This randomized phase III trial studies pemetrexed disodium to see how well it works compared with erlotinib hydrochloride as second-line therapy in treating patients with non-small cell lung cancer that has spread to other places in the body. Pemetrexed disodium and erlotinib hydrochloride may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether pemetrexed disodium is more effective than erlotinib hydrochloride in treating advanced non-small cell lung cancer.

PRIMARY OBJECTIVES: I. To evaluate whether there are differences in progression-free survival due to treatment with erlotinib (erlotinib hydrochloride) compared to pemetrexed (pemetrexed disodium) for subsets of previously treated non-small cell lung cancer (NSCLC) patients defined by epidermal growth factor receptor (EGFR)-fluorescent in situ hybridization (FISH) positive versus negativity. SECONDARY OBJECTIVES: I. Ascertain the presence or absence of true differences due to treatment in the objective clinical endpoints of overall survival, confirmed response rate, and adverse event profile for subsets of patients defined on the basis of the epidermal growth factor receptor (EGFR)-FISH positivity versus negativity. II. Ascertain the presence or absence of true differences due to treatment in objective clinical endpoints (i.e., progression free survival, overall survival, confirmed response rate, and adverse event profile) for subsets of patients defined on the basis of the epidermal growth factor receptor (EGFR) expression as measured by immunohistochemistry (IHC). III. Ascertain the presence or absence of true differences due to treatment in objective clinical endpoints (i.e., progression free survival, overall survival, confirmed response rate, and adverse event profile) for subsets of patients defined on the basis of the epidermal growth factor receptor (EGFR) gene mutation status (MUT). IV. To evaluate the prognostic effect of EGFR copy number as measured by FISH separately by treatment arm, i.e., is there a difference in outcome for FISH(+) patients receiving erlotinib compared to FISH (-) patients receiving erlotinib, and similarly is there a difference in outcome for FISH(+) patients receiving pemetrexed compared to FISH (-) patients receiving pemetrexed. V. To evaluate the prognostic effect of EGFR expression as measured by IHC separately by treatment arm, i.e., is there a difference in outcome for IHC(+) patients receiving erlotinib compared to IHC (-) patients receiving erlotinib, and similarly is there a difference in outcome for IHC(+) patients receiving pemetrexed compared to IHC (-)patients receiving pemetrexed. VI. To evaluate the prognostic effect of EGFR mutation status separately by treatment arm, i.e., is there a difference in outcome for MUT(+) patients receiving erlotinib compared to MUT(-) patients receiving erlotinib, and similarly is there a difference in outcome for MUT(+) patients receiving pemetrexed compared to MUT(-) patients receiving pemetrexed. VII. To prospectively test the hypothesis that functionally relevant polymorphisms in the genes encoding for pemetrexed targets, as well as genes encoding for one or more of the key enzymes involved in the transport, activation, and inactivation of pemetrexed, either singly or in combination, play a role in the efficacy and/or toxicity of pemetrexed. VIII. To prospectively test the hypothesis that functionally relevant polymorphisms in the EGFR gene as well as genes encoding for one or more of the key enzymes involved in the metabolism of erlotinib, either singly or in combination, play a role in the efficacy and/or toxicity of erlotinib. IX. Evaluate proteomic signatures in blood samples as predictors of survival and response to treatment with erlotinib. X. To evaluate expression of thymidylate synthase, dihydrofolate reductase, phosphoribosylglycineamide (GAR) formyltransferase, and methylthioadenosine phosphorylase gene expression in tumor samples, as measured by IHC or quantitative polymerase chain reaction, as predictors of survival and response to treatment with pemetrexed. XI. To evaluate proteomic signatures in blood samples of patients as predictors of response and survival to treatment with erlotinib. XII. To evaluate the following variables measured in tumor samples, as predictors of response and survival to treatment with pemetrexed: Expression of thymidylate synthase, dihydrofolate reductase and GAR formyltransferase genes methylthioadenosine phosphorylase expression by IHC or quantitative polymerase chain reaction (PCR). XIII. To evaluate the following variables measured in tumor samples, as predictors of response and survival to treatment with erlotinib: Rat sarcoma (Ras) mutational status, EGFR mutational status and, epithelial to mesenchymal transition (EMT) status (measured by E-cadherin expression and vimentin expression) by IHC. OUTLINE: Patients are randomized to 1 of 2 treatment arms. ARM I: Patients receive erlotinib hydrochloride orally (PO) once daily (QD) on days 1-21. Courses repeat every 21 days in the absence of disease progression or unacceptable toxicity. ARM II: Patients receive pemetrexed disodium intravenously (IV) over 10 minutes on day 1. Courses repeat every 21 days in the absence of disease progression or unacceptable toxicity. After completion of study therapy, patients are followed every 3 months until disease progression and then every 6 months for up to 5 years.