Index of Dispersion Test for Count Data

The index of dispersion test checks whether counts fit a Poisson distribution by referring the sum of squared deviations divided by the mean to a chi-square distribution with n - 1 degrees of freedom.

산포 지수 검정은 편차 제곱합을 평균으로 나눈 값을 자유도 n−1 의 카이제곱 분포와 비교해 계수 자료가 포아송 분포와 맞는지 판정하는 검정입니다.

Are my count data overdispersed relative to a Poisson distribution, and should I use a negative binomial model?

계수 자료가 포아송보다 과대산포되어 있는지, 음이항 모형을 써야 하는지 어떻게 확인하나요?

Under a Poisson model the variance equals the mean, so D = sum((x - mean)^2) / mean follows a chi-square distribution with n - 1 degrees of freedom; large values indicate overdispersion and small values underdispersion. A likelihood-ratio test of the negative binomial against the Poisson, with a halved p value because the dispersion parameter lies on the boundary, gives a second view.

When to use Index of Dispersion Test for Count Data

• You plan Poisson regression or a Poisson rate analysis and need to check equidispersion.
• You want to choose between Poisson and negative binomial models for counts without covariates.
• You want to compare observed and expected Poisson frequencies, including excess zeros.

When to use something else

• Counts depend on covariates; check dispersion in the fitted Poisson regression model.
• The data contain many more zeros than expected; consider zero-inflated Poisson or negative binomial models.

Assumptions and requirements

• Counts are non-negative integers.
• Observations are independent and share a common mean.
• The mean is not too small (roughly 1 or more) for the chi-square approximation.

What the analysis reports

• Mean, variance and variance-to-mean ratio
• Dispersion statistic D with one-sided and two-sided p values
• Negative binomial versus Poisson likelihood ratio, NB2 alpha and boundary-corrected p value
• Observed versus expected Poisson frequency table

How to report it

The index of dispersion test indicated overdispersion (mean = [value], variance = [value]), D = [value], chi2([df]) p = [value], and the negative binomial fitted better than the Poisson, LR = [value], p = [value].

Related methods

• Poisson Regression
• Negative Binomial Regression
• Zero Inflated Poisson Regression

Run Index of Dispersion Test for Count Data on Quorant