Solved – Interpretation of Interaction coefficients in GLM using categorical predictors (in R)

categorical datageneralized linear modelinteraction

I ran a Gamma GLM using 3 categorical predictors:

Year – with 4 classes
Organ – with 3 classes
Site – with 3 classes

My response variable is Biomass.

My model is:

GLM <- glm(biom ~ fyear + organ + site + year:organ + year:site + organ:site,
       data = data, family = Gamma(link = "log"))

The summary(GLM) gives me this (3 coefficients are not defined because of singularities):

                  Estimate   Std.Error t value Pr(>|t|)  
(Intercept)        3.34408    0.39101   8.552 7.89e-14 ***
year2             -0.55195    0.29480  -1.872  0.06382 .  
year3              0.65445    0.29480   2.220  0.02847 *  
year4             -0.20425    0.29616  -0.690  0.49186    
organ2             1.62266    0.39846   4.072 8.80e-05 ***
organ3             2.64728    0.33840   7.823 3.40e-12 ***
site2              1.01485    0.53400   1.900  0.05999 . 
site3              0.41056    0.52632   0.780  0.43703    
year2:organ2       0.03728    0.29480   0.126  0.89959    
year3:organ2      -0.03519    0.29480  -0.119  0.90520    
year4:organ2      -2.03455    0.30021  -6.777 6.34e-10 ***
year2:organ3          NA         NA      NA       NA    
year3:organ3          NA         NA      NA       NA    
year4:organ3          NA         NA      NA       NA    
year2:site2        0.78444    0.36105   2.173  0.03195 *  
year3:site2       -0.01524    0.36105  -0.042  0.96641    
year4:site2        0.28738    0.37216   0.772  0.44166    
year2:site3        1.04849    0.36105   2.904  0.00445 ** 
year3:site3        0.08768    0.36105   0.243  0.80858    
year4:site3        0.71053    0.36105   1.968  0.05159 .  
organ2:site2      -1.41692    0.48655  -2.912  0.00435 ** 
organ3:site2      -1.59445    0.48655  -3.277  0.00140 ** 
organ2:site3      -0.86975    0.47763  -1.821  0.07133 .  
organ3:site3      -1.30913    0.47763  -2.741  0.00715 **

The first coefficient (3.34408) is the intercept, so it stands for biomass for the year 1, site 1 and organ 1.
The second one (-0.55195) is the difference between the mean biomass of the year 2 and year 1.
The third one (0.65445) is the difference between the mean biomass of the year 3 and year 1.
the 4th (-0.20425) is the difference between the mean biomass of the year 4 and year 1.
The 5th (1.62266) is the difference between the mean biomass on organ 2 and organ 1.

…and so on until the 8th coefficient
After the main coefficients start the interactions.

What is their interpretation? YEAR2*ORGAN2 is the difference between what?
In addition, year 3 significantly differs from year 1 while year 4 doesn't. What happens between year 3 and year 4? Do they significantly differ?

Best Answer

Your interpretation should be slightly different:

The second one (-0.55195) is the difference between the mean biomass of the year 2 and year 1 For observations with intercept values on other categories, namely organ1 and year1.
The third one (0.65445) is the difference between the mean biomass of the year 3 and year 1 For observations with intercept values on other categories, namely organ1 and year1
etc.

The interaction effect is the difference in main effect with other categories on other variables. So the eight one (YEAR2*ORGAN2) is the difference between observations with year1 and organ1 versus year2 and organ2 (in addition to the main effect).

About your second question: you can't really answer that based on this table, since all first categories are the reference group. I think you should run a pairwise comparison of groups to answer that question.

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Solved – Interpreting coefficients of an interaction between categorical and continuous variable

Your interpretation of the model’s coefficients is not completely accurate. Let me first summarize the terms of the model.

Categorial variables (factors): $race$, $sex$, and $educa$

The factor race has four levels: $race = \{white, black, mexican, multi/other\}$.

The factor sexhas two levels: $sex = \{male, female\}$.

The factor educa has five levels: $educa = \{1, 2, 3, 4, 5\}$.

By default, R uses treatment contrasts for categorical variables. In these contrasts, the first value of the factor is used a reference level and the remaining values are tested against the reference. The maximum number of contrasts for a categorical variable equals the number of levels minus one.

The contrasts for race allow testing the following differences: $race = black\ vs. race = white$, $race = mexican\ vs. race = white$, and $race = multi/other\ vs. race = white$.

For the factor $educa$, the reference level is $1$, the pattern of contrasts is analogous. These effects can be interpreted as the difference in the dependent variable. In your example, the mean value of cog is $13.8266$ units higher for $educa = 2$ compared to $educa = 1$ (as.factor(educa)2).

One important note: If treatment contrasts for a categorical variable are present in a model, the estimation of further effects is based on the reference level of the categorical variable if interactions between further effects and the categorical variable are included too. If the variable is not part of an interaction, its coefficient corresponds to the average of the the individual slopes of subsets of this varible along all remaining categorical variables. The effects of $race$ and $educa$ correspond to average effects with respect to the factor levels of the other variables. To test overall effects of $race$, you would need to leave $educa$ and $sex$ out of the model.

Numeric variables: $lg\_hag$ and $pdg$

Both lg_hag and pdg are numeric variables hence the coefficients represent the change in the dependent variable associated with an increase of $1$ in the predictor.

In principle, the interpretation of these effects is straightforward. But note that if interations are present, the estimation of the coefficients is based on the references categories of the factors (if treatment contrasts are employed). Since $pdg$ is not part of an interaction, its coefficient corrsespods to the average slope of the variable with respect. The variable $lg\_hag$ is also part of an interaction with $educa$. Therefore, its effect holds for $educa = 1$, the base level.; it is not a test of an overall influence of the numeric variable $lg\_hag$ irrespective of the levels of the factors.

Interactions between categorical and numeric variables: $lg\_hag \times educa$

The model does not only include main effects but also interactions between the numeric variable $lg\_hag$ and the four contrasts associated with $educa$. These effects can be interpreted as the difference in the slopes of $lg\_hag$ between a certain level of $educa$ and the reference level ($educa = 1$).

For example, the coefficient of lg_hag:as.factor(educa)2 (-21.2224) means that slope of $lg\_hag$ is $21.2224$ units lower for $educa = 2$ compared to $educa = 1$.

Solved – Specification and interpretation of interaction terms using glm()

x/z expands to x + x:z and so far I have used this only to model nested random effects.

set.seed(42)
x <- rnorm(100)
z <- rnorm(100)
y <- sample(c(0,1),100,TRUE)

fit2 <- glm(y ~ x/z, family = "binomial") 
fit3 <- glm(y ~ x + z %in% x, family = "binomial")
identical(summary(fit2)$coefficients,summary(fit3)$coefficients)
#TRUE
fit4 <- glm(y ~ x + x:z, family = "binomial")
identical(summary(fit2)$coefficients,summary(fit4)$coefficients)
#TRUE

fit5 <- glm(y ~ I(x/z), family = "binomial")    
a <- x/z
fit6 <- glm(y ~ a, family = "binomial")
all.equal(summary(fit5)$coefficients,summary(fit6)$coefficients)
#[1] "Attributes: < Component 2: Component 1: 1 string mismatch >"
#which means that only the rownames don't match, but values are identical

Best Answer

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Solved – Interpreting coefficients of an interaction between categorical and continuous variable

Solved – Specification and interpretation of interaction terms using glm()

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