The state law requires that Wisconsin’s minimum wage “shall not be less than a living wage.” …
A living wage is defined under the law as “reasonable comfort, reasonable physical well-being, decency and moral well-being.”
…”The department has determined that there is no reasonable cause to believe that the wages paid to the complainants are not a living wage,” Robert Rodriguez, administrator of DWD’s Equal Rights Division, wrote in the denial letter.
The letter, released yesterday, is here. Interestingly, I have not been able to locate this document on the DWD website. In addition, I have not been able to locate any statistical or other quantitative analysis justifying this assessment (as of 10PM Central, on 10/7).
The standard critique — that there would be substantial job loss as a consequence of the a minimum wage hike, as argued by the Wisconsin Restaurant Association  — were trotted out. As I have noted before, the theoretical effects can go either way, and there is some evidence that the minimum wage increase effects are either small negative, or even possibly positive, on employment. 
Note that the Federal poverty threshold in 2014 for a two person household is $15730/year. If one were to work 40 hours/week for 52 weeks/year at the Wisconsin minimum wage, then gross income would equal $15080.
Update, 10/8 1:40PM Pacific: The irrepressible Rick Stryker observes that the Doyle administration determined $7.25 minimum wage as consistent with a living wage in July 2009. Well, lots has changed since 2009, including the fact that the price level has risen — a fact judiciously omitted by Mr. Stryker. Here is the minimum wage in 2009$ (and keep in mind, the CPI is plutocratic, so that it applies to upper income households). For instance, the CPI for food and beverages, which would be higher weighted for lower income households, has risen 11.9% vs. 10.9% for CPI.
Figure 3: Minimum wage in $/hour (blue) and in 2009$ (red). Deflation uses CPI-All.
Update: Here is the the 2009 Wisconsin Legislative Council Informational Memo regarding how DWD raised the minimum wage:
The Department of Workforce Development (DWD) is authorized by state statute to change the
state minimum wage through the rule promulgation process.
Despite some people’s assertions, the legislature did not pass legislation to effect this change.
You can also read more of the history of DWD’s decisionmaking here.
Update, 10/10, 9:25AM Pacific: The Legislative Council has provided a memo which outlines the authority granted to agencies to promulgate rules, including those relating to the minimum wage. The relevant portion:
The Wisconsin Constitution delegates to the legislative branch the power to make laws and assigns to the executive branch the power to enforce and execute laws. Although the legislature is highly capable of adopting public policies and setting the agenda for the state, it does not
always have the administrative expertise or resources to implement public policies. Also, the implementation and enforcement of the laws may require consideration of details and the adoption of procedures that cannot be foreseen when legislation is enacted. For these reasons the legislature has delegated to the executive branch the power to promulgate—to make known and put into effect—administrative rules.
Administrative rules have the effect of law and, as stated in the Wisconsin Statutes, are “issued by an agency to implement, interpret, or make specific legislation enforced or administered by the agency or to govern the organization or procedure of the agency.” State agencies are granted rule-making power to actualize the legislature’s public policy decisions. The administrative rules are compiled in the
Wisconsin Administrative Code…
As the history of minimum wage changes in Wisconsin makes clear, DWD can make the changes on its own, subject to clearance by the Legislature. Hence, assertions that the legislature must on its own initiate minimum wage changes, or change the determination of what constitutes a living wage, are incorrect. Authority is delegated by the Legislature to DWD (and the Legislature can override DWD).