# Interaction suppresses the main effect? How to interpret it?

I have a simple model without interaction and it stated significant effect for all the explanatory variables (continuous variable rok and categorical variables obdobi (levels hn and nehn) and kraj:

Call:
glm(formula = cbind(ml, ad) ~ rok + obdobi + kraj, family = "quasibinomial")

Deviance Residuals:
Min       1Q   Median       3Q      Max
-3.8007  -1.1716  -0.5117   1.0864   4.2184

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept) -107.60761   53.96993  -1.994  0.04674 *
rok            0.05381    0.02686   2.003  0.04576 *
obdobinehn    -0.26962    0.11646  -2.315  0.02104 *
krajJHC        0.68869    0.31009   2.221  0.02683 *
krajJHM       -0.26607    0.32166  -0.827  0.40855
krajLBK       -1.11305    0.61942  -1.797  0.07298 .
krajMSK       -0.61390    0.41828  -1.468  0.14285
krajOLK       -0.49704    0.36981  -1.344  0.17958
krajPAK       -1.18444    0.39401  -3.006  0.00279 **
krajPLK       -1.28668    0.49672  -2.590  0.00988 **
krajSTC        0.01872    0.31222   0.060  0.95220
krajULKV      -0.41950    0.69220  -0.606  0.54478
krajVYS       -1.17290    0.44614  -2.629  0.00884 **
krajZLK       -0.38170    0.40969  -0.932  0.35198
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasibinomial family taken to be 1.645035)

Null deviance: 1136.22  on 489  degrees of freedom
Residual deviance:  938.02  on 476  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 4


Then I added interaction obdobi:kraj:

Call:
glm(formula = cbind(ml, ad) ~ rok + obdobi + kraj + obdobi:kraj,
family = "quasibinomial")

Deviance Residuals:
Min       1Q   Median       3Q      Max
-3.4635  -1.1706  -0.4597   1.0275   4.6829

Coefficients:
Estimate Std. Error t value Pr(>|t|)
(Intercept)         -101.49501   54.53576  -1.861  0.06336 .
rok                    0.05102    0.02715   1.879  0.06086 .
obdobinehn            -1.11653    0.62058  -1.799  0.07264 .
krajJHC               -0.16805    0.51957  -0.323  0.74651
krajJHM               -0.77451    0.53738  -1.441  0.15018
krajLBK               -3.29567    1.42164  -2.318  0.02087 *
krajMSK               -0.73640    0.67267  -1.095  0.27420
krajOLK               -0.41582    0.68758  -0.605  0.54564
krajPAK               -1.50156    0.63871  -2.351  0.01914 *
krajPLK               -1.48611    0.75745  -1.962  0.05036 .
krajSTC               -0.34170    0.52059  -0.656  0.51191
krajULKV              -1.72550    1.02726  -1.680  0.09369 .
krajVYS               -1.93603    0.65862  -2.940  0.00345 **
krajZLK               -0.71065    0.65791  -1.080  0.28063
obdobinehn:krajJHC     1.44638    0.65507   2.208  0.02773 *
obdobinehn:krajJHM     0.82070    0.67910   1.209  0.22746
obdobinehn:krajLBK     3.31340    1.61026   2.058  0.04018 *
obdobinehn:krajMSK     0.12470    0.87281   0.143  0.88645
obdobinehn:krajOLK     0.04528    0.82529   0.055  0.95627
obdobinehn:krajPAK     0.48978    0.81921   0.598  0.55022
obdobinehn:krajPLK     0.23075    1.02316   0.226  0.82167
obdobinehn:krajSTC     0.50339    0.65976   0.763  0.44585
obdobinehn:krajULKV    2.49157    1.43679   1.734  0.08356 .
obdobinehn:krajVYS     1.48201    0.92082   1.609  0.10820
obdobinehn:krajZLK     0.49357    0.85087   0.580  0.56214
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

(Dispersion parameter for quasibinomial family taken to be 1.613648)

Null deviance: 1136.22  on 489  degrees of freedom
Residual deviance:  899.28  on 465  degrees of freedom
AIC: NA

Number of Fisher Scoring iterations: 4


Strange thing happened - the main effects rok and obdobi are no longer significant! How can this happen? How to interpret this fact? If the interaction obdobi:kraj has significant effect, then the obdobi also has significant effect, right?

Note that the second model differs significantly (tested by anova(..., test = "Chi")).

EDIT: added anova tables of the models (but since this is glm and not simple lm, mean sum of squares and p-values are missing and I don't know how to interpret it...)

> anova(model1)
Analysis of Deviance Table

Model: quasibinomial, link: logit

Terms added sequentially (first to last)

Df Deviance Resid. Df Resid. Dev
NULL                      489    1136.22
rok      1     3.06       488    1133.16
obdobi   1    11.20       487    1121.96
kraj    11   183.94       476     938.02

> anova(model2)
Analysis of Deviance Table

Model: quasibinomial, link: logit

Terms added sequentially (first to last)

Df Deviance Resid. Df Resid. Dev
NULL                           489    1136.22
rok           1     3.06       488    1133.16
obdobi        1    11.20       487    1121.96
kraj         11   183.94       476     938.02
obdobi:kraj  11    38.74       465     899.28

• Speaking of ANOVA, have you tried to look at the models separately from an ANOVA point of view and see what you get? e.g. anova(myadditivemodel) – John Sep 28 '11 at 22:52
• @John, I tried (I've added the output of anova(), see my updated post), but since this is glm and not simple lm, mean sum of squares and p-values are missing and I don't know how to interpret it... – Curious Sep 28 '11 at 23:06
• It looks like you can do anova(model2, test="F") to get p-values, but they are for tests of adjacent rows in a series of nested models. – Karl Sep 28 '11 at 23:42
• @Karl, I thought I have to use test="Chi" for glm! – Curious Sep 28 '11 at 23:53
• @Tomas yes you're probably right – Karl Sep 29 '11 at 0:01

The main effects went from "significant" to "not", but the evidence really didn't change all that much. For example, p=0.047 to p=0.063 for rok isn't, to me, a remarkable change. And a lack of evidence for a coefficient being non-zero isn't the same as saying it is 0.

In considering the coefficient for obdobinehn when the interaction is included, you need to pay careful attention to the factor contrasts that are being used, as the meaning of the coefficient changes and depends on those contrasts.

Note also that if a covariate is involved in an important interaction, then it does have an effect on the outcome, even if it shows no main effect.

I agree with John's comment that it's useful, with factor covariates, to look at an ANOVA table.

• Thanks, Karl, for response! 1) can you please explain the 2nd paragraph of your answer? 2) I updated my post and included the ANOVA tables, but since this is not a simple lm I don't know how to interpret them... Thanks! – Curious Sep 28 '11 at 23:15
• You have different options for how factors are converted to numeric columns. Type options("contrasts") to see what's being used. It's probably "contr.treatment". – Karl Sep 28 '11 at 23:28
• @Tomas, There are some potentially useful links about contrasts at this StackOverflow question. Also look at the help file for contr.treatment. – Karl Sep 28 '11 at 23:35
• yes, it is contr.treatment. Do you recommend other setting? Is it somehow important for the core of the question or is it just a convenience thing? (Just to be able to interpret the coefficients easily). Thanks – Curious Sep 28 '11 at 23:46
• @tomas, the meaning of that obd... coefficient, and so what it means for it to be zero, changes with different treatments. With the interaction, I think it's the effect of obd... within the first level of kraj. – Karl Sep 29 '11 at 0:05

You'll notice that in your ANOVA's (deviance tables) of the models there is no difference in the main effects with, or without the interaction. You don't have to know how to interpret the deviance table, just recognize that there's no difference!

Keep in mind that your "Estimate" column in the regression is about the magnitude of the slope and the associated tests are of that magnitude. When you add interactions you can change how the slope is calculated and change it's significance. That doesn't mean it went away, it just means it's qualified by an interaction.

So, a short answer is that, if you had a main effect without the interaction then you have a main effect. It's very common to do the additive and then additive+interaction models separately so you can see where your main effects are and then look at your interactions. The fact that it went away gives you some clues about the kind of interaction that you have but it's hard for someone to answer the whole thing with just what you've reported. Your next step is to start making some graphs. For example, make one with obdobinehn at different levels of kraj.

You should really look at a paper on interpreting these interaction effects. A complete answer for your query is far too difficult to do based on what you've provided, and even guidance about where to go next is very involved. Read the linked paper, see how far you get, and get back to the SE with more questions at that time.

• But those ANOVA tables are built up a term at a time with the interaction last, so the main effects rows are exactly the same and are in the absence of interaction. – Karl Sep 29 '11 at 3:01
• shhh... don't tell them that... it makes it look less magical. You could just consider it an example of an accepted practice where what I described happens. Look at the additive effects, record what is significant, and then look at interactions and ignore the additive effects changing (until you've looked at the interactions carefully and understand it). – John Sep 29 '11 at 3:16