Try
The worked binomial fit has β̂=(0,log 3), information I=diag(5/2,3/2) and inverse information diag(2/5,2/3). At the new predictor x=1, the fitted probability is 3/4. Can twelve trials justify an exact normal confidence statement merely because the optimizer found the right coefficients?
No exact normal law follows from these calculations. We can construct a conditional asymptotic Wald approximation, then state the assumptions and its small-sample limitation. The target row u=(1,1) gives η̂=uᵀβ̂=log 3 and estimated variance uᵀI⁻¹u=2/5+2/3=16/15.
| Quantity | Calculation | Meaning |
|---|
| Fitted link | log 3 | Estimated log odds at x=1 |
| Link standard error | 4/√15 | Model-based asymptotic scale |
| Nominal link interval | log 3 ± 1.96·4/√15 | Approximate 95% parameter interval |
| Probability interval | logistic of the two link endpoints | Transformed interval contained in (0,1) |
Calculate the endpoints, retaining the expression as the reproducible result. Compare this interval target with the outcome of one new attempt. A probability parameter and a future binary outcome are different inferential targets.
Predict
Suppose the same fitted mean is used for twenty repeated attempts by each of ten people. Predict whether treating all 200 attempts as independent could distort uncertainty. Then predict whether a Pearson residual pattern can tell you uniquely that the variance, rather than the mean, is wrong.
A good mean fit cannot establish independence. A pattern might reflect mean misspecification, dependence, a wrong variance relation, an outlier, or chance. Diagnostics identify questions for the investigation; they do not automatically select and validate the answer.
Experiment
Return to the GLM workbench. Complete the worked binomial fit and inspect its information and covariance alongside the existence and numerical status. Use Prediction row: predictor values only to compare x=0 and x=1. The intercept is supplied automatically. Use the displayed information to construct the link interval on paper; the workbench supplies point predictions, not a prediction interval. Label your interval as an asymptotic parameter calculation, not an exact finite-sample guarantee.
For the worked Poisson model, use prediction x=0 with Prediction exposure set to 3. Its fitted rate is 2, so its mean count is 6. At the exact fit, u=(1,0) gives log-rate variance 10/84=5/42. The fixed log exposure adds log 3 to the log mean but no estimated coefficient. A link interval for the mean can be obtained by exponentiating log 6±1.96√(5/42). This is not a prediction interval for the next count: that count also varies according to the conditional response distribution.
Record a frozen equation, target, exposure or trial total, and a loss rule on paper before checking new responses. The workbench is a temporary fitting tool; it does not turn the following public synthetic check into an untouched dataset. Do not claim that a displayed fit or written report has received independent human review.
Understand
Where inverse-information uncertainty comes from
Under a correctly specified independent-response model, finite interior true coefficients, an identifiable design and a suitable growing-information regime, a likelihood asymptotic theorem gives approximate normality of the MLE with covariance equal to inverse information. The theorem also needs regularity and control of individual observations; full rank in one small table is not a substitute. This lesson uses that theorem as a dependency, not a proof that twelve particular trials are enough. Review likelihood inference for the distinction between optimization and sampling uncertainty.
A first-order explanation starts from the score equation 0=U(β̂). Taylor expansion around the true β gives approximately I(β)(β̂−β)=U(β). The score is a sum of independent centered contributions with covariance I(β). A suitable central limit theorem and control of the Taylor remainder lead to the normal approximation. Replacing unknown information by information at β̂ requires consistency. These are substantial hypotheses; the algebra alone does not prove them.
For a fixed nonzero target row u, the estimated link variance is uᵀI(β̂)⁻¹u. Form a link interval and apply the monotone inverse link to its endpoints. This preserves probability or positive-mean ranges. A direct symmetric probability interval based on a first-order delta calculation may leave [0,1], which is one reason to be explicit about the construction. Neither transformation repairs separation, dependence or a wrong mean equation.
Residuals and dispersion
A Pearson residual divides an observed-minus-fitted count by its model standard deviation: (s−np)/√(np(1−p)) for binomial, or (y−μ)/√μ for Poisson. Squared Pearson residuals summarize discrepancy, but fitted residuals are not independent standard-normal observations. Their calibration reflects fitting, design, response discreteness and the adequacy of approximation. A nominal residual dimension of rows minus parameters does not itself establish a chi-square reference when cells are sparse.
For instance, four Poisson counts (0,0,4,4) fitted by an intercept have mean 2 and Pearson sum 8. The model residual dimension is 3, but that contrast is a diagnostic, not proof of a particular overdispersion mechanism. The variance may exceed the Poisson mean, observations may be dependent, or the mean equation may omit a relevant predictor. A dispersion correction or different family needs its own justification. For binomial grouping, the number of rows and the number of trials play different roles; do not silently replace one with the other.
Frozen checks answer a different question
A frozen model assigns probabilities (1/4,3/4) to two new groups of four attempts. Suppose the new counts are (2,2). Omitting constants common to both predictions, its negative log likelihood is 4 log(16/3). Revising both probabilities to 1/2 gives loss 8 log 2, lower by 4 log(4/3). If those outcomes caused the revision, the smaller loss is a development result on exposed data. Preserve the original result and collect another relevant evaluation record before making an independent-validation claim.
Exposure, group totals and target population belong in that record. Model calibration and predictive loss do not establish causal effects. An ordinary finite MLE, an exact small-table calculation and a small displayed standard error can all coexist with a misspecified collection mechanism.
Apply
A frozen Poisson rate is 2 events per hour. At a separately stated exposure of 3 hours, the observed count is 9. Compute the predicted count, Pearson contribution and the improvement in negative log likelihood if you revise the rate to 3 using this response. What must the next evaluation disclose?
Check the arithmetic and the exposure record
The original mean is 6, and the Pearson contribution is (9−6)²/6=3/2. The frozen loss, up to a shared factorial term, is 6−9 log 6. Revising the rate to 3 makes the mean 9 and loss 9−9 log 9, improving it by −3+9 log(3/2)>0. This uses the observed response to change the model. Record both rates, the same three-hour exposure and both losses, identify the revision as development, and use new appropriately collected outcomes for another check. One discrepancy cannot identify a population dispersion law.
Continue
Use the four mixed written tasks to connect family choice, vector derivatives, existence and conditional uncertainty. Return to finite-existence certificates when an optimizer status is ambiguous, or frozen regression validation to compare the same evidence discipline for another response model. A sustained empirical investigation still needs data provenance, a justified design, retained revisions and independently reviewed transfer evidence.
