It has been proposed that earthquakes are preceded by electromagnetic signals detectable from ground- and space-based measurements. Ionospheric anomalies, such as variations in the electron density a few days before earthquakes, are one of the precursory signals proposed. Since Global Positioning System (GPS) data can be used to measure the ionospheric total electron content (TEC), the technique has received attention as a potential tool to detect ionospheric perturbations related to earthquakes. Here, we analyze 2 years (2003--2004) of data from the Southern California Integrated GPS Network (SCIGN), a dense network of 265 continuous GPS stations centered on the Los Angeles basin, for possible precursors. This time period encompasses the December 2003, M6.6, San Simeon and September 2004, M6.0, Parkfield earthquakes. We produce TEC time series at all SCIGN sites and apply three different statistical tests to detect anomalous TEC signals preceding earthquakes. We find anomalous TEC signals but no statistically significant correlation, in time or in space, between these TEC anomalies and the occurrence of earthquakes in southern California for the 2003--2004 period. This result does not disprove the possibility of precursory phenomena but show the signal-to-noise ratio of a hypothetical TEC precursor signature is too low to be detected by the analysis techniques employed here. Precursors may still be revealed for future large earthquakes in well instrumented areas such as California and Japan, if the tests can be developed into techniques that can better separate external influences from the actual TEC signal.