A proportional hazard function together with partial likelihood estimation is the most common approach to the analysis of censored data. However, partial likelihood estimation is established on the grounds that the censoring is non-informative. The partial likelihood approach enjoys many good properties when the censoring is indeed non-informative. However, in reality, censoring can be informative. One pays a price in the efficiency of the estimator if partial likelihood estimation is used when the censoring is indeed informative. This problem is particularly acute in the nonparametric case. When censoring is informative, to make use of the information provided by the censoring times, it is better to take the local complete likelihood approach. Motivated by the data set about the first birth interval in Bangladesh, we propose here a varying-coefficient proportional hazard function to fit informatively censored data. We take the complete likelihood approach coupled with local linear modelling to estimate the functional coefficients involved in the model. Asymptotic properties of the proposed estimator are established, that show the proposed estimator is indeed more efficient than the maximum local partial likelihood estimator. A simulation study was conducted to demonstrate how much the proposed estimator improves the efficiency of the maximum local partial likelihood estimator when sample size is finite. In reality, we do not know whether censoring is informative or not, and a cross-validation based criterion is proposed to check whether the censoring is informative or not. Finally, the proposed varying-coefficient proportional hazard function, together with the proposed estimation method, is used to analyse the first birth interval in Bangladesh, leading to some interesting findings.