State space models (SSM) for binary time series data using a flexible skewed link functions are introduced in this paper. Commonly used logit, cloglog and loglog links are prone to link misspecification because of their fixed skewness. Here we introduce two flexible links as alternatives, they are the generalized extreme value (GEV) and the symmetric power logit (SPLOGIT) links. Markov chain Monte Carlo (MCMC) methods for Bayesian analysis of SSM with these links are implemented using the JAGS package, a freely available software. Model comparison relies on the deviance information criterion (DIC). The flexibility of the proposed model is illustrated to measure effects of deep brain stimulation (DBS) on attention of a macaque monkey performing a reaction-time task [19]. Empirical results showed that the flexible links fit better over the usual logit and cloglog links.