Interpretation of interaction terms of categorical variables and the calculation of margins mean

[Vincent Li]

Greetings!

I'll take a random-effect model as an example to elaborate my questions.


Regarding the first question:

xtreg srs i.condition##i.period, i(indi_num) re vce(robust)
margins i.condition#i.period,atmeans

Random-effects GLS regression Number of obs = 366
Group variable: indi_num Number of groups = 122

R-squared: Obs per group:
Within = 0.1207 min = 3
Between = 0.0129 avg = 3.0
Overall = 0.0273 max = 3

Wald chi2(8) = 22.56
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0040

(Std. err. adjusted for 122 clusters in indi_num)
----------------------------------------------------------------------------------
| Robust
srs | Coefficient std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
condition |
1 | -2.707317 4.5169 -0.60 0.549 -11.56028 6.145645
2 | .1810976 4.571382 0.04 0.968 -8.778647 9.140842
|
period |
1 | -8.97561 2.776295 -3.23 0.001 -14.41705 -3.534171
2 | -8.268293 2.256088 -3.66 0.000 -12.69014 -3.846442
|
condition#period |
1 1 | 6.609756 3.346904 1.97 0.048 .0499453 13.16957
1 2 | 3.439024 3.061285 1.12 0.261 -2.560983 9.439032
2 1 | 7.92561 3.131587 2.53 0.011 1.787812 14.06341
2 2 | 6.593293 2.895518 2.28 0.023 .9181814 12.2684
|
_cons | 94.2439 3.354506 28.09 0.000 87.66919 100.8186
-----------------+----------------------------------------------------------------
sigma_u | 18.959945
sigma_e | 8.8829199
rho | .82000724 (fraction of variance due to u_i)
----------------------------------------------------------------------------------





Adjusted predictions Number of obs = 366
Model VCE: Robust

Expression: Linear prediction, predict()
At: 0.condition = .3360656 (mean)
1.condition = .3360656 (mean)
2.condition = .3278689 (mean)
0.period = .3333333 (mean)
1.period = .3333333 (mean)
2.period = .3333333 (mean)

----------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
condition#period |
0 0 | 94.2439 3.354506 28.09 0.000 87.66919 100.8186
0 1 | 85.26829 3.258103 26.17 0.000 78.88253 91.65406
0 2 | 85.97561 3.112846 27.62 0.000 79.87454 92.07668
1 0 | 91.53659 3.024844 30.26 0.000 85.608 97.46517
1 1 | 89.17073 3.506125 25.43 0.000 82.29885 96.04261
1 2 | 86.70732 3.46236 25.04 0.000 79.92122 93.49342
2 0 | 94.425 3.105612 30.40 0.000 88.33811 100.5119
2 1 | 93.375 3.435914 27.18 0.000 86.64073 100.1093
2 2 | 92.75 3.332357 27.83 0.000 86.2187 99.2813
----------------------------------------------------------------------------------

combining with the outcomes of margins, the mean of 0.con#0.peorid=intercept=94.24.
for example, 1.con#0.peorid=94.24-2.17=91.53, 1.con#1.peorid=94.24-2.17-8.98+6.61=89.16
following this way, the difference between 1.con#0.peorid and 1.con#1.peorid should be -8.98+6.61. However, when interpreting the results of the random effect, our assessment of whether there is a significant difference between 1.con#0.peorid and 1.con#1.peorid depends solely on:

condition#period |
            1 1  |   6.609756   3.346904     1.97   0.048     .0499453    13.16957

Why is it? or did I misinterpret the outcomes?


For the second question, a between-subject var was added into the model:

xtreg srs age i.condition##i.period, i(indi_num) re vce(robust)
margins i.condition#i.period,atmeans

Random-effects GLS regression Number of obs = 366
Group variable: indi_num Number of groups = 122

R-squared: Obs per group:
Within = 0.1207 min = 3
Between = 0.0501 avg = 3.0
Overall = 0.0595 max = 3

Wald chi2(9) = 28.71
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0007

(Std. err. adjusted for 122 clusters in indi_num)
----------------------------------------------------------------------------------
| Robust
srs | Coefficient std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
age | 3.102243 1.449154 2.14 0.032 .2619534 5.942532
|
condition |
1 | -2.404659 4.427024 -0.54 0.587 -11.08147 6.272149
2 | .7410146 4.599746 0.16 0.872 -8.274322 9.756351
|
period |
1 | -8.97561 2.780192 -3.23 0.001 -14.42469 -3.526534
2 | -8.268293 2.259254 -3.66 0.000 -12.69635 -3.840236
|
condition#period |
1 1 | 6.609756 3.351601 1.97 0.049 .0407385 13.17877
1 2 | 3.439024 3.065581 1.12 0.262 -2.569404 9.447453
2 1 | 7.92561 3.135982 2.53 0.011 1.779198 14.07202
2 2 | 6.593293 2.899582 2.27 0.023 .9102163 12.27637
|
_cons | 73.20918 10.43768 7.01 0.000 52.75171 93.66665
-----------------+----------------------------------------------------------------
sigma_u | 18.657676
sigma_e | 8.8829199
rho | .81521436 (fraction of variance due to u_i)
----------------------------------------------------------------------------------









Adjusted predictions Number of obs = 366
Model VCE: Robust

Expression: Linear prediction, predict()
At: age = 6.688525 (mean)
0.condition = .3360656 (mean)
1.condition = .3360656 (mean)
2.condition = .3278689 (mean)
0.period = .3333333 (mean)
1.period = .3333333 (mean)
2.period = .3333333 (mean)

----------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
condition#period |
0 0 | 93.95861 3.403995 27.60 0.000 87.2869 100.6303
0 1 | 84.983 3.302808 25.73 0.000 78.50962 91.45639
0 2 | 85.69032 3.077888 27.84 0.000 79.65777 91.72287
1 0 | 91.55395 2.8207 32.46 0.000 86.02548 97.08242
1 1 | 89.1881 3.377072 26.41 0.000 82.56916 95.80704
1 2 | 86.72468 3.299201 26.29 0.000 80.25837 93.191
2 0 | 94.69963 3.085564 30.69 0.000 88.65203 100.7472
2 1 | 93.64963 3.452676 27.12 0.000 86.8825 100.4167
2 2 | 93.02463 3.333012 27.91 0.000 86.49204 99.55721
----------------------------------------------------------------------------------

How to calculate 0.con#0.period in margins table (93.96) with the intercept from the RE model when other variables are controlled for?


Thank you!

[Clyde Schechter]

However, when interpreting the results of the random effect, our assessment of whether there is a significant difference between 1.con#0.peorid and 1.con#1.peorid depends solely on:
Code:

condition#period | 1 1 | 6.609756 3.346904 1.97 0.048 .0499453 13.16957
Why is it? or did I misinterpret the outcomes?

No, that's not right. Your calculation that preceded that is incorrect. And the difference between 1.condition#0.period and 1.condition#1.period is 1.condition#1.period + 1.period.

How to calculate 0.con#0.period in margins table (93.96) with the intercept from the RE model when other variables are controlled for?

You can't do it just from the -xtreg, re- model outputs because it also depends on the distribution of the age variable in the estimation sample. And using the -atmeans- option here is not the correct way to adjust for covariate age because it also constrains the values of condition and period in the calculations. What you really wanted to run is -margins condition#period, at((mean) age)-. Had you done that, the predicted margin for 0.condition#0.period would be calculated as the value of _b[cons] + _b[age]*6.688525, the last of these numbers being the mean value of age in the estimation sample.

[Vincent Li]

Thanks Clyde for clarification.

Let's star from the second question.
I ran -margins i.condition#i.period, at((mean) age)-

The outcomes are:

Adjusted predictions Number of obs = 366
Model VCE: Robust

Expression: Linear prediction, predict()
At: age = 6.688525 (mean)

----------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
condition#period |
0 0 | 93.95861 3.403995 27.60 0.000 87.2869 100.6303
0 1 | 84.983 3.302808 25.73 0.000 78.50962 91.45639
0 2 | 85.69032 3.077888 27.84 0.000 79.65777 91.72287
1 0 | 91.55395 2.8207 32.46 0.000 86.02548 97.08242
1 1 | 89.1881 3.377072 26.41 0.000 82.56916 95.80704
1 2 | 86.72468 3.299201 26.29 0.000 80.25837 93.191
2 0 | 94.69963 3.085564 30.69 0.000 88.65203 100.7472
2 1 | 93.64963 3.452676 27.12 0.000 86.8825 100.4167
2 2 | 93.02463 3.333012 27.91 0.000 86.49204 99.55721
----------------------------------------------------------------------------------

The means for condition and period have not been estimated. However, the margins for i.condition#i.period are identical to those obtained after running -margins i.condition#i.period, atmeans-. Is this typical?
With the estimation, the margins of 0.condition#0.period=73.20918+6.688525*3.102243=93 .9586.


Regarding the first question, let's still use the first example without covariates.

xtreg srs i.condition##i.period, i(indi_num) re vce(robust)
margins i.condition#i.period,atmeans

Random-effects GLS regression Number of obs = 366
Group variable: indi_num Number of groups = 122

R-squared: Obs per group:
Within = 0.1207 min = 3
Between = 0.0129 avg = 3.0
Overall = 0.0273 max = 3

Wald chi2(8) = 22.56
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0040

(Std. err. adjusted for 122 clusters in indi_num)
----------------------------------------------------------------------------------
| Robust
srs | Coefficient std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
condition |
1 | -2.707317 4.5169 -0.60 0.549 -11.56028 6.145645
2 | .1810976 4.571382 0.04 0.968 -8.778647 9.140842
|
period |
1 | -8.97561 2.776295 -3.23 0.001 -14.41705 -3.534171
2 | -8.268293 2.256088 -3.66 0.000 -12.69014 -3.846442
|
condition#period |
1 1 | 6.609756 3.346904 1.97 0.048 .0499453 13.16957
1 2 | 3.439024 3.061285 1.12 0.261 -2.560983 9.439032
2 1 | 7.92561 3.131587 2.53 0.011 1.787812 14.06341
2 2 | 6.593293 2.895518 2.28 0.023 .9181814 12.2684
|
_cons | 94.2439 3.354506 28.09 0.000 87.66919 100.8186
-----------------+----------------------------------------------------------------
sigma_u | 18.959945
sigma_e | 8.8829199
rho | .82000724 (fraction of variance due to u_i)
----------------------------------------------------------------------------------






. margins i.condition#i.period

Adjusted predictions Number of obs = 366
Model VCE: Robust

Expression: Linear prediction, predict()

----------------------------------------------------------------------------------
| Delta-method
| Margin std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
condition#period |
0 0 | 94.2439 3.354506 28.09 0.000 87.66919 100.8186
0 1 | 85.26829 3.258103 26.17 0.000 78.88253 91.65406
0 2 | 85.97561 3.112846 27.62 0.000 79.87454 92.07668
1 0 | 91.53659 3.024844 30.26 0.000 85.608 97.46517
1 1 | 89.17073 3.506125 25.43 0.000 82.29885 96.04261
1 2 | 86.70732 3.46236 25.04 0.000 79.92122 93.49342
2 0 | 94.425 3.105612 30.40 0.000 88.33811 100.5119
2 1 | 93.375 3.435914 27.18 0.000 86.64073 100.1093
2 2 | 92.75 3.332357 27.83 0.000 86.2187 99.2813
----------------------------------------------------------------------------------

The formula is: SRSit=a_it+b_1i*condition+b_2it*period+b_3itcondition*period+e

0.condition*0.period=94.2439

1.condition*0.period=94.2439-2.707317*1=91.5366, which is consistent in margins

1.condition*1.period=1.condition*0.period+1.period +1.condition*1.period=94.2439-2.707317*1-8.97561*1+6.609756*1*1=89.1707, which is consistent in margins
That's also how I calculated above. Why is it incorrect?

the difference between 1.con#0.peorid and 1.con#1.peorid should be -8.98+6.61

-8.98+6.61 is 1.period+1.condition*1.period as you mentioned above.


When interpreting the outcomes of this part:

condition#period |
1 1 | 6.609756 3.346904 1.97 0.048 .0499453 13.16957
1 2 | 3.439024 3.061285 1.12 0.261 -2.560983 9.439032
2 1 | 7.92561 3.131587 2.53 0.011 1.787812 14.06341
2 2 | 6.593293 2.895518 2.28 0.023 .9181814 12.2684

Should it be interpreted that when condition = 1, the dependent variable will increase by 6.609756 on average as the period changes from point 0 to point 1? However, this interpretation seems inconsistent with the calculations, as the dependent variable should increase by -8.98 + 6.61.

How should I report the results for condition 1 and condition 2 based on the findings above? or should I report the change according to the margins?

[Vincent Li]

I followed your previous post:
https://www.statalist.org/forums/for...action-results

this time, I ran

xtreg srs i.condition##i.period, i(indi_num) re vce(robust)

Random-effects GLS regression Number of obs = 366
Group variable: indi_num Number of groups = 122

R-squared: Obs per group:
Within = 0.1207 min = 3
Between = 0.0129 avg = 3.0
Overall = 0.0273 max = 3

Wald chi2(8) = 22.56
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0040

(Std. err. adjusted for 122 clusters in indi_num)
----------------------------------------------------------------------------------
| Robust
srs | Coefficient std. err. z P>|z| [95% conf. interval]
-----------------+----------------------------------------------------------------
condition |
1 | -2.707317 4.5169 -0.60 0.549 -11.56028 6.145645
2 | .1810976 4.571382 0.04 0.968 -8.778647 9.140842
|
period |
1 | -8.97561 2.776295 -3.23 0.001 -14.41705 -3.534171
2 | -8.268293 2.256088 -3.66 0.000 -12.69014 -3.846442
|
condition#period |
1 1 | 6.609756 3.346904 1.97 0.048 .0499453 13.16957
1 2 | 3.439024 3.061285 1.12 0.261 -2.560983 9.439032
2 1 | 7.92561 3.131587 2.53 0.011 1.787812 14.06341
2 2 | 6.593293 2.895518 2.28 0.023 .9181814 12.2684
|
_cons | 94.2439 3.354506 28.09 0.000 87.66919 100.8186
-----------------+----------------------------------------------------------------
sigma_u | 18.959945
sigma_e | 8.8829199
rho | .82000724 (fraction of variance due to u_i)
----------------------------------------------------------------------------------

margins period, dydx(condition)

Conditional marginal effects Number of obs = 366
Model VCE: Robust

Expression: Linear prediction, predict()
dy/dx wrt: 1.condition 2.condition

------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
0.condition | (base outcome)
-------------+----------------------------------------------------------------
1.condition |
period |
0 | -2.707317 4.5169 -0.60 0.549 -11.56028 6.145645
1 | 3.902439 4.786246 0.82 0.415 -5.478431 13.28331
2 | .7317073 4.655937 0.16 0.875 -8.393762 9.857176
-------------+----------------------------------------------------------------
2.condition |
period |
0 | .1810976 4.571382 0.04 0.968 -8.778647 9.140842
1 | 8.106707 4.735055 1.71 0.087 -1.173829 17.38724
2 | 6.77439 4.560089 1.49 0.137 -2.16322 15.712
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

the outcomes of-margins period, dydx(condition)- are what I should refer to, right?

[Vincent Li]

When interpreting the outcomes of this part:
period |
1 | -8.97561 2.776295 -3.23 0.001 -14.41705 -3.534171
2 | -8.268293 2.256088 -3.66 0.000 -12.69014 -3.846442
condition#period |
1 1 | 6.609756 3.346904 1.97 0.048 .0499453 13.16957
1 2 | 3.439024 3.061285 1.12 0.261 -2.560983 9.439032
2 1 | 7.92561 3.131587 2.53 0.011 1.787812 14.06341
2 2 | 6.593293 2.895518 2.28 0.023 .9181814 12.2684

Should it be interpreted that when condition = 1, the dependent variable will increase by 6.609756 on average as the period changes from point 0 to point 1? However, this interpretation seems inconsistent with the calculations, as the dependent variable should increase by -8.98 + 6.61.

How should I report the results for condition 1 and condition 2 based on the findings above? or should I report the change according to the margins?

As for the interpretation, p value is also a part of it. Does the result indicate that, in condition 1, the dependent variable will increase by an average of 6.609756+(-8.98) (1.period) as the period changes from timepoint 0 to timepoint 1? but no sig dif between time point 0 and time point 2 (the change in Y should be 1,condition#2.period+1.period)? Should we use the p-value of the interaction term to assess the significance of the difference between time points 0 and 1, or between time points 0 and 2 in condition 1?

[Clyde Schechter]

1.

1.condition*1.period=1.condition*0.period+1.period +1.condition*1.period=94.2439-2.707317*1-8.97561*1+6.609756*1*1=89.1707, which is consistent in margins
That's also how I calculated above. Why is it incorrect?

No, it's not how you calculated above. It's correct as you have it here. But in the original question you omitted the 1.period term.

2.

Should it be interpreted that when condition = 1, the dependent variable will increase by 6.609756 on average as the period changes from point 0 to point 1?

No! The regression coefficients for the interaction terms are not interpreted that way. The interpretation is that the marginal effect of condition = 1 will increase by 6.609756 on average as the period changes from point 0 to point 1. It does not directly say anything about how the dependent variable changes.

3.

the outcomes of-margins period, dydx(condition)- are what I should refer to, right?

Refer to for what purpose? The results of -margins period, dydx(condition)- give you the marginal effects of condition, at each value of period. If that's what you want.

4.

Does the result indicate that, in condition 1, the dependent variable will increase by an average of 6.609756+(-8.98) (1.period) as the period changes from timepoint 0 to timepoint 1?

Yes. For a significance test of this difference, -lincom 1.period + 1.condition#1.period-.

[Vincent Li]

Thank you so much Clyde. Now I totally understand them.

the outcomes of-margins period, dydx(condition)- are what I should refer to, right?
Refer to for what purpose? The results of -margins period, dydx(condition)- give you the marginal effects of condition, at each value of period. If that's what you want.

No, that’s not what I’m looking for. I want to know the results of -margins condition, dydx(period)-. This will yield consistent results with -lincom 1.period + 1.condition#1.period- and -margins i.condition#i.period, pwcompare(pv), correct?"

margins condition, dydx(period)

Conditional marginal effects Number of obs = 366
Model VCE: Robust

Expression: Linear prediction, predict()
dy/dx wrt: 1.period 2.period

------------------------------------------------------------------------------
| Delta-method
| dy/dx std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
0.period | (base outcome)
-------------+----------------------------------------------------------------
1.period |
condition |
0 | -8.97561 2.776295 -3.23 0.001 -14.41705 -3.534171
1 | -2.365854 1.869211 -1.27 0.206 -6.02944 1.297733
2 | -1.05 1.4488 -0.72 0.469 -3.889596 1.789596
-------------+----------------------------------------------------------------
2.period |
condition |
0 | -8.268293 2.256088 -3.66 0.000 -12.69014 -3.846442
1 | -4.829268 2.069186 -2.33 0.020 -8.884799 -.7737381
2 | -1.675 1.814964 -0.92 0.356 -5.232263 1.882263
------------------------------------------------------------------------------
Note: dy/dx for factor levels is the discrete change from the base level.

lincom 1.period + 1.condition#1.period
lincom 2.period + 1.condition#2.period
lincom 1.period + 2.condition#1.period
lincom 2.period + 2.condition#2.period

lincom 1.period + 1.condition#1.period

( 1) 1.period + 1.period#1.condition = 0

------------------------------------------------------------------------------
srs | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
(1) | -2.365854 1.869211 -1.27 0.206 -6.02944 1.297733
------------------------------------------------------------------------------

. lincom 2.period + 1.condition#2.period

( 1) 2.period + 2.period#1.condition = 0

------------------------------------------------------------------------------
srs | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
(1) | -4.829268 2.069186 -2.33 0.020 -8.884799 -.7737381
------------------------------------------------------------------------------

. lincom 1.period + 2.condition#1.period

( 1) 1.period + 1.period#2.condition = 0

------------------------------------------------------------------------------
srs | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
(1) | -1.05 1.4488 -0.72 0.469 -3.889596 1.789596
------------------------------------------------------------------------------

. lincom 2.period + 2.condition#2.period

( 1) 2.period + 2.period#2.condition = 0

------------------------------------------------------------------------------
srs | Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
(1) | -1.675 1.814964 -0.92 0.356 -5.232263 1.882263
------------------------------------------------------------------------------

margins i.condition#i.period,pwcompare(pv)

Pairwise comparisons of adjusted predictions Number of obs = 366
Model VCE: Robust

Expression: Linear prediction, predict()

---------------------------------------------------------
| Delta-method Unadjusted
| Contrast std. err. z P>|z|
-----------------+---------------------------------------
condition#period |
(0 1) vs (0 0) | -8.97561 2.776295 -3.23 0.001
(0 2) vs (0 0) | -8.268293 2.256088 -3.66 0.000
(1 0) vs (0 0) | -2.707317 4.5169 -0.60 0.549
(1 1) vs (0 0) | -5.073171 4.852383 -1.05 0.296
(1 2) vs (0 0) | -7.536585 4.820856 -1.56 0.118
(2 0) vs (0 0) | .1810976 4.571382 0.04 0.968
(2 1) vs (0 0) | -.8689024 4.801897 -0.18 0.856
(2 2) vs (0 0) | -1.493902 4.728352 -0.32 0.752
(0 2) vs (0 1) | .7073171 1.979934 0.36 0.721
(1 0) vs (0 1) | 6.268293 4.445775 1.41 0.159
(1 1) vs (0 1) | 3.902439 4.786246 0.82 0.415
(1 2) vs (0 1) | 1.439024 4.75428 0.30 0.762
(2 0) vs (0 1) | 9.156707 4.501118 2.03 0.042
(2 1) vs (0 1) | 8.106707 4.735055 1.71 0.087
(2 2) vs (0 1) | 7.481707 4.660455 1.61 0.108
(1 0) vs (0 2) | 5.560976 4.340448 1.28 0.200
(1 1) vs (0 2) | 3.195122 4.688574 0.68 0.496
(1 2) vs (0 2) | .7317073 4.655937 0.16 0.875
(2 0) vs (0 2) | 8.44939 4.397117 1.92 0.055
(2 1) vs (0 2) | 7.39939 4.636304 1.60 0.110
(2 2) vs (0 2) | 6.77439 4.560089 1.49 0.137
(1 1) vs (1 0) | -2.365854 1.869211 -1.27 0.206
(1 2) vs (1 0) | -4.829268 2.069186 -2.33 0.020
(2 0) vs (1 0) | 2.888415 4.335263 0.67 0.505
(2 1) vs (1 0) | 1.838415 4.577683 0.40 0.688
(2 2) vs (1 0) | 1.213415 4.500476 0.27 0.787
(1 2) vs (1 1) | -2.463415 1.678131 -1.47 0.142
(2 0) vs (1 1) | 5.254268 4.683774 1.12 0.262
(2 1) vs (1 1) | 4.204268 4.909014 0.86 0.392
(2 2) vs (1 1) | 3.579268 4.837098 0.74 0.459
(2 0) vs (1 2) | 7.717683 4.651104 1.66 0.097
(2 1) vs (1 2) | 6.667683 4.877853 1.37 0.172
(2 2) vs (1 2) | 6.042683 4.80547 1.26 0.209
(2 1) vs (2 0) | -1.05 1.4488 -0.72 0.469
(2 2) vs (2 0) | -1.675 1.814964 -0.92 0.356
(2 2) vs (2 1) | -.625 1.487739 -0.42 0.674
---------------------------------------------------------