Using ANOVA on percentages? I have a table  with four groups (4 BMI groups) as the independent variable (factor). I have a dependent variable that is "percent mother smoking in pregnancy". 
Is it permissible to use ANOVA for this or do I have to use chi-square or some other test? 
 A: If you choose to do an ordinary ANOVA on proportional data, it is crucial to verify the assumption of homogeneous error variances. If (as is common with percentage data), the error variances are not constant, a more realistic alternative is to try beta regression, which can account for this heteroscedasticity in the model. Here is a paper discussing various alternative ways of dealing with a response variable that is a percentage or proportion:
http://www.ime.usp.br/~sferrari/beta.pdf
If you use R, the package betareg may be useful.
A: There is a difference between having a binary variable as your dependent variable and having a proportion as your dependent variable.


*

*Binary dependent variable: 


*

*This sounds like what you have. (i.e., each mother either smoked or she did not smoke)

*In this case I would not use ANOVA. Logistic regression with some form of coding (perhaps dummy coding) for the categorical predictor variable is the obvious choice if you are conceptualising the binary variable as the dependent variable (otherwise you could do chi-square).


*Proportion as dependent variable: 


*

*This does not sound like what you have. (i.e., you don't have data on the proportion of total waking time that a mother was smoking during pregnancy in a sample of smoking pregnant women).

*In this case, ANOVA and standard linear model approaches in general may or may not be  reasonable for your purposes. See @Ben Bolker's answer for a discussion of the issues.


A: It depends on how close the responses within different groups are to 0 or 100%.  If there are a lot of extreme values (i.e. many values piled up on 0 or 100%) this will be difficult.  (If you don't know the "denominators", i.e. the numbers of subjects from which the percentages are calculated, then you can't use contingency table approaches anyway.) If the values within groups are more reasonable, then you can transform the response variable (e.g. classical arcsine-square-root or perhaps logit transform). There are a variety of graphical (preferred) and null-hypothesis testing (less preferred) approaches for deciding whether your transformed data meet the assumptions of ANOVA adequately (homogeneity of variance and normality, the former more important than the latter). Graphical tests: boxplots (homogeneity of variance) and Q-Q plots (normality) [the latter should be done within groups, or on residuals].  Null-hypothesis tests: e.g. Bartlett or Fligner test (homogeneity of variance), Shapiro-Wilk, Jarque-Bera, etc.
A: You need to have the raw data, so that the response variable is 0/1 (not smoke, smoke).  Then you can use binary logistic regression.  It is not correct to group BMI into intervals.  The cutpoints are not correct, probably don't exist, and you are not officially testing whether BMI is associated with smoking.  You are currently testing whether BMI with much of its information discarded is associated with smoking.  You'll find that especially the outer BMI intervals are quite heterogeneous.
