# p-value of 1.00 for a dichotomous predictor -- possible reasons for this oddity?

In my logistic regression analysis, my dichotomous predictor variable 'A' gave a strange p value of 1.00 under Wald's test. The analysis was done with 2 continuous covariates, a continuous predictor variable and an interaction variable consisting of the continuous predictor and the dichotomous predictor. The DV is a dichotomous outcome.

Prior to the above analysis (some history on how i arrived at the logistic regression analysis), the DV was a continuous variable which was transformed into a dichotomous variable with values 0 and 1. The transformation was made because of the extremely skewed distribution. In the boxplot of the DV, 8 outliers were shown (out of a total of 98 persons) at the upper end of the graph and the rest were represented by a horizontal line at the lower end in the graph.

I am wondering if the above issue of the strange p value was due to the transformation of the DV. Hierarchichal multiple regressions were conducted with other similar predictors, covariates, the same DV and 'A', and 'A' showed a consistent significant value (all p < .01). It was only in the logistic regression that gave the strange p value.

If it is not due to the transformation of the DV, could it be because logistic regression was not the appropriate analytical method, or could this all be due to possibe statistical error (which seems in order though)? Thanks.

• What happens when you do the logistic regression with just the dichotomous predictor? If you don't get p=1 then, which variable do you have to add in order to make p=1? Oct 10, 2011 at 11:15
• When I did the logistic regression with just the dichotomous predictor, I've gotten p = 1 again, results are as follows: B = -20.44, SE = 5469.57, Wald's alpha = 0, Odds ratio = 0. From the values, there seemed to be some statistical errors. In my original logistic regression, there were 3 blocks and the dichotomous predictor was entered in block 2. Oct 10, 2011 at 23:54
• So when you crosstabulate the dichotomous predictor and dichotomous outcome variable, are they perfectly uncorrelated? Oct 10, 2011 at 23:59
• in addition, the correlation coefficient of the dichotomous variable with the DV was moderate at .45 (p < .01). have just checked on partial/quasi- complete separation as suggested by Peter below. the large values of the SE and B seemed to align with the partial separation theory, although I am not 100% sure on this yet. Oct 11, 2011 at 0:09

• The p-value is the probability of observing an effect as large or larger if the null hypothesis were true, so perhaps all other possible outcomes would imply a larger observed effect.

### Simulated data

For example, take this simulation in R, where I generate some data where x an y are orthogonal to one another.

> x <- c(rep(0, 50), rep(1, 50))
> y <- c(rep(0, 25), rep(1, 25), rep(0, 25), rep(1, 25))
> dat <- data.frame(x, y)
> table(x, y)
y
x    0  1
0 25 25
1 25 25


If we run a logistic regression, x would predict none of y and the p-value of the coefficient associated with X would be one.

> m1 <- glm(y ~ x, data =dat,  family=binomial)
> summary(m1)
...
Coefficients:
Estimate Std. Error z value Pr(>|z|)
(Intercept) -1.017e-17  2.828e-01       0        1
x            1.344e-18  4.000e-01       0        1
...


I can thing of two things to investigate - complete or semi-complete separation and collinearity. Both of these can do very odd things to p-values.