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I have plotted Kaplan-Meier curves of survival times for cancer patients with above and below average levels of Copy Number Variation (CNV), and performed a log-rank test for each cancer type.

I have also fit a univariate Cox Model for each cancer type, with CNV levels as the variable.

There are several cancers which reach statistical significance in the Cox Model and not for the Kaplan-Meier curves. There is one cancer where the situation is vice-versa.

What could be causing this effect and which test should I trust?

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Differences in "significance" between non-parametric tests (no assumptions about underlying distributions of variables, typically just based on relative ranks) and parametric tests (with assumptions about distributions, for example that they follow a normal distribution) are frequently found in other types of statistical analyses, with non-parametric tests typically having less power (if the assumptions of the parametric test hold). Here, K-M/log-rank tests are non-parametric (just looking at which events occur first), while the Cox model is semi-parametric (unspecified baseline hazard, but hazard ratios among groups assumed to be strictly proportional). My guess is that you are on the borderline of having enough events in your data sets to document a relation of survival to CNV.

Look at the shapes of the K-M curves to see what's going on, and do appropriate tests that the Cox proportional-hazards assumption is being met.

Which to "trust" depends on the use you intend to make of your analyses. If you are simply doing these analyses to determine whether to follow-up with an independent study, use your scientific judgment along with the hints these results provide. If you are intending to publish these results, you should probably trust neither on its own without further analyses. Studies such as this are notorious for finding "significant" results that are very dependent on the peculiarities of a particular data set. This site has many pages on how to overcome these limitations with techniques like cross-validation and bootstrapping; this page and its "Related" pages are good places to start.

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