We have performed 23 GHz VLA observations of 7 compact, luminous infrared galaxies, selected to have evidence of starburst activity. New and published multi-frequency data are combined to obtain the spectral energy distributions of all 7 galaxies from the near-infrared to the radio (at 1.4 GHz). These SEDs are compared with new models, for dust enshrouded galaxies, which account for both starburst and AGN components. In all 7 galaxies the starburst provides the dominant contribution to the infrared luminosity; in 4 sources no contribution from an AGN is required. Although AGN may contribute up to 50 percent of the total far--infrared emission, the starbursts always dominate in the radio. The SEDs of most of our sources are best fit with a very high optical depth of (>=50) at 1 micron. The scatter in the far-infrared/radio correlation, found among luminous IRAS sources, is due mainly to the different evolutionary status of their starburst components. The short time-scale of the star formation process amplifies the delay between the far-infrared and radio emission. This becomes more evident at low radio frequencies (below about 1 GHz) where synchrotron radiation is the dominant process. In the far-infrared (at wavelengths shorter than 100 micron) an additional source of scatter is provided by AGN, where present. AGN may be detected in the near-infrared by the absence of the knee, typical of stellar photospheres. However, near-infrared data alone cannot constrain the level at which AGN contribute because the interpretation of their observed properties, in this wave-band, depends strongly on model parameters.