Get overall tendency in the dependent variable, beyond the effect of the independent variable Hypothetical data-set:
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"?
 A: Everyone has to be (like? see?) one of the four colours, so the notion of a colourless "baseline happiness" isn't meaningful.† Whether you consider blue, say, as a reference level, & describe the effects of each other colours as a deviation from that, or consider the effects of each colour as a deviation from the mean proportion happy of 0.61, is an arbitrary choice resulting in substantively equivalent models. (See e.g. UCLA: Statistical Consulting Group, R Library: Contrast Coding Systems for categorical variables for some commonly used schemes.) So if someone says the proportion of  happy people is high just because a lot are blue & red, you have to ask what frequencies of blue & red they're contrasting the observed frequencies with, & why.
When you come to compare the happiness of different groups, including a dummy variable for "group" in the model does allow you to talk about something useful: the coefficient for that variable describes a difference between the groups that isn't attributable to their being a different mix of colours.
† Or if they don't have to, then you simply haven't measured the colourless "baseline happiness" when such people aren't in your sample..
