Plotting non-significant moderations I have a non-significant moderation. However, when I plot it, the lines are not parallel, even crossing at a certain point. Does this mean anything? Should I still present a plot that can be misleading, as it shows an interaction that is not significant?
 A: The non-significance tells you that the slopes of the lines are not significantly different from one another. Even if the lines cross, the two different slopes could be so mildly different from one another that they still are not significant from one another.
Another way of thinking of this is that they may look not parallel, but they aren't significantly non-parallel.
To illustrate this, I will (a) simulate two variables with a given correlation, (b) randomly place them into one of two groups, (c) plot the interaction between this random group variable and the two variables:
library(ggplot2) # load package for graphing
library(MASS) # load package to generate data
library(magrittr) # for wrangling data
mu <- rep(0,2) # generate vector of means; both variables will have mean of zero
Sigma <- matrix(c(1, .3, .3, 1), nrow=2, ncol=2) # generate correlation matrix between two variables
set.seed(1839) # set seed for replication
dat <- mvrnorm(n=100, mu=mu, Sigma=Sigma) %>% # create data frame based on these means and correlations
  as.data.frame() %>% # make it data frame
  set_colnames(c("x", "y")) %>% # rename variables
  mutate(group=as.factor(c(rep(0, 50), rep(1, 50)))) # making group variable

The interaction is not significant, obviously, because group was determined randomly. Remember both groups were generated from the exact same population:
summary(lm(y~x*group, dat)) # interaction is not significant

    Call:
lm(formula = y ~ x * group, data = dat)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.97233 -0.70442  0.07415  0.67710  2.20670 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)  
(Intercept) -0.02865    0.13177  -0.217   0.8284  
x            0.30753    0.13464   2.284   0.0246 *
group1      -0.14368    0.19317  -0.744   0.4588  
x:group1    -0.07746    0.21131  -0.367   0.7147  
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9315 on 96 degrees of freedom
Multiple R-squared:  0.08256,   Adjusted R-squared:  0.05389 
F-statistic: 2.879 on 3 and 96 DF,  p-value: 0.03994

But let's look at a plot with the two lines. You will see that, without the data points drawn in, the program automatically zooms in nice, and you will notice that the lines are NOT parallel. Indeed, they touch:
# plot without points
ggplot(dat, aes(x=x, y=y, colour=group)) +
  geom_smooth(method="lm", se=FALSE)


When you actually draw the points in, it looks a little less impressive:
# plot with points
ggplot(dat, aes(x=x, y=y, colour=group)) +
  geom_jitter() +
  geom_smooth(method="lm", se=FALSE)


So note that, even when the groups were generated from the same process, two lines can cross. The main point is that the interaction was not statistically significant—they were not significantly non-parallel, that is.
Does this mean anything? Maybe. The two lines could be just random noise—indeed, by using the common frequentist threshold of .05, you could chalk up the look of the lines to "just chance" (speaking loosely). If you really want to know if it means anything, you should think about power:


*

*How big was your N? If you had small N, maybe the lack of significance is due to small N. That is, maybe the interaction is really there, but the effect is too small to be detected by the amount of participants you have. So the next question to think about is...

*Is this a small effect? If you are looking for a small effect, maybe the lack of significance is that you have a small N for the size of your effect. You could try to design a study where the interaction effect would be stronger, or, like I said before, you could get more N.
You should look at your N and the $\Delta R^2$ due to the interaction effect?
Should you present the plot? I would argue no. You could basically just be showing people noise. But it depends on the size of the effect, your N, theoretical considerations, etc.
