The authors study some aspects of the left tail decreasing (LTD) notion of positive dependence, and of the LTD stochastic order that was introduced by J. Averous and J.-L. Dortet-Bernadet [Can. J. Stat. 28, No.1, 151-157 (2000; Zbl 1066.62527)]. In particular they show (though not by using the following terminology) that if two random vectors satisfy that some conditional distribution based on the first vector is smaller than an analogous conditional distribution based on the second vector, with respect to the likelihood ratio stochastic order, then the vectors are ordered with respect to the LTD stochastic order.