# Linear Mixed Model Interpretation

I'm working on analyzing some data that need to use lme model, but I'm not sure about interpreting the output.

Data looks like this:

> head(data)
Mouse.Number Mouse.Number.Unique Overall.Mouse Batch Day      Drug Tumor.Volume  X
1            1                   1             1    10 -35 Cetuximab         27.4 NA
2            1                   1             1    10 -31 Cetuximab         14.3 NA
3            1                   1             1    10 -28 Cetuximab         19.7 NA
4            1                   1             1    10 -24 Cetuximab         29.6 NA
5            1                   1             1    10 -21 Cetuximab         39.1 NA
6            1                   1             1    10 -17 Cetuximab         43.5 NA


Please ignore column Mouse.Number, Mouse.Number.Unique and X.

Explaining the data:

1. There are negative days, but ignore them (they are the values before applying the treatment)
2. I want to model Tumor.Volume ~ Day*Drug, and the random part is Day|Overall.Mouse. My code is

lme_pos_1 <- lme(Tumor.Volume ~ Day*Drug, random = ~Day|Overall.Mouse, data=data_pos_10)

Summary is:

> summary(lme_pos_1)
Linear mixed-effects model fit by REML
Data: data_pos_10
AIC      BIC    logLik
2364.189 2405.549 -1170.094

Random effects:
Formula: ~Day | Overall.Mouse
Structure: General positive-definite, Log-Cholesky parametrization
StdDev    Corr
(Intercept) 42.215374 (Intr)
Day          4.059765 0.455
Residual    23.012150

Fixed effects: Tumor.Volume ~ Day * Drug
Value Std.Error  DF   t-value p-value
(Intercept)         108.32758 12.319403 195  8.793249  0.0000
Day                  -1.99631  1.142348 195 -1.747546  0.0821
DrugDacomitinib      15.03261 26.337891  37  0.570760  0.5716
DrugErlotinib        41.38484 18.198080  37  2.274132  0.0289
DrugVehicle          43.08154 18.204460  37  2.366538  0.0233
Day:DrugDacomitinib   0.03427  2.874163 195  0.011923  0.9905
Day:DrugErlotinib    12.28038  1.797236 195  6.832924  0.0000
Day:DrugVehicle      13.71968  1.806300 195  7.595461  0.0000
Correlation:
(Intr) Day    DrgDcm DrgErl DrgVhc Dy:DrD Dy:DrE
Day                  0.382
DrugDacomitinib     -0.468 -0.179
DrugErlotinib       -0.677 -0.259  0.317
DrugVehicle         -0.677 -0.258  0.317  0.458
Day:DrugDacomitinib -0.152 -0.397  0.174  0.103  0.103
Day:DrugErlotinib   -0.243 -0.636  0.114  0.262  0.164  0.253
Day:DrugVehicle     -0.242 -0.632  0.113  0.164  0.258  0.251  0.402

Standardized Within-Group Residuals:
Min          Q1         Med          Q3         Max
-2.81026418 -0.52194774 -0.07643594  0.53258816  2.77830783

Number of Observations: 240
Number of Groups: 41


I'm not sure about how to get information from the output, like is this a good model, how can I know it? What values are important, etc.

Basically I want to know:

1. in random effects part, what does the standard deviation stands for?
2. what does the correlation part want to imply?
3. how to know if this is a good model or not, and how can I improve it? (I tried some qqnorm plot and residual plot, they all seem ok)

1: The standard deviation stands for the standard deviation of the fitted random intercept respectively the fitted random slope. I.e. the assumption of your fitted model is

$\left(\matrix{b_0 \\ b_1}\right) \sim \mathcal{N} \left(\left(\matrix{0\\0}\right), \left(\matrix{\tau_0 & \tau_{01} \\ \tau_{10} & \tau_1}\right)\right)$,

where $\sqrt{\tau_0}$ estimates to 42.215374, $\sqrt{\tau_1}$ estimates to 4.059765 and the correlation resulting from the estimated covariance $\tau_{01}$ of $b_0$ and $b_1$ estimates to 0.455. In other words, the values provide information on the variance of your data, which can be explained by the random effects. StdDev of Residual as the name suggests stands for the estimated residual standard deviation.

2: A similar question on stack overflow: How do I interpret the 'correlations of fixed effects' in my glmer output?

3: To evaluate whether your model is "a good model", you will need to consider the aim of your analysis.

If the aim of your model is to explain the variability of your data as much as possible, you may have a look at Proportion of explained variance in a mixed-effects model. In this case, you could maybe improve your model by adding additional covariates / random effects or check, if any of your covariates has a non-linear influence on Tumor.Volume. But be aware when comparing mixed models - changing the fixed effects in a mixed model will result in a different calculation of the likelihood and thus comparisons based on the likelihood produce invalid results (with REML estimation).

If your aim is to predict tumor volumes with further observations of covariates, you'll have to chose a criterion (e.g. some sort of prediction error) , which evaluates your prediction performance and, based on this, compare your models.

If your model is based on theoretical assumptions (i.e. you don't need to evaluate the fit or prediction), you just have to interpret the results (relevance / significance of estimations and so on).

• Could you explain to me how the random part works? I tried to look for some documents explaining this, but I can't. Basically what I know is that it is in the form of 1|subject, or *|subject, what does this mean, what does 1 stands for, and what if it is replaced by another variable *? Thanks!
– TYZ
Feb 19 '14 at 15:48
• In a lme-call, random = ~1|subject stands for a mixed model with random intercept. In this case a random intercept makes sense if you are assuming, that every subject has a specific tumor volume (e.g. because every mouse has a specific body size or is in different stage of tumor) at the beginning of the study.
– R大卫
Feb 19 '14 at 18:23
• On the other hand a call of the form *|subject stands for a mixed model with random intercept and random slope (e.g. a random slope for day). In this case, a random slope for day|subject makes sense if you are assuming, that, in addition to a random intercept, every subject has an individual pattern of tumor growth for each day.
– R大卫
Feb 19 '14 at 18:24
• I have another question: what if the lme is not convergent with random slope and random intercept (i.e. Day|Mouse), and if I use 1|Mouse, the result is not significant (where it should be significant). What other method can I use?
– TYZ
Feb 23 '14 at 15:58