Why Examine Coefficient-Specific Influence?
A GLM coefficient describes a particular model term: for example, the change in log odds associated with a predictor, or the log rate ratio for a one-unit increase in exposure. An observation may have an unusual effect on one coefficient without having a similarly large effect on every other coefficient. DFBETAs are designed to make that coefficient-by-coefficient effect visible.
The basic idea is to compare the fitted coefficient with the value obtained when one observation is omitted and the model is refitted. The comparison is specific to a coefficient, not a general verdict that an observation is “bad.” It is aligned with the SOA Exam PA (Predictive Analytics) syllabus.
The deletion unit is a row in the fitted data. In grouped binomial data, for instance, one row might represent many trials; deleting that row is not the same operation as deleting one individual trial. Keep the data’s unit of observation clear when interpreting a diagnostic.
What a DFBETAS Value Says
The numerator records how much the estimate changes, while the standard error in the denominator expresses that change in units of estimation uncertainty. A DFBETAS value of \(1.5\) means that the full-data coefficient is 1.5 deleted-fit standard errors above the estimate from the fit without that row. The sign tells the direction; the magnitude indicates the size relative to uncertainty.
DFBETAS are commonly used as screening diagnostics. A familiar rule of thumb flags values for which
This is a guideline, not a hypothesis test or a universal cutoff. Its value depends on the number of rows, and a flag calls for investigation rather than automatic deletion. With many coefficients and observations, some values may exceed a screening threshold by chance. Consider the coefficient’s meaning, the observation’s context, the model specification, and whether the fit remains comparable after deletion.
Worked Example: A Flag for One Logistic Regression Coefficient
A hypothetical insurer fits a logistic regression for whether a submitted service claim is referred for review. The model includes an intercept and a continuous measure of claim complexity. For one row, the full-data complexity coefficient is \(0.48\). After deleting that row and refitting the same model, the coefficient is \(0.18\), with deleted-fit standard error \(0.12\). There are \(100\) rows. Calculate the raw DFBETA and DFBETAS for complexity, apply the screening rule, and interpret the result.
Method. Compare the two estimates in the stated order, full fit minus deleted fit, then divide by the deleted-fit standard error.
Raw change: \(\operatorname{DFBETA}=0.48-0.18=0.30\).
Standardized change: \(\operatorname{DFBETAS}=0.30/0.12=2.50\).
Screening value: \(2/\sqrt{100}=2/10=0.20\). Since \(|2.50|>0.20\), this row is flagged by the rule of thumb for the complexity coefficient.
Interpretation and check. The full-data estimate is \(2.50\) deleted-fit standard errors greater than the estimate without this row. The positive sign agrees with \(0.48>0.18\). The result does not prove that the row is erroneous or that it should be removed; it identifies a coefficient-specific sensitivity to review. Common mistake avoided: treating \(2.50\) as a probability or as a percentage change. It is a standardized coefficient difference.
Why the Weighted Least-Squares Formula Helps Explain Influence
A GLM is not generally an ordinary least-squares model. However, at a converged fit, its iteratively reweighted least-squares (IRLS) calculations use a local weighted least-squares problem. This gives a useful approximation to how deleting a row can change the coefficient vector. It is a local approximation because a full GLM refit can change both the fitted means and the working weights.
For a weighted least-squares problem, write \(X\) for the design matrix, \(W\) for the diagonal matrix of working weights, and \(z\) for the working response. Let \(\hat{\boldsymbol{\beta}}=(X^\mathsf{T}WX)^{-1}X^\mathsf{T}Wz\), assuming \(X^\mathsf{T}WX\) is invertible. For row \(i\), let \(x_i^\mathsf{T}\) be its design row, \(w_i\) its weight, and \(e_i=z_i-x_i^\mathsf{T}\hat{\boldsymbol{\beta}}\) its working residual. Define its weighted leverage as \(h_i=w_i x_i^\mathsf{T}(X^\mathsf{T}WX)^{-1}x_i\).
Write \(A=X^\mathsf{T}WX\), so \(A\hat{\boldsymbol{\beta}}=X^\mathsf{T}Wz\).
The deleted information matrix is \(A-w_ix_ix_i^\mathsf{T}\), and the deleted right-hand side is \(X^\mathsf{T}Wz-w_ix_iz_i\).
Using the rank-one inverse identity for \(A-w_ix_ix_i^\mathsf{T}\), the coefficient difference is \(\hat{\boldsymbol{\beta}}-\hat{\boldsymbol{\beta}}_{(-i)}=A^{-1}x_iw_i(z_i-x_i^\mathsf{T}\hat{\boldsymbol{\beta}})/(1-w_ix_i^\mathsf{T}A^{-1}x_i)\). Substituting \(e_i\) and \(h_i\) gives the displayed formula.
The expression shows why a row can strongly affect a coefficient: its working residual can be large, its leverage can be high, or the direction \(A^{-1}x_i\) can align strongly with that coefficient. The denominator \(1-h_i\) also signals that the local deletion calculation can become unstable as leverage approaches one. In a GLM, use an actual refit for the defined DFBETAS when feasible; the working-fit expression explains the mechanism but does not replace that definition.
Worked Example: Standardizing a Factor-Coefficient Change
A hypothetical Poisson GLM models the number of equipment incidents per site, with a log link and a two-level maintenance indicator. The coefficient compares the “enhanced maintenance” level with the reference level. For one site row, the full-data coefficient is \(-0.35\). When that row is deleted, the estimate is \(-0.05\), and its deleted-fit standard error is \(0.10\). The data set has \(64\) rows. Calculate and interpret the diagnostic.
Raw change: \(\operatorname{DFBETA}=-0.35-(-0.05)=-0.30\).
Standardized change: \(\operatorname{DFBETAS}=-0.30/0.10=-3.00\).
Screening value: \(2/\sqrt{64}=2/8=0.25\). Thus \(|-3.00|>0.25\), so the row is flagged for this factor coefficient.
The negative sign means the full-data estimate is lower than the deleted-data estimate. Because the model uses a log link, the coefficient is on the log-mean scale; its DFBETAS value is not itself a rate ratio. The diagnostic indicates sensitivity of this particular coefficient and does not by itself establish a problem with the count distribution, exposure, or site data. Common mistake avoided: reversing the subtraction and reporting \(+3.00\); the magnitude would match, but the direction would be wrong.
Coefficient Influence Is Not Overall Model Influence
Each DFBETAS value concerns one coefficient. A row can have a large value for a particular slope and small values for the intercept and other slopes. Conversely, a row can produce modest standardized changes in several coefficients without any single coefficient’s DFBETAS appearing exceptional. Therefore, a DFBETAS table answers which coefficient estimates are sensitive to this row? It does not summarize all possible ways the fitted model may be affected.
This distinction matters in models with factors, interactions, or several related predictors. A row may influence a contrast for one factor level or an interaction coefficient while leaving the other terms relatively stable. Interpret the coefficient using its coding and role in the model; do not assume every coefficient corresponds to a standalone effect on the response scale.
| Diagnostic feature | What it describes | What it does not establish |
|---|---|---|
| Large \(|\operatorname{DFBETAS}_{ij}|\) | Row \(i\) changes coefficient \(j\) substantially relative to its deleted-fit standard error | That the row is incorrect, should be removed, or changes every coefficient substantially |
| Small \(|\operatorname{DFBETAS}_{ij}|\) | Little standardized change in coefficient \(j\) under this deletion comparison | That the row has no effect on overall fit or on other coefficients |
| Several notable values in one row | More than one coefficient is sensitive to that row | A single explanation for the changes; investigate coefficients and data context separately |
Worked Example: Reading a DFBETAS Row Without Overgeneralizing
A hypothetical logistic model for whether a policy renewal is completed includes an intercept, a continuous account-age predictor, and a binary contact-channel indicator. For a given row, the reported full and deleted-fit estimates and deleted-fit standard errors are as follows. The data set contains \(80\) rows.
| Coefficient | Full estimate | Deleted estimate | Deleted-fit SE |
|---|---|---|---|
| Intercept | 0.20 | 0.05 | 0.30 |
| Account age | 0.030 | 0.026 | 0.010 |
| Contact channel | -0.60 | -0.20 | 0.25 |
Intercept: The raw change is \(0.20-0.05=0.15\), so DFBETAS is \(0.15/0.30=0.50\).
Account age: The raw change is \(0.030-0.026=0.004\), so DFBETAS is \(0.004/0.010=0.40\).
Contact channel: The raw change is \(-0.60-(-0.20)=-0.40\), so DFBETAS is \(-0.40/0.25=-1.60\).
The screening value is \(2/\sqrt{80}\), approximately \(0.2236\). Each absolute DFBETAS exceeds that value, but the channel coefficient has the largest standardized change. Its negative sign indicates that the full-data channel coefficient is lower than the deleted-fit estimate. This does not imply that the observation has the same level of influence on overall model fit; that is a separate diagnostic question. Common mistake avoided: comparing the raw changes \(0.15\), \(0.004\), and \(-0.40\) directly. The coefficients have different scales, so standardization is essential to this comparison.
Practical Interpretation and Common Traps
A sound interpretation proceeds from the model definition to the coefficient-specific calculation, then to context. Confirm that the full and deleted fits use the same response, predictors, factor reference levels, offsets, weights, and link. If the deletion changes the usable data or the model cannot be refitted comparably, the resulting difference may not represent the intended one-row influence comparison.
- Do not confuse sign and magnitude. The sign gives direction under the stated full-minus-deleted convention; the absolute value is used for the common screening rule.
- Do not compare raw DFBETAs across unlike coefficients as if they were standardized. Raw changes depend on coefficient units; DFBETAS scale the change by a standard error.
- Do not treat a rule of thumb as a deletion rule. Investigate a flagged row, check its data and context, and consider the model’s purpose before taking action.
- Do not equate coefficient-specific influence with overall influence. A diagnostic focused on one coefficient cannot answer every question about the model fit.
- Check the standardization convention. Implementations can differ in whether the denominator uses a deleted-fit or full-fit standard error. Compare like with like, especially when reading software output.
DFBETAS are most useful as a map of sensitivity: they point to the coefficient and observation combination worth examining. They do not supply a causal explanation, validate the model assumptions, or decide whether a record should be removed. The interpretation should remain tied to the response, link, coefficient coding, and observation unit.
Check Your Understanding
For each question, use the full-minus-deleted convention and the deleted-fit standard error defined above.
- A Poisson GLM coefficient is \(0.72\) in the full fit and \(0.60\) after deleting one row. The deleted-fit standard error is \(0.08\). Find the raw DFBETA and DFBETAS, and state the direction of change. (Answer: DFBETA \(=0.12\); DFBETAS \(=1.50\); the full estimate is higher.)
- A logistic coefficient is \(-0.40\) in the full fit and \(-0.10\) in the deleted fit, with deleted-fit standard error \(0.15\). For \(n=100\), does it exceed the common screening threshold? (Answer: DFBETAS \(=-2.00\); threshold \(=0.20\); yes.)
- For \(n=49\), a row has DFBETAS \(0.25\) for one coefficient and \(-0.10\) for another. Which value exceeds \(2/\sqrt{n}\), and what should follow from that flag? (Answer: The threshold is \(2/7\approx0.2857\); neither value exceeds it, so neither is flagged by this rule.)
- A factor coefficient is \(-0.30\) in the full fit and \(-0.50\) after deleting a row. The deleted-fit standard error is \(0.10\). Calculate the standardized change and interpret its sign. (Answer: DFBETAS \(=2.00\); the full estimate is greater, or less negative, than the deleted estimate.)
- A row has DFBETAS values \(0.05\) for the intercept and \(1.80\) for one interaction coefficient. What is the appropriate coefficient-specific conclusion, and what cannot be concluded from these values alone? (Answer: The interaction coefficient is much more sensitive to this row than the intercept; these values alone do not establish the row’s overall influence or whether it should be removed.)