Dummy Interaction effect

gender, residence variables

So, in our example the dependent variable is: neurosis – How many symptoms of neurosis do you have out of the nine symptomes? So, this shows the number of symptoms that people experienced on themselves in the past one week. 

Our independent variables are: gender and residence (village or city).

The value of neurosis between man and women will be different in villages and cities. So, we keep the effect of gender, but we say that this effect will be altered, it will be different depending on where you live (in cities or in villages).

Maybe your gender is not that important if you live in a city or it is not that important if you live in a village. Maybe living in a city creates an atmosphere of gender inequality: men and women take their shares almost equally, the level of responsibility is equal, fathers can go and pick up their children from the nursery, they can do the grocery, while in the villages this might not be the case.

Research question: Is the difference between in the level of neurosis in men and women less in the cities than in the villages?

Steps:

1. Check the outliers: Check if there are any extreme values.
2. Create inter variable: You have to create a new variable. We usually name it as inter: we use gender multiplied by the type of residence. Check your new variable in the Variable View window.
3. Run the regression: Dependent variable – neurosis Independent variables: gender, type of residence and inter.
4. Interpret the results: The interpretation depends on what you coded 0 and 1. Before the interpretation make sure that you know how the values are coded. Best is if you write it down on a paper or on a note that: 0-Male , 1 – Female, Village – 0, City -1

Interpret the coefficients: b0,b1,b2

b0 is the constant from the coefficient table, and then you have b1 that reflects the effect of gender, b2 that reflects the effect of the residence and b3 that is the interaction effect/ coefficient

b0: average level of neurosis among men in villages (you have to see all the categories where the independent variables are 0: men-0, village-0).

b1: Here we see the values of this independent variable (men, women) and the 0 value of the other independent variable (0-Village). In the interpretation we move from the 0 category to the category coded 1. Difference in the average level of neurosis between men and women in villages – women’s level is higher. So, if we move from men (coded as 0) living in villages to women(coded as 1)  living in villages then the average level of neurosis will grow by 3.2 points. 

So, on average women in villages have 3.2 points higher level of neurosis than men in villages.

(If this were not be a dummy variable, but a scale variable then you would interpret it as “with every step we move from 0 towards the end of scale the level of neurosis will grow by  …”)

b2: Here we see the values of this independent variable (0-villages, 1-cities) and the  0 category of the other independent variable (0-men). We compare men in villages to men in cities. This is the difference in the average level of neurosis between men in villages and men in cities. And then you see the category coded 1- cities. If we compare the value of men in cities to the men in villages, men in cities (1-cities) have higher levels of neurosis. 

So, men in cities have 1.821 higher level of neurosis than men in villages.

Two ways of interpreting the b3

b3 shows the magnitude of the interaction effect. So, this is the interaction effect. It shows us how much  one independent variable alters the effect of the other independent variable in the model.

The interaction effect is asymmetrical, which means that we can interpret this in two ways. The result will be the same. Why? Because the value of the interactional effect only shows you the difference. It is the same if you look at it from one way or the other. So, it is the difference in the level of neurosis between women and men and since it is a minus it drops from villages to cities. Thus, because this coefficient is asymmetrical b3 will be – 1,342  no matter how you look at it.

1. We can say that the main cause of the change in the level of neurosis is gender but this is altered by the place of residence.
2. Or we can say that the main cause of the change in the level of neurosis is the place of residence and this is altered by gender.

So in the following you will see how the SPSS counts the b3 coefficient. This might help you understand this value.

While interpreting b3 we reorder our equation in such a way that the effect of one independent variable will depend on the other independent variable. And then we substitute the values of the latter variable into the formula.

First interpretation of b3

Here we say that gender is the main effect, but the power of gender is altered by the place of residence.

To understand the interpretation of the b3 coefficient let’s write down the regression equation:

Ŷ=b0+b1*gender+b2*residence+b3*inter

In this case let’s substitute the interaction variable with its constituing two original variables. So, in the place of “inter” let’s substitute the gender*residence:

Ŷ=b0+b1*gender+b2*residence+b3*gender*residence

Let’s reorder the equation in such a way that the effect of gender will depend on the residence:

Ŷ=b0+b2*residence+(b1+b3*residence)*gender

Then inside the parentheses let’s substitute the values of the residence, the 0 and 1 and let’s calculate the effect of the gender for every single value.

From the table we can see how the residence changes/alters the effect of the gender. The magnitude of the effect of gender for those who live in villages is 3,15. The magnitude of the effect of gender for those who live in cities is 1,81. Thus, on average women in villages have 3,15 points more neurotic symptoms than men living in villages. On the contrary, in the cities the difference between the two genders (between men and women) is 1,81. 

In cities the effect of gender is substantially weaker than in villages and exactly as much as b3 shows us. So, in cities the effect of gender is 1,34 points weaker than in villages.

Second interpretation of b3

If we want to look at it the other way around and we say that the main cause is residence which is altered by gender, we will get exactly the same value for b3: -1,342

So, while in our previous example we were interested in how the residence alters the effect of gender on the level of neurosis. But we can formulate our question in the opposite way too: How does the effect of the residence change/alter depending on the fact that we talk about men or women? So, how does the gender alter the effect of the residence on the level of neurosis?

Let’s reorder the equation in such a way that the effect of residence will depend on the gender:

Ŷ=b0+b1*gender+(b2+b3*gender)*residence

Here we will examine the hypothesis that the effect of the residence depends on the person’s gender. Let’s substitute the two values of the gender variable, 0 and 1 inside the parentheses.

As we can see it from the table, the effect of residence for men is 1,82 and the effect of residence for women is 0,48. B3 is the difference between the two. So, we can interpret b3 as: living in a city increases the symptoms of the neurosis by 1,82 points in men, while the increase is just 0,48 in women.

So, among women the effect of residence is 1,34 points weaker than among men.

General interpretation

0-Male , 1 – Female, Village – 0, City -1

b0: On average the level of the dependent variable “among” first independent variable (coded as 0-men) “in” the second independent variable (coded as 0-villages). So, here all the independent variables are coded as 0. Example: the average level of neurosis among men in villages.

b1: The difference of the average level of the dependent variable between the categories of this independent variable (men and women) and in the 0 category of the other independent variable (village). The category coded as 1 is b1 points higher/smaller than the one coded with 0. Example: On average women in villages have 3.2 points higher level of neurosis than men in villages. – So, here you change to the value of the 0 category in the other independent variable (residence – village is 0).

b2: The difference of the average level of the dependent variable between the categories of this variable in the 0 category of the other independent variable. The category coded as 1 in the other independent variable is higher/smaller than the one coded with 0. Example: Men in cities have 1.821 higher level of neurosis than men in villages. – So, here you change to the value of the 0 category in the other independent variable (gender – men is 0).

How to decide which way to interpret the b3?

It is important to know that the way to choose between these two types of interpretation depends on our research question. So, depending on our research question we have to choose the way of interpreting b3. But no matter how we choose the value of the interaction effect will be the same.

Another example: religion and gender

Hypothesis: Gender moderates the relationship between believe in God and satisfaction with your life.

1. Check the outliers: Check if there are any extreme values.
2. Create inter variable: You have to create a new variable. We usually name it as inter: gender multiplied by religion. Check your new variable in the Variable View window and label it.
3. Run the regression: Dependent variable: satisfaction with your life Independent variables: religion, gender and inter.
4. Interpret the results: The interpretation depends on what you coded 0 and 1. Before the interpretation make sure that you know how the values are coded. Best is if you write it down on a paper or on a note that: 0-does not believe in God , 1 – believe in God, 0 – female, 1 – male

0-does not believe in God, 1 – believe in God, 0 – female, 1 – male

b0: The average level of satisfaction with life for women who do not believe in God is 6.968.

b1: Women who believe in God score 0.059 less than women who do not believe in God. This is statistically not significant, so there is a 7,9% probability that we got this result by chance. (p=0,079)

b2: Men who do not believe in God score 0.142 less than women who do not believe in God. (p<0,05)

b3: Among those who believe in God the effect of gender is 0.107 higher than among those who do not believe in God. We could also say that among men the effect of religion is 0.107 higher than among women. (p<0,05)