When looking at the Federal debt there are a lot of ways to interpret the data. You can look at the dollar amount which is currently ~16 trillion or you can analyze it as a percent of the GDP. Are there other ways of interpreting debt ratios than the two mentioned? Is there a best or standard approach when analyzing debt at a governmental level?
There are definitely other ways.
As any time when dealing with time series data, a good analysis would be velocity - how fast is the debt changing, and what are the rates of change and how do THOSE change.
Additional analysis is separating one-time expenditures from recurring expenditures (e.g. debt incurred as a result of paying for WWII is a one-time event, whereas debt incurred as a result of paying for government pensions is a recurring one). This is a difficult analysis since the debt is not directly attributed to expenditures; but the math can be done (it's done for performance attribution of financial portfolios).
There are also non-financial facets of analyzing debt:
Geopolitical implications (debt to China vs. debt held by US public vs. debt held by UK government).
Debt reduction analysis with an eye on high level root cause (as someone wisely noted in another answer about which president had biggest debt reduction, very frequently the root causes of budgetary changes are inherited from prior administrations). For example, large scale defunding of military purchasing may reduce debt short term, but will predictably increase the debt later on when ramp-up is required.