9.9 Correlated groups of observations

In our modelling stratgey above, we mentioned the incorporation of population stratification if available. What does this mean. Our regression is seeking to capture the characteristics of particular patients. These characterisitcs are made manifest through the slopes of fitted lines, the estimated coefficients of particular variables. A goal is to estimate these characteristics as precisely as possible. Bias can be introduced when correlations between patients are not accounted for. Correlations may be as simple as being treated within the same hospital. By virtue of this fact, these patients may have commonalities not be captured by the observed variables.

Population characteristics can be incorporated into our models. We may not be intersted in capturing and measuring the effects themselves, but ensuring they are accounted for in the analysis.

One approach is to include grouping variables as random effects. These may be nested with each other, for example patients within hospitals within countries. These are added in additon to the fixed effects we have been dealing with up until now.

These models go under different names including mixed effects model, multilevel model, or hierarchical model.

Other approaches, such as generalized estimating equations are not dealt with here.

9.9.1 Simulate data

Our melanoma dataset doesn’t include any higher level structure, so we will simulate this for demonstration purposes. We have just randomly allocated 1 of 4 identifiers to each patient below.

9.9.2 Plot the data

This all might be better in linear regression. Bit difficult to demonstrate with binomial data We will speak in terms of ‘hospitals’ now, but the grouping variable(s) could clearly be anything.

The simplest random effects approach is a ‘random intercept model’. This allows the intercept of fitted lines to vary by hospital. The random intercept model constrains lines to be parallel, in a similar way to the additive models discussed above and in chapter 7.

It is harder to demonstrate with binomial data, but we can stratify the 5 year mortality by T-stage (considered as a continuous variable for this purpose). Note there were no deaths in ‘hospital 4’. We can model this accounting for interhospital variation below.

9.9.3 Mixed effects models in base R

There are a number of different packages offering mixed effects modelling in R, our preferred is lme4.

## Loading required package: Matrix
## 
## Attaching package: 'Matrix'
## The following objects are masked from 'package:tidyr':
## 
##     expand, pack, unpack
## Generalized linear mixed model fit by maximum likelihood (Laplace
##   Approximation) [glmerMod]
##  Family: binomial  ( logit )
## Formula: mort_5yr ~ t_stage.factor + (1 | hospital_id)
##    Data: .
## 
##      AIC      BIC   logLik deviance df.resid 
##    174.6    191.2    -82.3    164.6      200 
## 
## Scaled residuals: 
##     Min      1Q  Median      3Q     Max 
## -1.4507 -0.3930 -0.2891 -0.0640  3.4591 
## 
## Random effects:
##  Groups      Name        Variance Std.Dev.
##  hospital_id (Intercept) 2.619    1.618   
## Number of obs: 205, groups:  hospital_id, 4
## 
## Fixed effects:
##                  Estimate Std. Error z value Pr(>|z|)    
## (Intercept)      -3.13132    1.01451  -3.087  0.00203 ** 
## t_stage.factorT2  0.02256    0.74792   0.030  0.97593    
## t_stage.factorT3  1.82349    0.62920   2.898  0.00375 ** 
## t_stage.factorT4  2.61190    0.62806   4.159  3.2e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Correlation of Fixed Effects:
##             (Intr) t_s.T2 t_s.T3
## t_stg.fctT2 -0.365              
## t_stg.fctT3 -0.454  0.591       
## t_stg.fctT4 -0.459  0.590  0.718

The base R output is similar to glm(). It includes the standard deviation on the random effects intercept as well. Meaning the variation between hospitals being captured by the model.

The output can be examined using tidy() and glance() functions as above.

We find it more straightforward to use finalfit.

We can incorporate our (made-up) hospital identifier into our final model from above. Using keep_models = TRUE, we can compare univariable, multivariable and mixed effects models.

TABLE 3.4: Multilevel (mixed effects) logistic regression.
Dependent: 5-year survival No Yes OR (univariable) OR (multivariable) OR (multivariable reduced) OR (multilevel)
Ulcerated tumour Absent 105 (91.3) 10 (8.7)
Present 55 (61.1) 35 (38.9) 6.68 (3.18-15.18, p<0.001) 3.06 (1.25-7.93, p=0.017) 3.26 (1.35-8.39, p=0.011) 2.49 (0.94-6.59, p=0.065)
Age (years) (0,25] 10 (71.4) 4 (28.6)
(25,50] 62 (84.9) 11 (15.1) 0.44 (0.12-1.84, p=0.229) 0.37 (0.08-1.80, p=0.197)
(50,75] 79 (76.0) 25 (24.0) 0.79 (0.24-3.08, p=0.712) 0.60 (0.15-2.65, p=0.469)
(75,100] 9 (64.3) 5 (35.7) 1.39 (0.28-7.23, p=0.686) 0.61 (0.09-4.04, p=0.599)
Sex Female 105 (83.3) 21 (16.7)
Male 55 (69.6) 24 (30.4) 2.18 (1.12-4.30, p=0.023) 1.21 (0.54-2.68, p=0.633)
T-stage T1 52 (92.9) 4 (7.1)
T2 49 (92.5) 4 (7.5) 1.06 (0.24-4.71, p=0.936) 0.74 (0.15-3.50, p=0.697) 0.75 (0.16-3.45, p=0.700) 0.83 (0.18-3.73, p=0.807)
T3 36 (70.6) 15 (29.4) 5.42 (1.80-20.22, p=0.005) 2.91 (0.84-11.82, p=0.106) 2.96 (0.86-11.96, p=0.098) 3.83 (1.00-14.70, p=0.051)
T4 23 (51.1) 22 (48.9) 12.43 (4.21-46.26, p<0.001) 5.38 (1.43-23.52, p=0.016) 5.33 (1.48-22.56, p=0.014) 7.03 (1.71-28.86, p=0.007)
TABLE 3.4: Model metrics: multilevel (mixed effects) logistic regression.
Number in dataframe = 205, Number in model = 205, Missing = 0, AIC = 190, C-statistic = 0.802, H&L = Chi-sq(8) 13.87 (p=0.085)
Number in dataframe = 205, Number in model = 205, Missing = 0, AIC = 184.4, C-statistic = 0.791, H&L = Chi-sq(8) 0.43 (p=1.000)
Number in model = 205, Number of groups = 4, AIC = 173.2, C-statistic = 0.866

As can be seen, incorporating the (made-up) hospital identifier has altered our coefficients. It has also improved the model discrminiation with a c-statistic of 0.830 from 0.802. Note that the AIC should not be used to compare mixed effects models estimated in this way with glm() models (the former uses a restricted maximum likelihood [REML] approach by default, while glm() uses maximum likelihood).

Random slope models are an extension of the random intercept model. Here the gradient of the response to a particular variable is allowed to vary by hospital. For example, this can be included using random_effect = "(thickness | hospital_id)" where the gradient of the continuous variable tumour thickness was allow to vary by hospital.

As models get more complex, care has to be taken to ensure the underlying data is understood and assumptions are checked.

Mixed effects modelling is a book in itself and the purpose here is to introduce the concept and provide some approaches its incorporation. Clearly much is written else where for those who are enthusiastic to learn more.

9.9.4 Exercise

Repeat this for all the variables contained within the data, particular:

t_stage.factor, age, ulcer.factor, thickness and age.factor.

Write their odds ratios and 95% confidence intervals down for the next section!