Scale independent

Hypothesis: The higher the number of people living together in the same household the less likely they feel sad.

b0: the log odds for category 1 of the dependent variable when the value of the independent variable is 0/ when the independent variable equals 0.

Note: b0 is not always interpretable in real life. In our database the minimum interpretable value of this independent variable is 1 because 1 means that 1 person lives in this household, so the respondent lives alone. However, the program computes the constant value and we can use this value for other calculations.

Exp(b0): the odds for category 1 of the dependent variable when the independent variable is 0.

The odds of a person feeling sad in households consisting of 0 persons are 0,273. (Interpretable only mathematically but not in reality because there are no empty households in this database.)

b1: how much larger or smaller the log odds become as the independent variable increases by 1 unit.

b1: The logarithm of the odds for feeling sad is by 0,375 lower with every additional person who lives in the same household.

Note: here your independent variable is the number of people living in a household. So, 1 unit increase here means that you compare a person who lives in a household with x number of persons with a person who lives together with x+1 number of persons in the same household.

Exp(b1): The odds of feeling sad is 0,697 times as low with every additional person who lives in the same household. So, this means that the fewer persons live in the same household the higher the probability is for them to feel sad. In other words: The more people you live together with in the same household the more likely you are to not feel sad.

b0+b1: the log odds for category 1 of the dependent variable when the independent variable is 1.

b0+b1: The logarithm of the odds of feeling sad for a person who lives alone is (-0,375+(-1,375)=-1,675). The number of people living in the household=1.

Exp(b0)*Exp(b1): The odds of feeling sad for a person who lives alone are 0,188 (0,273*0,687=0,188).

So, now we know the odds of feeling sad for a person who lives alone but what are the odds of feeling sad for a person whose household consists of 2 or 3 … people? 2 or 3 or x people b0+X*b1, X is the value of the independent, which is the number of people living together.

For example, in a two people household, the log odds of feeling sad is b0+2*b1 =-0,375+(-1,375)=-3,125.

The odds of feeling sad in a two people household is Exp(b0)*Exp(b1)*Exp(b1) = Exp(b0)*Exp(b1) on the power of two = 0,687*0,273*0,273=0,05. So the odds for a member of a two people household to feel sad are very low. (When the odds are much higher than 1 then we can say very high.)

b1 coefficient > 0 -> a 1 unit increase in X increases the likelihood/probability that y=1 This means that an increase in X makes the outcome of 1 more likely.

b1 coefficient < 0 -> a 1 unit increase in X decreases the likelihood/probability that y=1.  This means that an increase in X makes the outcome of 1 less likely.

Exp(B): < 1 -> we say „as low as” : a 1 unit increase in X decreases the likelihood/probability that y=1.  This means that an increase in X makes the outcome of 1 less likely. It indicates a negative effect.

Exp(B): > 1 -> we say „as high as” a 1 unit increase in X increases the likelihood/probability that y=1 This means that an increase in X makes the outcome of 1 more likely. It indicates a positive effect.

Exp(B)=1 -> no effect

Conclusion: The results support the hypothesis stating that “The higher the number of people living together in the same household the less likely they feel sad.” We got to this conclusion because the p-value (p=0,001) is lower than 0,05 and the Exp(b1) ( which is lower than 0 ) show us that if a person lives together with 1 more person then she is less likely to feel sad. Because b1 is a negative number we say that the independent variable has a negative effect on the dependent variable. So, the increase in the number of people living together in the same household has a negative effect on the likelihood of feeling sad. (Note: We coded feeling sad as 1.)

Other questions:

1. What are the odds of feeling sad for a person who is living alone?
   Exp(b0)*Exp(b1): The odds of a person feeling sad among those who live alone are Exp(b0)*Exp(b1). Note: because living alone is not the 0 value of the independent variable!!!
2. How much larger or smaller are the odds of feeling sad for each additional person who is living in the same household?
   Exp(b1): The odds of feeling sad is multiplicated by 0,687 for each additional person who is living in the same household.
3. What are the odds of feeling sad for a person who is living with 1 other person in the same household? (The respondent is included in the number of people in the household. So, here we calculate the odds for 2 people.)
   Exp(b1)*Exp(b1)*Exp(b0)=0.687*0.687*0,273. So, this is Exp(b1) on the power of 2 * Exp(b0)
4. What are the odds of feeling sad for a person who is living with 2 other persons in the same household?(So, there are 3 persons living in this household.)
   Exp(b1)*Exp(b1)*Exp(b1)*Exp(b0)=0.687*0.687*0.687*0,273.
   So, this is Exp(b1) on the power of 3 * Exp(b0)
5. What are the odds of feeling sad for a person who is living with 3 other persons in the same household?
   Exp(b1)*Exp(b1)*Exp(b1)*Exp(b1)*Exp(b0)=0.687*0.687*0.687*0.687*0,273.
   So, this is Exp(b1) on the power of 4 * Exp(b0)