Kenya's incumbent President Uhuru Kenyatta has taken a strong lead as votes are counted after Tuesday's election.
With 80% of results in he has 55% of the vote against 44% for his rival, Raila Odinga, figures show.
However, the opposition coalition has rejected the partial results, saying election officials have not produced documents to prove their validity.
"They are fictitious, they are fake," said Mr Odinga.
"We have our projections from our agents which show we are ahead by far," he added.
The electoral commission is urging people to wait calmly for full results.
Many fear a repeat of the violence that followed the disputed 2007 election when more than 1,100 Kenyans died and 600,000 were displaced.
"During this critical phase, we urge all Kenyans to exercise restraint as we await official results from the polling stations and indeed as they start trickling in," the commission said.
Voting passed off largely peacefully and some polling stations remained open after the scheduled 17:00 (14:00 GMT) closing time.
Mr Kenyatta is hoping for a second term in office.
To win outright, a candidate needs more than 50% of the vote, and at least 25% in 24 of Kenya's 47 counties. If that threshold is not met, a run-off vote between the top two candidates will be triggered.
Voting for the national and local assemblies has also been taking place.
People started queuing early to ensure they could cast their vote. Long queues could be seen, and video footage at one polling station showed people injured after an apparent stampede.
There was also the failure of some voter-identification equipment and one in four polling stations were apparently without mobile phone coverage meaning that officials would have to drive to the nearest town to send results.
There were reports that one man had been killed in clashes in the Kilifi area.
But there was one heartening moment when a woman gave birth to a baby girl as she queued in West Pokot to cast her ballot. New mother Pauline Chemanang called the circumstances of the birth a "blessing" and called her baby Kura, Swahili for "ballot", according to local radio.
Casting his vote in his hometown of Gatundu, north of Nairobi, Mr Kenyatta said he would accept the outcome of the election.
"To my competitors, as I have always said, in the event that they lose, let us accept the will of the people. I am willing, myself, to accept the will of the people," he said.
Opposition leader Mr Odinga cast his ballot in the Nairobi slum of Kibera.
Speaking outside the voting centre, he told his supporters: "Let's turn out in large numbers and vote."