christoph - 3 months ago 27

R Question

I am trying to understand how R determines reference groups for interactions in a linear model. Consider the following:

`df <- structure(list(id = structure(c(1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L,`

2L, 2L, 2L, 2L, 3L, 3L, 3L, 3L, 3L, 3L, 4L, 4L, 4L, 4L, 4L, 4L,

5L, 5L, 5L, 5L, 5L, 5L), .Label = c("1", "2", "3", "4", "5"), class = "factor"),

year = structure(c(1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L,

1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L, 2L, 1L,

2L, 1L, 2L, 1L, 2L), .Label = c("1", "2"), class = "factor"),

treatment = structure(c(2L, 2L, 2L, 2L, 2L, 2L, 1L, 1L, 1L,

1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 2L, 2L, 2L, 2L, 2L, 2L,

2L, 2L, 2L, 2L, 2L, 2L), .Label = c("0", "1"), class = "factor"),

y = c(1.4068142116718, 2.67041187927052, 2.69166439169131,

3.56550324537293, 1.60021286173782, 4.26629963353237, 3.85741108250572,

5.15740731957689, 4.15629768365669, 6.14875441068499, 3.31277276551286,

3.47223277168367, 3.74152201649338, 4.02734382610191, 4.49388620764795,

5.6432833241724, 4.76639399631094, 4.16885857079297, 4.96830394378801,

5.6286092105837, 6.60521404151111, 5.51371821706176, 3.97244221149279,

5.68793413111161, 4.90457233598412, 6.02826151378941, 4.92468415350312,

8.23718422822134, 5.87695836962708, 7.47264895892575)), .Names = c("id",

"year", "treatment", "y"), row.names = c(NA, -30L), class = "data.frame")

lm(y ~ -1 + id + year + year:treatment, df)

#Coefficients:

# id1 id2 id3 id4

# 2.6585 3.9933 4.1161 5.3544

# id5 year2 year1:treatment1 year2:treatment1

# 6.1991 0.7149 -0.6317 NA

R tries to estimate the full set of interactions instead of consistently omitting a reference group. As a result, I am getting

`NA`

Also, R is inconsistent with which groups it drops. I would like to estimate a model with the same omitted group (

`year1`

`year1`

`year1:treatment1`

I understand that there are several workarounds for this problem (e.g. creating all the variables by hand and writing them out in the model's formula). But the actual models I am estimating are much more complicated versions of this problem and such a workaround would be cumbersome.

Answer

R tries to estimate the full set of interactions instead of consistently omitting a reference group. As a result, I am getting

`NA`

s in the results.

There is no causality between those two. You get `NA`

purely because your variables have nesting.

R is inconsistent with which groups it drops. I would like to estimate a model with the same omitted group (

`year1`

) in the main effect and interactions. How to force R to omit`year1`

and`year1:treatment1`

from the preceding model?

There is no inconsistency but `model.matrix`

has its own rule. You get seemingly "inconsistent" contrasts because you don't have main effect `treatment`

but only interaction `treatment:year`

.

In the following, I will first explain `NA`

coefficients, then the contrasts for interaction, and finally give some suggestions.

`NA`

coefficientsBy default, contrast treatment is used for contrasting factor, and by default of `contr.treatement`

, the first factor level is taken as reference level. Have a look at your data:

```
levels(df$id)
# [1] "1" "2" "3" "4" "5"
levels(df$year)
# [1] "1" "2"
levels(df$treatment)
# [1] "0" "1"
```

Now take a look at a simple linear model:

```
lm(y ~ id + year + treatment, df)
#Coefficients:
#(Intercept) id2 id3 id4 id5 year2
# 2.153 1.651 1.773 2.696 3.541 1.094
# treatment1
# NA
```

You can see that `id1`

, `year1`

and `treatment0`

are not there, as they are taken as reference.

Even without interaction, you already have an `NA`

coefficient. This implies that `treatment`

is nested in `span{id, year}`

. When you further include `treatment:year`

, such nesting still exists.

In fact, a further test shows that `treatment`

is nested in `id`

:

```
lm(y ~ id + treatment, df)
# Coefficients:
#(Intercept) id2 id3 id4 id5 treatment1
# 2.700 1.651 1.773 2.696 3.541 NA
```

In summary, variable `treatment`

is completely redundant for your modelling purpose. If you include `id`

, then there is no need to include `treatment`

or `treatment:*`

where `*`

can be any variable.

**It is very easy to get nesting when you have many factor variables in a regression model. Note, contrasts will not necessarily remove nesting, as contrast only recognises variable name, but not potential numerical feature.** Take the following example for how to cheat `contr.treatment`

:

```
df$ID <- df$id
lm(y ~ id + ID, df)
#Coefficients:
#(Intercept) id2 id3 id4 id5 ID2
# 2.700 1.651 1.773 2.696 3.541 NA
# ID3 ID4 ID5
# NA NA NA
```

Look, contrasts works as expected, but `ID`

is nested in `id`

so we have rank-deficiency.

We first remove the noise imposed by `NA`

, by dropping `id`

variable. Then, a regression model with `treatment`

and `year`

will be full-rank, so no `NA`

should be seen if contrasts is successful.

**Contrasts for interaction, or high-order effects, depends on the presence of low-order effects.** Compare the following models:

```
lm(y ~ year:treatment, df) ## no low-order effects at all
#Coefficients:
# (Intercept) treatment0:year1 treatment1:year1 treatment0:year2
# 5.4523 -1.3976 -1.3466 -0.6826
#treatment1:year2
# NA
lm(y ~ year + treatment + year:treatment, df) ## with all low-order effects
#Coefficients:
# (Intercept) treatment1 year2 treatment1:year2
# 4.05471 0.05094 0.71493 0.63170
```

In the first model, no contrasts is done, because there is no presence of main effects, but only the 2nd order effect. The `NA`

here is due to the absence of contrasts.

Now consider another set of examples, by including some but not all main effects:

```
lm(y ~ year + year:treatment, df) ## main effect `treatment` is missing
#Coefficients:
# (Intercept) year2 year1:treatment1 year2:treatment1
# 4.05471 0.71493 0.05094 0.68264
lm(y ~ treatment + year:treatment, df) ## main effect `year` is missing
#Coefficients:
# (Intercept) treatment1 treatment0:year2 treatment1:year2
# 4.05471 0.05094 0.71493 1.34663
```

Here, contrasts for interaction will happen to the variable whose main effect is missing. For example, in the first model, main effect `treatment`

is missing so interaction drops `treatement0`

; while in the second model, main effect `year`

is missing so interaction drops `year1`

.

Always include all low-order effects when specifying high-order effects. This not only gives easy-to-understand contrasts behaviour, but also has some other appealing statistical reason. You can read Including the interaction but not the main effects in a model on Cross Validated.

Another suggestion, is to always include intercept. In linear regression, a model with intercept yields unbiased estimate, and residuals will have 0 mean.