pollthe lab

Measurement models, matched to your data

Media research produces four dominant data shapes, binary attitudes, continuous composite scores, count data, and relationships between measures. Each has a conjugate model with an exact, closed-form posterior. Choose the shape that matches your study and update your belief live.

One prior caveat worth the ink: a Beta prior on a proportion and a Normal prior on a mean are both summaries of the same assumption, that you walked in with some belief. The default "weakly informative" priors shown here follow the modern default-prior literature (Gelman, Jakulin, Pittau, & Su, 2008, "A weakly informative default prior distribution for logistic and other regression models," Annals of Applied Statistics).

CONJUGATE · EXACT Beta-Binomial

Binary attitude item Beta(α₀,β₀) prior + Binomial likelihood

Any yes/no, agree/disagree, or "showed the effect" survey item. Example framing: 40 respondents saw a political ad; 18 reported a measurable post-test attitude shift.

Your prior

Prior mean belief θ 0.30

Prior strength (effective n) 6

Your data

Respondents (n)

Showed effect (x)

Prior Likelihood (normalized) Posterior

Naive proportion x/n

–

Posterior mean

–

Posterior SD

–

95% equal-tail CI

–

95% HDI

–

Info gained (KL)

–

calculateconjugate update

α₁ = α₀ + x β₁ = β₀ + (n − x)

mean = α₁/(α₁+β₁) var = α₁β₁ / [(α₁+β₁)²(α₁+β₁+1)] KL = D_KL(Beta₁ ‖ Beta₀)

movie Sequential update: watch it one respondent at a time

respondent 0 / 24

quizcheck yourself

You collect 10 more respondents with roughly the same success rate as your first batch. What happens to the posterior?

It resets and starts over from the new prior mean. It gets narrower: more concentrated around roughly the same mean, as α and β both grow with consistent evidence. It gets wider, because more data always increases uncertainty.

CONJUGATE · EXACT Normal-Normal

Continuous composite score Normal(μ₀,σ₀) prior + Normal likelihood

For means of Likert composites, attitude thermometers, or standardized effect scores. Example framing: a 5-item parasocial-interaction composite (1–7) among 40 respondents; your prior says the population mean is about 3.5 with SD 0.8.

Scale matters. Enter the prior mean and SD on the same scale as your data (raw units). If your outcome is z-scored, use a z-scored prior. The lab below shows the exact conjugate update in the units you choose.

Your prior

Prior mean μ₀ 3.50

Prior SD σ₀ 0.80

Your data

Sample mean ȳ

Data SD (σ, assumed known)

Sample size n

Prior Posterior

x-axis: score on the outcome scale (e.g., 1–7). The curves are the prior and posterior densities of the population mean μ.

Posterior mean μ₁

–

Posterior SD σ₁

–

95% equal-tail CI

–

95% HDI

–

Prior precision 1/σ₀²

–

Data precision n/σ²

–

calculateconjugate update (precision adds)

posterior precision = prior precision + data precision

μ₁ = (μ₀/σ₀² + n·ȳ/σ²) / (1/σ₀² + n/σ²) σ₁² = 1 / (1/σ₀² + n/σ²)

CONJUGATE · EXACT Gamma-Poisson

Exposure or event counts Gamma(α₀,β₀) prior + Poisson likelihood

For count data: weekly media exposures, tweets mentioning a brand, news items read, scandal mentions covered. Example framing: you believe viewers see roughly λ ≈ 2.0 health segments per month; you survey 10 respondents and count their exposures.

Your prior on the rate λ

Prior mean λ₀ 2.0

Prior SD 1.0

We parameterize the Gamma prior by its mean and SD; that sets α₀ = mean²/sd² and β₀ = mean/sd².

Your data (one count per respondent)

Counts, comma-separated (e.g. 3,1,4,2)

Prior (Gamma on λ) Posterior (Gamma on λ)

Sum of counts Σx

–

n respondents

–

Posterior mean λ

–

Posterior SD

–

95% equal-tail CI

–

95% HDI

–

Posterior predictive. The next respondent's count follows a Negative Binomial with r = α₁, p = β₁/(β₁+1), a compound Poisson-Gamma that naturally over-disperses. The blue mass under each future count is that predictive distribution.

calculateconjugate update

α₁ = α₀ + Σx β₁ = β₀ + n E[λ] = α₁/β₁

FISHER z · NORMAL APPROX Correlation

Relationship between two measures Uniform(ρ) prior, via Fisher z

For associations central to agenda-setting (media-issue salience ↔ public salience), cultivation (TV viewing ↔ fear of crime), or media credibility ↔ news engagement. A uniform prior on ρ ∈ (−1,1) is the natural "no prior directional belief" default; it becomes a Normal posterior on the Fisher z-transform.

Your data

Sample correlation r

Sample size n

Prior centre on ρ (flat = 0)

Prior SD on Fisher-z

z = arctanh(ρ); a SD of 1 is very diffuse. Set centre 0 & SD large for the uniform-like default.

Prior on ρ Posterior on ρ

Fisher z of r

–

Posterior mean ρ

–

Posterior SD

–

95% equal-tail CI ρ

–

Posterior mode ρ (approx)

–

Report, don't just threshold. The enormous modern criticism of binary "significant / not significant" thinking (Wasserstein & Lazar, 2016, The American Statistician) is precisely what a full posterior correlation interval addresses, you get the whole plausible spread of ρ, not a single 0/1 verdict.

calculatefisher z-transform

z = arctanh(r) = ½·ln[(1+r)/(1−r)] SE ≈ 1/√(n−3)

The correlation posterior is Normal on the z-scale with mean = prior z-precision-weighted and SD = 1/√(prior SD² + n−3), then transformed back to ρ. This is the classical normal-approximation Bayesian treatment (also the basis of confidence intervals for r in most textbooks).

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