Proportion / Percentage Regression Analysis methods I am attempting to help a colleague with data analysis and I'm a bit swamped as to what is appropriate. We are trying create a formula to explain why children in various provinces have stunted growth and we predict it is due to Vitamin A deficiency and damage from Diarrhea events (Simplified of course). 
I have proportional data for nine provinces on:


*

*the percentage of stunted children  

*the percent that have received Vitamin A recently

*the percent that have recently had diarrhea.


Example such as: 
Province 1   ---- 14.5% Stunted-----75.4% Vitamin A intake -----14% Diarrhea     

Since the data is not normal and also inherently bounded by 0 and 100%, I assume I should not be using linear regression. This website suggests using a probit or two-limit tobit model, but I've found suggestions to use GLMM, ancovas, or logistic regression.
I use normally use SPSS but can use R, JMP if needed. 
Any suggestions would be greatly appreciated. 
 A: Carry out a multiple regression with a logit transformed dependent variable. This may be able to identify what factors are associated with stunting, but there are limitations to how you can use and interpret the output.
The typical way to analyse this data would be to use multiple regression with the percentage of children with Vitamin A deficiency and the percentage of children that recently had diarrhoea both as predictor variables, and to use the percentage of stunted children as the response variable.
Because classical regression assumes that the errors of the response variable are normally distributed, convert the percentage of stunted children to a decimal (57% is 0.57) and then apply a logit transformation ( ln (p / ( 1- p ) ), where p is your proportion. (A logit transformation is very similar to a probit transformation.)
There is no need to transform the predictor variables on the basis that percentages are not normally distributed, only the response variable.
Regression, and other ANOVA based techniques, have a whole range of assumptions that must be met if the interpretation of the statistical outputs is to lead reliable conclusions. A nice guide to preliminary data exploration is given here http://onlinelibrary.wiley.com/doi/10.1111/j.2041-210X.2009.00001.x/full, and an explanation of regression diagnostics is given here http://www.nature.com/nmeth/journal/v13/n5/full/nmeth.3854.html. Both of these should help you identify whether the results of the multiple regression are useful.
With your data, you will not be able to explicitly test whether the relationships between Vitamin A deficiency, diarrhoea and stunting differ among the provinces. Of course, if there are one or two outliers then this could be taken as evidence that there are different relationships in these provinces compared to all the others.
If you do find a significant effect of either factor and want to use the regression model, then it only applies to the prediction of the proportion of stunting within a state, not to the individual children within that state. It is not clear how the results could be used to predict a child's risk of stunting given their nutritional and health history.
The multiple regression will treat each of the risk factors as being independent and additive in the effect on the risk of stunting, but the risks may not be additive. Again, the presence of outliers in the regression may indicate that this is occurring.
With the example you have presented, there is a strong risk that both your predictors are correlated which, firstly, decreases the likelihood that you will detect the effects of the predictors, and more importantly, will lead to biased estimates of the regression coefficients.
Regression techniques also assume that the predictors are measured without error. I assume this is not the case here. The use of classical (Model I) regression techniques on data where all variables are subject to measurement error is that, again, regression coefficients are biased. Some more information in this problem is given here https://www.ine.pt/revstat/pdf/rs100104.pdf.
Sokal and Rohlf (Biometry, Third Edition, Chapters 14, 15 and 16) explore the conceptual confusion between correlation (which your problem is, strictly speaking) and regression. Despite this, multiple regression is probably a sensible starting point and may give some useful indications of factors associated with stunting.
