What would be the effect of modeling a binary predictor in an OLS model as [-1, 1] instead of [0, 1]? I am using an OLS model to predict a continuous variable using several continuous predictors and one binary categorical predictor. I know that usually binary variables are modeled as [0, 1], but I am curious what the effect of giving a binary categorical predictor values of [-1, 1] would be. I had accidentally done the [-1, 1] coding in the first round of analysis, and after correcting the coding to [0, 1] I saw that the change in coding made the regression result similar but less extreme. I realize I don't fully understand what change this had on the model, and I'd appreciate insight into why it is so important to code the categorical variable as [0, 1].
[Edit] To clarify what I mean by the result being 'less extreme': I am fitting this model separately for each individual in our dataset, and then I do a wilcoxon signed-rank test across the coefficients from the binary variable (the coefficients do not meet the normality assumption for a t-test). The test is significant for the coefficients resulting from the [-1, 1] coding but not for the [0, 1] coding. Naturally I would like the test to be significant but now I feel that I cannot accept this result because it is arbitrarily affected by the coding. Is testing the coefficients resulting from the [-1, 1] coding somehow more or less legitimate? I am not sure which test result to accept, or whether I have approached the problem incorrectly.

Let me provide an explanation for this query with a simpler scenario.  Let's just take one scalar variable as the response variable, and one dichotomous variable as the predictor variable.  If you run OLS regression, you actually are just doing a 2 independent sample t-test (assuming homogeneity of variances between the two groups).  The key is to recognize that the dichotomous variable really is just a group-membership indicator variable.
Now, the results for the null hypothesis statistical test (NHST) are going to be the same as the results for the NHST asking if the slope of the OLS regression is different from zero.  (The t-ratio and p-values will be the same.)
So, here's the key to the OLS approach.  If you run the regression, you get something like
$$y = 12.34 + 5.67 D$$
(extracting the slope and intercept from the OLS output).  Here, I've used $D$ as the predictor variable (instead of $x$) because I want to indicate that it is a dichotomous variable (takes on only 1 of 2 values).
So, ¿what are the slope and intercept?  This is where the coding of {0,1} or {-1,1} comes into play.  If you use the former coding scheme of {0,1}, then the intercept of 12.34 is the mean of the $D=0$ group, and the slope is the difference in the group means.  So, if you plug in $D=1$, you can get the other group's mean from this equation:  12.34 + 5.67 = 18.01.
Now, if you use the latter coding scheme of {-1,1}, then the intercept is the GRAND mean (the mean if you put both data sets together).  In this case, the slope is half the difference between the group.  So, to get each of the groups means, you need to plug in the value for $D$ for each group: $D=-1$ would give a group mean of $12.34 - 5.67 = 6.67$ and $D=+1$ would give a group mean of $122.34 + 5.67 =18.01$.
Note, I'm assuming the same number of members in each group...but the ideas essentially can be adjusted to extend to account for unbalanced groupings.
The idea extends to more complicated models with other scalar predictors.
I hope this is helpful.
