What does it mean when a variable is significant as a binary variable but not as an ordinal variable? Let's say I am running Cox proportional regression model for a variable with 4 levels (strongly agree, agree, disagree, strongly disagree). When I code it as an ordinal variable with these 4 categories, it is an insignificant contributor. However, if I make this variable binary (agree vs disagree), it is significant. Is this still appropriate? When is it okay to convert an ordinal to a binary variable?  How do I go about interpreting this result? Thank you!

This could be a case of the difference between significance and non-significance itself being non-significant.  See this paper: http://www.stat.columbia.edu/~gelman/research/published/signif4.pdf
