High p-value in ANOVA for the interaction but with non-parallel interaction plot?

When I was dealing with my data, I got the p-value for my interaction term as 0.143, which in hypothesis testing, we cannot reject the null that there are no effects caused by the interaction.

In this case, the interaction plot should be more parallel as the interaction is not significant.

However, when I plot the interaction plot for my data, this came out:

This doesn't seem to be parallel at all, which confuses me given such high p-value.

What am I missing here? Thanks!

• As Robert mentions, sample size is the likely culprit. How many samples do you have?
– Dave
Feb 14, 2021 at 13:35
• @Dave There were like 48 samples, so I think the sample size is too small
– Edi
Feb 20, 2021 at 15:42

One possible explanation here is that you don't have sufficient statistical power to detect a "significant" interaction effect.

We can do a simple simulation to show this. We simulate data similar to that in the question:

set.seed(48)
dt <- expand.grid(car = c("A", "B", "C"), gas = c("petrol", "unleaded", "diesel", "biofuel"), reps = 1:2)
X <- model.matrix( ~ car * gas, dt)

betas <- c(25, 1, 1, 1, 1, 0, 0, 0, 0, 0, 0, 2)
dt$miles <- X %*% betas + rnorm(nrow(dt))  So we have simulated data where we expect an interaction carC:gasbiofuel but no other interaction, similar to the OP ggplot(dt, aes(y = miles, x = car, group = gas, color = gas)) + geom_smooth(method='lm', formula= y~x, se = FALSE)  However, if we run an ANOVA we find: summary(aov(miles ~ car * gas, dt)) ## Df Sum Sq Mean Sq F value Pr(>F) ## car 2 14.852 7.426 3.718 0.0554 . ## gas 3 8.947 2.982 1.493 0.2663 ## car:gas 6 12.189 2.032 1.017 0.4590  and we obtain a non-significant interaction term. However if we increase the sample size: dt <- expand.grid(car = c("A", "B", "C"), gas = c("petrol", "unleaded", "diesel", "biofuel"), reps = 1:10) X <- model.matrix( ~ car * gas, dt) dt$miles <- X %*% betas + rnorm(nrow(dt))

summary(aov(miles ~ car * gas, dt))

##              Df Sum Sq Mean Sq F value   Pr(>F)
## car           2  45.96  22.981  24.213 2.05e-09 ***
## gas           3  22.72   7.572   7.978 7.47e-05 ***
## car:gas       6  12.57   2.095   2.207   0.0477 *
## Residuals   108 102.50   0.949



and we obtain a significant interaction term.