Handle Overdispersed Counts with Negative Binomial

Detect Poisson overdispersion and refit with Negative Binomial to absorb extra variation.

Count data often violates the Poisson assumption that the variance equals the mean. When the observed variance is substantially larger (a condition called overdispersion) a Poisson GAM underestimates uncertainty and produces confidence intervals that are too narrow. The Negative Binomial family adds a dispersion parameter that absorbs the extra variation, giving honest standard errors and better-calibrated predictions.

Load data

Load the fish dataset and inspect it.

import whittaker as wk

data = wk.load_dataset("fish", as_frame=True)
data.columns.tolist()

['temperature', 'depth', 'count']

data.head()

	temperature	depth	count
0	13.232243	20.213893	7.0
1	13.969823	20.798765	14.0
2	24.284515	36.965959	3.0
3	9.838319	30.989895	2.0
4	20.002011	45.968733	4.0

Fit a Poisson model first

Start with Poisson to establish a baseline and check for overdispersion.

poisson_model = wk.GAM(
    "count ~ s(temperature) + s(depth)",
    family=wk.Poisson(),
).fit(data)

poisson_model.summary()

GAM fit summary
============================================================
Formula:    count ~ s(temperature) + s(depth)
Family:     Poisson(link='log')
Inference:  GCV
Observations: 300
Coefficients: 19

Parametric coefficients:
  Term                       Estimate    Std.Err    z value    p-value
  ------------------------ ---------- ---------- ---------- ----------
  (Intercept)                  1.5349     0.0289     53.128    < 1e-16

Approximate significance of smooth terms:
  Term                        EDF Ref.df     Chi.sq    p-value
  ------------------------ ------ ------ ---------- ----------
  s(temperature)             4.24      5    294.642    < 1e-16
  s(depth)                   1.74      2    194.880    < 1e-16

Total EDF:  6.98
Scale est:  1.000000
Deviance:   320.5707
Null dev:   843.4578
Dev. expl:  62.0%
GCV score:  1.120093
AIC:        1305.79
BIC:        1331.65

Look at the summary’s residual deviance relative to the residual degrees of freedom. A ratio well above 1 (sometimes called the dispersion ratio) signals overdispersion. When the ratio is large, Poisson confidence intervals are too narrow and p-values are anti-conservative.

Fit a Negative Binomial model

Refit with wk.NegativeBinomial(), which estimates the extra dispersion parameter automatically.

nb_model = wk.GAM(
    "count ~ s(temperature) + s(depth)",
    family=wk.NegativeBinomial(),
).fit(data)

nb_model.summary()

GAM fit summary
============================================================
Formula:    count ~ s(temperature) + s(depth)
Family:     NegativeBinomial(theta=329.8, link='log')
Inference:  GCV
Observations: 300
Coefficients: 19

Parametric coefficients:
  Term                       Estimate    Std.Err    z value    p-value
  ------------------------ ---------- ---------- ---------- ----------
  (Intercept)                  1.5347     0.0291     52.789    < 1e-16

Approximate significance of smooth terms:
  Term                        EDF Ref.df     Chi.sq    p-value
  ------------------------ ------ ------ ---------- ----------
  s(temperature)             4.24      5    290.499    < 1e-16
  s(depth)                   1.73      2    190.878    < 1e-16

Total EDF:  6.97
Scale est:  1.000000
Deviance:   315.5890
Null dev:   829.7217
Dev. expl:  62.0%
GCV score:  1.102575
AIC:        1305.71
BIC:        1331.51

Compare the deviance explained between the two models. The Negative Binomial typically shows a lower (more honest) deviance explained because it no longer tries to explain variance that is simply inherent noise. The estimated dispersion parameter appears in the summary. A value near zero indicates little overdispersion, a large value indicates substantial overdispersion.

Partial effects

wk.partial_effects(nb_model)

The partial effect plots show the relationship between each covariate and the log-expected count. Confidence bands will generally be wider than those from the Poisson model, reflecting the additional uncertainty that overdispersion introduces.

Diagnostics

wk.check(nb_model)

A well-fitting Negative Binomial model should show residuals with no strong pattern against fitted values, and a Q-Q plot closer to the diagonal than the Poisson model produced.