Why Measure Changes in Fitted Values?
A case can matter because removing it changes the fitted coefficients, but a coefficient change is not always the most direct measure of what that means for the model’s predictions. DFFITS focuses on the change in the fitted value for the case that was removed. In a generalized linear model (GLM), that change can be measured either on the linear-predictor scale or on the response scale. The distinction matters because the inverse link can transform the same change in the linear predictor into a small or large change in the fitted mean, depending on where the case lies.
A GLM does not have one universally adopted DFFITS formula for every response family and scale. We therefore define the standardization explicitly here. It compares the full-fit and case-deleted predictions at the case’s covariate values, then scales the change by an estimated standard error for the full-fit prediction. The result is a diagnostic, not a test statistic with a universal reference distribution or cutoff.
\(\displaystyle \operatorname{DFFITS}_{\eta,i}=\frac{\hat\eta_i-\hat\eta_{i(i)}}{\sqrt{\hat\phi_{(i)}\,x_i^\mathsf{T}(X^\mathsf{T}WX)^{-1}x_i}}\),
and response-scale DFFITS by
\(\displaystyle \operatorname{DFFITS}_{\mu,i}=\frac{\hat\mu_i-\hat\mu_{i(i)}}{|g^{-1\,\prime}(\hat\eta_i)|\sqrt{\hat\phi_{(i)}\,x_i^\mathsf{T}(X^\mathsf{T}WX)^{-1}x_i}}\).
Here \(W\) is the full-fit GLM iteratively reweighted least-squares weight matrix with the dispersion \(\phi\) excluded, and \(\hat\phi_{(i)}\) is the dispersion estimate from the case-deleted fit. For Poisson and binomial models with their usual dispersion fixed at 1, use \(\hat\phi_{(i)}=1\). The response-scale denominator uses the delta-method standard error of the full-fit response prediction. If the inverse-link derivative is zero, that delta-method standardization is degenerate and should not be used at that point.
In the usual independent-observation GLM with unit observation weights, \(W\) is diagonal, with \(i\)th diagonal entry \(w_i=g_i^2/V(\mu_i)\), where \(g_i=d\mu_i/d\eta_i\) and \(\operatorname{Var}(Y_i)=\phi V(\mu_i)\). The GLM leverage is \(h_{ii}=w_i x_i^\mathsf{T}(X^\mathsf{T}WX)^{-1}x_i\). If a model uses known observation weights or grouped responses, its variance and weight definitions must incorporate those features consistently.
How the Two Scales Differ
The linear-predictor version records the change before applying the inverse link. The response version records the actual change in the fitted mean, in units standardized by its delta-method uncertainty. The derivative in the response-scale denominator is evaluated at the full fit, so this definition has a clear local interpretation. For a nonlinear inverse link, it is an approximation to response-prediction uncertainty, while the numerator is the actual case-deleted response change.
Consequently, the two DFFITS values need not be equal. A log link converts a given change in \(\eta\) into a relative change in \(\mu\), while a logit link has derivative \(p(1-p)\), which becomes small near probabilities 0 and 1. Also, each scale’s standardizing uncertainty depends on the fitted model’s weights. A large raw change, by itself, does not establish that a case has a large standardized influence.
Worked Example: Exact Gaussian DFFITS in an Intercept-Only Model
Consider the Gaussian intercept-only model with observations \(1,2,3,8\), unknown variance, and identity link. We examine observation \(i\) with value 8. The full fitted mean is \(\hat\mu_i=\hat\eta_i=(1+2+3+8)/4=3.5\). After deleting 8, the fitted mean is \(\hat\mu_{i(i)}=2\), so the fitted-value change is \(1.5\).
For an intercept-only model, \(X^\mathsf{T}X=4\), and \(x_i=1\), so \(x_i^\mathsf{T}(X^\mathsf{T}X)^{-1}x_i=1/4\). The deleted observations are \(1,2,3\), whose mean is 2 and whose residual sum of squares is \(2\). Their residual degrees of freedom are \(3-1=2\), giving \(\hat\phi_{(i)}=s_{(i)}^2=1\). Thus the standard error in the DFFITS denominator is \(\sqrt{1/4}=0.5\), and \(\operatorname{DFFITS}_{\eta,i}=\operatorname{DFFITS}_{\mu,i}=1.5/0.5=3\).
This is the familiar linear-model DFFITS. The identity link makes the two scales identical; the next examples show why that equality does not generally hold in a GLM.
Worked Example: A Poisson Case Can Shift the Predictor More Than the Mean
Take a Poisson intercept-only model with counts \(1,2,3,10\) and a log link. The full mean is \(\hat\mu=16/4=4\), so \(\hat\eta=\log(4)\). Removing the observation 10 leaves mean \(\hat\mu_{i(i)}=6/3=2\) and predictor \(\hat\eta_{i(i)}=\log(2)\).
For a Poisson log-link model, the intercept information is the sum of the fitted means. The full information is \(16\), so the full-fit standard error of \(\hat\eta_i\) is \(1/\sqrt{16}=0.25\). The deleted dispersion is 1. Therefore \(\operatorname{DFFITS}_{\eta,i}=[\log(4)-\log(2)]/0.25=\log(2)/0.25\approx 2.773\).
For the response scale, the log-link derivative is \(d\mu/d\eta=\mu\), which at the full fit is 4. The standard error is therefore \(4(0.25)=1\). The response change is \(4-2=2\), giving \(\operatorname{DFFITS}_{\mu,i}=2/1=2\). Both diagnostics identify a substantial fitted-value change, but they express it on different scales.
Worked Example: A Binomial Change Near the Middle of the Probability Range
Suppose 10 independent Bernoulli observations are fitted by an intercept-only logistic model, with 4 successes. Examine a success. The full fitted probability is \(\hat p=4/10=0.4\), and the full predictor is \(\hat\eta=\log(0.4/0.6)\approx-0.4055\). After deleting that success, \(\hat p_{i(i)}=3/9=1/3\) and \(\hat\eta_{i(i)}=\log(1/2)\approx-0.6931\).
The binomial information for the full-fit intercept is \(10(0.4)(0.6)=2.4\). Thus the standard error for the linear predictor is \(1/\sqrt{2.4}\approx0.6455\), and \(\operatorname{DFFITS}_{\eta,i}=(-0.4055+0.6931)/0.6455\approx0.446\).
For the logistic inverse link, \(d p/d\eta=p(1-p)\), which at the full fit is \(0.4(0.6)=0.24\). The response-scale standard error is \(0.24/\sqrt{2.4}\approx0.1549\), while the response change is \(0.4-1/3\approx0.0667\). Hence \(\operatorname{DFFITS}_{\mu,i}\approx0.0667/0.1549\approx0.430\). These values are fairly close here; their closeness is not guaranteed for probabilities nearer 0 or 1.
One-Step Approximation and Its Interpretation
Exact case deletion requires refitting the model without each observation. A useful first-order approximation explains how leverage and response discrepancy combine. It is based on a Taylor expansion of the deleted score equations around the full maximum-likelihood estimate. It is not an exact deletion identity for a nonlinear GLM.
\(\displaystyle \hat\eta_i-\hat\eta_{i(i)}\ \approx\ \frac{h_{ii}}{1-h_{ii}}\frac{y_i-\hat\mu_i}{g_i}.\)
Using the full-fit weights and the deleted dispersion estimate in the DFFITS standardization gives
\(\displaystyle \operatorname{DFFITS}_{\eta,i}\ \approx\ \frac{y_i-\hat\mu_i}{\sqrt{\hat\phi_{(i)}V(\hat\mu_i)}}\frac{\sqrt{h_{ii}}}{1-h_{ii}}.\)
The approximation is local: it can be poor when deletion substantially changes the fit, the weights, or the dispersion.
At the full estimate, the score contribution from observation \(i\) is \(u_i=x_i g_i(y_i-\hat\mu_i)/\{\phi V(\hat\mu_i)\}\). The full score is zero, so the score for the remaining observations, evaluated at the full estimate, is \(-u_i\).
Let \(A=X^\mathsf{T}WX\). The full Fisher information is \(A/\phi\); removing case \(i\) gives \(I_{(i)}\approx[A-w_i x_i x_i^\mathsf{T}]/\phi\). A first-order Taylor expansion of the deleted score gives \(\hat\beta_{(i)}-\hat\beta\approx-I_{(i)}^{-1}u_i\).
Set \(q_i=x_i^\mathsf{T}A^{-1}x_i\), so \(h_{ii}=w_iq_i\). The Sherman–Morrison identity gives \(x_i^\mathsf{T}(A-w_i x_i x_i^\mathsf{T})^{-1}x_i=q_i/(1-h_{ii})\). Substituting \(w_i=g_i^2/V(\hat\mu_i)\) into \(x_i^\mathsf{T}(\hat\beta-\hat\beta_{(i)})\) yields the stated predictor-change approximation.
The full-fit predictor variance estimate is \(\hat\phi_{(i)}q_i=\hat\phi_{(i)}h_{ii}/w_i\). Dividing the approximate change by its square root gives the displayed DFFITS approximation.
The same local argument clarifies why a response-scale version can behave differently. For small changes, \(\hat\mu_i-\hat\mu_{i(i)}\approx g_i(\hat\eta_i-\hat\eta_{i(i)})\). When \(g_i>0\), the derivative also multiplies the response-scale standard error, so the first-order standardized changes are approximately equal on the two scales. In a finite deletion, however, the response change can be nonlinear in the predictor change, as the Poisson example illustrates.
Classical DFFITS as a Special Case
For a Gaussian linear model with identity link, known design matrix, and common variance, the deletion relation is exact. Let \(e_i=y_i-\hat y_i\) be the full-fit residual and \(h_{ii}=x_i^\mathsf{T}(X^\mathsf{T}X)^{-1}x_i\). If \(h_{ii}<1\), the Sherman–Morrison identity applied to deleting row \(i\) gives \(\hat y_i-\hat y_{i(i)}=h_{ii}e_i/(1-h_{ii})\).
If \(s_{(i)}^2\) is the residual variance estimate from the deleted fit, the classical externally studentized residual is \(t_i=e_i/[s_{(i)}\sqrt{1-h_{ii}}]\). The DFFITS denominator in our definition is \(s_{(i)}\sqrt{h_{ii}}\). Therefore
This derivation shows the connection with the standard linear-model diagnostic. It also explains why the GLM approximation involves both a residual-like discrepancy and a leverage factor. Unlike the Gaussian case, the GLM relation is generally only first-order because deleting an observation changes the fitted means and hence the weights.
Common Pitfalls and Practical Use
- Treating the scales as interchangeable. A change in \(\eta\) is not a change in \(\mu\). State which scale is reported and use its corresponding standard error.
- Using the inverse-link derivative at an unstated point. The response-scale definition here uses the derivative at the full fitted predictor. Using a deleted-fit derivative is a different convention and should be identified explicitly.
- Calling the approximation an exact GLM identity. The one-step formula uses a local score expansion and fixed full-fit weights. If deletion strongly changes the fit, compute the deleted fit directly.
- Interpreting leverage as influence by itself. The approximation combines leverage with the response discrepancy. High leverage does not guarantee a large DFFITS value, and a large residual alone does not determine the deletion effect.
- Applying one cutoff to every model. Scale, sample size, response family, and the chosen standardization affect interpretation. Treat DFFITS as a screening diagnostic and inspect the case and model, rather than relying on a universal threshold.
Exact computation requires the deleted fit to exist and the relevant information matrix to be nonsingular. In particular, \(h_{ii}<1\) is needed for the one-step expression. A failed or boundary case-deleted fit is not a finite ordinary DFFITS value; it is a signal to report the fitting problem rather than silently substitute the approximation.
Check Your Understanding
Answer each question by identifying the scale or assumption involved.
- In the DFFITS definition given here, what changes between the linear-predictor and response-scale numerators?
- For a log-link model, why can a fixed change in the linear predictor correspond to different absolute changes in the fitted mean at different starting means?
- What two factors appear in the one-step approximation to a GLM predictor change?
- Why is the GLM one-step deletion expression generally approximate even though the Gaussian linear-model deletion identity is exact?
- What condition on \(h_{ii}\) is required for the displayed one-step expression, and what does a failure of that condition imply for its use?