I am currently writing a proposal and am confused regarding which statistical method I should use. I aim to examine the motivation of runners in participating a charity marathon. It will use runners categorized into 8 motivations and 4 types of running groups. Hence, there is two categorization involved. I will know what the 8 motivations and 4 running groups are and I will know the classifications to each runner. I just want to try to relate the motivations and the groups. I am still new to statistics and am confuse if do I use two different statistics or can I use one?
You can do this with the chi-squared test of independence for a 2-way contingency table. You simply collect information on which group a person is from and which motivation they have for a group of runners:
Group Motivation A c B h ... D f
Then you form a contingency table with the counts for every combination:
Motivation Group a b c d e f g h A 3 5 1 9 ... B 1 ... C ... D ...
Then run the chi-squared test. If $p<\alpha$ the results are statistically significant, which means that certain groups seem to be more likely to have certain motivations than others than you would expect from chance alone.
Here are a couple of extra notes:
With so many groups and motivations ($32$ total combinations), you will need a lot of data. A basic rule of thumb is that you want the expected count (not necessarily the observed count) to be 5 in each cell. That means you would need at least $160$ runners to meet that criterion. However, $N=160$ doesn't necessarily mean you would have good statistical power (that is, your results could easily be non-significant even though the groups differ in motivation). Depending on how strong an association you want to be able to detect, you will probably need a lot of data. If you won't be able to get a lot of data, one thing you could try is to see if you can group the motivations (and/or the groups) into a smaller number of more overarching categories before collecting data. For example, maybe motivations
a-dare of a similar type, and so are
e-h; then you would have a $4\times2$ table instead of a $4\times8$ table.
Under the hopeful assumption that your results are significant, you may have trouble figuring out what they mean. Again, this is because you have so many groups and motivations. To enhance interpretation, you could compute the row-wise percentages for each cell (each row adds up to 100%). You could also try plotting the table, some options for plotting this type of data can be found here and here.