Logit model probabilities categorical variable I have run a logit model such that $Y$ ~ $X$, where $X$ is a categorical variable with 4 alternatives. I have dummy coded $X$ so I can perform regression, this means that category 1 is the base category. 
I would like to compare the probability of each alternative $X$ including the base category. 
The point of my research is to select "the worst" level of $X$ in the sense that it is the less likely to happen, or the most negative one. However, I am not sure on how to proceed so I thought on looking at probabilities. 
Any suggestion on how to decide which of the four alternatives of $X$ is "the worst"? 
 A: You should use marginal predictions. The way they work, is that they use the estimated equation to calculate the predicted value of the logit with the other variables held at some value (which is usually the mean or the observed value) for each observation. The logit is then converted into probabilities and that value is averaged for each category in your X variable.  This way you get the average predicted probability for each category of X with all other variables held constant.
The standard errors of the marginal predictions are calculated using the delta method. If the standard error of the predicted logit is:
$$ s_{p_{j}} = \sqrt(x_{j}Vx'_{x})$$
Where $V$ is the estimated variance matrix for the model and $x_j$ is the observation for which you are getting the error.
You can then get the standard errors for the predicted probabilities by:
$$SE = p_j * (1-p_j) * s_{p_{j}}$$
Where $p_j$ is the predicted probability for case $j$.
A: If X is ordinal, you can try applying a PCA transform after the dummy coding. Retain only the first PC coordinates as a single continuous variable instead of the untransformed dummies. Plot the three dummies along the first two PCs first to check whether they follow any interpretable or meaningful order. If so, then this approach, which is slightly different from what you've requested, will provide a seamless impact of each category or even a subgroup of categories on Y. While you will only use the more abstract PC as a regressor, you can always trace back its meaning to the dummy variables.
E.g. If you have 3 dummies taking the values, very happy, happy, unhappy, the reference being very unhappy, then the 1st PC could translate the degree of happiness or unhappiness or other, depending on their order along the said PC.
More details on this approach can be found in Gordon Linoff's Data Mining book.
