For nonexperimental data, causal direction can often be inferred if information about time is available. This is because (according to many, though not all, theories) causes must precede their effects temporally. This can be determined by statistical time series models, for instance, or with a statistical test based on the idea of Granger causality , or by direct experimental manipulation. The use of temporal data can permit statistical tests of a pre-existing theory of causal direction. For instance, our degree of confidence in the direction and nature of causality is much greater when supported by cross-correlations , ARIMA models, or cross-spectral analysis using vector time series data than by cross-sectional data .
There are reams of literature addressing whether these two definitions are the same and, if not, to which of them Hume gives primacy. Robinson is perhaps the staunchest proponent of the position that the two are nonequivalent, arguing that there is an nonequivalence in meaning and that they fail to capture the same extension. Two objects can be constantly conjoined without our mind determining that one causes the other, and it seems possible that we can be determined that one object causes another without their being constantly conjoined. But if the definitions fail in this way, then it is problematic that Hume maintains that both are adequate definitions of causation. Some scholars have argued for ways of squaring the two definitions (Don Garrett, for instance, argues that the two are equivalent if they are both read objectively or both read subjectively), while others have given reason to think that seeking to fit or eliminate definitions may be a misguided project.