The aim of this investigation is to present a new analysis of short-term variations in total solar irradiance by developing regression models and to extend these to epochs when irradiance measurements were not available. In our models the sunspot area is used to quantify sunspot darkening while facular brightening is calculated using facular area, 10.7 cm radio flux and Mg II core-to-wing ratio. Models developed with various proxies are compared with a view to identify the role of key parameters in solar variability. We also study the relationship between different facular proxies and show that the facular area and 10.7 cm radio flux do not vary linearly with the Mg II core-to-wing ratio. We emphasize that the facular term in current empirical models (using facular area or radio flux proxies) on short time scale needs to have a nonlinear component in order to obtain a better correlation with observed irradiance. Our analysis demonstrates that the correlation for daily variations in solar irradiance improves by 10% using a quadratic term in the model based on radio flux as a facular proxy, which is a significant improvement on earlier models. On the other hand, the correlation remains unchanged in the model using Mg II core-to-wing ratio. Thus we point out that various proxies for facular brightenings contribute differently to solar irradiance. We estimate the solar irradiance variations at epochs before irradiance observation began, in particular to the start of the radio flux measurements, and find that there is no drastic increase in radiative output during the most active solar cycle 19 while for cycle 20 we observe a much lower irradiance during maximum.