# Two slopes between 0 and 1, neither different from 0, could they be different from each other?

I'm reading a paper that does not report the coefficients from two OLS regressions. In both cases there is 1 response variable and 1 predictor variable. The predictor variable is the same in both cases. I know the subject matter of the paper well, which leads me to believe that the means of the slopes are almost certainly between 0 and 1. Although the slopes for these two regressions are not reported, the author does report that neither slope is significantly different from 0 (p ≥ 0.05).

If neither slope is different from 0, but both slopes are between 0 and 1, could the slopes be different from each other?

To try to figure this out, I did a quick test in R. I used two slopes that were very different (0.99 and 0.01), but chose s.e.'s for each that would make them barely "insignificant". To compare the slopes, I used the formula from the answer to THIS question.

pnorm(
(0.99 - 0.01)/ #difference between means
sqrt(0.61^2 + 0.0062^2), #sqrt of sum squares of s.e.'s
lower.tail=FALSE
)


OK, so this quick-and-dirty test suggests that the two slopes in the author's analysis can't be different.

Is it necessarily true that if two slopes are between 0 and 1, and neither different from 0, that they cannot be significantly different from each other?

It depends on how you do the testing. For instance, consider this model for the three variables $x$, $y_0$, and $y_1$:

$$\cases{ \mathbb{E}[y_0] = \beta_{0} + \beta_{1}x \\ y_1 = y_0 + \gamma x }$$

where $\beta_0$, $\beta_1$, and $\gamma$ are parameters and $\gamma$ is the difference in slopes. Evidently

$$\mathbb{E}[y_1] = \mathbb{E}[y_0 + \gamma x] = (\beta_{0} + \beta_{1}x) + \gamma x = \beta_0 + \beta_2 x$$

with $\beta_2 = \beta_1 + \gamma$. Then it is possible for estimates $\widehat{\beta}_1$ and $\widehat{\beta}_2$ to be indistinguishable from zero while determining, via regressing $y_1-y_0 = \gamma x$ against $x$, that $\widehat{\gamma}$ differs significantly from zero. The key idea is that $y_1$ and $y_0$ are not independent.

As an example here are simulated data in R:

n <- 10
x <- 1:n
delta <- x / n^2
y.0 <- residuals(lm(rnorm(n) ~ x)) + delta/2
y.1 <- y.0 + delta
pairs(cbind(x, delta, y.0, y.1))


Neither of the regressions $y_i \sim x$ is significant ($\widehat{\beta}_0 = 1/200,$ $p=0.937$ and $\widehat{\beta}_1 = 3/200$, $p=0.812$):

> summary(lm(y.0 ~ x))

Estimate Std. Error t value Pr(>|t|)
(Intercept)  0.00000    0.37903   0.000    1.000
x            0.00500    0.06109   0.082    0.937

> summary(lm(y.1 ~ x))

Estimate Std. Error t value Pr(>|t|)
(Intercept) 5.266e-17  3.790e-01   0.000    1.000
x           1.500e-02  6.109e-02   0.246    0.812


The regression of $y_1 - y_0 \sim x$ is extremely significant ($\widehat{\gamma} = 1/100,$ $p \lt 2\times 10^{-16}$):

> summary(lm((y.1 - y.0) ~ x))

Estimate Std. Error   t value Pr(>|t|)
(Intercept) 2.633e-17  1.415e-17 1.860e+00   0.0999 .
x           1.000e-02  2.281e-18 4.384e+15   <2e-16 ****

• Just to be sure I get the idea, you're saying that the slopes could be different if the y-axes in the two regressions are correlated, right? Thanks a ton for the answer, this is definitely a good piece of information to keep in mind! – rbatt Aug 13 '13 at 20:04
• I think I can agree with that, provided we understand "y-axes" to mean "dependent variables." This phenomenon is akin to the difference between an unpaired and paired t-test: the test in your question is a legitimate one, analogous to an unpaired t-test, but it lacks power to detect true differences in slope when the variables are (conditionally) positively correlated. Testing the slope of the regressed difference is the analog of performing a t-test of a difference of paired variables to see whether it is zero. – whuber Aug 13 '13 at 20:07
• That is what I meant, thanks for the additional explanation :) – rbatt Aug 13 '13 at 20:46