Interpreting interaction coefficients for a binary regression model

[benchammer]

Hello,

I'm having a bit of trouble interpreting the coefficients in a binary regression model that contains interaction effects. Was wondering if anyone could please help!

The model has a dependent binary variable called sanctions, where sanctions = 1 means that a country has received diplomatic sanctions and sanctions = 0 means it has not. The explanatory variables are:
Free_election: Binary variable where 1 = the country has free elections and 0 means it doesn't. The coefficient is 5.510.
Free_expression: can take values from 1 to 10. The coefficient is -0.29.
Free_election*free_expression: Interaction between the 2 explanatory variables. The coefficient is -0.9.

Am I correct in saying that the effect of free_elections is to increase the log(odds) of sanctions = 1 by a factor of 4.61 (as 5.510-0.9 = 4.61)? Could I also correctly say that an increase of 1 in free expression reduces the log(odds) by a factor of -1.19 (as -0.9 - 0.29 = -1.19)?

Thanks in advance for any help, really struggling with these interactions!

Ben

[Bruce Weaver]

I assume "binary regression" means binary logistic regression.  

You did not say what the constant is, but in order to illustrate, assume it is 1.  Here are the coefficients:

                B
Constant 1
X1            5.51
X2           -0.29
X1*X2      -0.90
                                               
                 X2
             0        1        Diffs
X1   0     1       0.71    -0.29
      1     6.51   5.32    -1.19   -0.9
                                               
Diffs       5.51   4.61
                 -0.9

The constant gives the fitted value of the log-odds(Y=1) when X1 and X2 are both equal to 0.  It is in the top left cell of the 2x2 table above.  

The coefficient for X1 gives the (simple main) effect of X1 when X2 = 0--i.e., it s the difference between the top left and bottom left cells in the 2x2 table:  6.51 - 1 = 5.51.

The coefficient for X2 gives the (simple main) effect of X2 when X1 = 0--i.e., it s the difference between the top left and top right cells in the 2x2 table:  0.71 - 1 = -0.29.

The bottom right cell value = the sum of all the coefficients.

The coefficient for X1*X2 = the difference between the differences.  It can be obtained two ways:
1)  4.61 - 5.51 = -0.9
2)  -1.19 - (-0.29) = -0.9

When you exponentiate, differences become ratios, and differences of differences become ratios of ratios.

The Sage publication by James Jaccard (on interactions in logistic regression) is a useful reference for these types of models.

HTH.

benchammer wrote

Hello,

I'm having a bit of trouble interpreting the coefficients in a binary regression model that contains interaction effects. Was wondering if anyone could please help!

The model has a dependent binary variable called sanctions, where sanctions = 1 means that a country has received diplomatic sanctions and sanctions = 0 means it has not. The explanatory variables are:
Free_election: Binary variable where 1 = the country has free elections and 0 means it doesn't. The coefficient is 5.510.
Free_expression: can take values from 1 to 10. The coefficient is -0.29.
Free_election*free_expression: Interaction between the 2 explanatory variables. The coefficient is -0.9.

Am I correct in saying that the effect of free_elections is to increase the log(odds) of sanctions = 1 by a factor of 4.61 (as 5.510-0.9 = 4.61)? Could I also correctly say that an increase of 1 in free expression reduces the log(odds) by a factor of -1.19 (as -0.9 - 0.29 = -1.19)?

Thanks in advance for any help, really struggling with these interactions!

Ben

[MaxJasper]

5.51 is for f.elect= 0

 

-.29 is for 1 unit increase in f.exp between 1 to 10

 

change in log(odds) = -5.51 [0 to1] -.29 * {1,2,3...10} + .9 [0 to 1] * {1,2,3,...10}

 

 

Hence for a country {f.elect=0, f,exp=3} :: log(odds) = 5.51 -.29 * 3 -.9*0*3 = 4.23

 

This is when your f.exp is a scale variable as you have only one coeff for it.

 

Max.

 

 

Hello,

I'm having a bit of trouble interpreting the coefficients in a binary regression model that contains interaction effects. Was wondering if anyone could please help!

The model has a dependent binary variable called sanctions, where sanctions = 1 means that a country has received diplomatic sanctions and sanctions = 0 means it has not. The explanatory variables are:

• Free_election: Binary variable where 1 = the country has free elections and 0 means it doesn't. The coefficient is 5.510.
• Free_expression: can take values from 1 to 10. The coefficient is -0.29.
• Free_election*free_expression: Interaction between the 2 explanatory variables. The coefficient is -0.9.

Am I correct in saying that the effect of free_elections is to increase the log(odds) of sanctions = 1 by a factor of 4.61 (as 5.510-0.9 = 4.61)? Could I also correctly say that an increase of 1 in free expression reduces the log(odds) by a factor of -1.19 (as -0.9 - 0.29 = -1.19)?

Thanks in advance for any help, really struggling with these interactions!

Ben

 

[Bruce Weaver]

After reading Max's response, I noticed that I missed the 1-10 range for one of the variables.  My earlier reply treated both of them as dichotomous.  Sorry about that.

Bruce Weaver wrote

I assume "binary regression" means binary logistic regression.  

You did not say what the constant is, but in order to illustrate, assume it is 1.  Here are the coefficients:

                B
Constant 1
X1            5.51
X2           -0.29
X1*X2      -0.90
                                               
                 X2
             0        1        Diffs
X1   0     1       0.71    -0.29
      1     6.51   5.32    -1.19   -0.9
                                               
Diffs       5.51   4.61
                 -0.9

The constant gives the fitted value of the log-odds(Y=1) when X1 and X2 are both equal to 0.  It is in the top left cell of the 2x2 table above.  

The coefficient for X1 gives the (simple main) effect of X1 when X2 = 0--i.e., it s the difference between the top left and bottom left cells in the 2x2 table:  6.51 - 1 = 5.51.

The coefficient for X2 gives the (simple main) effect of X2 when X1 = 0--i.e., it s the difference between the top left and top right cells in the 2x2 table:  0.71 - 1 = -0.29.

The bottom right cell value = the sum of all the coefficients.

The coefficient for X1*X2 = the difference between the differences.  It can be obtained two ways:
1)  4.61 - 5.51 = -0.9
2)  -1.19 - (-0.29) = -0.9

When you exponentiate, differences become ratios, and differences of differences become ratios of ratios.

The Sage publication by James Jaccard (on interactions in logistic regression) is a useful reference for these types of models.

HTH.

benchammer wrote

Hello,

I'm having a bit of trouble interpreting the coefficients in a binary regression model that contains interaction effects. Was wondering if anyone could please help!

The model has a dependent binary variable called sanctions, where sanctions = 1 means that a country has received diplomatic sanctions and sanctions = 0 means it has not. The explanatory variables are:
Free_election: Binary variable where 1 = the country has free elections and 0 means it doesn't. The coefficient is 5.510.
Free_expression: can take values from 1 to 10. The coefficient is -0.29.
Free_election*free_expression: Interaction between the 2 explanatory variables. The coefficient is -0.9.

Am I correct in saying that the effect of free_elections is to increase the log(odds) of sanctions = 1 by a factor of 4.61 (as 5.510-0.9 = 4.61)? Could I also correctly say that an increase of 1 in free expression reduces the log(odds) by a factor of -1.19 (as -0.9 - 0.29 = -1.19)?

Thanks in advance for any help, really struggling with these interactions!

Ben

[benchammer]

Thanks for your help guys, I think I understand it now. The James Jaccard book seems very good.

Max: The coefficient of 5.51 is for f.elect = 1.

Thanks again,
Ben