What statistical method can be used in this case?
I think the issue here is simply the definition of "ancova". Under one definition, there is that assumption of homoscedascity of slopes. But you could simply do the analysis as a "general linear model" and not call it "ancova". In this case, you include an interaction term of the covariate and the categorical independent variable. Something like Y = IV + Covariate + IVxCovariate + errors.
I thought this article covered the issue well: https://www.theanalysisfactor.com/ancova-assumptions-when-slopes-are-unequal/
Sal Mangiafico
How can I interpret the IV effect on the Y, in this case?
Rather than having a single F-test to compare the group means with the covariate equal to its mean, you will have to use multiple F-tests to compare the group means at selected values of the covariates--e.g., at low, medium and high values of the covariate. Many authors suggest the following:
- Low = mean - SD
- Medium = mean
- High = mean + SD
But these are just suggestions, and are not carved in stone. You could just as well use the quartiles of the covariate distribution, or any other values that make sense in the particular context. And you could look at more than 3 values of the covariate if it makes sense in the context.
If you say what software you use, someone may be able to give more specific advice about how to do this.
HTH.
If you conduct the analysis of variance for the model I described, you will get an F-test for the categorical IV. This may be satisfactory for your purposes, or you may want to try something like what Bruce Weaver suggested.
Let's start with a hypothetical example. Let's say we're measuring the weight of male and female individuals of some species and we're using the age of the individuals as the covariate. For the data:
Male = (100, 99, 104, 108, 107, 110, 112, 111, 120, 115)
Female = (100, 104, 101, 102, 105, 103, 105, 106, 107, 107)
Ages = (20, 23, 26, 30, 35, 40, 45, 50, 55, 60)
We get the following anova (type II SS):
Sum Sq Df F value Pr(>F)
Sex 105.800 1 25.704 0.0001137 ***
Age 300.284 1 72.955 2.348e-07 ***
Sex:Age 68.260 1 16.584 0.0008865 ***
Residuals 65.856 16
And the included plot.
From the anova results we can see that overall males weight more than females. But, really, because the interaction is significant, it's the interaction that is really interesting. To me, presenting the plot really tells the story.
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