Why DC cannot shoot a picture with shallow depth of field

I carry an Optio S3 to make ad hoc photography.

According to the specification, the tiny camera has an aperture f2.6 - f4.8

However, even at an aperture f2.6, I can never shoot a picture with sufficient shallow depth of field (DOF), except in macro mode of course.

I start to ask why.

The most authoritative reference (but hard to understand) about Depth of Field is at Wiki (link http://en.wikipedia.org/wiki/Depth_of_field)

It gives complicated formula to show the relation of DOF vs. format size (though roughly speaking, DOF is inversely proportional to format size). Since my DC has a small CCD, the DOF is less shallow than the conventional SLR.

However, I try to associate the problem with APS-C or Four Thirds or micro Four Thirds.

In these cases, usually, we will use a crop factor to represent the effective focal length at full-frame (FF, at 35mm) format size . For instance, even you use a lens with 18mm on DSLR with crop factor 1.6, its effective angle of view is just same as 28.8mm at FF.

Please note that aperture is actually stated in F-number (the ratio of focal length to effective aperture diameter).

Then why not use similar the crop factor to calculate the effective DOF?

For instance, my Optio S3 has the focal length 5.8mm (equivalent to 35mm at FF). This means that the crop factor is about 6X.

Then even at aperture f2.6, the "equivalent" aperture at FF is already 6x2.6 = 15.6

This explains why I cannot shoot at shallow DOF even with the largest aperture (smallest F-number) because its FF-equivalent aperture at 15.6!