Small-sample properties of the likelihood ratio test, the Wald test and the score test about significance of the effect in the Cox proportional hazards model for the right-censored and left-truncated data are investigated. These are large-sample tests, and, therefore, these are only approximate tests and they do not necessary maintain chosen significance level. Consequently, the p-value can be inaccurate as well. Higher order approximations of the likelihood function based on the Barndorff-Nielsen formula and the Lugannani-Rice formula are used in order to improve the accuracy of statistical inferences. The accuracy of these tests together with proposed approximations are compared by means of simulations under conditions of decreasing the sample size, and increasing proportion of right-censored and left-truncated data in the Cox model with the exponential and the Weibull distribution of the baseline hazard function. The results show that higher order approximations based on the Lugannani-Rice and the Barndorff-Nielsen formulas in combination with the Wald statistic improve the accuracy.