I am working on a study that examines word processing with native and non-native speakers. We have three independent variables: Groups (NSs and NNS), word types (five conditions), and word relatedness (related and unrelated words), and one dependent variable; Reaction Time.
I have successfully ran linear mixed effects model in R using (lme4) package and I was able to understand the output. However, there is something that I do not understand, the intercepts (reference level). Could someone explain what does the intercept or reference means? Does it mean all levels of the variables are compared to it? If so, should I adjust the intercept level or let lme4 uses the default?
I am struggling to make this decision and I need reasons to back up any choice I make.
Note that that the Word Types are not related to each other, but they are compared with regard to Word Relatedness. Interestingly, when I change the intercept level the whole output changes. If you see below the Related:WordType also changes when the intercept is changed — shouldn't this be the same in the two outputs?
> contrasts(NS$WordType) # default intercept by R
2 3 4 5
1 0 0 0 0
2 1 0 0 0
3 0 1 0 0
4 0 0 1 0
5 0 0 0 1
Linear mixed model fit by REML ['lmerMod']
Formula: RT ~ Related * WordType * (1 | Item) + (1 | Subject)
Data: NS
REML criterion at convergence: 59570.13
Random effects:
Groups Name Variance Std.Dev.
Item (Intercept) 1099 33.15
Subject (Intercept) 4002 63.26
Residual 16153 127.10
Number of obs: 4737, groups: Item, 120; Subject, 43
Fixed effects:
Estimate Std. Error t value
(Intercept) 681.373 13.179 51.70
Related2 13.655 8.379 1.63
WordType2 1.220 12.729 0.10
WordType3 5.700 12.709 0.45
WordType4 5.199 12.696 0.41
WordType5 -48.050 12.690 -3.79
Related2:WordType2 -20.925 11.862 -1.76
Related2:WordType3 -6.780 11.869 -0.57
Related2:WordType4 -19.746 11.854 -1.67
Related2:WordType5 28.870 11.859 2.43
NS$WordType <- relevel(NS$WordType,"4") # Changed the intercept to word type 4
contrasts(NS$PrimeType)
1 2 3 5
4 0 0 0 0
1 1 0 0 0
2 0 1 0 0
3 0 0 1 0
5 0 0 0 1
Linear mixed model fit by REML ['lmerMod']
Formula: RT ~ Related * WordType * (1 | Item) + (1 | Subject)
Data: NS
REML criterion at convergence: 59570.13
Random effects:
Groups Name Variance Std.Dev.
Item (Intercept) 1099 33.15
Subject (Intercept) 4002 63.26
Residual 16153 127.10
Number of obs: 4737, groups: Item, 120; Subject, 43
Fixed effects:
Estimate Std. Error t value
(Intercept) 686.5723 13.1779 52.10
Related2 -6.0911 8.3859 -0.73
WordType1 -5.1994 12.6957 -0.41
WordType2 -3.9794 12.7283 -0.31
WordType3 0.5006 12.7083 0.04
WordType5 -53.2497 12.6895 -4.20
Related2:WordType1 19.7457 11.8541 1.67
Related2:WordType2 -1.1791 11.8656 -0.10
Related2:WordType3 12.9659 11.8734 1.09
Related2:WordType5 48.6158 11.8662 4.10
I have also came across afex package — first intended to calculate p-values. I noticed it controlled and chooses a different default when running the model.
> contrasts(NS$WordType) # default intercept by afex
[,1] [,2] [,3] [,4]
1 1 0 0 0
2 0 1 0 0
3 0 0 1 0
4 0 0 0 1
5 -1 -1 -1 -1
Linear mixed model fit by REML ['lmerMod']
Formula: RT ~ Related * WordType * (1 | Item) + (1 | Subject)
Data: NS
REML criterion at convergence: 59583.5
Random effects:
Groups Name Variance Std.Dev.
Item (Intercept) 1099 33.15
Subject (Intercept) 4002 63.26
Residual 16153 127.10
Number of obs: 4737, groups: Item, 120; Subject, 43
Fixed effects:
Estimate Std. Error t value
(Intercept) 679.1560 10.2793 66.07
Related1 -4.9693 1.8766 -2.65
WordType1 9.0442 7.0943 1.27
WordType2 -0.1982 7.0930 -0.03
WordType3 11.3543 7.1003 1.60
WordType4 4.3708 7.0972 0.62
Related1:WordType1 -1.8580 3.7485 -0.50
Related1:WordType2 8.6044 3.7544 2.29
Related1:WordType3 1.5319 3.7579 0.41
Related1:WordType4 8.0148 3.7509 2.14
When I saw the negative numbers in the contrasts, I wonder if I could produce the same output as afex when I adjust the intercepts.
> NS$WordType <- relevel(NS$WordType,"5") # changed the intercept to word type 5
> NS$Related <- relevel(NS$Related, "2") # changed word relatedness to 2 (unrelated)
> contrasts(NS$WordType) # both adjusted to mimic afex output
1 2 3 4
5 0 0 0 0
1 1 0 0 0
2 0 1 0 0
3 0 0 1 0
4 0 0 0 1
Linear mixed model fit by REML ['lmerMod']
Formula: RT ~ Related * WordType * (1 | Item) + (1 | Subject)
Data: NS
REML criterion at convergence: 59570.13
Random effects:
Groups Name Variance Std.Dev.
Item (Intercept) 1099 33.15
Subject (Intercept) 4002 63.26
Residual 16153 127.10
Number of obs: 4737, groups: Item, 120; Subject, 43
Fixed effects:
Estimate Std. Error t value
(Intercept) 675.8473 13.1857 51.26
Related1 -42.5247 8.3944 -5.07
WordType1 19.1802 12.6920 1.51
WordType2 -0.5246 12.6622 -0.04
WordType3 18.1004 12.7000 1.43
WordType4 4.6339 12.7001 0.36
Related1:WordType1 28.8701 11.8595 2.43
Related1:WordType2 49.7949 11.8727 4.19
Related1:WordType3 35.6499 11.8805 3.00
Related1:WordType4 48.6158 11.8662 4.10
As you can see, the output is different from afex output.
To summarize, my questions are as follows:
What is the intercept in Linear mixed effects models?
What do I need to know in order to make a wise decision in adjusting the intercept?
Why afex package differ in choosing the intercept? Would it be the "best" approach in choosing the intercept for this type of design?
Best Answer
The intercept is the predicted value of the dependent variable when all the independent variables are 0. Since all your IVs are categorical, the meaning of an IV being 0 depends entirely on the coding of the variable, and the default is not necessarily going to be the most useful. You can figure out what each program uses as a default from the documentation.
The reasons to select a particular reference level could be substantive (e.g. one group is likely to be seen as the "default") or statistical (e.g. one reasonable choice is to use the level with the lowest (or highest) level of the predicted value, to make all the comparisons to that level.
When a categorical variable has only two levels, changing the default will simply flip the sign (e.g. if boys are 5 inches taller than girls, then girls are 5 inches shorter than boys) but when there are more levels the changes are not as intuitively obvious.