Compare and contrast the use of the Chisquared test, Fisher’s Exact Test and logistic regression when analysing data.
All these tests are widely used in the statistical reporting of data and give a representation of the likelihood that a given spread of data occurs by chance.
The Chisquare(d) statistic is used when comparing categorical data (e.g. counts). Often, these data are simply displayed in a “contingency table” with R rows and C columns. It’s use is less appropriate where total numbers are small (e.g. N <20) or smallest expected value is less than 5.
Fisher’s Exact test is used when comparing categorical data (e.g. counts), but is only generally applicable in a 2 x 2 contingence table (2 columns and 2 rows). It is specifically indicated when total numbers are small (e.g. N <20) or smallest expected value is less than 5.
Logistic regression is used when comparing a binary outcome (e.g. yes/no, lived/died) with other potential variables. Logistic regression is most commonly used to perform multivariable analysis (“controlling for” various factors), and these variables can be either categorical (e.g. gender), orcontinuous (e.g. weight), or any combination of these. The standard ICU mortality predictions are based on logistic regression analysis.
When one is invited to "compare and contrast" things, one is well served by a table structure.
First, the prose form: much of what follows is heavily borrowed from LITFL.
Additional reading can be done, if one wishes to actually understand these concepts.
I recommend the following free online resources:
Additionally, I invite everybody to visit this page, where the author Steve Simon (presumably, somebody qualified in statistics) responds to an email he received which asked him to comment on the differences between a Chisquare test, Fisher's Exact test, and logistic regression.
A statistical test commonly used to compare observed data with data we would expect to obtain according to a specific hypothesis. The chisquare test can be used to test for the "goodness to fit" between observed and expected data.
Another test like the Chisquare test, to compare observed data with expected data.
Now that the prose is finished, let us tabulate the differences and similarities between these tests.
Chi Square  Fisher's Exact Test  Logistic regression  
Application  "give a representation of the likelihood that a given spread of data occurs by chance"  
Specific uses 
Nominal data: large samples 
Nominal data: small samples 
Binary variables 
Advantages 



Limitations 



The ideal reference for this is the BMJ, with their combination of rich statistics info and OldWorld credibility. I link to the relevant sections of their Statistics at Square One, by T D V Swinscow.