Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I have statistics of $100$ questions which can be answered either "yes" or "no":
1) $63.3 - 36.7$ ($63.3\%$ respondents answered "yes" and $36.7\%$ answered "no")
2) $30.1 - 69.9$
...
100) $88.0 - 12.0$

That $100$ answers would be our sample.

Then I ask the same $100$ questions to $101^{st}$ respondent and get new set of answers:
1) yes
2) yes
...
100) no

What I need to do is some how calculate "correlation" value between this respondent and overall sample. Any method will be OK, I just need to get some number. Therefore there may be different "right" answers.

Thanks in advance.

share|improve this question
    
I think what you are asking for is the sample correlation coefficient. –  Rasmus Aug 16 '10 at 12:28
    
This is not, directly, the correlation coefficient, but you can write down some relevant measures using correlation coefficients. Asking on stats.stackexchange.com will get answers from statisticians. –  T.. Aug 17 '10 at 7:14
    
Thanks, I'll continue my little research. Never thought it would require such an immerse into statistics :-) –  Stanislav Shabalin Aug 17 '10 at 17:26

1 Answer 1

up vote 2 down vote accepted

[In short, any of the two quantities will satisfy your need- $d$ (easy to calculate), or a monotonic function of $d$ $P(D\lt d)$ (needs bootstrapping to calculate). Both share the property that smaller they are, bigger the chance that new point comes from surveyed data. are described below. For a detailed discussion, read on.]

The problem can essentially be reformulated as a statistical hypothesis testing problem, and we need to test if the new observation comes from the surveyed population.

I will outline the construction of one such test. For this I will assume the questions are independent.

$H_0$: New observation comes from the population. Assuming independence, the population can be characterized by a vector of probabilities $(p_1,p_2,...,p_n)$ for the $n$ questions. These $p_i$'s correspond to your calculated % of "yes".

$H_1$: New observation does not come from the population.

Suppose the new observation is $B=(B_1,...,B_n)$, where each $B_i$ is 1 if "yes" and 0 if "no" was entered.

A reasonable test statistic would be $D=\sum (B_i-p_i)^2$, which looks at the total squared deviation from the expected answers.

You can then judge if you want to label the new sample as coming from your population by evaluating $P(D\lt d)$ - if this probability is <5%, you can say with 95% confidence that it comes from the sample.

Computing the exact distribution of $D$ under $H_0$ will be challenging, to determine $P(D\lt d)$ you can use bootstrapping - i.e. simulating observations under $H_0$ a million times and checking what proportion of times your simulated $D$ is less than observed $d$.

If you need a 0 to 1 valued 'correlation' like statistic, you can look at $1-P(D\lt d)$.

share|improve this answer

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.