Sense of Community (SOC) is vital to individual and collective well-being. Although social interactions have moved increasingly online, still little is known about the specific relationships between the nature of these interactions and Sense of Virtual Community (SOVC). This study addresses this gap by exploring how conversational structure and linguistic style predict SOVC in online communities, using a large-scale survey of 2,826 Reddit users across 281 varied subreddits. We develop a hierarchical model to predict self-reported SOVC based on automatically quantifiable and highly generalizable features that are agnostic to community topic and that describe both individual users and entire communities. We identify specific interaction patterns (e.g., reciprocal reply chains, use of prosocial language) associated with stronger communities and identify three primary dimensions of SOVC within Reddit -- Membership & Belonging, Cooperation & Shared Values, and Connection & Influence. This study provides the first quantitative evidence linking patterns of social interaction to SOVC and highlights actionable strategies for fostering stronger community attachment, using an approach that can generalize readily across community topics, languages, and platforms. These insights offer theoretical implications for the study of online communities and practical suggestions for the design of features to help more individuals experience the positive benefits of online community participation.