(CNN) Taiwan President Tsai Ing-wen announced her resignation Saturday as head of the Democratic Progressive Party after the party suffered a "crushing defeat" in local government elections, a vote with significant implications for the island's relationship with mainland China.

The local elections, as well as a failed referendum on whether the island's Olympic athletes would compete under the name "Taiwan," were both seen as bellwether contests for public support on independence.

Voters Saturday expressed dissatisfaction with Tsai's harder-line approach to relations with the Beijing government and continued sluggish economic performance.

Analysts say that most support the status quo, reaping economic benefits from mainland China without being governed by it.

Tsai's Democratic Progressive Party has traditionally leaned in favor of formal independence from China, compared to Taiwan's other main political party, the Kuomintang, as the Nationalists are known locally.

Cross-strait relations have cooled under Tsai's leadership and the Chinese government under President Xi Jinping has become more tenacious in asserting its claim over the island, which Beijing considers a breakaway province that must be reunited with the mainland.

Taiwan's Central News Agency reported Tsai apologized to supporters for the ruling party's "disappointing performance" in a speech at its headquarters, and she took full responsibility for the party's losses.

Although she resigned as the party head, Tsai -- the first woman to be elected President of Taiwan -- will serve out the remainder of her term. But the Saturday vote could hamper her chances at winning another term during the next presidential election, scheduled for January 2020.

Tsai said while her administration is moving in the right direction, the election results were proof that the Taiwanese people are setting a higher standard for their government.

The traditionally pro-independence DPP is expected to undergo a "major reshuffle" soon, Tsai said, to prepare for the challenges ahead.

Premier Lai Ching-te echoed Tsai in a Facebook post, saying the election results indicated the public's dissatisfaction with the government's performance.

Lai had offered to resign earlier Saturday and take ownership of the party's defeat, but Tsai asked him to stay on as premier to ensure the continuity of the government's policies and initiatives.

Tsai was elected President in a landslide victory in January 2016 after a political career mostly as an outsider. She joined the Democratic Progressive Party in 2004 and was its chairwoman by 2008. She lost her first bid for the presidency in 2012 and resigned as the party head before taking the mantle back up in 2014.

Tsai is a lawyer by training and studied at National Taiwan University before continuing her studies by earning a master of law degree from Cornell University and then a doctorate from the London School of Economics.

Ma Xiaoguang, a spokesman for China's Taiwan Affairs Office, said on Sunday: "The results reflect Taiwan people's strong wishes to continue enjoying benefits of peaceful cross-strait relations and to keep improving the economy and their livelihood.

"We will continue to firmly oppose 'Taiwan independence' forces and their activities and unite the people of Taiwan to take a path of peaceful development in cross-strait relations."

Regarding the decision striking down the Olympic name change, Ma said "the result shows the lack of public support for the gambit of using the interest of Taiwan athletes as a wager."

Taiwan currently competes using a special Olympic flag and anthem and participates under the name "Chinese Taipei," as decided under a 1979 resolution by the International Olympic Committee.

According to Taiwan's Central News Agency, a US State Department spokesperson said in an email to the news agency, "The United States congratulates the people of Taiwan for once again demonstrating the strength of their vibrant democratic system through a successful round of elections."

The spokesperson said the United States looks forward to working with both old and new counterparts "to continue our fruitful cooperation on a wide range of issues of mutual concern," CNA said.

Despite the lack of formal diplomatic ties, Taiwan remains an important American ally in the region, and the Trump administration has sought closer ties between Washington and Taipei.

America's alliance with Taiwan has long been lambasted by officials in Beijing.

Decisions on same-sex marriage

Also in Saturday referendums, voters in Taiwan opposed changing the current civil law to accommodate same-sex marriage , dealing a heavy blow to the island's LGBT community and its allies.

The three referendum questions initiated by groups that opposed marriage equality passed, while those put forth by same-sex marriage advocates did not.

For instance, the majority vote was yes on a question that asked, "Do you agree that Civil Code regulations should restrict marriage to being between a man and a woman?"

The votes came after Taiwan's high court passed a resolution in May 2017 ruling it was unconstitutional to ban same-sex marriage.

Taiwan lawmakers had a two-year deadline to enshrine marriage equality into law, but a deadlock gave opponents an opportunity to put the issue to the public in the form of referendum questions.

A majority of the island's voters, however, did say "yes" to a referendum question -- also put forth by opponents of marriage equality -- that asked if the legislature should enact a new law to protect the rights and interests of same-sex couples.