GLM for choice data

Notes about analyses: In the following analyses, Effort is squared. Effort and Reward are numerics that are scaled and mean-centred. Models incorporate a treatment-coded factor level structure for Group and deviation-coded factor level (contr.sum) structure for recipient. Below shows summary table restricted to group effects for maximal models incorporating trial number, followed by the full output for all 3 models.

	HC vs PDoff			HC vs PDon			PDon vs PDoff
Predictors	Log-Odds	CI	p	Log-Odds	CI	p	Log-Odds	CI	p
(Intercept)	4.97	3.90 – 6.03	<0.001	4.78	3.73 – 5.83	<0.001	3.88	2.95 – 4.81	<0.001
Effort	-1.51	-2.02 – -1.00	<0.001	-1.51	-2.01 – -1.00	<0.001	-1.22	-1.61 – -0.82	<0.001
Reward	2.32	1.72 – 2.92	<0.001	2.11	1.50 – 2.71	<0.001	1.37	0.97 – 1.77	<0.001
Group [Comparator]	-0.00	-1.49 – 1.48	0.996	0.07	-1.41 – 1.55	0.927	0.16	-0.02 – 0.35	0.080
Recipient s2z1	0.82	0.29 – 1.35	0.002	0.57	0.17 – 0.97	0.005	0.80	0.43 – 1.17	<0.001
Trial number r	0.48	0.16 – 0.81	0.004	0.51	0.19 – 0.82	0.002	-0.06	-0.28 – 0.16	0.585
Effort × Reward	-0.08	-0.24 – 0.08	0.316	-0.07	-0.22 – 0.09	0.403	0.13	0.00 – 0.26	0.050
Effort × Group [Comparator]	0.17	-0.56 – 0.89	0.649	0.07	-0.65 – 0.79	0.852	-0.02	-0.20 – 0.16	0.822
Reward × Group [Comparator]	-0.37	-1.20 – 0.46	0.378	-0.24	-1.09 – 0.61	0.580	0.21	0.03 – 0.38	0.020
Effort × Recipient s2z1	0.12	-0.05 – 0.28	0.168	0.16	0.01 – 0.31	0.042	0.13	-0.00 – 0.25	0.058
Reward × Recipient s2z1	0.39	0.08 – 0.69	0.012	0.22	0.06 – 0.37	0.006	0.09	-0.04 – 0.22	0.180
Group [Comparator] × Recipient s2z1	0.40	-0.30 – 1.11	0.263	0.12	-0.44 – 0.67	0.675	-0.07	-0.26 – 0.11	0.447
Effort × Trial number r	-0.32	-0.47 – -0.17	<0.001	-0.33	-0.47 – -0.18	<0.001	-0.16	-0.28 – -0.03	0.015
Reward × Trial number r	0.55	0.39 – 0.71	<0.001	0.57	0.42 – 0.72	<0.001	0.16	0.03 – 0.29	0.013
Group [Comparator] × Trial number r	-0.46	-0.89 – -0.02	0.038	-0.37	-0.78 – 0.05	0.086	-0.02	-0.20 – 0.16	0.828
Recipient s2z1 × Trial number r	0.07	-0.08 – 0.22	0.383	0.09	-0.06 – 0.23	0.239	0.12	-0.00 – 0.25	0.057
(Effort × Reward) × Group [Comparator]	0.20	-0.03 – 0.42	0.086	0.12	-0.11 – 0.34	0.304	-0.17	-0.35 – -0.00	0.045
Effort × Reward × Recipient s2z1	0.10	-0.05 – 0.24	0.195	0.10	-0.04 – 0.23	0.176	0.06	-0.06 – 0.18	0.343
Effort × Group [Comparator] × Recipient s2z1	0.01	-0.22 – 0.25	0.912	-0.04	-0.26 – 0.18	0.709	-0.09	-0.27 – 0.08	0.294
Reward × Group [Comparator] × Recipient s2z1	0.01	-0.36 – 0.39	0.948	-0.18	-0.40 – 0.04	0.105	-0.03	-0.20 – 0.14	0.721
(Effort × Reward) × Trial number r	-0.11	-0.25 – 0.03	0.133	-0.11	-0.25 – 0.04	0.140	0.05	-0.08 – 0.18	0.461
(Effort × Group [Comparator]) × Trial number r	0.10	-0.10 – 0.31	0.321	-0.16	-0.37 – 0.04	0.122	-0.20	-0.38 – -0.03	0.021
(Reward × Group [Comparator]) × Trial number r	-0.38	-0.59 – -0.17	<0.001	-0.37	-0.59 – -0.15	0.001	-0.04	-0.22 – 0.14	0.677
(Effort × Recipient s2z1) × Trial number r	0.04	-0.09 – 0.18	0.521	0.05	-0.08 – 0.18	0.474	0.14	0.02 – 0.26	0.022
(Reward × Recipient s2z1) × Trial number r	-0.01	-0.15 – 0.13	0.900	0.00	-0.14 – 0.14	0.981	0.06	-0.06 – 0.19	0.320
(Group [Comparator] × Recipient s2z1) × Trial number r	0.01	-0.20 – 0.22	0.912	0.06	-0.15 – 0.27	0.564	-0.05	-0.22 – 0.13	0.616
Effort × Reward × Group [Comparator] × Recipient s2z1	-0.04	-0.25 – 0.16	0.686	-0.04	-0.24 – 0.16	0.696	-0.00	-0.18 – 0.17	0.963
(Effort × Reward × Group [Comparator]) × Trial number r	0.20	-0.00 – 0.41	0.055	0.08	-0.14 – 0.29	0.483	-0.10	-0.28 – 0.09	0.301
(Effort × Reward × Recipient s2z1) × Trial number r	-0.05	-0.19 – 0.09	0.472	-0.06	-0.19 – 0.08	0.429	0.08	-0.05 – 0.21	0.232
(Effort × Group [Comparator] × Recipient s2z1) × Trial number r	0.15	-0.05 – 0.34	0.140	0.03	-0.17 – 0.22	0.802	-0.07	-0.25 – 0.10	0.404
(Reward × Group [Comparator] × Recipient s2z1) × Trial number r	0.03	-0.17 – 0.23	0.794	0.06	-0.14 – 0.26	0.543	-0.02	-0.20 – 0.16	0.861
(Effort × Reward × Group [Comparator] × Recipient s2z1) × Trial number r	0.16	-0.04 – 0.36	0.124	-0.10	-0.30 – 0.11	0.363	-0.18	-0.37 – 0.00	0.051
Random Effects
σ²	3.29			3.29			3.29
τ₀₀	9.84 _ID			9.90 _ID			7.61 _ID
τ₁₁	2.10 _{ID.scale(Effort)}			2.07 _{ID.scale(Effort)}			1.16 _{ID.scale(Effort)}
	2.71 _{ID.scale(Reward)}			2.99 _{ID.scale(Reward)}			1.13 _{ID.scale(Reward)}
	0.57 _{ID.scale(Trial.number.r)}			1.09 _{ID.Recipient_s2z1}			0.96 _{ID.Recipient_s2z1}
	1.72 _{ID.Recipient_s2z1}			0.52 _{ID.scale(Trial.number.r)}			0.15 _{ID.scale(Trial.number.r)}
	0.26 _{ID.scale(Reward):Recipient_s2z1}
ρ₀₁	0.25			0.17			0.12
	0.40			0.32			0.29
	0.37			-0.40			-0.21
	-0.09			0.51			0.23
	0.11
ICC	0.84			0.83			0.77
N	80 _ID			80 _ID			38 _ID
Observations	11802			11869			11207
Marginal R² / Conditional R²	0.284 / 0.885			0.260 / 0.877			0.231 / 0.822

Below shows a graph of real choice behaviour, separated by effort and reward levels

Emmip graphs of interactions including trial number:

HC versus PDoff: Emmip graphs

HC versus PDon: Emmip graphs

PDon versus PDoff: emmip graphs

Plots of actual data showing interactions:

HC vs PDoff

HC vs PDon

PDon vs PDoff

Credits won during task:

-I have included summary stats and a histogram of credits won.

-I have also included the model output for the GLM of credits won - as a simple linear model; lmer(total_credits ~ Recipient_s2z * Group + (1|ID), and with gamma log link function (glmer(total_credits ~ Recipient_s2z * Group + (1|ID), family = Gamma(link = ‘log’)) - as with Jo’s lesion data, it is the only model to converge without singular fit and of anova(m.gam.log, m.ig.inv, m.ig.invquad, m.ig.log) also provides the best fit to the data.

Group	Recipient_s2z	variable	n	mean	sd
HC	self	total_credits	42	407	45.6
HC	other	total_credits	42	378	72.1
PD off	self	total_credits	38	401	51.8
PD off	other	total_credits	38	354	100.5
PD on	self	total_credits	38	402	51.2
PD on	other	total_credits	38	357	92.2

Credits won: PDon vs PDoff

total_credits ~ Recipient_s2z*Group + (1|ID) (normal model followed by gamma log link)

	total_credits
Predictors	Estimates	CI	p
(Intercept)	377.87	357.06 – 398.68	<0.001
Recipient s2z1	23.37	9.89 – 36.85	0.001
Group [PD on]	1.80	-17.26 – 20.86	0.852
Recipient s2z1 × Group [PD on]	-0.67	-19.73 – 18.39	0.945
Random Effects
σ²	3534.63
τ₀₀ _ID	2446.92
ICC	0.41
N _ID	38
Observations	152
Marginal R² / Conditional R²	0.082 / 0.458

	total_credits
Predictors	Estimates	CI	p
(Intercept)	370.71	339.34 – 404.97	<0.001
Recipient s2z1	1.08	1.01 – 1.15	0.019
Group [PD on]	1.01	0.92 – 1.10	0.871
Recipient s2z1 × Group [PD on]	1.00	0.91 – 1.09	0.937
Random Effects
σ²	0.04
τ₀₀ _ID	0.01
ICC	0.17
N _ID	38
Observations	152
Marginal R² / Conditional R²	0.095 / 0.248

Credits won: HC vs PDoff

total_credits ~ Recipient_s2z*Group + (1|ID): (normal model followed by gamma log link)

	total_credits
Predictors	Estimates	CI	p
(Intercept)	392.75	375.08 – 410.42	<0.001
Recipient s2z1	14.37	2.24 – 26.50	0.021
Group [PD off]	-14.88	-40.53 – 10.76	0.253
Recipient s2z1 × Group [PD off]	9.00	-8.60 – 26.60	0.314
Random Effects
σ²	3167.43
τ₀₀ _ID	1778.21
ICC	0.36
N _ID	80
Observations	160
Marginal R² / Conditional R²	0.079 / 0.410

	total_credits
Predictors	Estimates	CI	p
(Intercept)	389.93	363.70 – 418.05	<0.001
Recipient s2z1	1.04	0.98 – 1.10	0.163
Group [PD off]	0.96	0.86 – 1.06	0.379
Recipient s2z1 × Group [PD off]	1.03	0.95 – 1.12	0.463
Random Effects
σ²	0.03
τ₀₀ _ID	0.00
ICC	0.12
N _ID	80
Observations	160
Marginal R² / Conditional R²	0.085 / 0.195

Results of computational modelling

Next I have plotted computational modelling parameters from the EM hierarchical models. Note that in MLE models, two k one beta had lowest summed BIC and won (over two k two beta) in 81% participants, although two k two beta was best by AIC. By EM, the two k two beta model wins on summed BIC and on exceedance probability.

Net I have analysed associations of k values with deomgraphic factors in linear effects models.

First I have analysed UPDRS III and (self and other) k values in a linear mixed effects model (lmer):

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	0.00	-0.30 – 0.30	1.000	0.00	-0.32 – 0.32	1.000
UPDRS	0.20	-0.04 – 0.43	0.099	-0.01	-0.22 – 0.21	0.943
Random Effects
σ²	0.22			0.15
τ₀₀	0.75 _ID			0.88 _ID
ICC	0.77			0.86
N	38 _ID			38 _ID
Observations	76			76
Marginal R² / Conditional R²	0.038 / 0.781			0.000 / 0.857

Next looking at levodopa equivalent dose:

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.02	-0.33 – 0.29	0.893	0.01	-0.32 – 0.33	0.974
LED	0.21	-0.10 – 0.52	0.175	-0.02	-0.35 – 0.31	0.918
Random Effects
σ²	0.21			0.15
τ₀₀	0.78 _ID			0.92 _ID
ICC	0.79			0.86
N	38 _ID			38 _ID
Observations	74			74
Marginal R² / Conditional R²	0.044 / 0.798			0.000 / 0.861

Looking across all groups:

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.01	-0.20 – 0.18	0.936	0.00	-0.18 – 0.19	0.970
Age	-0.10	-0.30 – 0.09	0.296	0.12	-0.06 – 0.31	0.190
Random Effects
σ²	0.86			1.01
τ₀₀	0.15 _ID			0.00 _ID
ICC	0.15			0.00
N	80 _ID			80 _ID
Observations	116			116
Marginal R² / Conditional R²	0.010 / 0.158			0.015 / 0.020

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	0.02	-0.20 – 0.24	0.865	-0.03	-0.24 – 0.18	0.768
Gender [1]	-0.04	-0.26 – 0.18	0.706	0.07	-0.14 – 0.27	0.535
Random Effects
σ²	0.86			1.01
τ₀₀	0.15 _ID			0.00 _ID
ICC	0.15
N	80 _ID			80 _ID
Observations	118			118
Marginal R² / Conditional R²	0.001 / 0.152			0.003 / NA

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.00	-0.19 – 0.19	0.988	0.00	-0.18 – 0.18	1.000
AMI Total	-0.12	-0.31 – 0.07	0.214	0.05	-0.13 – 0.23	0.594
Random Effects
σ²	0.85			1.01
τ₀₀	0.15 _ID			0.00 _ID
ICC	0.15
N	80 _ID			80 _ID
Observations	118			118
Marginal R² / Conditional R²	0.014 / 0.158			0.002 / NA

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.00	-0.19 – 0.19	0.989	0.00	-0.18 – 0.18	1.000
AMI Social	-0.11	-0.30 – 0.08	0.248	0.01	-0.17 – 0.20	0.886
Random Effects
σ²	0.85			1.01
τ₀₀	0.14 _ID			0.00 _ID
ICC	0.14
N	80 _ID			80 _ID
Observations	118			118
Marginal R² / Conditional R²	0.012 / 0.154			0.000 / NA

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.00	-0.19 – 0.19	0.986	0.00	-0.18 – 0.18	1.000
AMI Behavioural	-0.09	-0.28 – 0.10	0.343	-0.07	-0.25 – 0.11	0.454
Random Effects
σ²	0.86			1.00
τ₀₀	0.14 _ID			0.00 _ID
ICC	0.14
N	80 _ID			80 _ID
Observations	118			118
Marginal R² / Conditional R²	0.008 / 0.144			0.005 / NA

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	-0.00	-0.19 – 0.19	0.996	0.00	-0.18 – 0.18	1.000
AMI Emotional	-0.03	-0.22 – 0.17	0.796	0.19	0.01 – 0.37	0.040
Random Effects
σ²	0.85			0.97
τ₀₀	0.16 _ID			0.00 _ID
ICC	0.16
N	80 _ID			80 _ID
Observations	118			118
Marginal R² / Conditional R²	0.001 / 0.156			0.036 / NA

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	0.01	-0.19 – 0.20	0.948	0.00	-0.18 – 0.19	0.964
LARS TOTAL	-0.06	-0.26 – 0.13	0.536	-0.01	-0.19 – 0.18	0.953
Random Effects
σ²	0.86			1.02
τ₀₀	0.15 _ID			0.00 _ID
ICC	0.15
N	80 _ID			80 _ID
Observations	117			117
Marginal R² / Conditional R²	0.004 / 0.149			0.000 / NA

	scale(as.numeric(PM_self_k))			scale(as.numeric(PM_other_k))
Predictors	Estimates	CI	p	Estimates	CI	p
(Intercept)	0.01	-0.19 – 0.20	0.955	0.00	-0.18 – 0.19	0.963
LARS F E	0.08	-0.12 – 0.27	0.441	-0.03	-0.22 – 0.15	0.731
Random Effects
σ²	0.85			1.01
τ₀₀	0.16 _ID			0.00 _ID
ICC	0.16
N	80 _ID			80 _ID
Observations	117			117
Marginal R² / Conditional R²	0.006 / 0.161			0.001 / NA

In view of the positive model result between AMI emotional and other k, I have plotted these: