Identifying Break-Points in Binomial Responses (Alternative to Multiple Chi-Squared Tests?) So I have a research question whereby I have subjects' age and a yes or no response to a question. Ages range from 45-90 approx. I am currently looking at whether age has an affect on these yes or no responses (I have an underlying hypothesis as to why this may be the case).
I have run chi-squared tests and it appears that 55 seems to be a significant age as if I run >55, <= 55 there is no significance and if I run >=55, < 55 there is significance. I was wondering if there is a method I can use to better tease out where these breakpoints in the data lie? And what should I be looking out for?
For instance,
[1] "Test for > 55, <= 55"
     Yes No
>55   69 50
<=55  28 16
p-value = 0.5139

[1] "Test for >= 55, < 55"
     Yes No
>=55  74 60
<55   23  6
p-value = 0.01658

Thanks.
 A: When 55 is taken as a class by itself, the P-value is also high
(see below), so I suspect the effect is real.
I don't know your reasons for splitting around 55, but as a
general rule it would be better to have more age groups
and to have them more even spread--perhaps three groups
with approximately the same number in each. I suspect
you could even have four age groups without spreading
the data too thinly. Ideally, no group should be so small
that the expected mean squares in all cells are above
five.
In explaining significance, you should look for cells that
have contributions to the total chi-squared value that
are above 2 (or especially 3). For example, in my printout
below the main 'action' is that those 55 and older have
many fewer No's than would be expected if age and opinion
were independent attributes of the population.
 Expected counts are printed below observed counts
 Chi-Square contributions are printed below expected counts

            Opinion
           ---------
  Age     Yes     No  Total
  -------------------------     
  < 55     69     50    119
        70.82  48.18
        0.047  0.068

    55      5     10     15
         8.93   6.07
        1.727  2.538

  > 55     23      6     29
        17.26  11.74
        1.911  2.808
 --------------------------
 Total     97     66    163

 Chi-Sq = 9.099, DF = 2, P-Value = 0.011

Addendum: The best approach now would be to pick 3 or 4 age intervals with roughly equal total counts in each. In your case, maybe four intervals of about 40 students each. Maybe three intervals one $\ge 55$ and about a 50-50 split of younger subjects (roughly 53-55 subjects per group.). 
You say you had reasons from the start to expect an effect a priori and those reasons might inform the choice of cut points. In general, deliberately choosing cut points during analysis to get the smallest P-value is 'data dredging' and may even be deceptive. 
