## Confounders in Time-Series Regression

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The book covers, among other topics, linear, logistic, and Poisson regression, generalized linear models, and hypothesis testing and shows examples where these techniques are applied using Stata. This text is suitable for a graduate level course in epidemiology or biostatistics; each chapter contains an applied exercise where the reader can implement the tools just covered on a practical problem and bibliographical references for those interested in exploring the topics more in depth.

The last chapter implements the solutions to all the exercises in the book using Stata, as well as using other packages. Stata: Data Analysis and Statistical Software. Go Stata. Purchase Products Training Support Company. We now repeat an identical exact search but this time without the previous restriction on the location of arcs. This allows us to determine the best multivariate regression model of the data, that is, we consider all variables simultaneously.

In a graphical model the standard way to interpret the results relative to a single variable is to compute its Markov blanket [ 29 ]. To predict values for any variable in a DAG, then all we need to know are the variables in its Markov blanket, and all other variables in the graph can be discarded. Conversely, each variable in the Markov blanket is needed because each provides knowledge about the variable of interest.

This then suggests that b5 should be included along with these other variables for further investigation into their potential epidemiological significance with response g5. Globally optimal multivariable regression model with g2 as the response variable and globally optimal multivariate regression model of all 17 variables. Markov blanket for variable g2 are those variables in grey. Globally optimal multivariable regression model with b3 as the response variable and globally optimal multivariate regression model of all 17 variables.

This is a generalised linear model with logit link function. Markov blanket for variable b3 are those variables in grey. As the multivariate model permits arcs both to and from the response variable this is perhaps no surprise, although there is no reason that this need always be the case. What may be rather more surprising is that arcs identified in the multivariable model may not be identified in the multivariate model. The multivariable model suggests b4 is worthy of further investigation. In contrast, the full multivariate model suggests that in fact b4 is only indirectly related with g2, and this indirect dependence is also remote in the graph, i.

In other words there is little statistical evidence to support epidemiological investigation of b4. This result cannot be dismissed by arguing that the multivariable model is somehow more parsimonious, because the same model selection metric is used in all model comparisons. A guide to the relative size and interpretation of differences in log marginal likelihoods can be found in Table 2. In summary, therefore, the data supports that the multivariate models are simply a better fit to the data in our examples. The key difference between the results here is that the multivariable model implies that there are three variables worthy of further investigation.

In contrast, the multivariate model has ten variables in the Markov blanket for b3, six of which are directly connected with b3. To complete our case study analyses, and further emphasize that our proposed multivariate regression approach is simply a generalization of usual multivariable regression, it is readily possible to compute odds ratios and mean effects of the parameters arcs in our graphical model. This is a log odds ratio as we have a logistic regression between b5 and g5 in this part of the graph. The latter two intervals are for the mean effect rather than log odds as these are Gaussian regressions.

## Regression Analysis of Aggregate Continuous Data : Epidemiology

A key point of note here is that nodes with the same parents have identical parameter estimates in each model e. The multivariate model is simply a collection of multivariable models and so the parameter estimates will be the same given the same parents. The difference is that the former is more flexible and allows any node to have parents, unlike in a GLM type model.

Generally speaking —and as we have seen in our case study examples —this means that the parents, and therefore parameter estimates, may be different for at least some variables nodes in the data e. When the analytical task is to identify statistical dependencies with one or more response variables, then both theoretically and as demonstrated in the above empirical examples, the more general additive Bayesian network structure discovery approach appears clearly preferable. In particular, the multivariable approach is just a special case of the multivariate approach, i.

Hence, there is nothing to lose, at least in statistical terms, by adopting the more general approach.

Moreover, the far simpler multivariable approach may identify covariates which are not supported by the multivariate model, e. A possible explanation for such contradictions is the Yule-Simpson paradox, in that we are trying to describe observations from a complex disease system of inter-dependent variables through a multivariable model, which may just be too simplistic for this particular application. By using a multivariate regression approach the trade-off being made with classical multivariable regression is that the former may provide potentially more information about the disease system under study, in terms of identifying statistical dependencies.

This may lead to new and novel findings. A brief contrast may be made here with historical approaches such as path analyses [ 31 ], which were applied reasonably commonly during the s to address a range of chronic and environmental diseases [ 32 — 34 ], and this approach still appears occasionally in the epidemiological literature. The key distinction between path analyses and additive Bayesian network structure discovery is that the former is explicitly causal, where some or all, of the graph structure is determined apriori via expert opinion. The latter asserts only the presence of statistical dependency, and while it can include prior expert opinion into the structural search it is a Bayesian approach after all the default usage is to allow the data itself to identify an optimal graph structure.

The advantage of allowing the study data itself to identify an optimal graph is that this may include arcs which an expert may not, and may not include arcs which an expert would assert must be present in the given disease system. The epidemiological challenge is then to explain such discrepancies which may result in gaining new insight into the disease system. Reliable, easy to use software is essential for facilitating the uptake of any new data analytic technique into the epidemiological community.

In order to apply additive Bayesian network structure discovery to study data appropriate software is required. In practice, however, other approaches are necessary because the central task in Bayesian network analyses is structure learning which involves fitting and comparing a great many different multivariate models.

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In programs such as OpenBUGS and JAGS it is simply computationally impractical to fit every model via Markov chain Monte Carlo simulation, in addition to the difficulty in reliably estimating the marginal likelihood for each model. Instead, programs which employ analytical approximations, i. Laplace approximations [ 37 , 38 ], are preferable and indeed arguably necessary. The abn library also includes wrappers to allow INLA to be used for all numerical estimation.

While these software libraries are all for use with R it is also possible for R to be accessed from within other popular statistical software such as SAS via IML Studio. The main limitation when applying additive Bayesian network structure discovery to epidemiological data is computational feasibility. The number of variables which can be included in a Bayesian networks analysis is limited.

As a guide, this might be less than about 25 variables for exact structural search techniques and perhaps up to 40 for heuristic approaches e. Inclusion of more variables is possible with access to specialist computing facilities and expertise. This means that including additional variables, such as interaction terms, which can be done easily enough just as in standard regression modelling, requires careful consideration. Each term adds to the number of variables in the model, and therefore adds considerably to the computational time required to perform structural searches.

There are a number of ad-hoc ways to address the computational demands. For example by splitting variables into smaller thematic groups for analyses. This may then suggest that some variables can be dropped, reducing the computationally burden to a more manageable level. For larger problems more variables , model averaging using order-based Markov chain Monte Carlo is an option [ 19 ] as it can cope with many more variables e. Such averaging approaches randomly sample from the posterior landscape of possible graphs strictly speaking, node orderings , with better fitting graphs being sampled more often than poorer fitting graphs, and during this sampling i.

This is an approach used in bioinformatics for sequence analyses e. Access to scientific computing facilities, while not essential, is highly beneficial. An additional severe computational drain is addressing over-fitting, which is an ever present problem in model selection [ 41 ], irrespective of whether using exact or heuristic searches.

Good practice in structure discovery is to either utilize some form of model averaging, for example using majority consensus graphs as the optimal model [ 9 , 42 ], or else using parametric bootstrapping approaches [ 43 ] applied to the globally best graph [ 11 ]. The majority consensus approach is similar to that used in phylogenetics with tree structures, except here a majority consensus graph is created from all arcs which appeared in at least a majority of heuristic search results. This provides an alternative way to estimate relative support for individual arcs other than by Markov chain Monte Carlo, which can be highly problematic when dealing with graph structures see [ 19 ].

A single exact search for a model comprising of 20 variables may take 24 hours to complete on a modern desktop, and this may need to be repeated many hundreds times during model averaging or bootstrapping to ensure robust results. Missing observations are a common feature of field studies and epidemiological data.

### Relative Importance of the Independent Variables

In standard regression modelling, observations with missing values are usually dropped from analyses as it is essential to maintain identical observations when comparing different models. In graphical regression modelling this is also the easiest course of action. There are, however, a number of established algorithms for fitting graphical models in the presence of missing values due to the joint probabilistic nature of these models. These are elegant conceptual solutions, although they do still assume that values are missing at random, but such approaches are numerically highly complex.

It is unclear whether these would be feasible in the context of structure discovery, as when there are missing values in the data the graph can no longer be split into conditionally independent computational units i. This is a very considerable complication, both in terms of implementation and computing time. Approaches have been developed for structure discovery in the presence of missing values, such as Structural-EM [ 47 ], although implemented in models with a simpler parameterisation than those presented here.

The implementation of such approaches is an area of future work, but further highlights the considerable existing theory and potential of graphical regression and structure discovery approaches in analyses of epidemiological data. The wide utilization of regression modelling in epidemiological analyses means that outputs from such analyses have a ready application in disease control and prevention programs.

Up until recently, such applications have been constrained by the use of multivariable regression. Extending multivariable regression to full multivariate regression —utilizing additive Bayesian network structure discovery —offers the epidemiologist potentially far greater insight into the complex inter-relationships between variables within a disease system. The main constraint in the use of this methodology is its considerable computational demands, but given the ever increasing availability of cheap computing power this technique is increasingly feasible for use in a wide range of studies.

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