Your arguments are question begging all the way through: e.g. predation is ‘considered’ to be a minor factor, so evidence concerning predation can be given little weight. Food and habitat are surely much more important, so changes in these factors ‘must’ be main cause of sparrow decline. These are just assertions. My question is why do you believe this? Is your belief based on evidence, or are you just following received wisdom?
Predation will be given as much weight as habitat and food when someone can show a proper correlation based on accurate data.
Having now looked into your methodology in more depth...
Firstly, GBFS data:
This survey relies on peak counts over a long period. You might recall a headline last year re Blue Tits declining by 42% in gardens over 40 yrs? This result was from the GBFS
http://www.bto.org/news-events/press-releases/feeling-blue-garden-acrobat-takes-tumble
Yet every other survey has shown that Blue Tits have increased strongly over the same period:
http://www.bto.org/birdtrends2004/wcrbluti.htm
This is important, as it shows that GBFS can throw up potentially spurious results (What you should have done was confirm the trends using CBC/BBS data before progressing with the model).
Blue Tits are likely to have declined in GBFS due to changes in birdfeeders and food, meaning less birds are visible at one time but NOT that less birds are present in the area overall. So this means that GBFS cannot be relied on to reflect local populations - it can show the opposite population trend to what is actually happening!
Second, you compared urban and rural gardens over time, but your treatment of change (e.g. when rural gardens are built on and become urban) was completely unsuitable - you merely looked at a modern map to see if the garden was urban or rural TODAY, not in 1970. You state that you just lowered the urban threshold in order to account for this. There is a massive problem here, as your study period (1970 onwards) was a period of massive urbanisation (e.g. vast council estate creation, new towns), so it is probable that some of your urban gardens started off as totally rural. Your fudge of lowering the threshold for 'urban' overall does not work, as it still takes no account of this change. You say data was not available to check this - it is. There is ample OS data documenting urbanisation, but you seem not to know about/bothered with it. This is a critical flaw, due to sample sizes (see below).
So that's two big data quality issues. But there is another.
Third, your sample sizes in each Sparrowhawk recovery zone, for rural and urban gardens, are small and heavily unbalanced. E.g. In zone 4 you have about 27 vs 8, in zone 3 about 25 vs 5. Anything less than 30 is considered 'with a pinch of salt', yet most of your samples are below this standard threshold, and you are even unable to say if any of them changed classification over time, which would whittle your samples down even more.
So the data you used:
1. can generate spurious results (that disagree with all other surveys).
2. was not controlled for changes in birdfeeding methods/gardens that may affect peak count (cf Blue Tits).
3. likely contained errors of urban/rural classification, and did not adequately control for landscape change.
4. had small and unbalanced sample sizes.
As such, your patterns of sparrow decline are highly questionable, especially as you attempted to use a complex spatial pattern. In other words, you may have shoe-horned inadequate data into your model, and failed to check if it was accurate.
As such, a model that just generates a correlation is of very limited use, as you cannot be certain that you are correlating 'real' patterns, rather than patterns in the limited and questionable data.