You mention dependent variables, it means there are independent variables in your data. If your target is find the relationship among the dependent variable and use linear regression modeling, then you do not need do the transforamtion at begining. Please NOTE:

The assumption of identical of indepednet normal distribution is for residuals rather than the dependent variable. The raw data of outcome might not be normal or unimod.

If we only have y and x: If the independent variable X is binary and has significant effect on the dependent variable Y, the dependent variable will be bimodal. It can be easily shown by simulation.

You need do the regression first,  plot the residual with prediction and independent variable, to check if transformation is needed.

If homogeneity of variance holds, but not random distributed around zero, varies by independent factor, make transformation on independent factor.

If homogeneity of variance doesn’t hold, transform dependent variable.

If bimoal is shown, please check if there is an extra binary predictor, which is not in the model.

Please give more details on spatial factor.  State, or ZIP code, Region, how many levels. Say if you only have a few level, you need check both mean and variance. If variance is not equal across them, weighted analyses by 1 over variance can be used. …

GEE is for: if there are several records for each individual. Say the testing score for each student in several exams, we can assume the students are independent; however the exam records for each student are not. If we want use all exam records in the model, those record are correlated  with unknown structure,  we should use GEE.

Best,

Yuanzhang Li

A truly bimodal variable must have each mode addressed separately. You must identify the contribution from each mode.

Dear Daniel,

Although I do not understand your problem, the fact you mention spatial correlation makes me to assume you are dealing with a dicotomous variable sampled in space points. There are kriging techniques for dealing with these problems. A keyword may be "indicator kriging". See for intance a couple of papers I co-authored

Tolosana-Delgado, R., V. Pawlowsky-Glahn, and J. J. Egozcue, 2008a: Indicator

kriging without order relation violations. Mathematical Geosciences,

40(3), 327-347.

Tolosana-Delgado, R., V. Pawlowsky-Glahn, and J. J. Egozcue, 2008b: Simplicial

indicator kriging. Journal of China University of Geosciences,

19(1), 65-71.

On the other hand, the origin of two modes in a probability density may be diverse. One frequen case is that the distribution is produced by a mixture of populations. Another way of obtaining two modes appears when the support of the variable is restricted to an interval. The case of the beta distribution (one dimensional case of the Dirichlet distribution) is typical: the variable is supported in (0,1), for large variances, the probability density has two modes in the extremes of the interval. If this is the case, the problem is not the fact of the two modes, but the scale you are considering for your data. In the univariate case in (0,1) you can take logit transformation, i.e. x_i is transformed into log[x_i/(1-x_i)] which is likely to be distributed as an unimodal variable (with good luck, it can be approximately normal --logistic-normal distribution or normal on the simplex).

If your distribution is multivariate, you have to be very careful with the sample space of the data. Are the data compositional? If so, there are transformations which are generalisations of the logit transform that can solve your problem, see Isometric Log-Ratio (ILR) transforms. You can get information from

Buccianti, A., Mateu-Figueras, G. and Pawlowsky-Glahn, V., (eds) (2006) Compositional Data Analysis in the Geosciences: From Theory to Practice. Geological Society, London, Special Publications

Pawlowsky-Glahn, V. and Buccianti

A. (eds.), (2011) Compositional Data Analysis: Theory and Applications, Wiley,

Chichester UK.

Mateu-Figueras, G., V. Pawlowsky-Glahn, and J. J. Egozcue, 2013: The

normal distribution in some constrained sample spaces. SORT - Statistics

and Operations Research Transactions, 37(1), 29-56.

Three questions: 1) Is it possible to transform a bimodal variable into normal or other 'more friendly' distribution variables? 2) If not, what statistical analysis can be done for a bimodal dependent variable to include spatial correlation? 3) Are Generelized Estimating Equations fit for this purpose? My answers are:

1) No If you want to transform it into a normal "friendly distribution" it will be incoherent.

2) It is necessary to know the sample and if it comes from two different sectors. For example if human height is measured and there are two sectors: male and female, each one has its own mode, and if you want to mix them you need the fractions of males and females in real society and each sector data set. Then the mixing produces a bimodal distribution which can not be normal, but it may be analized.

3) I do not know GEE well, so I can not anwer. In any case if you may give a data set of 20 or 40 points it is possible to analyze and graph it. What graphs do you want to obtain?

Probably this does not help you too much, so excuse me because questions are too general.

emilio

 You mention dependent variables, it means there are independent variables in your data. If your target is find the relationship among the dependent variable and use linear regression modeling, then you do not need do the transforamtion at begining. Please NOTE:

The assumption of identical of indepednet normal distribution is for residuals rather than the dependent variable. The raw data of outcome might not be normal or unimod.

If we only have y and x: If the independent variable X is binary and has significant effect on the dependent variable Y, the dependent variable will be bimodal. It can be easily shown by simulation.

You need do the regression first,  plot the residual with prediction and independent variable, to check if transformation is needed.

If homogeneity of variance holds, but not random distributed around zero, varies by independent factor, make transformation on independent factor.

If homogeneity of variance doesn’t hold, transform dependent variable.

If bimoal is shown, please check if there is an extra binary predictor, which is not in the model.

Please give more details on spatial factor.  State, or ZIP code, Region, how many levels. Say if you only have a few level, you need check both mean and variance. If variance is not equal across them, weighted analyses by 1 over variance can be used. …

GEE is for: if there are several records for each individual. Say the testing score for each student in several exams, we can assume the students are independent; however the exam records for each student are not. If we want use all exam records in the model, those record are correlated  with unknown structure,  we should use GEE.

Best,

Yuanzhang Li

Have your tried to fit a Gaussian mixture? This is, a convex linear combination of normal densities. It can be done by maximum likelihood by the EM algorithm.