3.1.1 Non-participation in case-control studies

Since non-participation is a recognised problem in case-control studies as summarised by [2], analysis methods which can incorporate information relating to non-participation should be adopted. Non-participation in a case-control study may be just one variable from a missed question or an unavailable piece of data, it may be several variables if an individual is too ill to participate in an interview but their medical records are available, it could be the majority of variables when only an individual’s disease status and basic demographics are available, or it may extend to all information except their disease status. Usually when individuals decline to participate in case-control studies they are excluded from the main analysis, and when variables are missing from an individual, those individuals are omitted from any analyses requiring such variables, through complete case analysis, which can lead to biased results.

Conclusions regarding missing data have been successfully reported using CEGs as developed in [11] and this same approach can be used to tackle non-participation in case-control studies, since non-participation is one explanation for why data may be missing. However much data are missing, and by whichever means the data are missing, CEGs can be used by including an additional edge in the event tree for each variable which has missing data. Although, the more variables which have missing data, the more edges there will be in the event tree, thus extending the tree and possibly, although not always, complicating the CEG.

In a CEG, conclusions can be drawn using both the available and missing variables, including statements about the missingness mechanisms [11]. Missingness is often described using the three standard definitions given by Rubin; missing completely at random (MCAR), missing at random (MAR) and missing not at random (MNAR) [12]. MAR means that any systematic difference between the missing and observed data can be explained by the observed data [13]. Knowledge about the missingness mechanism may be required to know the suitability of methods designed to adjust for non-participation. For example, often imputation requires the data to be MAR as demonstrated by [13] while other population-based methods can be used when data are MNAR such as that developed by [14]. If information regarding the missingness mechanism can be obtained using CEGs, this can assist adjustment choice.

Figure 1

Non-participation staged tree. s denotes a situation and l denotes a leaf.

Hypothetical example

Let there be 100 participants in a case-control study. Some are full-participants (62%) providing their disease status, gender and smoking habits. Some are partial-participants (8%) providing their disease status and gender, but declining to provide their more personal smoking habits. Finally there are non-participants (30%) whose disease status is known from general practitioner records or a disease registry. Figure 1 shows the staged event tree for these data and Table 1 shows the data used. Uniform priors are assumed, meaning that it is not known whether a path containing male, female or unknown is more likely to be taken, and this also applies to smoker/non-smoker/unknown and case/control. For all examples here we use the Bayesian agglomerative hierarchical clustering (AHC) algorithm from [15], to group vertices from the tree into stages as defined for CEGs [7]. Very simply, the Bayesian AHC algorithm is a clustering algorithm which starts with all vertices in the tree from a given variable separate and merges (or agglomerates) them, and is hierarchical since clusters have sub-clusters, which in turn have sub clusters and so on. The Bayesian element allows prior knowledge to be incorporated into the algorithm. More formally, the AHC algorithm is a local greedy search algorithm for finding the maximum a posteriori CEG. The algorithm starts with the finest partition of the vertices in the tree and seeks to combine vertices at each iteration which will result in the highest scoring CEG [15]. The initial CEG formed is identical to the event tree except the leaves are collapsed into one terminal vertex, and scored using the posterior probability of the CEG given the data. The algorithm continues by testing each pair of situations in the initial CEG from the same variable which have the same number of edges, to find which pair of situations in the same stage maximises the ratio of the scores from this CEG and the initial CEG. This is repeated until the coarsest partition has been achieved. The CEG with the highest score is then selected and the algorithm returns a list of vertices in the same stage.

Vertices in Figure 1 in the same stage are shown by colouring their corresponding edges in the same colour. For example situations s1 and s2 are in the same stage and hence their edges are assigned the same colours. This means it is plausible that the distribution of smoking may have come from the same population for males and for females, even though the numbers assigned to the smoking edges for males and females differs. Vertices are then said to be in the same position if each corresponding vertex along the remainder of the two paths is in the same stage, and vertices in the same position are collapsed over to form the CEG. Situations s1 and s2 are also in the same position as their subtrees are assigned the same colours, hence s1 and s2 are in the same position and therefore collapsed over to form the CEG shown in Figure 2. Full details for stages and positions are available in [7]. The CEGs in this paper are ordinal CEGs meaning that vertices from the same variable are ordered vertically by percentage of those possessing the outcome of interest. Each vertex has been labelled with the percentage of individuals who have the outcome of interest, from all those who have taken the same path. For example, the root vertex w0 shows the entire sample and from Table 1 there are 15 cases, hence 15% of the individuals at w0 are cases. At w2, there are 30 individuals who have unknown gender, 4 of which are cases (4/30≈13%). The calculation for w1 is a little more complicated since there are two paths, male and female, which lead to this vertex; 40 females and 30 males, of which 5 and 6 respectively are cases. Therefore the percentage of cases at w1 is 11/70≈16%. All percentages are calculated in the same way, which highest percentages placed above lower percentages in the CEG. Note that w0 and w∞ will always be the total number of individuals possessing the outcome of interest, and hence always be the same value.

Figure 2

Chain event graph for non-participation. Percentage of cases shown at each position. Colouring is not required since all stages and positions (W) are equal. There is only one edge from w2 as unknown gender results from non-participation, which also results in an unknown smoking category.

The CEG in Figure 2 suggests that there are similarities between males and females with respect to the disease of interest, since both genders lead to position w1. The missing gender category resulting from non-participation may be due to data MNAR since it leads to position w2 where individuals have a lower probability of disease (13%) than males and females (16%). Smokers generally have an increased probability of the disease (w3=36%) than non-smokers (w4=12%), and the unknown smoking status resulting from partial-participants or non-participants leads to both position w3 and position w4. Given known gender, the unknown smoking values may be mainly smokers, since both the smoker and unknown categories lead to w3, hence suggesting the values may be MNAR.

Note that all 100 participants have been used to draw these conclusions about the associations between gender, smoking and the disease of interest, as well as conclusions about the missingness mechanisms. No individuals have been lost through complete case analysis or similar approaches. In this instance, since the data are likely to be MNAR, a population-based method could be appropriate to investigate non-participation such as that by [14], hence not introducing bias by using standard multiple imputation and assuming the data are MAR [13].

3.1.2 Associations by disease severity

Case-control studies usually record a binary outcome of case or control. However, diseases often have different severities such as terminal or not, and this level of detail is likely to be clinically useful and hence recorded in the medical notes. The tree and corresponding CEGs can therefore have additional edges denoting the possible severities of the disease such as control, mild case or severe case.

The CEG can be formed in the usual manner as developed by [7], and as before conclusions can be drawn regarding the combinations of variables associated with the range of severity outcomes. It is possible that only individuals with a particular characteristic or combination of characteristics are able to possess the most severe category of the disease and this information may otherwise be hidden in a standard case-control analysis. The case categories can of course be collapsed and the data analysed with a binary outcome for comparison with any previous analyses.

It may also be that the mechanism differs by disease severity. For instance, there may be more missingness amongst cases with the more severe version of the disease and the missingness may be in variables requiring input from the patient. These differences may be useful to highlight where missingness is occurring and in subsequent studies, different data collection strategies may be adopted, such as collection for all participants only through medical records. An example is given in Appendix A.1.

3.2.1 Recruitment by data collection method

The effectiveness of different recruitment techniques or data collection strategies (such as web surveys, postal surveys, and electronic reminders) may be of interest. Rather than use a CEG which shows personal characteristics along each path, a CEG can be developed which contains solely information about data collection approaches. With the binary disease status forming the final vertices in the event tree, the CEG can be used to determine which approaches are more associated with cases and which are more associated with controls. Although this approach could highlight successful methods for recruiting case and controls respectively, adopting these approaches would assume a causal effect rather than an association, and could result in bias by recruiting the disease groups in different ways. Therefore, this approach should be used to identify bias which can then be adjusted for, rather than to suggest recruitment techniques. An example is given in Appendix B.1.

3.2.2 Participation as the outcome of interest

Typically in CEGs the final vertices of the event tree represent the outcome of interest. For case-control studies this is the disease status of the individuals and thus presents two options; case or control. If the outcome of interest is instead non-participation, the event tree can be restructured such that the final vertices represent participation (yes/no). Data collection techniques or individual characteristics can then form the paths in the tree. The CEG highlights which combinations of techniques or characteristics result in comparable probabilities of participants, and an ordinal CEG as introduced in [9] can be used to order the combinations of categories from those associated with the lowest probability of participation to the highest. This approach provides a summary of the participation rates as well as providing information on which factors, or their proxies, are associated with participation, and which should be included as good practice in the reporting of a study requiring participants. These findings can be used to form part of an investigation into possible selection bias in the study. Selection bias occurs in case-control studies when selection is conditioned on and affected by both the exposure and disease of interest [16]. Therefore if participation is affected by both the disease status of the individual and the exposure of interest in the study, then selection bias may be present.

This use of CEGs could be extended to more than two participation categories. For example the final vertices could have edges representing “no participation”, “partial participation” and “full participation”, where partial participation relates to those willing to give demographic data but not sensitive data, or for those willing to participate in a questionnaire but not in a subsequent interview. A similar approach could be used for recruitment phases, where each variable represents a study phase such as first contact, reminder, second reminder and so on, with the outcome of interest being whether the individual participated, refused or ignored, along with their reason for non-participation if given.

Figure 3

Participation staged tree.

Figure 4

Chain event graph for participation. Percentage of participating individuals shown at each position (W).

Figure 5

An example of an asymmetric tree.

1.2 Hypothetical example

Let there be 50 copies of a survey distributed by mail and 100 distributed using a cheaper web option. Reminders are sent to 40% of the mail recipients and 50% of the web recipients, since electronic means are cheaper than postage costs. Some individuals return a completed survey, while others do not. Figure 3 shows these variables in a tree along with the number of individuals taking each edge; the corresponding CEG is shown in Figure 4.

The CEG shows that distribution by web and mail generally result in the same probability of receiving a completed survey (50%), but reminders are associated with increased participation rates. Those designing the survey may consequently choose to distribute web rather than mail surveys to save costs, but to include reminders to those who do not participate in the first phase. This is important, since often those who respond to reminders differ from those who responded to the initial survey and differ again to those who do not respond at all. It also assumes a causal association. Therefore, while this tactic appears to increase equality and reduce bias by increasing participation rates, it is possible that this may in fact lead to increased bias by recruiting different participant characteristics, possibly in each disease group. Therefore, this CEG may need to be used in conjunction with a CEG similar to that introduced in the next section (in Section 3.2.3), to compare the characteristics of the individuals. Knowing the recruitment potential of different techniques is informative and the associations found may be causal, hence these associations can be explored further. Additionally, potential biases can be unveiled by investigating the characteristics of those who are recruited using different techniques, particularly if these characteristics differ by disease group.

In addition, CEGs can be used for asymmetric problems, meaning that a particular decision at one variable can affect the choices available for later variables. This allows the tree here to be restructured such that the chronological ordering is web/mail, participant/non-participant, reminder/no reminder (only for those who were non-participants after the first phase) and finally participant/non-participant. Therefore the paths through the tree would be of varying lengths, as shown in Figure 5. It may be important to distinguish between participants from the first phase and participants recruited after the reminder phase, and this can be achieved by using different category names for the two groups of participants.

3.2.3 Participation by disease group

The factors associated with participation are required for the case and control group separately, since it is expected that the two disease groups will have different reasons for choosing to participate and since differences between these comparison groups can introduce bias. The CEGs used thus far have ordered the variables chronologically. However, different orderings can be adopted by CEGs depending upon their use [17]. If knowledge regarding the factors associated with participation for the cases and controls is required separately, disease status can be placed as the first variable in the tree, and participation as the final variable. This ensures the cases and controls are reported separately regarding participation. An example is given in Appendix C.1.

3.2.4 Amalgamated case-control participation data

If there are a series of case-control studies with similar variables recorded, the data from each study can be combined into one larger analysis using CEGs, where the outcome would be the disease status. Alternatively, the characteristics of cases or controls who are more likely to participate in a study of a particular topic may be of interest, or the most successful recruitment techniques may be sought. These findings can be achieved by having participation as the outcome variable, rather than disease status, as was demonstrated in Figure 4.

CEGs can be used in the same way as in Section 3.2.2 but the data are fed in from several studies. This may be particularly useful for studies of sensitive topics or those investigating very rare diseases, where the number of participants may be smaller. Conclusions can be drawn about the combination of factors associated with participation as demonstrated in Section 3.2.2. Patterns in the data can be used to form hypotheses regarding ways in which to increase participation in under-represented categories, or to inform future studies, although increased participation would not necessarily result in reduced bias. As CEGs can incorporate missing data as shown in [11], studies which record similar but not identical variables can be combined directly, with unrecorded variables included as an additional edge labelled ‘unrecorded’. An example is shown in Appendix D.1.

This approach could also be used with the disease status as the outcome of interest, and with missing data as a result of non-participation or otherwise, included as extra edges. In this instance this adaptation could provide a full analysis for case-control studies. If desired, data missing from non-participation and data missing for other reasons, could be given separate edges in the event tree, such that differences between the types of missingness can be represented.

3.3.1 Data reliability

CEGs have been used previously with prospective data but rarely with retrospective data, which can have limitations such as recall bias. This feature of the case-control study data can be incorporated into the CEG framework to enhance the authenticity of the analysis and to include additional information about the reliability of the data obtained.

Data in retrospective studies may be recorded using a variety of means such as medical records, through interview, or using national databases. Some sources may be cross-checked and verified with other authorities, while other sources may depend upon a single handwritten report, such as in older medical records or in areas without electronic databases. In some instances the only source will be the memory of those present and will require the individual to recall specific details. Recall bias is known to differ between participants of different disease groups in case-control studies as stated in [18] and hence data reliability may differ by both source and disease status.

One way in which to allow for potentially less reliable study data is to form a CEG which has a greater dependency on prior knowledge, provided these data are collected from a more reliable source. Since CEG learning is Bayesian and combines prior knowledge with data, non-uniform priors can be specified during the agglomerative hierarchical clustering (AHC) algorithm phase to achieve this. The examples preceding this have used uniform priors since no additional information has been available from experts or previous studies about which paths are more likely. The equivalent sample size explained in [8] is also specified and if a large equivalent sample size is used this suggests stronger prior beliefs and hence allows the priors to play a more dominant role, rather than depending strongly on the data.

The resulting tree will be structurally identical, but labelled with priors (or left unlabelled). The CEG may differ according to whether uniform or non-uniform priors are used, and this will depend upon the priors assigned and the data collected. This approach could be particularly useful in studies which suffer from non-participation, since the true population distributions of variables may not be apparent from the study data and this could potentially affect the conclusions generated.

Figure 6

Data reliability: Staged tree with uniform priors.

Figure 7

Data reliability: Chain event graph formed from uniform priors. W denotes positions.

Figure 8

Data reliability: Staged tree with non-uniform priors. The numbers indicate priors rather than individuals; the number of individuals are shown in Figure 6.

Figure 9

Data reliability: Chain event graph formed from non-uniform priors. W denotes positions.

Hypothetical example

Let there be a hypothetical study with two binary exposures, one of which is gender, plus a binary disease. First the analysis will be conducted using uniform priors and then with non-uniform priors constructed using hypothetical expert knowledge. Figure 6 shows the staged tree formed with uniform priors and Figure 7 shows the corresponding CEG. Figure 7 shows that males are more associated with case status than females (80% compared with 27%) and that for males, being exposed or not leads to the same position in the CEG (w3) with 80% of the individuals being cases. However females differ by exposure, with 13% of exposed females (w4) being cases and 80% of non-exposed females being cases (w3).

Figure 8 shows the staged tree which uses the same data, but where non-uniform priors are allocated. The priors are the smallest possible such that each value is integer and these priors are shown along the edges of the tree in Figure 8. The priors have been assigned using the hypothetical prior knowledge of half males and half females in the population, with the exposure being as common amongst males as it is females, and with around 20% of the population being exposed. The ratio of cases to controls in the study has been maintained. The same data as in Figure 6 but with the priors in Figure 8, results in the CEG in Figure 9.

Figure 9 differs from Figure 7 in that position w3 in Figure 7 is split into two positions (w3 and w4) in Figure 9, hence priors can affect the CEG produced. This split separates unexposed males, from exposed males and unexposed females, but returns two positions (w3 and w4) with the same proportion of cases (80%). Otherwise the CEGs are comparable and similar conclusions can be drawn.

In this example, the vertical ordering in the CEG is unchanged since the original position (Figure 7, w3) and the two new positions (Figure 9, w3 and w4) each have 80% of individuals who are cases. However it is possible that in some instances the splitting of positions could result in the reordering of the variables associated with these positions, especially if the percentage of (in this example) cases is similar amongst the positions. The equivalent sample size corresponds to the strength of the prior beliefs and could also affect the CEG, hence it is advised to check the robustness of the CEG with respect to changes in the equivalent sample size [9].

3.3.2 Subset-chain event graphs

CEGs are usually simple for a small number of variables, but quickly become complicated and more difficult to read when there are a large number of variables and/or each variable has a large number of categories. This includes where there are a large number of variables with missing data, where each variable includes an additional edge denoting missingness. Here, for these instances which can occur in case-control studies, subset-chain event graphs (subset-CEGs) are proposed as a new variant of CEGs.

A subset-CEG is simply a subset of variables displayed in a CEG which relate to a particular aspect of the data, which can later be interpreted alongside other subset-CEGs. One such CEG could be constructed for individual characteristics and another for environmental factors, with the number of subset-CEGs dictated by the number of variables and categories. If desired, one final CEG can be constructed at the end of the analysis which contains all variables found to be important in the subset-CEGs.

Figure 10

A staged tree used to form a subset-chain event graph.

Figure 11

An example of a subset-chain event graph. Percentage of participating individuals shown at each position (W). Colouring is not required since stages and positions are equal.

Figure 12

A staged tree with variables selected using subset-chain event graphs.

Figure 13

A final chain event graph with variables selected using subset-chain event graphs. Percentage of participating individuals shown at each position (W).

Figure 14

A final chain event graph with variables selected using subset-chain event graphs, age and reminder variables swapped. Percentage of participating individuals shown at each position (W). Example colouring has been used to highlight which positions were in the same stage, as the staged tree is not shown.

Figure 15

Example of a grid to position vertices vertically with respect to their percentage in an ordinal chain event graph. Each vertical line in the grid represents 10%.

Hypothetical example

Let there be a study where a total of 150 male and female individuals, who can be classed as either old or young by a given cut-off age, are asked to participate in a study by web or mail, with some receiving a reminder to participate and others not. The characteristics of the individuals are given in the staged tree in Figure 10 and the corresponding CEG is shown in Figure 11. The recruitment details for the study are as were shown in Figures 3 and 4.

Figure 11 shows that participation does not differ between males and females, but is more likely from those who are old than those who are young. Figure 4 showed reminders to be associated with participation, but not the survey delivery mode and, if desired, these variables can be used to construct a final CEG for the dataset. Figure 12 shows the staged tree for the variables of age and reminders on participation, and Figure 13 shows the corresponding CEG. The CEG suggests that old individuals are more likely to participate than young, and that reminders are associated with increased participation in old individuals, but are not as effective for young individuals.

Since the natural ordering of age and reminders is unclear, the analysis was also rerun with the reminder variable before age. This resulted in the CEG shown in Figure 14 which shows reminders to be associated with increased participation and old individuals to be more likely to participate than young individuals. Again reminders are not as effective for young individuals as they are old individuals. Therefore the same conclusions are drawn, regardless of the ordering of the age and reminder variables. This should be expected, since age is not affected by reminders and the allocation of reminders is not determined by age, hence the ordering of these two variables is less important than in other scenarios.

Here, subset-CEGs have been used to simplify the analysis into smaller steps and use variables thought to be associated with the outcome to form a final CEG. Each subset-CEG shows a different aspect of the study, which might have been missed in a full CEG. This also improves the readability of the CEG. Another approach to improve readability, may be to present the CEGs against a grid representing the percentage of individuals at each vertex who have the outcome of interest (participation or disease status here). Rather than display the percentages within the vertices, here it is proposed that the vertices could be placed vertically against the grid to show their relative positioning. Figure 14 has been redrawn using a grid and is shown in Figure 15 as an example. This improves readability for spatial readers, but may cause the edges to be less clear, and could result in fewer planar graphs and hence a graph which is more difficult to interpret.