Ferdinand Christian Baur was by any reasonable measure a genius. He was, however, a genius operating at the level of his time. That time was dominated, especially in Germany, by the towering figure of Georg Hegel, Baur's older contemporary. When he turned to thinking about the dates of the New Testament, Baur developed a quasi-Hegelian narrative of dialectical development in the early Christian movement. He found warrant for that narrative in the Clementine literature, a large body of mostly romantic literature centred upon the person of Clement of Rome, remembered as the third (sometimes fourth) bishop of Rome after Peter. Baur made two crucial decisions with regard to this literature: first, he argued that much of it dated to the second century; and he saw in the literature's depiction of a struggle between Peter and Simon Magus coded reference to a struggle between Petrine and Pauline Christianity. Thus, he argued, Christian life during the first and the earlier part of the second centuries was dominated by a fundamental opposition between two contradictory forms of Christianity. This opposition was eventually reconciled, with the Gospel of John the fullest exemplar of Christianity after such reconciliation. The Acts of the Apostles, in showing Peter and Paul standing together in chapter fifteen and more generally showing Paul getting on with the apostles in Jerusalem, anachronistically projects the conditions of that reconciliation back on to the first Christian generation. Baur's understanding of the time that this development must have taken, and his reading of the Clementine material, leads him to the argument that the Gospel and Letters of John could not have been written until c. 170 or so.
This is where appeal to data comes in. Lightfoot demonstrated the implausibility of this narrative by showing that it simply could not withstand the data. There are some problems that should be obvious from the off. For instance, Gal. 2:1-10 presents Paul and Peter (and James and John) reaching agreement, comparable to what we find in Acts 15 (and Galatians was one of the five texts Baur allowed that Paul had written). In point of fact, it seems only by an argument from silence that one can say that the presentation of their relationship in Acts is more irenic than it was in reality. Luke might very well omit conflicts that occurred, but what he reports in terms of agreements are little if at all greater than in Paul's own writings. But Lightfoot's appeal to the data went well beyond this sort of observation, and in the process gave birth to some of the most prominent critical tools still used in the study of the Apostolic Fathers.
The Apostolic Fathers refer to a group of non-Christian Christian texts that are generally believed to have been written in the late first and early second centuries. They include, for instance, 1 and 2 Clement, letters attributed to the above-mentioned Clement of Rome. Let's begin with 1 Clement. Lightfoot showed that it was almost certainly written by the end of the first century (he places it around 95). And that's a problem for Baur, because 1 Clement talks about Paul and Peter together, with no hint of conflict between them...at precisely the time that Baur holds that there remains an unreconciled conflict between the two. And Baur faces even greater problems by the time that Lightfoot is done working through another set of core documents among the Apostolic Fathers, the writings of Ignatius.
The letters of Ignatius were a mess. Eusebius, in the fourth-century, provided a list of seven letters written by Ignatius, but we had more than those seven. Moreover, what we had existed in multiple recensions. Lightfoot patiently worked through the material, persuasively showing that the so-called "Middle Recension" (i.e. not the shortest, but not also the longest) of the seven letters mentioned by Eusebius were those written by Ignatius. He moreover argued persuasively that they were written no later than 117 (the probable latest date of Ignatius' death, as our best data suggests that he died under the emperor Trajan). Now, this is again significant, as Lightfoot is also able to show that these letters are loaded with Johannine imagery. He concludes that Ignatius likely knew John's Gospel. And that's a huge problem for Baur, given that the argues that John's Gospel and letters could not have been written for several decades after that point!
Of the above arguments advanced by Lightfoot, Ignatius' knowledge of John's Gospel is probably the only one that might be seriously contested in 2017, with some people arguing that Ignatius knew not the gospel itself but rather the sort of tradition that is reflected by the gospel. But this does not seem likely to provide much aid to Baur. If already by 117 there existed something close enough to John's Gospel that a researcher as competent as Lightfoot could conclude that it is John's Gospel, the suggestion that John's Gospel or letters could not have been writing for another forty or fifty years would seem to push the limits of the data. Baur's argument isn't that it wasn't written until then, but rather that it contains theology that Christians could not have produced prior to that time. Yet, if much of that theology is present in Ignatius by 117, then that constitutes a probably fatal blow to Baur and his Tübingen school.
In Method, Lonergan specifically refers readers to the account of Lightfoot's refutation of Baur given in Stephen Neill's The Interpretation of the New Testament (1861-1961). With that in mind, I will quote from Neill (here from p. 60 the revised edition of this work, which Tom Wright undertook in 1986):
Hardly less crushing was the blow delivered to the Tübingen theories by the genuine Clement and Ignatius. If the theories were correct, certain phenomena should have been observable in these letters. In point of fact, not merely is none of these phenomena to be observed, but what is to be found is so contradictory of what to be expected as to raise the question whether any of these phenomena were ever at any time found in the Christian world.This perfectly encapsulates what Lonergan means by "appeal to data": our hypothesis tells us what to expect in the data and what not to expect, so when the data contains what we should not expect and none of what we should, our hypothesis is probably a non-starter.