I-priors and interactions

This contains the R code for our paper.

Wicher Bergsma, Haziq Jamil (2023). Additive interaction modelling using I-priors.

The key documents are as follows.

02-sims.R and 03-sims_analysis.R contain the code for the simulations study detailed in Section 5 of the manuscript.
01-cow.R contains the code for the functional response model in the application section (Section 6) of the manuscript.

In this README, the main summary of findings are presented. Full details of the simulation results are found in the pdf files simres_corr.pdf and simres_uncorr.pdf.

Please install the developmental version of the {iprior} package.

Simulation study

Data pairs $(y_i,x_i)$, where $x_i\in\mathbb R^3$ for $i=1,\dots,n$, were simulated according to the following model

$$ y_i = \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i1}x_{i2} + \beta_5 x_{i1}x_{i3} + \beta_6 x_{i2}x_{i3} + \beta_7 x_{i1}x_{i2}x_{i3} + \epsilon_i $$

where $\epsilon_i\sim N(0,\sigma^2)$ such that $\text{Corr}(x_{ij},x_{ik})=\rho$, for $j\neq k$. The simulation settings were $n=100$, $\sigma=3$, and $\rho\in\{0,0.5\}$. The coefficients were varied according to the table below

	x1	x2	x3	x1x2	x1x3	x2x3	x1x2x3	code
1	1	0	0	0.0	0.0	0.0	0.00	1000000
2	1	1	0	0.0	0.0	0.0	0.00	1100000
3	1	1	0	0.5	0.0	0.0	0.00	1101000
4	1	1	1	0.0	0.0	0.0	0.00	1110000
5	1	1	1	0.0	0.5	0.0	0.00	1110100
6	1	1	1	0.5	0.5	0.0	0.00	1111100
7	1	1	1	0.5	0.5	0.5	0.00	1111110
8	1	1	1	0.5	0.5	0.5	0.25	1111111

For each set of true values of the coefficients, the four methods proposed the likeliest model to have generated the data set, from a search of hierarchically nested interaction models. This was replicated a total of $B=10,000$ times for each true value set.

The results below show proportion of times that each method selected the true model (higher is better). The geometric mean was used as a summary measure of the simulation runs.

Uncorrelated errors

	mod	iprior	lasso	spikeslab	gprior
1	1000000	0.69	0.21	0.56	0.45
2	1100000	0.55	0.33	0.32	0.29
3	1101000	0.52	0.11	0.02	0.04
4	1110000	0.33	0.37	0.18	0.13
5	1110100	0.32	0.15	0.01	0.01
6	1111100	0.26	0.09	0.00	0.00
7	1111110	0.16	0.08	0.00	0.00
8	1111111	0.19	0.06	0.00	0.98

Correlated errors

	mod	iprior	lasso	spikeslab	gprior
1	1000000	0.64	0.20	0.55	0.46
2	1100000	0.54	0.43	0.11	0.27
3	1101000	0.48	0.09	0.01	0.09
4	1110000	0.43	0.52	0.01	0.16
5	1110100	0.31	0.10	0.00	0.07
6	1111100	0.27	0.10	0.00	0.00
7	1111110	0.18	0.13	0.00	0.00
8	1111111	0.43	0.16	0.00	0.78

Summary of results

Functional response model

The model fitted was an I-prior model with ANOVA kernel (Pearson & fBm). The results are tabulated below.

Fixed hurst = 0.5

model	formula	loglik	error	lambda	psi	hurst	k	AIC	BIC
1	`time`	-2789.23	16.25	0.837	0.00375	0.5	2	5582.46	5591.45
2	`group * time`	-2789.20	16.24	0.019,-0.836	0.00375	0.5	3	5584.40	5597.88
3	`id * time`	-2295.16	2.89	-0.203,-0.088	0.07384	0.5	3	4596.33	4609.81
4	`(group + id) * time`	-2270.85	2.62	-1.019,-0.187,-0.085	0.08711	0.5	4	4549.70	4567.67
5	`group * id * time`	-2249.00	3.09	-1.057,4.918,0.047	0.06538	0.5	4	4506.00	4523.97

Fixed hurst = 0.3

model	formula	loglik	error	lambda	psi	hurst	k	AIC	BIC
1	`time`	-2792.78	16.22	4.465	0.00375	0.3	2	5589.56	5598.54
2	`group * time`	-2792.73	16.20	0.03,-4.462	0.00376	0.3	3	5591.47	5604.94
3	`id * time`	-2266.39	1.62	-0.163,-0.381	0.13445	0.3	3	4538.79	4552.26
4	`(group + id) * time`	-2242.30	1.44	0.708,-0.152,-0.36	0.15896	0.3	4	4492.60	4510.57
5	`group * id * time`	-2238.78	2.16	-1.184,-1.265,0.248	0.09450	0.3	4	4485.56	4503.53

Estimated hurst value

model	formula	loglik	error	lambda	psi	hurst	k	AIC	BIC
1	`time`	-2788.77	16.28	0.347	0.00374	0.62	3	5583.53	5597.01
2	`group * time`	-2788.75	16.27	0.013,-0.35	0.00375	0.61	4	5585.49	5603.46
3	`id * time`	-2253.21	0.16	-0.065,0.83	1.24112	0.17	4	4514.43	4532.40
4	`(group + id) * time`	-2231.13	0.13	0.102,0.058,0.745	1.59394	0.18	5	4472.27	4494.73
5	`group * id * time`	-2232.78	0.18	0.11,0.058,0.751	1.19639	0.18	5	4475.55	4498.01

Outro

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Name		Name	Last commit message	Last commit date
Latest commit History 39 Commits
figure		figure
.gitignore		.gitignore
00-common.R		00-common.R
01-cow.R		01-cow.R
02-sims.R		02-sims.R
03-sims_analysis.R		03-sims_analysis.R
README.Rmd		README.Rmd
README.md		README.md
cattle.csv		cattle.csv
cowres.RData		cowres.RData
iprior-interaction.Rproj		iprior-interaction.Rproj
model_codes.xlsx		model_codes.xlsx
simres.RData		simres.RData
simres_corr.Rmd		simres_corr.Rmd
simres_corr.pdf		simres_corr.pdf
simres_uncorr.RData		simres_uncorr.RData
simres_uncorr.Rmd		simres_uncorr.Rmd
simres_uncorr.pdf		simres_uncorr.pdf

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I-priors and interactions

Simulation study

Uncorrelated errors

Correlated errors

Summary of results

Functional response model

Fixed hurst = 0.5

Fixed hurst = 0.3

Estimated hurst value

Outro

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