Case Deletion and Influential Observations

Why Compare a GLM With and Without a Case?

A large residual or high leverage can flag an observation for investigation, but neither feature alone establishes how much that observation affects the fitted model. Case deletion makes the question concrete: fit the specified generalized linear model (GLM) to all observations, refit it after omitting one flagged case, and compare the estimates and predictions. The resulting differences are a sensitivity diagnostic, not a rule for deciding that the case is invalid.

The comparison is most informative when the two fits differ only in whether the flagged case is included. Keep the response definition, predictor set, link, factor coding, offsets, weights, and fitting choices consistent. Otherwise, the change cannot be attributed clearly to that case. A case-deleted fit can also be useful for assessing prediction sensitivity: compare predictions for the observations retained in the refit, and, where relevant, calculate a prediction for the omitted case using its recorded predictors.

Definition: For a GLM fitted to \(n\) cases, the case-deleted fit for case \(i\) is the refit of the same model using the other \(n-1\) cases. Write its coefficient estimate as \(\hat{\beta}_{(-i)}\). If the original estimate is \(\hat{\beta}\), the coefficient change is \(\hat{\beta}_{(-i)}-\hat{\beta}\). For an observation \(j\), compare its original fitted linear predictor \(\hat{\eta}_j\) and fitted mean \(\hat{\mu}_j\) with the corresponding quantities from the case-deleted fit, \(\hat{\eta}_{j(-i)}\) and \(\hat{\mu}_{j(-i)}\).

For a GLM with linear predictor \(\eta_j=x_j^\mathsf{T}\beta\) and link function \(g\), the fitted mean is \(g^{-1}(\eta_j)\). If the model includes a fixed offset \(o_j\), the predictor is \(x_j^\mathsf{T}\beta+o_j\). With the same predictors and offset in both fits, the change in the linear predictor for case \(j\) is

\hat{\eta}_{j(-i)}-\hat{\eta}_j = x_j^\mathsf{T}\bigl(\hat{\beta}_{(-i)}-\hat{\beta}\bigr).

The corresponding change in fitted mean is \(g^{-1}(\hat{\eta}_{j(-i)})-g^{-1}(\hat{\eta}_j)\). It need not be a constant multiple of the predictor change: the inverse link can translate the same link-scale shift into different response-scale shifts at different starting values. This is why it is useful to report both coefficient changes and changes in fitted values relevant to the application.

What the Comparison Does—and Does Not—Show

A substantial change in one or more coefficients indicates that the fitted relationship depends noticeably on the case. A noticeable change in fitted means indicates that predictions for some observations depend on it. Those are related but distinct findings. For example, a coefficient might change considerably while predictions for the cases of practical interest change very little; conversely, a modest coefficient change can matter for observations whose predictors make them especially sensitive to it.

The size of a raw coefficient change depends on the predictor’s units and coding. Interpret it in context: a change in a log-odds coefficient is not a change in probability of the same numerical size, and a change in a log-mean coefficient is not a change in expected count of that size. Earlier material on DFBETAs provides a standardized way to assess coefficient changes. Case-deleted refits complement such measures by making the resulting changes in fitted values and response-scale interpretation explicit.

There is no universal threshold that makes an observation “influential” for every model or purpose. Judge the differences against the scale of the coefficients, the application’s prediction needs, and the uncertainty in the model. Also check that both fits produce usable estimates. A comparison is not meaningful if the refit does not converge or if it estimates a fundamentally different model.

Worked GLM Case-Deletion Comparisons

Worked Example: A High Count in an Intercept-Only Poisson Model

A hypothetical portfolio has four equal-exposure cells with claim counts \(1,2,3,\) and \(10\). Assume the counts are independent and each follows a Poisson distribution with the same mean \(\mu\), so \(Y_j\sim\operatorname{Poisson}(\mu)\). Fit an intercept-only Poisson GLM with a log link. Compare its coefficient and fitted means with those from the refit that omits the cell with count \(10\).

Method. For an intercept-only model, \(\eta=\beta_0=\log(\mu)\), so the fitted mean is \(\hat{\mu}=\exp(\hat{\beta}_0)\). First calculate the maximum-likelihood estimate from all four counts:

\hat{\mu} = \frac{1+2+3+10}{4} = \frac{16}{4} = 4, \qquad \hat{\beta}_0=\log(4)\approx 1.3863.

The original fit therefore assigns fitted mean \(4\) to each cell. After deleting the count \(10\), the remaining total is \(1+2+3=6\), and three cells remain:

\hat{\mu}_{(-i)} = \frac{6}{3} = 2, \qquad \hat{\beta}_{0(-i)}=\log(2)\approx 0.6931.

The intercept change is \(\log(2)-\log(4)=\log(1/2)\approx-0.6931\). Each retained cell’s fitted mean changes from \(4\) to \(2\), a decrease of \(2\). The result shows that the count of \(10\) has a substantial effect on this small sample’s estimated common mean. It does not establish that the count is erroneous; the count and the model assumptions should be investigated before making any data-handling decision.

Check. The refitted mean multiplied by the three retained cells is \(2(3)=6\), their observed total. The original mean multiplied by all four cells is \(4(4)=16\), the original observed total.

Worked Example: Removing an Event From an Intercept-Only Logistic Model

In a hypothetical group of ten independent policy reviews, three outcomes are “event” and seven are “no event.” Assume each outcome is Bernoulli with the same event probability \(p\). Fit an intercept-only logistic regression and compare it with the refit after omitting one of the event cases.

Method. In this model, \(\beta_0=\log\{p/(1-p)\}\). With all ten reviews, \(\hat{p}=3/10=0.3000\), so

\hat{\beta}_0 = \log\left(\frac{3/10}{7/10}\right) = \log\left(\frac{3}{7}\right) \approx -0.8473.

After removing one event, two events and seven non-events remain, for nine reviews. The refitted event probability and intercept are

\hat{p}_{(-i)}=\frac{2}{9}\approx 0.2222, \qquad \hat{\beta}_{0(-i)} = \log\left(\frac{2/9}{7/9}\right) = \log\left(\frac{2}{7}\right) \approx -1.2528.

Thus, the intercept changes by \(\log(2/7)-\log(3/7)=\log(2/3)\approx-0.4055\). Every case in an intercept-only model has the same fitted probability: it changes from \(0.3000\) to \(0.2222\), a decrease of \(0.0778\), or about \(7.78\) percentage points. The coefficient change is on the log-odds scale; it is not a decrease of \(0.4055\) in probability.

Check. The refitted probability is \(2/9\), matching the two events among the nine remaining reviews. The log-odds change is negative because omitting an event lowers the event proportion.

Worked Example: A Case Changes a Group Effect in Logistic Regression

A hypothetical binary-outcome study has two groups, A and B, with four independent Bernoulli observations in each group. Group A has one event and three non-events; group B has three events and one non-event. Fit a logistic regression with an intercept and a group indicator \(x\), where \(x=0\) for A and \(x=1\) for B. Compare the estimates before and after deleting one event from group B.

Method. This model assigns a separate event probability to each group. In group A, \(\hat{p}_A=1/4\), so the intercept is the group A log-odds:

\hat{\beta}_0 = \log\left(\frac{1/4}{3/4}\right) = \log(1/3) \approx -1.0986.

In group B, \(\hat{p}_B=3/4\), with log-odds \(\log(3)\). The group coefficient is the difference between the group B and group A log-odds:

\hat{\beta}_1 = \log(3)-\log(1/3) = \log(9) \approx 2.1972.

Now remove one event from group B. Group A is unchanged, while group B has two events and one non-event, so \(\hat{p}_{B(-i)}=2/3\) and its log-odds are \(\log(2)\). The refitted group coefficient is therefore

\hat{\beta}_{1(-i)} = \log(2)-\log(1/3) = \log(6) \approx 1.7918.

The group coefficient decreases by \(\log(6)-\log(9)=\log(2/3)\approx-0.4055\). Group A’s fitted probability remains \(1/4=0.2500\); group B’s falls from \(3/4=0.7500\) to \(2/3\approx0.6667\). This example shows why it is useful to compare fitted values by group as well as the coefficient: the coefficient expresses a difference in log-odds, while the fitted probabilities describe the modeled outcomes directly.

Check. The refitted group B probability is \(2/3\), exactly the observed event fraction among its three retained cases. The group A fraction remains \(1/4\).

Why the Intercept-Only Poisson Estimate Is the Mean

The first example used the fact that the maximum-likelihood estimate of the common Poisson mean equals the sample mean. This follows directly from the model likelihood. Suppose \(Y_1,\ldots,Y_n\) are independent with \(Y_j\sim\operatorname{Poisson}(\mu)\), where \(\mu>0\). Ignoring terms that do not depend on \(\mu\), the log-likelihood is

\ell(\mu) = \left(\sum_{j=1}^{n}Y_j\right)\log(\mu)-n\mu+\text{constant}.

Differentiate with respect to \(\mu\), set the derivative equal to zero, and solve:

\frac{d\ell}{d\mu} = \frac{\sum_{j=1}^{n}Y_j}{\mu}-n = 0 \quad\Longrightarrow\quad \hat{\mu} = \frac{\sum_{j=1}^{n}Y_j}{n}.

The second derivative is \(-\sum_jY_j/\mu^2\), which is negative when the total count is positive, confirming a maximum at this estimate. If the total count is zero, the likelihood increases toward its boundary maximum at \(\mu=0\); a finite log-intercept estimate does not exist. Thus, the formula and finite log-intercept used in the example apply when the observed total is positive. Deleting a case changes both the total and the number of observations in this expression, which is why the mean—and therefore the log-intercept—can change.

Common Mistakes and a Careful Workflow

A case-deletion result is a diagnostic comparison, not an automatic data-cleaning instruction. An unusual but valid observation may contain important information about the population. Conversely, a data-entry or coding error should be corrected on its merits, not merely because removing it makes a coefficient more convenient.

Do not equate a large residual with influence. A residual describes disagreement between an observation and a fitted value. Case deletion directly checks how much the fitted model changes when that observation is omitted.
Do not compare unlike fits. Changing predictors, coding, offsets, or the response definition at the same time as deleting the case confounds the comparison.
Do not read link-scale changes as response-scale changes. A change in a logistic coefficient is a change in log-odds, not the same-sized change in probability.
Do not assume that the omitted case’s refitted prediction is a fitted value from the refit. That case was not used to estimate the refitted model; its predicted response is an out-of-fit prediction.
Do not delete a valid case just to stabilize a preferred conclusion. Report the sensitivity, investigate the case, and explain any justified correction or exclusion.

A disciplined comparison can be organized in four steps:

Fit and record the original model.
Keep the model form and estimation setup clear, and record the coefficients and fitted values relevant to the question.

Refit without one flagged case.
Change only the inclusion of that case. Check that the refit returns usable estimates.

Compare on appropriate scales.
Calculate coefficient changes and compare fitted linear predictors or response-scale means for observations of interest.

Interpret, investigate, and report.
Describe what changes and why it matters. Investigate data quality and model adequacy without treating influence alone as grounds for deletion.

For an actuarial analysis, the practical question is often whether a case materially changes a rate estimate, a risk comparison, or predictions used in a decision. State which quantity changes, by how much, and for which observations or groups. That makes the sensitivity assessment interpretable to a technical or non-technical audience. This approach is aligned with the SOA Exam PA (Predictive Analytics) syllabus.

Check Your Understanding

For each problem, compare the full fit with the specified case-deleted fit. Unless stated otherwise, assume independent observations and the GLM model described.

An intercept-only Poisson GLM with a log link is fitted to counts \(2,2,4,\) and \(12\), each with equal exposure. Find the full and case-deleted mean estimates and the intercept change when the count \(12\) is omitted. (Answer: full mean \(5\), deleted mean \(8/3\), intercept change \(\log(8/15)\approx-0.6286\).)
Twelve independent Bernoulli outcomes include four events and eight non-events. In an intercept-only logistic model, omit one non-event. Find the fitted probabilities before and after deletion and the intercept change. (Answer: probabilities \(1/3\) and \(4/11\); intercept change \(\log(8/7)\approx0.1335\).)
A logistic model uses an intercept and an indicator for group B. Group A has two events and three non-events; group B has four events and one non-event. Delete one event from group B. Find the group coefficient before and after deletion and its change. (Answer: coefficient \(\log(6)\approx1.7918\) before, \(\log(9/2)\approx1.5041\) after, change \(\log(3/4)\approx-0.2877\).)
An intercept-only Poisson log-link model uses exposure as an offset. The observed count and exposure pairs are \((2,1),(6,3),(9,3)\), with the offset equal to the log of exposure. Find the rate estimate and log-rate coefficient before and after omitting \((9,3)\). (Answer: rates \(17/7\) and \(2\); log-rate coefficient change \(\log(14/17)\approx-0.1942\).)
A flagged observation has a large residual, but deleting it barely changes the coefficients or the fitted values relevant to the analysis. What does this comparison establish, and what does it not establish? (Answer: it indicates that the fitted results examined are not very sensitive to that case under the specified refit; it does not establish that the case is erroneous or should be deleted.)

What You'll Learn