Developing causal models from observational longitudinal studies is an important, ubiquitous problem in many disciplines. A disadvantage of current causal discover algorithms, however, is the inherent instability in structure estimation. With nite data samples small changes in the data can lead to completely di erent optimal structures. The present work presents a new causal discovery algorithm for longitudinal data that is robust for nite data samples. We validate our approach on a simulated data set and real-world data on Chronic Fatigue Syndrome patients.
Developing causal models from observational longitudinal studies is an
important, ubiquitous problem in many disciplines, which has led to the development
of a variety of causal discovery algorithms in the literature [1{5]. A disadvantage
of current causal discovery algorithms, however, is the inherent instability in
structure learning. With nite data samples small changes in the data can lead
to completely di erent optimal structures, since errors made by the discovery
algorithm may be propagated and lead to further errors [
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We use a SEM for causal modeling. The general form of the equations is
xi = fi(pai; "i); i = 1; : : : ; n:
(1)
where pai denotes the parents which represent the set of variables considered to
be direct causes of Xi and "i represents errors on account of omitted factors that
are assumed to be mutually independent [
We use the method we developed in [
The method in [
To score our models on longitudinal data we use data reshaping. In the reshaped data, the rst n data points contain the relations that occur in the rst two time slices t0 and t1. The next n data points contain the relations that occur in time slices t1 and t2. The i-th subset of n data points contain used to generate longitudinal data. It contains four continuous variables (X1; : : : ; X4) in three di erent time slices t0; : : : ; t2. the relations in time slices ti 1 and ti. The reshaped data then allows us to use standard SEM software to compute the scores. 3
For this experiment, we generated a longitudinal data set with 400 instances from a causal graph as depicted in Figure 1b. The data set consists of three time slices with four continuous variables for each time slice.1 When we searched over SEM models we added prior knowledge that variables X1 and X2 do not cause variable X3 directly. We performed the search over 200 subsets.
As the true model is known, we measure the performance of our method by
means of the Receiver Operating Characteristic (ROC) [
xed to a value ( sel 2 f0:3; 0:6; 0:8; 0:9g) while
bic is varied. We compute the True Positive Rate (TPR) and the False Positive Rate (FPR) from the CPDAG of the true model. As for an example, in the case of edge stability, a true positive means that an edge that appears within the top-left region bounded by sel and bic also exists in the CPDAG of the true model. Figure 2 portrays the ROC curves for both edge and causal path stability. Generally we can see that higher values of sel tend to give better ROC curves. This suggests that our approach is able to nd the underlying structure with high reliability scores. A notable point is that the ROC curves stop at a TPR and/or FPR value lower than 1. Since some of the edges and paths are disallowed (i.e., no edges in time-slice ti 1 and no paths from ti to ti 1) some of the edges and causal paths in the stability graphs end up with a selection probability of 0 and the result is that the ROC curves cannot reach the upper right corner with TPR = FPR = 1.
1Available at http://bit.ly/1L6dBOo 0.3 0.6 0.8 0.9 0 . 1 8 . 0 6 R .0 P T .4 0 2 . 0 0 . 0 0.3 0.6 0.8 0.9 0.0 0.1 0.2 0.4 0.5 0.6
0.7 0.3
FPR
(a)
For an application to real-world data, we consider a data set about Chronic
Fatigue Syndrome (CFS) which consists of 183 subjects and ve time slices
with six discrete variables [
Figure 3a shows that nineteen relevant edges were found, consisting of eleven intra-slice and eight inter-slice relationships which among of these, six are between the same variables and two are between di erent variables. Figure 3b shows that thirty-two relevant causal paths were found, consisting of twelve intra-slice and twenty inter-slice relationships which among of these, six are between the same variables and fourteen are between di erent variables. For a more intuitive representation, we combine the stability graphs into a model using the following procedure. First, the nodes are linked according to the nineteen relevant edges. Second, edges are oriented according to our background knowledge. Eight of the inter-slice relationships are oriented from time slice ti 1 to ti and ve of the intra-slice edges can be oriented since it is known that the variable fatigue does not directly cause any other variable. Third, the edges are oriented according to the relevant causal paths, which results in another twenty-eight directed edges. The inferred model is shown in Figure 4. Each edge is annotated with a reliability score which is the maximum score obtained in the top-left region of the edge stability graph.
pbic 0 2 4 6 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50
Model complexity psel psel 0 2 4 6 8 11 14 17 20 23 26 29 32 35 38 41 44 47 50
Model complexity
(b)
From the stability graphs we can see that the most stable causal relations are
the inter-slice relations between the same variables followed by some of the
intraslice causal relations. Almost all of the inter-slice relations between di erent
variables are not considered relevant. A directed edge X ! Y in Figure 4 indicates
that a change in variable X causes a change in variable Y . In the intra-slice causal
relationships, we found that all variables are direct causes for fatigue severity.
We also found all variables, except fatigue, to be direct causes for the perceived
activity. Furthermore, the variable control is a direct cause for both focusing
on the symptoms and physical functioning. Generally the inter-slice
relationships show direct causes between the same variables. In addition, the variables
pActivity and control indicate a stronger direct cause for fatigue severity and
focusing on symptoms, respectively, as they contribute a direct cause in both
time slices. The inferred model is consistent with results reported in the medical
literature [
fatigue( i−1)
1 pActivity( i−1) 1 oActivity( i−1) 1 focusing( i−1) 1 functioning( i−1) 1 control( i−1) 1 0.93 0.64 fatigue( i)
1 pActivity( i) 0.71 oActivity( i)
0.91 focusing( i)
0.99 functioning( i)
1 control( i) 0.96 1 1 0.61 0.99 1 Causal discovery from longitudinal data is an important, ubiquitous problem in science. Current causal discovery algorithms, however, have di culty dealing with the inherent instability in structure estimation. The present work introduces a new discovery algorithm for longitudinal data that is robust for nite samples. Experimental results on both arti cial and real-world data sets show that the method results in reliable structure estimates. Future research will aim to estimate the size of causal e ects.
The research leading to these results has received funding from the DGHE of Indonesia and the European Community's Seventh Framework Programme (FP7/2007-2013) under grant agreement n 305697.