How do you handle a situation where your analysis contradicts what the client believes or wants to hear?

Verify before you present. This is the decisive step. Check the data definitions, the date ranges, the segmentation, and whether the pattern holds across cuts. An unwelcome finding that turns out to be an error destroys credibility permanently, and every subsequent finding is discounted.

Then present it properly:

• Lead with the evidence, not the conclusion's implications. Walk through how you got there so the client can follow and challenge the logic rather than the messenger.
• Separate observation from interpretation. Be explicit about which parts are established by the data and which are your judgement.
• Frame it forward. "This channel is not returning its cost, and redirecting that spend could fund X" is actionable. "This has been failing for two years" invites defensiveness.
• Give the client the finding privately first where the politics warrant it. Being surprised in front of their own leadership rarely produces a good outcome.

If the client rejects it: propose a limited test rather than escalating the argument. Letting evidence settle it preserves the relationship and usually resolves the disagreement.

Note: What you must not do is soften a genuine finding into ambiguity. Clients pay for an independent view, and delivering only what is comfortable is the fastest way to become worthless to them.