Interpretation of logistic regression intercept with one dummy coded categorical variable

Question

Feel free to critique my overall approach instead of answering my question directly.

I want to look at bivariate relationships among a binary outcome and multiple predictor variables before conducting multiple regression. EDIT: [The data are multiply imputed] and the categorical predictors have been dummy coded. Therefore, for each nominal categorical variable in the original data, there are k-1 dummy variables, with the omitted category serving as the reference group.

It seemed to me that there was no meaningful "bivariate" relationship between the outcome and a single [dummy] variable from a collection of related variables (this would change the comparison category to "everything else," which could not be the case in a multiple regression). For this reason, I did not use bivariate correlations. Instead, I did a series of "bivariate" logistic regressions including in a single regression all k-1 dummies of a variable. So, like: outcome = race_black race_other (race_white omitted). In any case, the reference category cannot be examined directly since it was omitted in the multiple imputation process and has missing values.

I figured if any coefficient of a binary dummy were significant, I would include the whole variable (group of k-1 dummies) in the omnibus regression.

After laboriously compiling a table of the results of these "bivariate" regressions without including the intercept, it occurred to me that (MAYBE?) the intercept is the "dummy" for the omitted category. If it is significant, is it in fact indicating that the omitted group is different from the mean on the outcome? If that's true, should I include the variable in my omnibus regression even if none of the explicit category variables are significant predictors?

I'm worried that I am thinking wrongly about the comparison groups and meanings of significant coefficients.

Answer 1

I think you are making this hard on yourself. Make sure race is a factor variable so that the software provides the overall $\chi^2$ of association with $k-1$ d.f. for $k$ categories. Coding doesn't affect the value of $\chi^2$. Don't use a stepwise process for making inference about the importance of race. Use the overall "chunk" test as described above, which has a built-in perfect multiplicity adjustment besides being invariant to coding. In R this would look like (for a binary or ordinal logistic model predicting $Y$):

require(rms)
f <- lrm(Y ~ rcs(age, 4) + race)
anova(f)   # 3 d.f. test for age, k-1 for race
# also prints 2 d.f. test of linearity in age
# age fit is restricted cubic spline with 4 default knots

When doing multiple imputation with the Hmisc package aregImpute function or with the mice package, you would substitute the following for the 2nd line above:

f <- fit.mult.impute(Y ~ rcs(age, 4) + race, lrm, impute_object, n.impute=20)

which would adjust the covariance matrix for multiple imputation [n.impute recommended to be the percent of observations that have any variable missing].

Answer 2

Seems an odd thing to be trying to do: see @Frank's answer.

But to answer the question in the title, if you're using the following coding scheme for conditions A, B, & C:

A: $\newcommand{\logit}{\operatorname{logit}}x_1=0, x_2=0$

B: $x_1=1, x_2=0$

C: $x_1=0, x_2=1$

then if you use the model with intercept $$\logit(\pi)= \beta_ 0 + \beta_1 x_1 + \beta_2 x_2$$ you have independent parameters to estimate for each condition:

A: $\logit(\pi)= \beta_ 0$

B: $\logit(\pi)= \beta_ 0 + \beta_1$

C: $\logit(\pi)= \beta_ 0 + \beta_2$

But if you use the model without intercept

$$\logit(\pi)= \beta_1 x_1 + \beta_2 x_2$$

then you're forcing the probability $\pi$ to be $\frac{1}{2}$ in condition A:

A: $\logit(\pi)= 0 $

B: $\logit(\pi)= \beta_1$

C: $\logit(\pi)= \beta_2$

Surely not what you want to do.

[Response to comment: Sorry; I understood "regression without [...] intercept" rather than "table [...] without [...] intercept". In any case, you can see that the intercept $\beta_0$ predicts the log odds for an individual in category A, so testing whether its estimate is significantly different from zero is simply testing whether the probability for an individual in category A is different from $\frac{1}{2}$—not usually an interesting null hypothesis.]

Answer 3

@FrankHarrell explained how to test whether race has an effect. You may also be interested in the size of that overall effect, especially since often it would be very implausible that the null hypothesis is true, that is, that a variable like race has exactly no effect whatsoever. So if we reject the null hypothesis we do not add much to our knowledge; we already knew that the null hypothesis was false before we analysed the data. If we fail to reject the null hypothesis, then that just meant that our dataset was not large enough to detect the effect that really exists and that to is not an overly informative conclussion. Instead a single summary measure of how large the effect of race could add a lot more, and this is what a sheaf coefficient was designed to do. In fact race/ethnicity is one of the examples used in the original article.

Heise, David R. (1972). Employing nominal variables, induced variables, and block variables in path analysis. Sociological Methods & Research, 1(2): 147--173. link