Same data, new method, same outcome – can I publish it? I applied a new method to a data set, which had already been analyzed in a research paper. My method is generally considered better, but it only slightly modifies the results and does not affect the conclusion of the paper aside from improving the accuracy of results by a small amount. The paper is about a classification problem. 

Is it possible to write a research paper about this?
I'd say it is difficult to answer this question in general. You should ask yourself if you are contributing something significantly novel to the academic discussion. For example, if you reanalyse the dataset and get completely different or new results, then it is probably worth publishing. On the other hand, if you only slightly modify the statistical method and arrive at the same results, it is probably not worth it. Before beginning the analysis, you could ask yourself if this dataset is particularly big or of good quality or if there are other, better datasets out there (in which case a reanalysis of this particular one might be unproductive).

To give an example, in the UK there have been many papers on the effectiveness of breast cancer screening that essentially analysed the same dataset using different methods. These methods were fairly complex and the various approaches have significantly advances our understanding of the advantages and possible detriments of breast cancer screening. In this case, publishing multiple papers was justified.