Economies of scale and scope indicates that a large company will in general be more profitable than a smaller one. Is there any statistical material that supports this? Of course, the interesting comparison is to look at profit/income ratio for large companies compared to small ones.
EDIT: So far this question has gotten two answers regarding the factors involved and how they interact, but this question is really asking about statistical material and therefore references to such material.
There is no general theory to support the notion that larger companies will be more profitable than smaller companies. Economies of scale are not always positive, one can have diseconomies of scale too. It is more common to talk about an optimal firm size, even going back to Stigler's (1958) "The Economies of Scale." Intuitively, if economies of scale extended indefinitely, then natural monopolies would dominate all industries in the long run.
A profit ratio, unfortunately, wouldn't quite get at scale economies. Consider, for example, that the denominator of your metric would be profit+cost and that you are trying to get at the cost reduction that derives from scale. Then, you are measuring the size of a company by the exact metric that should be reduced if scale economies exist, so the calculation would be a bit confounded.
It is my understanding that such assessments are usually conducted at the industry level by determining whether the industry is becoming increasingly concentrated among fewer firms over time. (Again see Stigler). If concentration is increasing, there is an implication that, at current firm sizes, there are economies of scale in the industry.
This isn't as rigorous as it should be, but may offer some useful insight into how big and small companies differ operationally.
Putting Apple aside, larger companies tend to sell larger volumes of products (even if they're MRI devices, or turbines) relative to what smaller companies can sell (obviously, in absolute terms as well). They are also able to negotiate volume discounts as well as payment terms. This allows them to finance sales through their supply chain. However, their large direct competitors are able to do the same thing as well. Competitive forces then drive prices down.
Smaller businesses, without these advantages of scale, tend to have to charge higher margins since they have to pay directly (and, if their clients are large businesses, finance the sale). Small businesses still have higher proportional costs of operation. Sadly, my reference here is a study I performed for the South African Revenue Service about ten years ago, and not available online. However, the time taken by a small business to manage admin, tax, HR is a greater proportion of revenue than for larger companies.
If the small business is a start-up with big investment from venture finance, then they could subsidise their selling price, run at a loss and try and gain scale. Funnily enough, there is a fantastic article on this by Joel Spolsky (Ben and Jerry's vs. Amazon)
For the average highly-competitive smaller company, the best choice is to chase design/quality/premium markets in order to justify the higher margins they have to charge.
And that's what makes Apple interesting as a case study. They were a small company in the presence of giants (Intel, Microsoft, IBM). They were "forced" to concentrate on design and premium markets in order to justify their need for higher margins. It almost didn't work but then they broke through. Now they're in the unique position of having gained scale but are still small enough relative to other electronics manufacturers to continue charging that premium (by volume their sales are still relatively small but their margins make them a giant).
This type of variation from market to market makes developing some sort of generalised solution very unlikely but the general requirement holds: that smaller companies must charge higher margins in order to create equivalent profits to larger companies which must gain scale through volume.
Small companies could have growth prospects. Large companies may not have that many. So look at ROE of companies by quatile to determine which companies have better growth.
