# Using a data-adaptive target parameter and CV-TMLE in survival analysis

This post is part of our Q&A series.

A question from graduate students in our Fall 2020 offering of “Biostatistical Methods: Survival Analysis and Causality” at UC Berkeley:

Question:

Hi Mark,

Within the field of industrial hygiene and occupational epidemiology there is interest in linking possible occupational exposures to deleterious health outcomes, most usually various cancers. Obviously in such a setting, it is nearly impossible without individual chemical biomarkers to have causal identifiability for a specific exposure (for example lead, pesticides, benzene, etc.) on some outcome. This is because it is impossible to control for other simultaneous exposures that may confound the relationship or may interact with other exposures to cause more severe outcomes. As such, rather than creating a job exposure matrix, giving a particular occupation a nominal risk score for a particular exposure (for example, a risk score 0-3 for lead exposure) it may be more interesting to look at the occupation itself. For a given outcome, say Leukemia, it may be of more interest to find the set of occupations that have the highest rates of Leukemia. This would then become a data-adaptive parameter such that occupational specific hazards would need to be calculated for each occupation category within a training data sample. The set of occupations that maximize the hazard difference can be captured as a rule. For example, in a training set it may be determined that high risk occupations are: teachers, librarians, manufacturers, and truckers compared to low risk occupations academic professionals, financial workers, and tech workers. This rule can then be carried over to a validation set where the hazard difference can be calculated for this rule which compares the occupations. Investigation of how occupational exposure may occur can happen post-hoc. For example, given teachers exposure to products such as markers, paper, plastics in binders, etc. over years, this may lead to certain cancers due to bioaccumulation of toxic chemicals in these products. Most occupational exposure focuses on manufacturing/transportation workers but other occupations may be overlooked. This data-adaptive procedure could allow for a more exploratory approach, to “listen to the data” and then investigate what exposures may be occurring in high-risk categories post-hoc. From an application standpoint, this seems to require a). creating job categories that are not too granular, b). calculating the occupation specific hazard for each occupation c). finding the set of occupations such that the sum of the hazards in that set maximize the difference d). creating a rule to indicate that set, e). calculating the risk for this rule in a validation set. The questions we have for this procedure are: is it appropriate in this setting to sum over hazards in order to identify the occupational set that has the largest risk difference compared to low-risk occupations? Given that we would likely use cross-validated survival TMLE (for right censored data with no time-varying covariates), can we derive proper statistical inference for such an approach?

Best, D.M., S.F., and A.M.

Hi D.M., S.F., and A.M.,

Sounds like your data is of type $O=(W, \text{min}(T, C), \Delta = \mathbb{I}(T \leq C))$ with one of covariates W_1 being type of occupation. You are interested in estimating occupation-specific risk $\psi(k) = S(t_0 \mid W_1 = k) = \mathbb{P}(T > t_0 \mid W_1 = k)$ for each occupation $k$. For that we could use TMLE $\psi_n(k)$ for each $k$. As a whole vector, the TMLE $\psi_n$ is asymptotically linear with a vector influence curve, allowing for formal statistical inference based on working model \psi_n \sim \text{N}(\psi_0, \Sigma) with $\Sigma$ covariance matrix of the influence curves. This allows you to construct a simultaneous confidence interval and carry out multiple testing procedures controlling particular family-wise error rates, for example. In this case, all these estimators would be based on $k$-specific disjoint samples, so that the estimators $\psi_n(k)$ are independent across $k$, although one could use pooling when estimating the conditional hazards of failure and censoring.

I am not completely clear why you prefer to not do this. It sounds that you like to group various occupations in groups and do that in a data adaptive manner. So, now you create presumably (based on training sample, given a sample split) disjoint and complementary sets $S_1, \ldots, S_J$ of occupations, and are concerned with inference for $\psi(S_j) = S(t_0 \mid W_1 \in S_j)$. This would be a vector of data dependent target parameters due to $\hat{S} = (\hat{S}_1, \ldots, \hat{S}_J)$ being a data-dependent partitioning. If $\hat{S}$ is a type of selector that optimizes contrast in risk, like $\psi(S_1)$ is group with maximal risk, etc., then inference treating these $J$ parameters as a priori specified will be finite-sample biased. On the other hand, if you use simultaneous inference based on all occupations as above, then all $K$ confidence intervals can be interpreted as true uniformly in all $K$ occupations, so that one is able to look at subgroups, etc., all still resulting in statements with 95% confidence (i.e., across repeated sampling, 95% of times your statements would be correct). In that case, the multiple testing adjustment was already made across all occupations.

The advantage of the data-dependent clustering and using CV-TMLE with multi-fold sample splitting is that it will obtain the power as if these (smaller number) $J$ target parameters where a priori specified, so much smaller multiple testing adjustment.

The price you pay is that the target parameters CV-TMLE targets are an average over sample splits of subset (selected on that training sample) specific risks, where the first subset $\hat{S}_1$ might be different from one training sample to another. So you really only obtain inference for $1/V \sum_v \Psi(P_0)(\hat{S}_1(P_{n,v})$ for the data adaptively selected target subset $\hat{S}_1(P_{n,v})$. If it happens to be very stable, then $\hat{S}_1(P_{n,v})$ might be similar across splits, in which case you will be fine with the interpretation. Either way, it provides powerful inference for that average across sample splits of split specific target parameters: rejection of the null would imply at least one of the sample split-specific target parameters is significant. That is the trade-off relative to doing simultaneous inference across all occupations.

This CV-TMLE approach applies to any data adaptive partitioning $\hat{S}$ into subgroups of occupations, so, yes, based on summing the individual occupation risks or whatever is fine. For example, you could create a dissimilarity matrix $d(i,j)$ of dimension $K$-by-$K$, where $K$ is total number of professions, and $d(i,j) = \psi(i) - \psi(j)$ and apply a clustering algorithm such as PAM (partitioning around medoids) specifying a desired number of clusters. This would then create groups with similar risks. Average silhouette width could be used to decide on number of clusters, as part of this data adaptive partitioning method, $\hat{S}$ aims to make the clusters as different as possible.

Best Wishes,

Mark

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