Repeated Observations And Claim-Count Dependence

Why Repeated Policy-Period Rows Need Attention

A frequency data set may contain one row for each policy-period rather than one row per policy. A policy observed over several years therefore contributes several counts. Treating every row as independent can be inappropriate when the periods share risk characteristics that are not represented in the model. A policy with persistently higher-than-average risk, for example, may tend to have more claims in each of its observed years.

This issue is separate from whether an exposure offset is correctly specified. An offset can scale the expected count for the length of a policy-period; it does not make repeated counts from one policy independent. The relevant questions are what is the observational unit for the dependence structure, what assumptions are being made about counts within that unit, and how the dependence affects uncertainty.

A common model makes counts independent conditional on a policy-specific risk effect, while allowing them to be dependent after that unobserved effect is averaged out. Conditional independence is not the same as marginal independence. This distinction is useful for understanding why repeated rows can contain less independent information than the row count suggests.

A Shared-Risk Count Model

Consider a policy \(i\) observed in periods \(t=1,\ldots,T_i\). Let \(N_{it}\) be its claim count, \(e_{it}\) its exposure, and \(\lambda\) the baseline claim rate. Suppose the policy has a positive shared risk multiplier \(\Theta_i\). The following model assumes \(E[\Theta_i]=1\), \(\operatorname{Var}(\Theta_i)=\alpha\), and conditional independence of its period counts given \(\Theta_i\):

For the gamma-frailty version of this model, take \(\Theta_i\sim\operatorname{Gamma}(1/\alpha,\text{ rate }1/\alpha)\), with \(\alpha>0\). This parameterization gives the stated mean and variance. Different policies are assumed independent of one another, including their risk effects. This is a model for illustrating shared heterogeneity; it is not a necessary description of every form of claim-count dependence.

Conditional independence means that after fixing the policy’s risk effect, the period counts are independent Poisson variables. But a high value of \(\Theta_i\) raises the expected count in every period, and a low value lowers each one. Averaging over the distribution of \(\Theta_i\) therefore generally creates dependence between the period counts.

Deriving the Marginal Variance and Covariance

Write \(\mu_{it}=e_{it}\lambda\). By the law of total expectation, the marginal expected count is \(E[N_{it}]=\mu_{it}\), since \(E[\Theta_i]=1\). The law of total variance separates the Poisson variation conditional on risk from the variation in the conditional mean:

For two distinct periods \(s\ne t\), conditional independence gives a zero conditional covariance. The law of total covariance then leaves the covariance of the conditional means:

Thus, in this shared-multiplier model, distinct periods have positive covariance when \(\alpha>0\) and both have positive expected counts. This result is specific to the model. More generally, a dependence structure can be present without positive covariance, so checking covariance alone cannot establish independence.

Let \(N_i=\sum_t N_{it}\) be the policy’s total count and \(M_i=\sum_t\mu_{it}\) its expected total count. Adding the period variances and all pairwise covariance terms yields:

The equality follows because \((\sum_t\mu_{it})^2=\sum_t\mu_{it}^2+2\sum_{s<t}\mu_{is}\mu_{it}\). The same result follows by conditioning directly on \(\Theta_i\): given the shared multiplier, the total is Poisson with mean \(M_i\Theta_i\).

Implications for Standard Errors

Suppose the target is a common annual rate and there are \(m\) independent policies, each observed for two one-year periods under the shared-risk model. The rate estimator is total claims divided by total exposure, \(\widehat{\lambda}=\sum_i(N_{i1}+N_{i2})/(2m)\). If the model’s mean is correct, its expectation is \(\lambda\). Dependence within a policy affects its variance even though different policies are independent.

For each policy, the total expected count is \(2\lambda\), so its total-count variance is \(2\lambda+\alpha(2\lambda)^2\). Summing over independent policies and dividing by the squared total exposure gives:

The first term is the variance that would arise under independent Poisson counts in all policy-period rows. The second term is the extra contribution from shared risk. This difference matters for uncertainty statements and tests even when the rate estimate itself remains centered on the true rate under the stated mean model.

Dependence Is Not the Same as Covariance

A common exam trap is to define dependent observations as observations with nonzero covariance. That definition is too narrow: zero covariance does not, in general, imply independence. The following discrete example makes the distinction explicit.

Choosing an Inference Strategy

When policy-period rows are clustered, the standard error must reflect the unit across which observations may be dependent. Two common approaches are to specify a joint model for the counts, such as a shared-risk model, or to retain a suitable mean model and use a cluster-robust variance estimator. The choice depends on the intended assumptions and analysis; neither approach makes arbitrary dependence disappear.

In a generalized linear model, a cluster-robust sandwich variance uses policy-level score contributions rather than treating each period’s score as an independent contribution. In schematic form, if \(U_i\) is the total score contribution from policy \(i\), and \(A\) is the negative derivative of the total score with respect to the model parameters, the unadjusted sandwich estimator is:

This estimator allows dependence within a policy cluster, but relies on independence across clusters and generally requires a sufficiently large number of independent clusters for its usual large-sample justification. With few clusters, its finite-sample performance may be poor, and a small-sample correction or a more explicitly specified model may be needed. A sandwich estimator adjusts estimated uncertainty; it does not by itself correct a misspecified mean structure or supply a model for the dependence.

Before fitting or interpreting repeated policy-period data, make the observational and dependence units explicit. If rows are years and clusters are policies, ask whether the working likelihood treats years from one policy as independent, whether that assumption is credible, and whether the standard errors account for the clustering. This is aligned with the CAS Exam 8 (Advanced Ratemaking) syllabus because ratemaking conclusions depend on both the fitted mean and an appropriate assessment of its uncertainty.

Common Mistakes and Practical Checks

• Counting rows as independent policies. Repeated years from one policy may share risk effects. Identify the cluster before interpreting a row-based standard error.
• Assuming an offset resolves dependence. An exposure offset adjusts the expected count for exposure length; it does not change the dependence structure among repeated observations.
• Adding period variances but omitting covariance terms. For a total count, include twice each distinct-period covariance. In the shared-gamma model, this produces \(M_i+\alpha M_i^2\).
• Equating dependence with nonzero covariance. Nonzero covariance implies dependence when defined, but dependent variables can have zero covariance.
• Assuming a robust standard error repairs every model problem. Cluster-robust inference addresses within-cluster dependence under its conditions; it does not automatically repair an incorrect mean model or a small number of clusters.