# Multiple linear regression

y = b_{0} + b_{1}*X_{1} + b_{2}*X_{2} + … + b_{n}*X_{n}

X_{1},X_{2},…X_{n}: independent variables, explanatory variables

y: dependent variable

b_{0}: constant – the point where the line crosses the vertical axes (Y), y-intercept, constant term

b_{1},b_{2},…b_{n}: coefficients

## Check beforehand

Here we have to watch out for multicollinearity between the independent variables. Multicollinearity is something that we do not want.

**Multicollinearity**: meaning are the independent variables capturing the same constract? Are they really measuring the same thing? Is there correlation between the independent variables? Is the correlation between the independent variables high? If yes, then you might want to drop the independent variables.

## Tables in SPSS:

You have to look at the **standardized coefficients** to know how much greater is one independent variable then the other. This puts them on the same scale. The standardized coefficients give a better idea about their value relative to each other then the Unstandardized coefficients.

Correlations-Part: this is the unique contribution

Sig: You want to see if there is a significant unique contribution. If it is below 0.05 then it is significant. Then we say (a)independent variable is a good predictor.

if not then we say that it is not significant. In this case we say that (a)independent variable is not a good predictor of the dependent variable.

Tolerance: how much variability for this specific independent variable is not explained by other independent variables in the model. If it is above 0.0

VIF: variance inflation factor. You want this to be less then 10.

Residual Statistics

We want the Std. Residual (Standard Residual) to fall between 3 and -3. This shows the same like the Scatterplot.

Video about more then one independent variable:Click here

## Exercise 1 – Multiple Linear Regression

We want to analyze if education, religion and age have an effect on how much power people feel over their life. In our previous example there was a significant effect of having a university degree on the power over life and now we would like to check wethear this relationship remains after involving other variables.

Hypothesis: Controlling for religion and age the positive effect of university degree remains significant on the power over life.

Dependent: power over life

Independents: education, religion, age

2,8% of the variability in the level of how much power do people feel over their life is explained by the education, religion and age. (p<0,001)

**Constant: when all the independent variable is 0 then this is the average of the dependent variable** **(7.003).**

In a hypothetical case where education, religion and age is 0, the level of power that people feel over their life is 7.003.

Controlled for religion and age, people who have a university degree gave 0.782 point higher on average for the level of how much power they feel they have over their life than those who do not have university degree. So, the ones who have a university degree feel more power over their life then those who do not have university degree (p<0,001).

So in our previous example the effect of education was … but after controlling for religion and age the effect is smaller but still significant.

Thus, the results support or refute the hypothesis?

Exercise:

State a hypothesis.

Run a linear regression.

Interpret b0 and b1 and the Sig. level.

Does the result support or refute your hypothesis?

Export the result (Coefficient Table only) into a WORD document and upload it to the Moodle with the syntax file. Both the files should start with your name: “Firstname-Lastname-dummy”

## Exercise 2 – Multiple Linear regression

We want to analyze if education, age, gender, living in Budapest or not have an effect on how much power people feel over their life.

The hypothesis: Controlling for age, gender, residence, the positive effect of university degree remains significant on the power over life.

Dependent variable: power over life

Independent variables: education, age, gender, residence (living in Budapest or not)

2,9% of the variability in the power over life is explained by the education, age, gender and by the fact that if the person currently lives in Budapest or not. (p<0,001)

If hypothetically the education, age, gender and current location would be 0 then people on average on a 1 to 10 scale, where 10 indicates absolute power over life, would give a 7.062 point.

Controlling for age, gender, resicency, those who have a university degree on average feel 0,793 points more power over their life than those who do not have (p<0,001). Thus, our result supports our hypothesis.

If someone gets one year older then he will feel 0,014 less power over his/her life then one year before (p<0,001).

Gender (p=0,356) and the fact the someone lives in Budapest or not (p=0,643) does not have an effect on how much power do they feel over their life.