Let’s consider the following regression model, conceptually identical with your example: 

Y = B0 + B1X1+ B2X2 + B3X1*X2

The interactive effect between X1 and X2 on Y corresponds to the B3 slope. If B3 is reliable (or "statistically significant”), it means that the effect of X2 on Y depends on the level of X1 (or otherwise, but it’s exactly the same, the effect of X1 on Y depends on the level of X2). 

Now, to answer your question : 

If B3 is positive (the interactive effect is positive), then it means that the more positive X2 is, the more positive becomes the effect of X1 on Y (or alternatively, the more negative X2 is, the more negative effect of X1 on Y becomes).

Conversely, if B3 is negative, then the more positive X2 is, the more negative the effect of X1 on Y becomes (or alternatively, the more negative X2 is, the more positive effect of X1 on Y becomes). 

Once again, it’s totally up to you, which variable among X1 and X2 is “the independent variable”, and which one is the “moderator”. Put it differently, the reasoning here applies both if you consider the effects at high/low levels of X1, or high/low levels of X2.

Hope it’s clear. 

Let’s consider the following regression model, conceptually identical with your example: 

Y = B0 + B1X1+ B2X2 + B3X1*X2

The interactive effect between X1 and X2 on Y corresponds to the B3 slope. If B3 is reliable (or "statistically significant”), it means that the effect of X2 on Y depends on the level of X1 (or otherwise, but it’s exactly the same, the effect of X1 on Y depends on the level of X2). 

Now, to answer your question : 

If B3 is positive (the interactive effect is positive), then it means that the more positive X2 is, the more positive becomes the effect of X1 on Y (or alternatively, the more negative X2 is, the more negative effect of X1 on Y becomes).

Conversely, if B3 is negative, then the more positive X2 is, the more negative the effect of X1 on Y becomes (or alternatively, the more negative X2 is, the more positive effect of X1 on Y becomes). 

Once again, it’s totally up to you, which variable among X1 and X2 is “the independent variable”, and which one is the “moderator”. Put it differently, the reasoning here applies both if you consider the effects at high/low levels of X1, or high/low levels of X2.

Hope it’s clear. 

Thanks Wojciech! Your explanation is simple and clear. Appreciate your prompt answer. 

One simple (probably obvious) comment to Wojciech's explanation. Overall, I do agree with the given reasoning, but:

Once again, it’s totally up to you, which variable among X1 and X2 is “the independent variable”, and which one is the “moderator”.

In a given model, all the variables should be pre-defined based on the available literature. This means that before you start your calculations (or even data collection) you should know in advance which variable is the independent one and which is being introduced into the model as the moderator. Although from statistical perspective (as Wojciech mentioned) it does not seem to be important, still it makes difference from theoretical perspective.   

Thanks Lukasz, I am well aware of that point, and I completely understood where Wojciech was coming from. Thanks for the heads up!

What if B1 and B2 are significant but B3 is not?

How can the interaction of two significant variables not be significant?

Dear Shaun,

if B1 and B2 are significant in a regression model but B3 is not significant, then that means that the respective effects of B1 on Y and B2 on Y are (in practical terms) independent of each other. For instance, no matter how high / low your participant / observation is on B1, it does not alter the effect of B2 on Y (conversely to an interactive effect, where you have this sort of dependancy).

Note that an interactive effect (here, B3) does not stemm automatically from the presence of two other variables being significant (B1 and B2). Very often you have predictors in your regression model that act independently on the predicted variable, and there is nothing unusual about it. 

Dear Will, 

with a negative effect of B1 on Y, and a negative effect of B3, this means that X2 (which is associated with B2, but it’s not B2 by itself) exacerbates the negative effect of B1 on Y. Let’s take an example : 

Y is lifetime (an objectively good thing to have more of)

X1 (associated with B1) is the number of cigarettes one smokes per day

X2 (associated with B2) is the quantity of alcohol one consumes per day.

X1*X2 (associated with B3) is their interaction

As smoking has a negative effect on life expectancy (the effect B1 being negative on Y), that means that the more you smoke (as X1 gets more positive), the less you live (Y  gets less positive).

As the interactive effect of smoking and drinking on life expectancy is negative (the effect of B3 being negative on Y), this means that the more you drink (the more X2 is positive), the more deleterious cigarettes become (B1 becomes more negative). For instance, each smoked cigarette makes you live 2 minutes less as you drink no alcohol, but if you drink one glass of wine per day, each smoked cigarette makes you live 6 minutes less. 

Hope to be clear.

Dear Wojciech, your explication is very good. Congratulation. I propose you other case:

Imagine that B1 is negative and B3 is positive.... How do you interpret it? 

Greetings,

Antonio

If B3 is positive (the interactive effect is positive), then it means that the more positive X2 is, the more positive becomes the effect of X1 on Y (or alternatively, the more negative X2 is, the more negative effect of X1 on Y becomes).

If X1 is negatively related to Y and X2 is positively related to Y, How do we interpret the result where B3 (interaction) is negative? Can we say that X2 is strengthening the negative effect of X1 on Y (although more X2 supposed to lead to a higher Y)? Can anyone shed some light on this? Thanks! 

@Wojciech Świątkowski : Very Helpful explanation. Thanks

@Shih-chi Chiu: I think what in your mind is correct. Although the expression that "X2 is strengthening the negative effect of X1 on Y" is a bit awkward, I agree with what you mean. Let's put it in a different way, maybe it's better to use Wojciech's example again. In case of drinking more alcohol, the negative effect of cigarette on average life span is exaggerated. In this sense, a negative interaction is "strengthening" the negative effect of X1 on Y.

Hope my explanation helps. 

I am encountering the same experience. But it seems the point of focus is the interaction. Though it is still unclear how can B3 be positive if B1 and B2 are negative?

Any possible example? Please assist. Thanks

Please let clarify if the functional form of the model is like

Q= f(y, z, and y.z)

now the problem is when we estimate the model , we got significant positive results of Y and Z on Q. However, using both in interaction term its give us negative and significant results. Is it possible like it, and how can explain this sort of results. Thanks

I want to examine a theory that says that there is an association between A and B, and C is a moderator. However, based on the theory, C (which is the moderator) affects the relation between A and B negatively! In other words, in the case of C, the relationship between A and B becomes weaker.

My question is:

How can I phrase my hypothesis in such a case?

What if B0 is negative?

Dear All ladies and gentlemen

could you please help me to answer my question? my hypothesis was that nurse work satisfaction had positive significance on nursing care quality, but result found that nurse work satisfaction had negative significance on nursing care quality, would you explain why it changed?

Valuable discussion. Tqvm