Generalized linear mixed-model (GLMM) trees: A flexible decision-tree method for multilevel and longitudinal data

Figures & data

Table I. Descriptive statistics of the full sample (N = 3,256).

		M (SD) or %	Min	Max
Age		11.33 (3.42)	4.00	18.00
Gender	Female	48%
Ethnicity	White	69%
	Mixed	5%
	Asian	8%
	Black or Black British	6%
	Other	5%
	Not reported or missing	13%
Case characteristics	Hyperactivity	10%
	Emotional problems	55%
	Conduct problems	20%
	Eating disorder	5%
	Self-harm	8%
	Autism	9%
	Special education needs	8%
	Other presenting problems	22%
	Infrequent characteristics	12%
Total difficulties T1		18.49 (7.13)	0.00	39.00
Total difficulties T2		15.48 (7.87)	0.00	38.00
Unadjusted treatment outcome		0.00 (1.00)	−4.51	5.12
Adjusted treatment outcome		0.16 (1.21)	−6.44	5.86

Table II. Statistically significant predictors of treatment outcome according to the original analyses of Edbrooke-Childs et al. (2017).

Unadjusted treatment outcome	Adjusted treatment outcome
Age	Autism
Ethnicity	Infrequent case characteristics
Eating disorder	Disorder duration
Hyperactivity
Autism
Infrequent case characteristics

Figure 1. GLMM tree for the unadjusted treatment outcome. Higher values represent poorer treatment outcomes. Panels depict subgroup sizes (N) and estimated fixed-effects means (M) with standard errors (SE). Note that SEs account for random variability between treatment centres, but not for the searching of the tree structure.

Figure 2. Random-effects predictions of the GLMM tree for the unadjusted treatment outcome. The y-axis represents indicators for service provider. The x-axis represents the predicted value of the random intercept, where higher values represent poorer treatment outcomes. Blue dots represent point predictions, black lines represent point predictions ±1.96 times the standard error. Note that these standard errors do not account for the searching of the tree structure.

Figure 3. GLMM tree for the adjusted treatment outcome. Lower values reflect poorer treatment outcomes. Panels depict subgroups sizes (N), estimated fixed-effects means (M) and their respective standard errors (SE). Ethnicity was coded White (1), Mixed (2), Asian (3), Black or Black British (4), Other (5), Not reported or missing (6). Note that SEs account for random variability between treatment centres, but not for the searching of the tree structure.

Figure 4. Random-effects predictions of the GLMM tree for the adjusted treatment outcome. The y-axis represents indicators for service provider. The x-axis represents the predicted value of the random intercept, where lower values represent poorer treatment outcomes. Blue dots represent point predictions, black lines represent point predictions ±1.96 times the standard error. Note that these standard errors do not account for the searching of the tree structure.

Table III. Prediction error of GLMM trees, traditional GLMMs and random forests, estimated through 10-fold cross validation.

	Unadjusted treatment outcome		Adjusted treatment outcome
Method	R	MSE (se)	R	MSE (se)
GLMM tree	.252	0.931 (0.028)	.179	1.421 (0.043)
Traditional GLMM	.253	0.930 (0.027)	.205	1.406 (0.043)
Random forest	.195	0.956 (0.028)	.049	1.464 (0.044)

Note. MSE denotes mean squared error; se denotes standard error; R is the square root of the treatment outcome variance explained by the predicted values; it can be interpreted as the correlation between predicted and observed treatment outcomes.

Edbrooke-Childs, J., Macdougall, A., Hayes, D., Jacob, J., Wolpert, M., & Deighton, J. (2017). Service-level variation, patient-level factors, and treatment outcome in those seen by child mental health services. European Child & Adolescent Psychiatry, 26(6), 715–722. https://doi.org/10.1007/s00787-016-0939-x

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