p-value in odds ratio and regression I have a few questions about the p-value. 
Firstly: Let's say we have a p-value of 0.52 in a contingency table analysis with a sample size of 60. If the sample size would be 90 with the same sample odds ratio, what would the relative size of the p-value for the chi-square test be, compared to the sample of 60? And why?? 
and secondly: In a regression analysis, if R-squared is 0.28, and we did the analysis again (where sample size stays the same, and R-squared stayed the same) but with more independent variables, what would happen to the obtained p-value ofthe observed test statistic? 
p-values sure are tricky things ;)
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Let's say we have a p-value of 0.52 in a contingency table analysis with a sample size of 60. If the sample size would be 90 with the same sample odds ratio, what would the relative size of the p-value for the chi-square test be, compared to the sample of 60? And why??

I assume you mean in a 2x2 table (since you use 'the' with odds ratio). 
There's simply not enough information to tell -- you haven't specified enough things. So this question is not answerable as stated. It could go up or down.
I'll do an example to show it, but you should try different tables to investigate it yourself.
A p-value of 0.52 (to 2dp) corresponds to a chi-square value of between 0.404060 and 0.423895.
Here's an example that shows that the p-value might go up or down when you keep the OR the same yet increase the total sample size in the table (because of the need for whole numbers I won't get the odds ratios perfect, but they'll be pretty close):
Table A (n=60):
23   9  | 32
17  11  | 28
--------+---
40  20  | 60

chi-squared = 0.4102 (with Yates' continuity correction)
OR = 1.65

Table B1 (n=90):
32  13  | 45
27  18  | 45
--------+---
59  31  | 90

chi-squared = 0.7873 (with Yates' continuity correction)
OR = 1.64

Table B2 (n=90):
42   6  | 48
34   8  | 42
--------+---
76  14  | 90

chi-squared = 0.3176 (with Yates' continuity correction)
OR = 1.65

So in B1 the chi-square went up, in B2 it went down, but the OR was effectively constant. If you're not using Yates' correction, you'll want to find your own examples.
If what I did doesn't fit your situation you left something important out!

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In a regression analysis, if R-squared is 0.28, and we did the analysis again (where sample size stays the same, and R-squared stayed the same) but with more independent variables, what would happen to the obtained p-value of the observed test statistic?

I assume you mean the overall F for "the observed test statistic"
For the relation between $F$ and $R^2$ we have:
$F = \frac{ R^2 }{ 1- R^2} \times \frac{ {n-p} }{ {p-1} }$
where $p$ here is the number of parameters including the constant.
From here you should be able to work out what happens to $F$ when p goes up, but beware ... note that just because F moves in some direction doesn't necessarily mean the p-value does what you expect, because the df also change. I suggest you take some typical examples for n and p and then consider what happens for n very small (but still bigger than the increased p) -- it might just be possible to make the p-value go the other direction.
