Multi-state models for investigating possible stages leading to bipolar disorder
- Charles DG Keown-Stoneman^{1}Email author,
- Julie Horrocks^{1},
- Gerarda A Darlington^{1},
- Sarah Goodday^{2},
- Paul Grof^{4} and
- Anne Duffy^{3}
https://doi.org/10.1186/s40345-014-0019-4
© Keown-Stoneman et al.; licensee Springer. 2015
Received: 28 August 2014
Accepted: 30 December 2014
Published: 24 February 2015
Abstract
Background
It has been proposed that bipolar disorder onsets in a predictable progressive sequence of clinical stages. However, there is some debate in regard to a statistical approach to test this hypothesis. The objective of this paper is to investigate two different analysis strategies to determine the best suited model to assess the longitudinal progression of clinical stages in the development of bipolar disorder.
Methods
Data previously collected on 229 subjects at high risk of developing bipolar disorder were used for the statistical analysis. We investigate two statistical approaches for analyzing the relationship between the proposed stages of bipolar disorder: 1) the early stages are considered as time-varying covariates affecting the hazard of bipolar disorder in a Cox proportional hazards model, 2) the early stages are explicitly modelled as states in a non-parametric multi-state model.
Results
We found from the Cox model thatthere was evidence that the hazard of bipolar disorder is increased by the onset of major depressive disorder. From the multi-state model, in high-risk offspring the probability of bipolar disorder by age 29 was estimated as 0.2321. Cumulative incidence functions representing the probability of bipolar disorder given major depressive disorder at or before age 18 were estimated using both approaches and found to be similar.
Conclusions
Both the Cox model and multi-state model are useful approaches to the modelling of antecedent risk syndromes. They lead to similar cumulative incidence functions but otherwise each method offers a different advantage.
Keywords
Background
In this developmental clinical staging model, non-mood disorders include primarily anxiety and sleep disorders, and in a subgroup of offspring from lithium non-responsive parents, attention deficit hyperactivity disorders (ADHD), and learning disabilities. Minor mood disorders comprise depressive disorders not otherwise specified, dysthymia, cyclothymia and adjustment disorders. Major mood disorder refers to major depressive disorder (single episode or recurrent). The last category, bipolar disorders, includes bipolar not otherwise specified, bipolar I and bipolar II disorders, and in the offspring of lithium non-responders schizoaffective disorder. Understanding the natural history of psychiatric disease is a critical advance supporting the identification of associated markers of illness risk and development. This approach has been very successful in other complex heritable medical diseases such as cancer and cardiovascular illness (Carlson et al. 2009; Thompson et al. 2007; Yancy et al. 2013). It is pertinent to test and develop the most rigorous analytic techniques to test the latter association as they involve the use of complex longitudinal data, and appropriate statistical analysis of data obtained from longitudinal investigations remains an important but challenging problem.
The Cox proportionalhazard model Cox (1972) is a common and familiar choice among investigators to examine longitudinal time to event data. However, there is concern over the validity of Cox models in the presence of time-varying covariates (Kalbfleisch and Prentice 2002). It has been suggested that multi-state models can be used as an alternative to Cox models to address some of these limitations (Cortese and Andersen 2010).
The clinical staging model proposed by Duffy et al. (2010) has previously been investigated using parametric multi-state models (Duffy et al. 2014). In order to assess the effectiveness of multi-state models when compared to Cox models we fit a non-parametric multi-state model with states representing the proposed stages of bipolar disorder and compared this to a Cox proportional hazards model in which the preceding stage of major depressive disorder is treated as a time-varying covariate (Cox 1972). The focus of this analysis is the effect of major depressive disorder on the risk of developing bipolar disorder in order to more easily demonstrate the statistical comparability between Cox models and multi-state models.
Methods
Data background
In compliance with the Helsinki Declaration, this research was approved by local research ethics boards in Ottawa, Halifax, and Calgary. As part of an ongoing longitudinal study, data were collected on 229 offspring from well characterized families with one parent having a confirmed history of bipolar disorder based on the best estimate procedure and blind consensus review (Duffy et al. 2007). In all families that were included, the other parent had no lifetime history of major affective disorder, schizophrenia, schizoaffective disorder, substance use disorder or personality disorder at baseline (Duffy et al. 2007). Bipolar disorder is highly heritable, therefore these offspring were at a much higher risk of developing bipolar disorder than the general public (Duffy et al. 2007; McGuffin et al. 2003). All offspring were recruited into the study between the ages of 7 and 25 from eligible families in which the proband parent was receiving long-term treatment in a mood disorders specialty clinic in either Ottawa or Halifax (Duffy et al. 2007). A more detailed description of the collection methods can be found in Duffy et al. (2007).
Number of subjects that experienced the different diagnoses from the full dataset with 229 subjects
No Diagnoses | Non-mood | Minor Mood | Major Mood | Bipolar |
---|---|---|---|---|
61 | 92 | 91 | 83 | 21 |
Number of subjects that experienced the different diagnoses from the subset of the dataset with 99 subjects
No Diagnoses | Non-mood | Minor Mood | Major Mood | Bipolar |
---|---|---|---|---|
21 | 45 | 40 | 38 | 10 |
Cox models
where α _{0}(t) is a baseline hazard function that is often unspecified, β _{1},β _{2},…,β _{ p } are regression coefficients, and x _{1i }(t),x _{2i }(t),…,x _{ pi }(t) are covariates for subject i. Note that the covariates can change with time. We can interpret the effect of a covariate as follows: A one unit increase in x _{ i1}(t) is estimated to multiply the hazard by \(e^{\beta _{1}}\phantom {\dot {i}\!}\), holding all other covariates constant.
where δ(s) is the number of events at time s, Y _{ i }(s) is an indicator variable which is 1 if subject i is at risk at time s, and x _{ i }(s) is the value of the covariate for subject i at time s.
Kalbfleisch and Prentice (2002) identified the importance of distinguishing between internal and external covariates when using a Cox model. External covariates are completely determined at the start of the study (e.g. a person’s sex or age) or are random but external to the subject (e.g. the weather). Internal covariates are random measurements on the subject that vary with time and may be affected if the event of interest occurs. For example, if the event of interest is death, then blood pressure would be an internal time-varying covariate, as it is a characteristic of the subject that varies with time and is affected by the event of interest, in the sense that it cannot be measured after death. Modelling of internal covariates can be difficult as the effect of the covariate on the event of interest may be complicated by the effect of time. For example, the effect of a covariate may not be observed until a time lag has passed (Kalbfleisch and Prentice 2002). Assuming that the effect of the internal covariate is correctly specified, including the correct lag and functional form, the interpretation of a regression coefficient in (1) is still valid (Kalbfleisch and Prentice 2002). However, estimates of the cumulative incidence function corresponding to (2) are considered invalid in the presence of internal covariates because the distribution of the covariate may depend on the event time (Kalbfleisch and Prentice 2002). Multi-state models are sometimes suggested as a solution to this problem (Cortese and Andersen 2010).
In this paper, we will consider Cox models with time to bipolar disorder as the response, and preceding diagnoses as time-varying covariates. Preceding diagnoses are considered internal covariates because they are measured on the subject and vary with time, and because once a diagnosis is received, the subject cannot subsequently receive a diagnosis of a less severe disorder, in accordance with diagnostic convention (American Psychiatric Association 2000). However, there is a problem with this approach, namely that traditional survival analysis allows only one outcome. On the other hand, we are interested in building a more complicated model. Therefore, in the next section we discuss multi-state models that allow multiple outcomes.
In this paper, preceding diagnoses are coded so that they take a value of 0 before and 1 after the diagnosis. The Cox models were fit using the coxph function from the survival R package using R version 3.0.2 for Windows 64bit (R Core Team 2013).
Multi-state model
where P _{ hj }(s,t) is the probability of being in state j at time t given a subject was in state h at time s.
The state probabilities can be expressed in matrix form as
where T _{ Z } is the time of transition into state Z from any other state.
It has previously been noted that the choice of a state structure for a multi-state model is important, and not unique for each situation (Hougaard 1999; Keown-Stoneman 2013). A good state structure can simplify calculations and make interpretation of the model more straightforward. The state structure must be complicated enough to accommodate all the pathways observed in the data, yet simple enough to allow meaningful inference.
The mstate package for R was used to fit all multi-state models for this paper (de Wreede et al. 2010; R Core Team 2013). Using the mstate package, one can estimate cumulative transition rates Λ(u), and state probabilities P(s,t).
Results and discussion
Cox model
Results from the Cox model
Covariate | Parameter | Standard | Z | p-value | Hazard |
---|---|---|---|---|---|
estimate | error | ratio | |||
Major mood | 1.6233 | 0.7147 | 2.271 | 0.0231 | 5.070 |
From the Cox model, there is evidence that the hazard of bipolar disorder is increased by the onset of major depressive disorder (p = 0.0231). In particular, it is estimated that having a diagnosis of major depressive disorder multiplies the hazard of bipolar disorder by 5.070 (Table 3).
Multi-state model
Conclusions
Comparison of models
When assessing a temporal association between two variables (e.g. depression and subsequent bipolar disorder), it is unclear how two popular methods, Cox proportional hazards and multi-state models compare. Both the Cox proportional hazards model and the multi-state model have benefits in the analysis of bipolar disorder. One advantage of treating previous events as states in a multi-state model approach over treating them as covariates in a Cox model is that the multi-state model does not assume proportional hazards. However, the Cox model has a more straightforward interpretation of covariate effects than non-parametric multi-state models and allows tests to be performed to determine if the effect of a covariate is significantly different from zero. A disadvantage of the multi-state model approach is that it cannot handle continuous internal covariates as states unless they are first transformed into categories, as they must be represented by discrete states. Also, since the hazard function is estimated separately for each transition, prediction based on non-parametric multi-state models may be more susceptible to bias due to sparse data.
As stated above, the focus of the Cox model is usually regression coefficients, while the focus of a multi-state model is state (stage) probabilities, hence the two approaches are generally not directly comparable. However, they will both produce conditional cumulative incidence functions (CIF). Here we estimated the conditional CIF of bipolar disorder after age 18 given a high-risk subject had major depressive disorder prior to age 18. The selection of major depressive disorder and 18 years old was arbitrarily chosen as an example of a conditional CIF. Since we assume the Markov property that the hazard of transitioning is independent of the length of stay in the current state, and that the subject has not been diagnosed with bipolar disorder before age 18, it is inconsequential when major depressive disorder occurred, as long as it was before age 18.
where \(\hat {A}(t|x(t)=1,t\ge 18)\) is the estimated cumulative hazard of bipolar disorder from time t onward, given major mood was diagnosed prior to age 18. This was estimated from a Cox model fit to the entire dataset, then the estimated changes in the cumulative hazard were used to construct the probability of bipolar disorder beyond age 18 given major mood was present.
Using the multi-state model approach we obtained estimates of the probability of high-risk offspring being in the various proposed stages of bipolar disorder. As an example analysis, when major mood disorder was assumed to have occurred prior to age 18, the Cox model estimate of the probability of bipolar disorder appears to be very similar to the multi-state model estimate (Figure 5). Using the same methods as in Figure 4, 95% confidence intervals were obtained for the multi-state model CIF in Figure 5 (Kalbfleisch and Prentice 2002).
In summary, if assessing the association between longitudinal variables is the primary interest for researchers then Cox models with time-varying covariates are recommended. If predicting the probability of entering future states, as in CIFs, is the primary interest then multi-state models are preferred. However, we have shown that Cox models can also be used to construct conditional CIFs. As shown in Figure 3 it is possible to include all of the stages of bipolar disorder as states in a multi-state model. Although this paper only included a Cox model with major mood disorder as a time-varying covariate for simplicity, it is also possible to include time-varying covariates that represent the presence of the two other previous stages, non-mood and minor mood. It would also be possible to test for interactions between the different stages.
It should be noted that multi-state models can also allow for other covariates to affect the transition intensities between states. Although the choice of using one randomly selected offspring from each family did not have a considerable effect on the results (data not shown) further work into the modelling of bipolar disorder could include more complex analyses that account for familial correlation.
Although they are unlikely to completely replace the Cox model, the multi-state model is another useful tool for investigating the relationships among different events over time.
Declarations
Acknowledgements
This work was supported by a Canadian Institutes of Health Research (CIHR) Operating Grant (MOP 102761).
Authors’ Affiliations
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