Deviance, Residuals, and Goodness of Fit

Diagnosing where a fitted probability model disagrees with the data

Statistics

Regression

Diagnostics

Goodness of Fit

Likelihood deviance, Pearson and deviance residuals, goodness-of-fit calibration, influence, simulation diagnostics, and bioinformatics interpretation.

Author

Jiahong Sun

Published

October 22, 2026

Modified

October 22, 2026

1 Fit Is More Than a Coefficient Table

A regression model specifies a conditional distribution for each outcome,

\[ Y_i\mid\mathbf x_i\sim F_i\{\mu_i(\boldsymbol\beta),\phi\}, \qquad i=1,\ldots,n, \]

not just a line through observed points. Here $\mu_i$ is the conditional mean, $\boldsymbol\beta$ controls its systematic structure, and $\phi$ is a dispersion parameter when the family has one. Diagnostics ask whether the fitted model reproduces important features of the data-generating process: the mean, variance, support, tails, dependence, and observation mechanism.

Three questions should remain separate:

Relative fit: does one specified model fit better than another?
Absolute adequacy: could the observed discrepancies plausibly arise under the fitted model?
Scientific validity: do the experimental units, measurements, covariates, and target population justify the intended interpretation?

Deviance and residuals address the first two questions. They cannot by themselves establish randomization, absence of confounding, correct sampling, or biological importance.

2 Deviance as Likelihood Discrepancy

Let $\ell(\widehat{\boldsymbol\beta},\phi;\mathbf y)$ be the maximized log-likelihood of the fitted mean model at a common dispersion $\phi$. A saturated model uses enough mean parameters to reproduce every observed response. For unit-dispersion families such as Poisson and binomial, the model deviance is

\[ \boxed{ D(\mathbf y;\hat{\boldsymbol\mu}) =2\{\ell(\text{saturated};\mathbf y)- \ell(\widehat{\boldsymbol\beta};\mathbf y)\}. } \]

For a general exponential-dispersion family, the conventional unscaled GLM deviance is $2\phi$ times this log-likelihood difference. Software conventions must therefore be checked when dispersion is not fixed at one.

The saturated model is a benchmark, not usually a scientifically useful model. Because it fits at least as well as the candidate, $D\geq0$. Smaller deviance means less likelihood loss relative to saturation for the same outcomes, weights, likelihood, and dispersion convention. Deviances from unrelated families or different datasets need not be comparable.

For independent exponential-family observations, deviance decomposes as

$$ D=_{i=1}^n d_i,

where $d_i\geq0$ is the unit’s deviance contribution. This decomposition helps locate lack of fit, but a large $d_i$ may indicate an unusual valid observation, a recording error, an inadequate mean, the wrong response family, or unmodeled dependence. The number alone does not identify the cause.

2.1 Poisson and binomial forms

For a Poisson response with fitted mean $\hat\mu_i$,

\[ d_i=2\left[ y_i\log\left(\frac{y_i}{\hat\mu_i}\right) -(y_i-\hat\mu_i) \right], \]

with $y_i\log(y_i/\hat\mu_i)=0$ when $y_i=0$. For a binomial count $y_i$ out of $m_i$ trials with fitted probability $\hat p_i$,

\[ d_i=2\left[ y_i\log\left(\frac{y_i}{m_i\hat p_i}\right) +(m_i-y_i)\log\left{ \frac{m_i-y_i}{m_i(1-\hat p_i)} \right} \right], \]

using the continuous limit for zero terms. These are likelihood discrepancies, not squared Euclidean distances.

3 Residuals Expose Structure Hidden by a Total

A single deviance can hide compensating discrepancies. Residuals attach a sign and scale to unit-level disagreement.

3.1 Response and Pearson residuals

The response residual is

\[ r_i^{\mathrm{response}}=y_i-\hat\mu_i. \]

It retains the outcome’s units, but its spread changes with the model variance. The Pearson residual adjusts for that mean–variance relationship:

\[ \boxed{ r_i^{P}=\frac{y_i-\hat\mu_i} {\sqrt{\hat\phi V(\hat\mu_i)/w_i}}. } \]

Here $V(\mu)$ is the family’s variance function and $w_i$ is a known prior weight. Pearson residuals are not automatically independent or standard normal; fitting consumes information, leverage changes their variance, and discrete outcomes remain discrete.

3.2 Deviance residuals

The signed square root of the deviance contribution is

\[ \boxed{ r_i^D=\operatorname{sign}(y_i-\hat\mu_i)\sqrt{d_i}, \qquad \sum_i(r_i^D)^2=D. } \]

Deviance residuals put likelihood discrepancy on an additive scale and often behave more symmetrically than raw residuals. Standardization by an appropriate leverage factor can make observations more comparable. It does not make the residuals exactly Gaussian or erase estimation uncertainty.

3.3 Randomized quantile residuals

For a fitted continuous conditional CDF $\hat F_i$, the probability integral transform suggests

\[ r_i^Q=\Phi^{-1}\{\hat F_i(y_i)\}. \]

For a discrete response, draw

\[ U_i\sim\operatorname{Uniform} \left\{\hat F_i(y_i^-),\hat F_i(y_i)\right\}, \qquad r_i^Q=\Phi^{-1}(U_i). \]

Under a correctly specified model with known parameters, these randomized quantile residuals have a standard-normal reference distribution. With fitted parameters they are only approximately so, and different randomizations give slightly different plots. Repeating the randomization or using simulation envelopes avoids overinterpreting one realization.

4 Worked Contributions for Count Data

Suppose four independent count units have observed counts $\mathbf y=(0,1,4,8)$ and fitted Poisson means $\hat{\boldsymbol\mu}=(1,2,3,7)$. Their Pearson contributions sum to

\[ X_P^2=\sum_i\frac{(y_i-\hat\mu_i)^2}{\hat\mu_i} =1+\frac12+\frac13+\frac17\approx1.98. \]

Their deviance is

\[ \begin{aligned} D=2\sum_i\left[ y_i\log(y_i/\hat\mu_i)-(y_i-\hat\mu_i) \right] \approx3.05. \end{aligned} \]

The zero count contributes $2$ to the deviance but $1$ to $X_P^2$; the two discrepancy measures weight departures differently. These totals illustrate the calculation only. Without knowing how the means were estimated, the number of fitted parameters, and the sampling design, comparing either total with a reference distribution would be unjustified.

5 Goodness-of-fit Calibration

For a regular GLM with independent units and an adequate model, residual deviance or Pearson’s statistic is sometimes compared with a chi-squared distribution having approximately

\[ \operatorname{df}_{\mathrm{residual}}=n-\operatorname{rank}(\mathbf X) \]

degrees of freedom. This approximation is most credible when unit contributions contain enough information and fitted values are not near boundaries. It can be poor for sparse binary data, many small expected counts, high leverage, estimated dispersion, penalized fits, random effects, selection, or a parameter dimension that is not small relative to $n$.

Grouped binomial data illustrate a crucial distinction. If several independent units share a covariate pattern, their grouped counts provide replication for assessing the fitted probability. With one Bernoulli observation at each unique covariate pattern, the saturated benchmark uses one parameter per observation; the residual deviance does not become a reliable omnibus test merely because software prints residual degrees of freedom.

A large goodness-of-fit statistic signals disagreement but not its mechanism. A small statistic does not prove the model: power may be weak, discrepancies may occur in features the statistic barely weights, or overdispersion may have been absorbed by an inflated scale estimate.

6 A Diagnostic Workflow

Diagnostics are most informative when tied to possible failures rather than treated as a ritual collection of plots.

Diagnostic question	Useful evidence	What a pattern can suggest
Is the conditional mean adequate?	Residuals against fitted values and each predictor	Curvature, missing interaction, wrong link
Is the variance model adequate?	Residual spread by fitted mean; Pearson statistic; simulations	Overdispersion, underdispersion, heterogeneity
Are tails and zeros represented?	Quantile residuals, rootograms, replicated maxima and zero counts	Heavy tails, zero mechanism, truncation
Is dependence left over?	Residuals by time, space, donor, batch, plate, or family	Clustering or autocorrelation
Does one unit dominate?	Leverage, standardized residuals, case-deletion changes	Extrapolation, data error, influential valid unit
Is the observation process represented?	Residuals by depth, detection limit, censoring, quality	Missing offset, selection, censoring, measurement error

Plots should retain design labels. A residual pattern by sequencing lane or collection site can be invisible after pooling. Conversely, searching hundreds of residual plots and reporting only the most striking one creates a selection problem of its own.

7 Leverage, Outliers, and Influence

These terms answer different questions:

Leverage measures how unusual a unit’s predictor pattern is in the fitted geometry.
An outlier has an outcome poorly predicted by the model at its predictors.
Influence measures how much an estimate, fitted value, or scientific conclusion changes when a unit is removed or downweighted.

A high-leverage unit can have a modest residual because it pulls the fit toward itself. A large residual at low leverage may have little effect on the target contrast. Inspect standardized residuals together with leverage and case-deletion changes in the actual estimand. Deleting a valid influential observation simply because it changes the answer understates uncertainty; verify the record, explain its design role, and report a principled sensitivity analysis.

8 Bioinformatics Example: RNA-seq Count Models

Consider 30 RNA-seq libraries from 30 independently sampled donors, balanced across two conditions and three processing batches. For one gene, a negative-binomial GLM might specify

\[ Y_i\sim\operatorname{NB}(\mu_i,\alpha), \qquad \log\mu_i=\log s_i+\beta_0+\beta_1A_i+\boldsymbol\gamma^\mathsf T\mathbf B_i, \]

where $s_i$ is a normalization factor, $A_i$ is condition, and $\mathbf B_i$ represents batch. Donors are the experimental units. Reads are technical measurements, and genes are parallel outcomes; neither creates additional condition replicates.

Useful checks include residuals against fitted abundance and normalization factor, residual distributions within batches, sample-level influence on the treatment log fold change, and simulated zero and upper-tail counts. A pattern at low fitted means may reflect discreteness rather than a missing biological effect, so simulation under the fitted count model is more informative than a normal Q–Q plot of raw residuals.

Statistically, a diagnostic departure says that some aspect of the fitted observation model does not reproduce the counts. Biologically, it does not by itself show differential expression or identify a pathway. Possible causes include cell-composition changes, batch imbalance, sample quality, mapping ambiguity, an incorrect dispersion model, or a true mean structure omitted from the design. Technical replicates can characterize assay variation, but only independent biological units support population-level treatment inference.

9 Assumptions and Sensitivity

Correct unit of independence: residual calibration inherits the model’s factorization. Repeated cells, reads, visits, or aliquots from one specimen cannot be diagnosed as if they were independent biological units.
Represented support and observation process: censoring, truncation, detection limits, excess zeros, and uncertain denominators can distort every residual type.
Stable estimation: boundary estimates, separation, weak identification, and heavy penalization change familiar reference distributions.
Prespecified scientific target: a better residual plot does not justify changing the estimand after seeing the answer.
Adequate calibration: when asymptotic references are doubtful, simulate complete datasets from the fitted model, refit each dataset, and compare the observed diagnostic with its refitted reference distribution.

Sensitivity analysis should vary plausible mean structures, variance families, dependence structures, observation mechanisms, and influential units. The goal is not to find one specification that passes every threshold, but to learn which scientific conclusions depend on which assumptions.

10 Connections and Take-home Messages

Generalized linear models defines the family, link, linear predictor, likelihood, and dispersion used here.
Link functions and exponential-family distributions develops variance functions and link-scale structure.
Linear-model assumptions and diagnostics connects residual checks to Gaussian linear models.
Diagnosing and modeling overdispersion explains why excess conditional variability needs a mechanism-specific response.
Likelihood and log-likelihood supplies the likelihood foundation for deviance.
Likelihood-based model comparison uses differences in maximized likelihood for carefully comparable models.

Deviance measures likelihood loss relative to a saturated benchmark; it is model-relative, not a universal score of scientific validity.
Residuals localize disagreement, but their scale and reference behavior depend on the fitted family, leverage, discreteness, and estimation.
A goodness-of-fit p-value is only as trustworthy as its independence, sparsity, parameter-count, and calibration assumptions.
Diagnostics should be plotted against design variables and paired with leverage, influence, and simulation-based checks.
In bioinformatics, diagnose at the biological experimental-unit level and distinguish statistical model failure from biological explanation.

11 References

Nelder, J. A., and Wedderburn, R. W. M. (1972). Generalized linear models. Journal of the Royal Statistical Society: Series A, 135(3), 370–384. https://doi.org/10.2307/2344614
McCullagh, P., and Nelder, J. A. (1989). Generalized Linear Models (2nd ed.). Chapman and Hall/CRC. https://doi.org/10.1201/9780203753736
Pregibon, D. (1981). Logistic regression diagnostics. The Annals of Statistics, 9(4), 705–724. https://doi.org/10.1214/aos/1176345513
Williams, D. A. (1987). Generalized linear model diagnostics using the deviance and single case deletions. Applied Statistics, 36(2), 181–191. https://doi.org/10.2307/2347550
Dunn, P. K., and Smyth, G. K. (1996). Randomized quantile residuals. Journal of Computational and Graphical Statistics, 5(3), 236–244. https://doi.org/10.1080/10618600.1996.10474708
R Core Team. glm, residuals.glm, and anova.glm: official R statistical software documentation. https://stat.ethz.ch/R-manual/R-devel/library/stats/html/glm.html

--- title: "Deviance, Residuals, and Goodness of Fit" subtitle: "Diagnosing where a fitted probability model disagrees with the data" description: "Likelihood deviance, Pearson and deviance residuals, goodness-of-fit calibration, influence, simulation diagnostics, and bioinformatics interpretation." author: "Jiahong Sun" date: "2026-10-22" date-modified: "2026-10-22" categories: [Statistics, Regression, Diagnostics, Goodness of Fit] toc: true number-sections: true code-fold: show draft: false --- # Fit Is More Than a Coefficient Table A regression model specifies a conditional distribution for each outcome, $$ Y_i\mid\mathbf x_i\sim F_i\{\mu_i(\boldsymbol\beta),\phi\}, \qquad i=1,\ldots,n, $$ not just a line through observed points. Here $\mu_i$ is the conditional mean, $\boldsymbol\beta$ controls its systematic structure, and $\phi$ is a dispersion parameter when the family has one. Diagnostics ask whether the fitted model reproduces important features of the data-generating process: the mean, variance, support, tails, dependence, and observation mechanism. Three questions should remain separate: 1. **Relative fit:** does one specified model fit better than another? 2. **Absolute adequacy:** could the observed discrepancies plausibly arise under the fitted model? 3. **Scientific validity:** do the experimental units, measurements, covariates, and target population justify the intended interpretation? Deviance and residuals address the first two questions. They cannot by themselves establish randomization, absence of confounding, correct sampling, or biological importance. # Deviance as Likelihood Discrepancy Let $\ell(\widehat{\boldsymbol\beta},\phi;\mathbf y)$ be the maximized log-likelihood of the fitted mean model at a common dispersion $\phi$. A **saturated model** uses enough mean parameters to reproduce every observed response. For unit-dispersion families such as Poisson and binomial, the model deviance is $$ \boxed{ D(\mathbf y;\hat{\boldsymbol\mu}) =2\{\ell(\text{saturated};\mathbf y)- \ell(\widehat{\boldsymbol\beta};\mathbf y)\}. } $$ For a general exponential-dispersion family, the conventional **unscaled** GLM deviance is $2\phi$ times this log-likelihood difference. Software conventions must therefore be checked when dispersion is not fixed at one. The saturated model is a benchmark, not usually a scientifically useful model. Because it fits at least as well as the candidate, $D\geq0$. Smaller deviance means less likelihood loss relative to saturation **for the same outcomes, weights, likelihood, and dispersion convention**. Deviances from unrelated families or different datasets need not be comparable. For independent exponential-family observations, deviance decomposes as $$ D=\sum_{i=1}^n d_i, $$ where $d_i\geq0$ is the unit's deviance contribution. This decomposition helps locate lack of fit, but a large $d_i$ may indicate an unusual valid observation, a recording error, an inadequate mean, the wrong response family, or unmodeled dependence. The number alone does not identify the cause. ## Poisson and binomial forms For a Poisson response with fitted mean $\hat\mu_i$, $$ d_i=2\left[ y_i\log\left(\frac{y_i}{\hat\mu_i}\right) -(y_i-\hat\mu_i) \right], $$ with $y_i\log(y_i/\hat\mu_i)=0$ when $y_i=0$. For a binomial count $y_i$ out of $m_i$ trials with fitted probability $\hat p_i$, $$ d_i=2\left[ y_i\log\left(\frac{y_i}{m_i\hat p_i}\right) +(m_i-y_i)\log\left{ \frac{m_i-y_i}{m_i(1-\hat p_i)} \right} \right], $$ using the continuous limit for zero terms. These are likelihood discrepancies, not squared Euclidean distances. # Residuals Expose Structure Hidden by a Total A single deviance can hide compensating discrepancies. Residuals attach a sign and scale to unit-level disagreement. ## Response and Pearson residuals The response residual is $$ r_i^{\mathrm{response}}=y_i-\hat\mu_i. $$ It retains the outcome's units, but its spread changes with the model variance. The Pearson residual adjusts for that mean--variance relationship: $$ \boxed{ r_i^{P}=\frac{y_i-\hat\mu_i} {\sqrt{\hat\phi V(\hat\mu_i)/w_i}}. } $$ Here $V(\mu)$ is the family's variance function and $w_i$ is a known prior weight. Pearson residuals are not automatically independent or standard normal; fitting consumes information, leverage changes their variance, and discrete outcomes remain discrete. ## Deviance residuals The signed square root of the deviance contribution is $$ \boxed{ r_i^D=\operatorname{sign}(y_i-\hat\mu_i)\sqrt{d_i}, \qquad \sum_i(r_i^D)^2=D. } $$ Deviance residuals put likelihood discrepancy on an additive scale and often behave more symmetrically than raw residuals. Standardization by an appropriate leverage factor can make observations more comparable. It does not make the residuals exactly Gaussian or erase estimation uncertainty. ## Randomized quantile residuals For a fitted continuous conditional CDF $\hat F_i$, the probability integral transform suggests $$ r_i^Q=\Phi^{-1}\{\hat F_i(y_i)\}. $$ For a discrete response, draw $$ U_i\sim\operatorname{Uniform} \left\{\hat F_i(y_i^-),\hat F_i(y_i)\right\}, \qquad r_i^Q=\Phi^{-1}(U_i). $$ Under a correctly specified model with known parameters, these randomized quantile residuals have a standard-normal reference distribution. With fitted parameters they are only approximately so, and different randomizations give slightly different plots. Repeating the randomization or using simulation envelopes avoids overinterpreting one realization. # Worked Contributions for Count Data Suppose four independent count units have observed counts $\mathbf y=(0,1,4,8)$ and fitted Poisson means $\hat{\boldsymbol\mu}=(1,2,3,7)$. Their Pearson contributions sum to $$ X_P^2=\sum_i\frac{(y_i-\hat\mu_i)^2}{\hat\mu_i} =1+\frac12+\frac13+\frac17\approx1.98. $$ Their deviance is $$ \begin{aligned} D=2\sum_i\left[ y_i\log(y_i/\hat\mu_i)-(y_i-\hat\mu_i) \right] \approx3.05. \end{aligned} $$ The zero count contributes $2$ to the deviance but $1$ to $X_P^2$; the two discrepancy measures weight departures differently. These totals illustrate the calculation only. Without knowing how the means were estimated, the number of fitted parameters, and the sampling design, comparing either total with a reference distribution would be unjustified. # Goodness-of-fit Calibration For a regular GLM with independent units and an adequate model, residual deviance or Pearson's statistic is sometimes compared with a chi-squared distribution having approximately $$ \operatorname{df}_{\mathrm{residual}}=n-\operatorname{rank}(\mathbf X) $$ degrees of freedom. This approximation is most credible when unit contributions contain enough information and fitted values are not near boundaries. It can be poor for sparse binary data, many small expected counts, high leverage, estimated dispersion, penalized fits, random effects, selection, or a parameter dimension that is not small relative to $n$. Grouped binomial data illustrate a crucial distinction. If several independent units share a covariate pattern, their grouped counts provide replication for assessing the fitted probability. With one Bernoulli observation at each unique covariate pattern, the saturated benchmark uses one parameter per observation; the residual deviance does not become a reliable omnibus test merely because software prints residual degrees of freedom. A large goodness-of-fit statistic signals disagreement but not its mechanism. A small statistic does not prove the model: power may be weak, discrepancies may occur in features the statistic barely weights, or overdispersion may have been absorbed by an inflated scale estimate. # A Diagnostic Workflow Diagnostics are most informative when tied to possible failures rather than treated as a ritual collection of plots. | Diagnostic question | Useful evidence | What a pattern can suggest | |---|---|---| | Is the conditional mean adequate? | Residuals against fitted values and each predictor | Curvature, missing interaction, wrong link | | Is the variance model adequate? | Residual spread by fitted mean; Pearson statistic; simulations | Overdispersion, underdispersion, heterogeneity | | Are tails and zeros represented? | Quantile residuals, rootograms, replicated maxima and zero counts | Heavy tails, zero mechanism, truncation | | Is dependence left over? | Residuals by time, space, donor, batch, plate, or family | Clustering or autocorrelation | | Does one unit dominate? | Leverage, standardized residuals, case-deletion changes | Extrapolation, data error, influential valid unit | | Is the observation process represented? | Residuals by depth, detection limit, censoring, quality | Missing offset, selection, censoring, measurement error | Plots should retain design labels. A residual pattern by sequencing lane or collection site can be invisible after pooling. Conversely, searching hundreds of residual plots and reporting only the most striking one creates a selection problem of its own. # Leverage, Outliers, and Influence These terms answer different questions: - **Leverage** measures how unusual a unit's predictor pattern is in the fitted geometry. - An **outlier** has an outcome poorly predicted by the model at its predictors. - **Influence** measures how much an estimate, fitted value, or scientific conclusion changes when a unit is removed or downweighted. A high-leverage unit can have a modest residual because it pulls the fit toward itself. A large residual at low leverage may have little effect on the target contrast. Inspect standardized residuals together with leverage and case-deletion changes in the actual estimand. Deleting a valid influential observation simply because it changes the answer understates uncertainty; verify the record, explain its design role, and report a principled sensitivity analysis. # Bioinformatics Example: RNA-seq Count Models Consider 30 RNA-seq libraries from 30 independently sampled donors, balanced across two conditions and three processing batches. For one gene, a negative-binomial GLM might specify $$ Y_i\sim\operatorname{NB}(\mu_i,\alpha), \qquad \log\mu_i=\log s_i+\beta_0+\beta_1A_i+\boldsymbol\gamma^\mathsf T\mathbf B_i, $$ where $s_i$ is a normalization factor, $A_i$ is condition, and $\mathbf B_i$ represents batch. Donors are the experimental units. Reads are technical measurements, and genes are parallel outcomes; neither creates additional condition replicates. Useful checks include residuals against fitted abundance and normalization factor, residual distributions within batches, sample-level influence on the treatment log fold change, and simulated zero and upper-tail counts. A pattern at low fitted means may reflect discreteness rather than a missing biological effect, so simulation under the fitted count model is more informative than a normal Q--Q plot of raw residuals. Statistically, a diagnostic departure says that some aspect of the fitted observation model does not reproduce the counts. Biologically, it does not by itself show differential expression or identify a pathway. Possible causes include cell-composition changes, batch imbalance, sample quality, mapping ambiguity, an incorrect dispersion model, or a true mean structure omitted from the design. Technical replicates can characterize assay variation, but only independent biological units support population-level treatment inference. # Assumptions and Sensitivity 1. **Correct unit of independence:** residual calibration inherits the model's factorization. Repeated cells, reads, visits, or aliquots from one specimen cannot be diagnosed as if they were independent biological units. 2. **Represented support and observation process:** censoring, truncation, detection limits, excess zeros, and uncertain denominators can distort every residual type. 3. **Stable estimation:** boundary estimates, separation, weak identification, and heavy penalization change familiar reference distributions. 4. **Prespecified scientific target:** a better residual plot does not justify changing the estimand after seeing the answer. 5. **Adequate calibration:** when asymptotic references are doubtful, simulate complete datasets from the fitted model, refit each dataset, and compare the observed diagnostic with its refitted reference distribution. Sensitivity analysis should vary plausible mean structures, variance families, dependence structures, observation mechanisms, and influential units. The goal is not to find one specification that passes every threshold, but to learn which scientific conclusions depend on which assumptions. # Connections and Take-home Messages - [Generalized linear models](generalized-linear-models.qmd) defines the family, link, linear predictor, likelihood, and dispersion used here. - [Link functions and exponential-family distributions](link-functions-and-exponential-family-distributions.qmd) develops variance functions and link-scale structure. - [Linear-model assumptions and diagnostics](linear-model-assumptions-and-diagnostics.qmd) connects residual checks to Gaussian linear models. - [Diagnosing and modeling overdispersion](diagnosing-and-modeling-overdispersion.qmd) explains why excess conditional variability needs a mechanism-specific response. - [Likelihood and log-likelihood](../statistical-inference/likelihood-and-log-likelihood.qmd) supplies the likelihood foundation for deviance. - [Likelihood-based model comparison](likelihood-based-model-comparison.qmd) uses differences in maximized likelihood for carefully comparable models. 1. Deviance measures likelihood loss relative to a saturated benchmark; it is model-relative, not a universal score of scientific validity. 2. Residuals localize disagreement, but their scale and reference behavior depend on the fitted family, leverage, discreteness, and estimation. 3. A goodness-of-fit p-value is only as trustworthy as its independence, sparsity, parameter-count, and calibration assumptions. 4. Diagnostics should be plotted against design variables and paired with leverage, influence, and simulation-based checks. 5. In bioinformatics, diagnose at the biological experimental-unit level and distinguish statistical model failure from biological explanation. # References - Nelder, J. A., and Wedderburn, R. W. M. (1972). Generalized linear models. *Journal of the Royal Statistical Society: Series A*, 135(3), 370--384. <https://doi.org/10.2307/2344614> - McCullagh, P., and Nelder, J. A. (1989). *Generalized Linear Models* (2nd ed.). Chapman and Hall/CRC. <https://doi.org/10.1201/9780203753736> - Pregibon, D. (1981). Logistic regression diagnostics. *The Annals of Statistics*, 9(4), 705--724. <https://doi.org/10.1214/aos/1176345513> - Williams, D. A. (1987). Generalized linear model diagnostics using the deviance and single case deletions. *Applied Statistics*, 36(2), 181--191. <https://doi.org/10.2307/2347550> - Dunn, P. K., and Smyth, G. K. (1996). Randomized quantile residuals. *Journal of Computational and Graphical Statistics*, 5(3), 236--244. <https://doi.org/10.1080/10618600.1996.10474708> - R Core Team. `glm`, `residuals.glm`, and `anova.glm`: official R statistical software documentation. <https://stat.ethz.ch/R-manual/R-devel/library/stats/html/glm.html>