There's a dependent binomial variable 'happiness', with $0 = unhappy$ and $1 = happy$. Then there's an independent categorical variable 'color' with the levels $blue, red, green, pink$.
We know that each color has a strong influence on the level of happiness, and we can measure that. Imagine that $blue$ and $red$ gave more happiness and $green$ and $pink$ less happiness. But now someone says "there's an overall tendency towards unhappiness in this data, beyond and in addition to the effect of color". How can I test that?
A clarification of what I have in mind:
In the hypothetical data set above, say that the average happiness $=0.61$. At the same time, however, this is because there just happens to be a lot of $blue$ and $red$ among the colors, which we know cause happiness. In a different population with the same distribution and effect of colors, the average happiness $=0.72$. The reason why the average happiness in these two populations is different, therefore, must be because their "baseline happiness" is different. If the only information we have is the data set for the population where the average happiness $=0.61$, is there any way to detect this "baseline happiness"?