classrooms or stores rather than deeper sampling within each, and to explain to a sponsor why a large-sounding sample delivers a modest effective size.
## Defining the intraclass correlation Split the variation in an outcome into two pieces: variation between clusters and variation within them. Writing the classroom-level offset as `u` and the pupil-level noise as `e`, the intraclass correlation is `ICC = var(u) / (var(u) + var(e))` It has two readings that must both be available to you in an interview. As a **variance share**, it is the fraction of total outcome variance attributable to differences between classrooms. As a **correlation**, it is the expected correlation between two randomly chosen pupils in the same classroom. Both readings are the same number: 0.3 here means 30% of the variance is between classrooms and any two classmates correlate 0.3. An ICC of 0 means classrooms are interchangeable and clustering is harmless. An ICC of 1 means every pupil in a classroom is a perfect copy of the classroom, so a classroom of any size carries exactly one observation's worth of information. ## The design effect For a mean estimated from `G` clusters of equal size `m`, the variance is inflated relative to a simple independent sample by `DEFF = 1 + (m - 1) * ICC` The `m - 1` is not a typo and is the single most common arithmetic slip on this question: a pupil is not correlated with itself in the sense that matters here, so it is the other `m - 1` classmates who add redundancy. With `m = 20` and `ICC = 0.3`: `DEFF = 1 + 19 * 0.3 = 1 + 5.7 = 6.7` ## Effective sample size Effective sample size is the actual number divided by the design effect: `n_eff = n / DEFF` With 40 classrooms of 20 pupils, `n = 800` and `n_eff = 800 / 6.7`, about 119. That is the number to quote: this study, on paper an 800-pupil study, carries the precision of roughly 119 independently sampled pupils. Standard errors scale with `1 / sqrt(n_eff)`, so they are `sqrt(6.7)` — about 2.59 — times wider than an analysis that pretended the pupils were independent, and any p-value computed on the naive assumption is badly optimistic. ## Why more pupils per classroom stops helping Rewrite effective sample size in terms of clusters. With `n = G * m`, `n_eff = G * m / (1 + (m - 1) * ICC)` Let `m` grow without limit and the expression tends to `G / ICC`. With 40 classrooms and an ICC of 0.3 the ceiling is `40 / 0.3`, about 133 effective pupils, no matter how large the classrooms become. The 20-pupil design already delivers 119 of those 133, roughly 90% of everything more pupils could ever buy. This is the design lesson that separates a candidate who has planned a study from one who has only analysed data: with meaningful clustering, power comes from **more clusters**, and enlarging clusters hits a wall fast. The mirror image matters too. A tiny ICC becomes serious when clusters are large: with `ICC = 0.01` and `m = 1000`, `DEFF = 1 + 999 * 0.01`, about 11, so a sample of a million rows carries the weight of roughly ninety thousand. Never dismiss an intraclass correlation as small without multiplying it by the cluster size. ## Caveats worth stating The formula assumes equal cluster sizes; with unequal sizes the inflation is larger than the formula using the average size suggests, because a few large clusters dominate. It is derived for a mean, so it is a planning approximation rather than an exact statement for an arbitrary regression coefficient — the inflation for a coefficient also depends on how the predictor is distributed within and between clusters, and a predictor constant within a cluster suffers close to the full design effect. And an intraclass correlation is an estimate with its own uncertainty, so a sample-planning exercise should test a range rather than a single value. ## Where the number comes from in practice An intraclass correlation is usually taken either from a pilot or from a comparable historical dataset, estimated as the between-cluster share of variance in a model with a classroom-level random intercept. It is outcome-specific: attainment, attendance and satisfaction from the same pupils can have quite different intraclass correlations, so one number should not be reused across metrics without checking.
- What happens to the design effect calculation when classroom sizes are unequal?The simple formula understates the inflation. Plugging the average size into 1 + (m - 1) * ICC is optimistic because large clusters contribute disproportionately to the variance, so a size-weighted version of the effective cluster size is needed. In planning work the safe move is to compute the design effect with a size measure that reflects the spread, or to simulate the actual size distribution rather than trusting the mean.
- An intraclass correlation of 0.01 sounds negligible — when is it not?Whenever clusters are large. The design effect multiplies the correlation by the cluster size minus one, so an ICC of 0.01 with 1000 observations per cluster gives roughly 1 + 999 * 0.01, about 11, and a million rows carry the weight of ninety thousand. Small correlations are only harmless alongside small clusters, and dismissing one without multiplying by cluster size is a classic error.
- For a fixed budget, would you add classrooms or add pupils per classroom?Add classrooms whenever the intraclass correlation is meaningful. Effective sample size tends to the number of clusters divided by the ICC as cluster size grows, so enlarging classrooms hits a hard ceiling while each new classroom raises it. Extra pupils per classroom are worth buying only when they are much cheaper than a new classroom or when the ICC is very close to zero.
Twenty pupils in one classroom are like twenty photographs taken from nearly the same spot. You have twenty images but nowhere near twenty viewpoints.
saying these in an interview costs you the question
- Uses cluster size m instead of m minus one
- Treats the intraclass correlation as an R-squared
- Calls an ICC of 0.01 negligible without checking cluster size
- Says more pupils per classroom always adds power
- Applies the equal-size formula to wildly unequal clusters