If you look at the summary of your fixed effects portion of the model, you can label each row as follows:
Value Std.Error DF t-value p-value
beta0 (Intercept) 204417.8 109088.33 168 1.87387 0.0627
beta1 Exposure2 -192542.9 58653.05 168 -3.28274 0.0013
beta2 Exposure3 -332725.9 58653.05 168 -5.67278 0.0000
beta3 SexM -232599.7 104210.57 22 -2.23202 0.0361
beta4 GenotypeA -184772.3 125723.74 22 -1.46967 0.1558
beta5 GenotypeB -40073.2 128715.16 22 -0.31133 0.7585
beta6 Basal 1650.4 48.83 41 33.79680 0.0000
beta7 Exposure2:SexM 135000.3 58133.66 168 2.32224 0.0214
beta8 Exposure3:SexM 203637.4 58133.66 168 3.50292 0.0006
beta9 Exposure2:GenotypeA 106377.5 71472.21 168 1.48838 0.1385
beta10 Exposure3:GenotypeA 159548.1 71472.21 168 2.23231 0.0269
beta11 Exposure2:GenotypeB 101246.7 70397.58 168 1.43821 0.1522
beta12 Exposure3:GenotypeB 191111.2 70397.58 168 2.71474 0.0073
So the fixed effects portion of your model looks like this:
beta0 + beta1*Exposure2 + beta2*Exposure3 + beta3*SexM +
beta4*GenotypeA + beta5*GenotypeB + beta6*Basal +
beta7*Exposure2*SexM + beta8*Exposure3*SexM +
beta9*Exposure2*GenotypeA + beta10*Exposure3*GenotypeA +
beta11*Exposure2*GenotypeB + beta12*Exposure3*GenotypeB
where the symbol * denotes multiplication and all the beta's are true fixed effects (i.e., unknown but estimable from the data via the values listed in the Value column of your fixed effects summary).
If you are interested in describing the effect of Sex, for instance, all you have to do is to find all the terms in the model which include the dummy variable SexM (which is equal to 1 for Males and 0 for Females) and group them together. The coefficient of SexM obtained after this grouping denotes the effect of Sex:
(beta3 + beta7*Exp2 + beta8*Exp3)*SexM (1)
From the above, we can see that the effect of Sex depends on the value of Exp. Recall that Exp2 and Exp3 are dummy variables defined as follows: Exp2 = 1 if Exp = 2 and 0 otherwise; Exp3 = 1 if Exp = 3 and 0 otherwise. We can exploit this to spell out the effect of SexM for:
Exp = 1 (that is, for Exp2 = 0 and Exp3 = 0);
Exp = 2 (that is, for Exp2 = 1 and Exp3 = 0);
Exp = 3 (that is, for Exp2 = 0 and Exp3 = 1).
When Exp = 1, substituting Exp2 = 0 and Exp3 = 0 in expression (1) above yields that the coefficient of SexM is beta3. This is the effect of Sex on your outcome variable when Exp = 1, all else being equal.
When Exp = 2, substituting Exp2 = 1 and Exp3 = 0 in expression (1) above yields that the coefficient of SexM is beta3 + beta7. This is the effect of Sex on your outcome variable when Exp = 2, all else being equal.
When Exp = 3, substituting Exp2 = 0 and Exp3 = 1 in expression (1) above yields that the coefficient of SexM is beta3 + beta8. This is the effect of Sex on your outcome variable when Exp = 3, all else being equal.
So if you wanted to set up linear combinations of parameters which encapsulate the effect of SexM for each value of Exp, all you need to do is to specify these combinations so they reflect the parameters you are interested in: beta3, beta3 + beta7 and beta3 + beta8. (Again, the betas are true fixed effects, not estimated fixed effects.)
Since the fixed effects portion of your model includes the parameters beta0 through beta12, you are going to set up each combination via a row vector which includes 13 components. The first component of this vector corresponds to beta0, the second component corresponds to beta1, ..., the last component corresponds to beta12.
In R, beta3 is a combination of all the fixed effects model parameters with weights given by the components of the row vector c1:
c1 <- rep(0, 13)
names(c1) <- paste0("beta",0:12)
c1[names(c1)=="beta3"] <- 1
c1
In other words, beta3 = 0*beta0 + 0*beta1 + 0*beta2 + 1*beta3 + 0*beta4 + 0*beta5 + 0*beta6 + 0*beta7 + 0*beta8 + 0*beta9 + 0*beta10 + 0*beta11 + 0*beta12.
The linear combination of model parameters beta0 through beta12 that will encapsulate the parameter beta3 + beta7 is given by:
c2 <- rep(0, 13)
names(c2) <- paste0("beta",0:12)
c2[names(c2)=="beta3"] <- 1
c2[names(c2)=="beta7"] <- 1
c2
In other words, beta3 + beta7 = 0*beta0 + 0*beta1 + 0*beta2 + 1*beta3 + 0*beta4 + 0*beta5 + 0*beta6 + 1*beta7 + 0*beta8 + 0*beta9 + 0*beta10 + 0*beta11 + 0*beta12.
The linear combination of model parameters beta0 through beta12 that will encapsulate the parameter beta3 + beta8 is given by:
c3 <- rep(0, 13)
names(c3) <- paste0("beta",0:12)
c3[names(c3)=="beta3"] <- 1
c3[names(c3)=="beta8"] <- 1
c3
such that beta3 + beta8 = 0*beta0 + 0*beta1 + 0*beta2 + 1*beta3 + 0*beta4 + 0*beta5 + 0*beta6 + 0*beta7 + 1*beta8 + 0*beta9 + 0*beta10 + 0*beta11 + 0*beta12.
If you now want to simultaneously test hypotheses such as:
H0: beta3 = 0 versus Ha: beta3 != 0
H0: beta3 + beta7 = 0 versus Ha: beta3 + beta7 != 0
H0: beta3 + beta8 = 0 versus Ha: beta3 + beta8 != 0
you can achieve this by using the multcomp package in R. (Here, != 0 stands for "not equal to zero".) These hypotheses will enable you to determine whether Sex has an effect for any of the specific levels of Exp (i.e., if males differ from females with respect to the average response, all else being the same). This package can also be used to compute simultaneous confidence intervals for beta3, beta3 + beta7 and beta3 + beta8.
All you need to do to use multcomp in conjunction with your lme model is something like this:
library(multcomp)
c <- rbind(c1, c2, c3)
g <- glht(model, linfct = c)
s <- summary(g, test=adjusted("holm"))
s
ci <- confint(summary(g, test=adjusted("holm")))
ci
If you don't want to adjust p-values or confidence levels for multiplicity, just use adjusted("none") in the above.
Of course, there are R packages which will do all of the above for you with a minimal number of commands. But it helps to know how to test your own simple effects (e.g., beta3, beta3 + beta7, beta3 + beta8) when interested in probing a significant interaction.
In the approach I presented here, contrasts are specified via linear combinations of all model parameters. There are other ways to specify contrasts, which I will leave to others on this forum to address.
