How To Interpret Dummy Variables In Regression

[Tea Rochlitz]

Thuswhen we have an intercept in the regression model and we want to avoid perfect multicollinearity, we create only one dummy to encode a categorical variable that has two categories.

I am wondering how I can interpret the estimated coefficient for variable B. I thought I could run a simple regression analysis, but I would also like to get an estimated effect of variable A while B is zero (which is what the coefficient for variable A represents, if I am understanding correctly).

The way you are interpreting the coefficients is not quite right. The general interpretation of the coefficient on a dummy variable in a multiple regression is "the expected (or average) difference in the dependent variable between those with $1$ and those with $0$ values of that dummy variable, holding other independent variables constant.

Variable A can be present (i.e., 1) only when Variable B is present (1). I am wondering how I can interpret the estimated coefficient for variable B, because the coefficient for B represent the presence of B while A is 0, which logically does not make sense.

I need to check the influence of a independent variable on three dependant variables in two different categories. That's why I added a dummy variable which is coded with 1's and 0's (1 for the first category and 0 for the second) as an independent variable.

Unfortunately I cannot really handle the output. I think that something went wrong.. or maybe it didn't? Maybe you can help me and say if I did the regression model right and how I can interpret the output. I'll put a screenshot beneath. Unfortunately it's in german, but actually the tools is quit the same as in english.. if you have question, just ask and I will try to answer as well as I can.

If your dummy variable is truly a separation of categories then you try using it as a "By" variable as opposed to an independent variable. You will get two separate fits. If you believe it should be an independent variable then make sure it is coded as Nominal and then you will see the effect in your fit model.

thank you very much for your response! I already tried the function "By" and I will use it if I don't find a good way to use it as a independant variable. As you said I coded it as nominal and it really looked better and I had a different graph for each group.. so, I think that's the solution.

Perhaps regression will not yield well with your data set. Based on your output report, it does not look like you have a good model yet. Have you tried neural networks and/or partitioning? You might get better predictive models this way. Discriminant Analysis might also be useful !

The coefficient on an indicator variable is an estimate of the average DIFFERENCE in the dependent variable for the group identified by the indicator variable (after taking into account other variables in the regression) and

OK, so how do we interpret this coefficient of 0.0075 on female? As we said before, it is the average DIFFERENCE in the dependent variable (Whether the person votes) with respect to a REFERENCE GROUP. The reference group is the group for whom the indicator is always equal to zero, which in this case is the set of male voters.

First, we see that the coefficient on democrat is -0.08. That means that the DIFFERENCE in turnout between Democrats and the reference group (here, Republicans) is 8%. So Democrats have 8 percentage point lower turnout on average in this data than Republicans.

Second, we see that the coefficient on unaffiliated is -0.06. That means that the DIFFERENCE in turnout between Unaffiliated voters and the reference group (here, Republicans) is 6%. So Unaffiliated voters have 6 percent point lower turnout on average in this data than Republicans.

Moreover, the p-value on these indicator variables tells us if these differences are significant. And indeed, they show clearly that the difference between Democrats and Republicans, and the difference between Unaffiliated voters and Republicans are both significant.

Interactions (at least when you interact an indicator variable with a continuous variable), just like a regular indicator variables, report differences between a group and the reference group. The difference is that instead of reporting the difference in average value of the dependent variable between the indicated group and the reference group, the coefficient on an interaction term is the average DIFFERENCE in the SLOPE associated with the continuous variable between the indicated group and the reference group.

Technically, dummy variables are dichotomous, quantitative variables. Their range of values is small; they can take on only two quantitative values. As a practical matter, regression results are easiest to interpret when dummy variables are limited to two specific values, 1 or 0. Typically, 1 represents the presence of a qualitative attribute, and 0 represents the absence.

The number of dummy variables required to represent a particular categorical variable depends on the number of values that the categorical variable can assume. To represent a categorical variable that can assume k different values, a researcher would need to define k - 1 dummy variables.

For example, suppose we are interested in political affiliation, a categorical variable that might assume three values - Republican, Democrat, or Independent. We could represent political affiliation with two dummy variables:

In this example, notice that we don't have to create a dummy variable to represent the "Independent" category of political affiliation. If X1 equals zero and X2 equals zero, we know the voter is neither Republican nor Democrat. Therefore, voter must be Independent.

When defining dummy variables, a common mistake is to define too many variables. If a categorical variable can take on k values, it is tempting to define k dummy variables. Resist this urge. Remember, you only need k - 1 dummy variables.

A kth dummy variable is redundant; it carries no new information. And it creates a severe multicollinearity problem for the analysis. Using k dummy variables when only k - 1 dummy variables are required is known as the dummy variable trap. Avoid this trap!

The value of the categorical variable that is not represented explicitly by a dummy variable is called the reference group. In this example, the reference group consists of Independent voters.

In analysis, each dummy variable is compared with the reference group. In this example, a positive regression coefficient means that income is higher for the dummy variable political affiliation than for the reference group; a negative regression coefficient means that income is lower. If the regression coefficient is statistically significant, the income discrepancy with the reference group is also statistically significant.

In this section, we work through a simple example to illustrate the use of dummy variables in regression analysis. The example begins with two independent variables - one quantitative and one categorical. Notice that once the categorical variable is expressed in dummy form, the analysis proceeds in routine fashion. The dummy variable is treated just like any other quantitative variable.

The first thing we need to do is to express gender as one or more dummy variables. How many dummy variables will we need to fully capture all of the information inherent in the categorical variable Gender? To answer that question, we look at the number of values (k) Gender can assume. We will need k - 1 dummy variables to represent Gender. Since Gender can assume two values (male or female), we will only need one dummy variable to represent Gender.

Note that X1 identifies male students explicitly. Non-male students are the reference group. This was a arbitrary choice. The analysis works just as well if you use X1 to identify female students and make non-female students the reference group.

At this point, we conduct a routine regression analysis. No special tweaks are required to handle the dummy variable. So, we begin by specifying our regression equation. For this problem, the equation is:

Values for IQ and X1 are known inputs from the data table. The only unknowns on the right side of the equation are the regression coefficients, which we will estimate through least-squares regression.

The first task in our analysis is to assign values to coefficients in our regression equation. Excel does all the hard work behind the scenes, and displays the result in a regression coefficients table:

The coefficient of muliple determination is 0.810. For our sample problem, this means 81% of test score variation can be explained by IQ and by gender. Translation: Our equation fits the data pretty well.

Before we conduct those tests, however, we need to assess multicollinearity between independent variables. If multicollinearity is high, significance tests on regression coefficient can be misleading. But if multicollinearity is low, the same tests can be informative.

To measure multicollinearity for this problem, we can try to predict IQ based on Gender. That is, we regress IQ against Gender. The resulting coefficient of multiple determination (R2k) is an indicator of multicollinearity. When R2k is greater than 0.75, multicollinearity is a problem.

With multiple regression, there is more than one independent variable; so it is natural to ask whether a particular independent variable contributes significantly to the regression after effects of other variables are taken into account. The answer to this question can be found in the regression coefficients table:

The regression coefficients table shows the following information for each coefficient: its value, its standard error, a t-statistic, and the significance of the t-statistic. In this example, the t-statistics for IQ and gender are both statistically significant at the 0.05 level. This means that IQ predicts test score beyond chance levels, even after the effect of gender is taken into account. And gender predicts test score beyond chance levels, even after the effect of IQ is taken into account.

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