# How to interpret the intercept of a multiple regression analysis after accounting for continuous and categorical variables

I'm attempting to run a regression analysis on a continuous outcome variable y, using a primary predictor variable that represents a treatment (x1, dummy coded with the control group=0, treatment group=1), as well as some other categorical and continuous variables that, for various reasons, I am including in the model as control variables. What I want to do is to be able to take the coefficient result for x1/the treatment group and and interpret it as the mean difference in y between the treatment group and the control group, holding all of the confounding variables constant, using the intercept to represent the mean for the control group.

I understand that the intercept is the mean of y when all X = 0, and thus it represents the mean of y at the reference category for all categorical variables. I want to be able to take the difference for the treatment effect and calculate that difference as a percentage from Y. Here is my code and output:

> model <- lm(y ~ x1 + x2 + x3 + x4 + x6 + x7, data=df)
> summary(model)

Call:
lm(formula = y ~ x1 + x2 + x3 + x4 + x6 + x7, data = df)

Residuals:
Min      1Q  Median      3Q     Max
-111.58  -33.77   -8.21   19.02  447.32

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)  90.2631    10.8291   8.335 1.50e-14 ***
x1Group 2   -31.0268    10.0981  -3.073 0.002433 **
x2HS         11.6441    10.7229   1.086 0.278887
x3yes        -2.1632     9.8294  -0.220 0.826047
x4           -1.3587     0.4511  -3.012 0.002944 **
x6yes        43.9864    11.2346   3.915 0.000126 ***
x7           14.8018     2.8292   5.232 4.39e-07 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 60.73 on 191 degrees of freedom
(1 observation deleted due to missingness)
Multiple R-squared:  0.3439,    Adjusted R-squared:  0.3199
F-statistic:  14.3 on 7 and 191 DF,  p-value: 6.751e-15


In this case, I would want to be able to say that, given the mean of y = 90.26 and the mean difference for x1 = -31.03 (resulting in an estimated mean of x1 at 58.23) the treatment is associated with a 36% difference in y between the treatment and control groups (58.23/90.26=0.64; 1-0.64=36).

My concern is that the intercept coefficient changes based on which reference category I have chosen for the confounding categorical variables. Assuming that the reference category is college for x2/education category, would my 36% reduction only apply to patients with a college education and "no" responses for x3 and x6? For my purposes, the categories are arbitrary. I just need to include them in the model because they're associated with y, and are imbalanced between the treatment and control groups due to some issues with randomization.