<p>In genome-wide association studies (GWAS), "generalization" is the replication of genotype-phenotype association in a population with different ancestry than the population in which it was first identified. Current practices for declaring generalizations rely on testing associations while controlling the family-wise error rate (FWER) in the discovery study, then separately controlling error measures in the follow-up study. This approach does not guarantee control over the FWER or false discovery rate (FDR) of the generalization null hypotheses. It also fails to leverage the two-stage design to increase power for detecting generalized associations. We provide a formal statistical framework for quantifying the evidence of generalization that accounts for the (in)consistency between the directions of associations in the discovery and follow-up studies. We develop the directional generalization FWER (FWER ) and FDR (FDR ) controlling r-values, which are used to declare associations as generalized. This framework extends to generalization testing when applied to a published list of Single Nucleotide Polymorphism-(SNP)-trait associations. Our methods control FWER or FDR under various SNP selection rules based on P-values in the discovery study. We find that it is often beneficial to use a more lenient P-value threshold than the genome-wide significance threshold. In a GWAS of total cholesterol in the Hispanic Community Health Study/Study of Latinos (HCHS/SOL), when testing all SNPs with P-values <5×10-8 (15 genomic regions) for generalization in a large GWAS of whites, we generalized SNPs from 15 regions. But when testing all SNPs with P-values <6.6×10-5 (89 regions), we generalized SNPs from 27 regions.</p>