When running a mixed-effects model with categorical predictors, you may wish to test the fixed effects of the model. When your model includes categorical variables with three or more levels or interactions, this requires a multiple degrees of freedom test. In other software packages like SAS, Type III tests of fixed effects are presented with the regression output. In R, this is not the case. However, we can use contrast and ANOVA-type commands to extract these effects.

We will use the dataset **hsbdemo** and the R packages **foreign** (to
read in the data) and **nlme** (to run a mixed-effects model). We will run a random-intercept model where read
is predicted by **female**, **prog**, and **math**.

library(foreign) library(nlme)d1 <- read.dta("https://stats.idre.ucla.edu/stat/data/hsbdemo.dta") attach(d1)

In order to use contrast commands, we will need to convert our string factor variables to integer factor variables starting at 1.

female12 <- as.factor(1*(female=="male") + 2*(female=="female"))prog123 <- as.factor(1*(prog=="general") + 2*(prog=="academic") + 3*(prog=="vocational"))

Next, before running the model, we define the contrasts for our factor variables.
For a simple model without interactions, we can use the **contr.treatment **
command which also allows us to specify a baseline (which is helpful if you are
attempting to match model results seen in a publication or another software). We provide the number of
levels and the level we wish to use as the baseline. This command generates a
corresponding matrix that we then indicate as the contrast matrix for a given
variable using the **contrasts** command. These matrices correspond to “dummy
coding” the categorical variables. For further information on dummy
coding, see FAQ: What is dummy coding?.

a <- contr.treatment(2, base = 2, contrasts = TRUE) class(a)[1] "matrix"a1 1 1 2 0b <- contr.treatment(3, base = 3, contrasts = TRUE) b1 2 1 1 0 2 0 1 3 0 0contrasts(female12) <- a contrasts(prog123) <- b

We can now run our model and see that the coefficients of the model are reported according to the baselines indicated in these contrasts.

m1.lme <- lme(read ~ female12 + prog123 + math, random =~ 1|cid, method = "ML") summary(m1.lme)Linear mixed-effects model fit by maximum likelihood Data: NULL AIC BIC logLik 1289.254 1312.342 -637.6271 Random effects: Formula: ~1 | cid (Intercept) Residual StdDev: 11.00646 4.809769 Fixed effects: read ~ female12 + prog123 + math Value Std.Error DF t-value p-value (Intercept) 70.08683 4.776200 176 14.674184 0.0000 female121 1.35292 0.731138 176 1.850427 0.0659 prog1231 -2.10673 1.006942 176 -2.092208 0.0379 prog1232 -2.04756 0.968723 176 -2.113675 0.0360 math -0.33022 0.074898 176 -4.408968 0.0000 Correlation: (Intr) fml121 pr1231 pr1232 female121 -0.035 prog1231 -0.231 -0.022 prog1232 -0.223 -0.001 0.415 math -0.843 -0.039 0.195 0.188 Standardized Within-Group Residuals: Min Q1 Med Q3 Max -2.5019691 -0.6206737 0.0354358 0.7148373 2.3331462 Number of Observations: 200 Number of Groups: 20

To see the Type III tests of fixed effects, we use the **anova.lme**
command. To indicate Type III tests, we provide **type = “marginal”**.

anova.lme(m1.lme, type = "marginal", adjustSigma = F)numDF denDF F-value p-value (Intercept) 1 176 220.85299 <.0001 female12 1 176 3.51188 0.0626 prog123 2 176 3.20534 0.0429 math 1 176 19.93743 <.0001

If we have a more complex model that includes interactions of categorical
variables, dummy coding no longer captures the main effects of these categorical
variables but rather the simple effects. To test for main effects in our Type
III tests, we will need to use a different contrast command that defines our
categorical variables using “effect coding”. The contrast
matrices under effect coding are no longer composed of only zeroes and ones;
each column in the contrast matrix now sums to zero. We can create these effect coding
matrices in R with the **contr.sum** function.

contrasts(female12) <- contr.sum contrasts(female12)[,1] 1 1 2 -1contrasts(prog123) <- contr.sum contrasts(prog123)[,1] [,2] 0 1 0 1 0 1 2 -1 -1

For more information on effect coding, see FAQ: What is effect coding?. We can run our model to estimate coefficients based on this coding, then generate the desired tests after running the model.

m1.lme2 <- lme(read ~ female12*prog123 + math, random =~ 1|cid, method = "ML") summary(m1.lme2)Linear mixed-effects model fit by maximum likelihood Data: NULL AIC BIC logLik 1288.429 1318.114 -635.2145 Random effects: Formula: ~1 | cid (Intercept) Residual StdDev: 10.88791 4.751459 Fixed effects: read ~ female12 * prog123 + math Value Std.Error DF t-value p-value (Intercept) 68.77214 4.674240 174 14.713011 0.0000 female121 0.94010 0.386053 174 2.435143 0.0159 prog1231 -0.69864 0.613890 174 -1.138046 0.2567 prog1232 -0.62045 0.590308 174 -1.051065 0.2947 math -0.31881 0.075398 174 -4.228375 0.0000 female121:prog1231 -0.02636 0.566363 174 -0.046538 0.9629 female121:prog1232 0.98816 0.584777 174 1.689812 0.0929 Correlation: (Intr) fml121 pr1231 pr1232 math f121:1231 female121 0.035 prog1231 -0.064 0.001 prog1232 -0.076 0.009 -0.581 math -0.844 -0.033 0.093 0.110 female121:prog1231 0.129 0.123 0.137 -0.097 -0.149 female121:prog1232 -0.127 0.163 -0.072 0.086 0.148 -0.644 Standardized Within-Group Residuals: Min Q1 Med Q3 Max -2.40201967 -0.66416223 0.05083885 0.71444086 2.32978676anova.lme(m1.lme2, type = "marginal", adjustSigma = F)numDF denDF F-value p-value (Intercept) 1 174 224.32402 <.0001 female12 1 174 6.14499 0.0141 prog123 2 174 2.96372 0.0542 math 1 174 18.52762 <.0001 female12:prog123 2 174 2.44197 0.0900

The table output from the **anova.lme** command contains the Type III tests of fixed effects for this model.