# Analysis of variance with two empty groups: alternative?

I have treated a species with two treatments:

• salinity: five different concentrations

• a pollutant (control and two different concentrations)

organisms grown with the two lowermost concentrations of salinity have been treated only with two concentrations of pollutant (control and the lowest one).

Trying to make a two-way ANOVA the software (PAST), as espected, replies that "one or more group is empty". I think it refers to the fact that the highest concentration of the pollutant has not been used with the two lowermost salinities*.

QUESTION: does an alternative test other than two-way ANOVA exist to verify the possible statistical significance of the differences observed and that can be used with such cases where one or more groups are empty?

*: it is not a mistake in th designation of the experiment. It could not be tested.

Instead of trying to force your data into a standard ANOVA framework, approach this as a multiple regression analysis instead.

You specify salinity and pollutant concentrations as the independent variables in the model, along with an interaction between them (the product of their values). The multiple regression framework allows you to treat those as continuous predictors instead of as levels of categorical predictors and gets around the "empty cell" problem.

I'm not familiar with PAST software, but most statistical software can handle multiple regression.

One warning: ANOVA and standard multiple regression assume that you have a continuous outcome variable. If your outcomes are relatively small numbers of counts of organisms, or fractions of organisms that die, you probably should be using a generalized linear model that is appropriate for your type of data. You would examine the individual salinity and pollutant variables and their interactions in the same way as for standard multiple regression, but the analysis takes into account the variability expected in count or binary (alive/dead) outcomes rather than assuming a normal distribution of unexplained variance.

• First of all thanks. I did the multiple regression and it gave results that are possibly interesting. I also tried the generaized linear model (the dependent variable is a percentage and it has N=39) but with PAST it seems possible with one independent variable at a time. Is that plausible? Sorry if these questions may seem banal. Jun 8 at 21:19
• I have read that multiple regression needs to meet five assumptions: statology.org/multiple-linear-regression-assumptions So I have verified that data do not meet these assumptions. So I wonder if an alternative solution exists. Jun 9 at 7:34
• @dandelion it depends on which assumptions were violated and how bad the violations are. If you didn’t meet the assumptions of multiple regression you maybe didn’t meet the assumptions of ANOVA, either. That’s particularly true if you need a generalized multiple regression model but use standard multiple regression instead. The page you linked to has some suggestions. A search on this site will turn up many more. Or ask a new question with specific details about your data and what you tried so far.
– EdM
Jun 9 at 12:56