Intermittent hypoxia and cerebellar development

Variable	Description
Mouse	Mouse unique identifier
Condition	Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)
Z	Bregma coordinates (Ant, Med, Post)
A_DD	Purkinje dendrite area (10^(-4) μm^(2))
A_DD_per_cell	Purkinje dendrite area (10^(-4) μm^(2)) per Purkinje cell
A_GLUD2	Gluδ2-labelled Parallel fiber area in the ML (10^(-5) μm^(2))
Vol_GLUD2	Gluδ2-labelled Parallel fiber volume in the ML (10^(-5) μm^(3))

Variable	Description
Mouse	Mouse unique identifier
Condition	Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)
Z	Bregma coordinates (Ant, Med, Post)
A_DD	Purkinje dendrite area (10^(-4) μm^(2))
A_DD_per_cell	Purkinje dendrite area (10^(-4) μm^(2)) per Purkinje cell
A_GLUD2	Gluδ2-labelled Parallel fiber area in the ML (10^(-5) μm^(2))
Vol_GLUD2	Gluδ2-labelled Parallel fiber volume in the ML (10^(-5) μm^(3))

22.1 Purkinje Dendrite Area (per cell)

22.1.1 Data Exploration

❖ Distribution:

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.933	0.266	0.071	0.286	0.362	0.502	1.54	0.299	−0.502	30
IH	0.857	0.199	0.04	0.232	0.353	0.31	1.16	−0.679	0.377	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.933	0.266	0.071	0.286	0.362	0.502	1.54	0.299	−0.502	30
IH	0.857	0.199	0.04	0.232	0.353	0.31	1.16	−0.679	0.377	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.858	0.27	0.073	0.315	0.302	0.53	1.92	2.094	7.745	29
IH	0.958	0.258	0.067	0.269	0.382	0.412	1.643	0.267	0.586	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.858	0.27	0.073	0.315	0.302	0.53	1.92	2.094	7.745	29
IH	0.958	0.258	0.067	0.269	0.382	0.412	1.643	0.267	0.586	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.993	0.391	0.153	0.394	0.518	0.513	2.062	1.251	1.525	24
IH	1.064	0.51	0.26	0.479	0.502	0.475	3.1	2.605	9.381	27

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.993	0.391	0.153	0.394	0.518	0.513	2.062	1.251	1.525	24
IH	1.064	0.51	0.26	0.479	0.502	0.475	3.1	2.605	9.381	27

22.1.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)
Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_DD_per_cell ~ Condition * Z + (1 | Mouse), 
    data = data, family = Gamma("log"), REML = TRUE, ziformula = ~0, 
    dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
93.22	94.13	118.21	0.13	0.06	0.08	0.31	0.31

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
93.22	94.13	118.21	0.13	0.06	0.08	0.31	0.31

❖ Residuals:

performance::check_model(
  mod, panel = FALSE,
  check = c("pp_check", "qq", "reqq", "linearity", "homogeneity")
)

❖ Predictions:

Simulating data from the model for pseudo “Posterior Predictive” plots.

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

❖ Model call:

```{r}
glmmTMB(formula = A_DD_per_cell ~ Condition * Z + (1 | Mouse), 
    data = data, family = gaussian("log"), REML = TRUE, ziformula = ~0, 
    dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
137.15	138.05	162.14	0.14	0.06	0.09	0.31	0.32

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
137.15	138.05	162.14	0.14	0.06	0.09	0.31	0.32

❖ Residuals:

performance::check_model(
  mod, panel = FALSE,
  check = c("pp_check", "qq", "reqq", "linearity", "homogeneity")
)

❖ Predictions:

Simulating data from the model for pseudo “Posterior Predictive” plots.

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

22.1.3 Effects Analysis

Chosen model: Gamma (‘log’)

```{r}
glmmTMB(formula = A_DD_per_cell ~ Condition * Z + (1 | Mouse), 
    data = data, family = Gamma("log"), REML = TRUE, ziformula = ~0, 
    dispformula = ~1)
```

22.1.3.1 Coefficients

❖ All effects (Wald):

parameters::parameters(
  mod, component = "conditional", effects = "fixed",
  exponentiate = should_exp(mod), p_adjust = "none", summary = TRUE, digits = 3
)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	0.937	0.041	(0.86, 1.02)	-1.497	0.134
Condition1	0.982	0.043	(0.90, 1.07)	-0.421	0.674
Z1	0.955	0.032	(0.89, 1.02)	-1.393	0.164
Z2	0.967	0.032	(0.91, 1.03)	-0.991	0.322
Condition1 * Z1	1.071	0.036	(1.00, 1.14)	2.043	0.041 *
Condition1 * Z2	0.954	0.032	(0.89, 1.02)	-1.400	0.162
Model: A_DD_per_cell ~ Condition * Z (168 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	0.937	0.041	(0.86, 1.02)	-1.497	0.134
Condition1	0.982	0.043	(0.90, 1.07)	-0.421	0.674
Z1	0.955	0.032	(0.89, 1.02)	-1.393	0.164
Z2	0.967	0.032	(0.91, 1.03)	-0.991	0.322
Condition1 * Z1	1.071	0.036	(1.00, 1.14)	2.043	0.041 *
Condition1 * Z2	0.954	0.032	(0.89, 1.02)	-1.400	0.162
Model: A_DD_per_cell ~ Condition * Z (168 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	0.18	1	0.670
Z	5.31	2	0.070
Condition:Z	4.43	2	0.110

term	statistic	df	p.value
Condition	0.18	1	0.670
Z	5.31	2	0.070
Condition:Z	4.43	2	0.110

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	61.92	77.54	-25.96	51.92
mod_full	8	63.51	88.50	-23.75	47.51	4.41	3	0.220

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	61.92	77.54	-25.96	51.92
mod_full	8	63.51	88.50	-23.75	47.51	4.41	3	0.220

Important

Our LRT() method removes the predictor plus all its interactions

22.1.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

❖ Marginal Means:

emmeans(mod, specs = pred, type = "response")

Condition	response	SE	df	lower.CL	upper.CL
N	0.92	0.057	166	0.815	1.039
IH	0.955	0.058	166	0.846	1.077

Condition	response	SE	df	lower.CL	upper.CL
N	0.92	0.057	166	0.815	1.039
IH	0.955	0.058	166	0.846	1.077

- Results are averaged over the levels of: Z
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = pred, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.964	0.083	166	0.813	1.144	1	−0.421	0.674

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.964	0.083	166	0.813	1.144	1	−0.421	0.674

- Results are averaged over the levels of: Z
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

❖ Marginal Means:

emmeans(mod, specs = pred, type = "response")

Z	response	SE	df	lower.CL	upper.CL
Ant	0.895	0.048	166	0.804	0.995
Med	0.907	0.049	166	0.815	1.009
Post	1.015	0.057	166	0.908	1.135

Z	response	SE	df	lower.CL	upper.CL
Ant	0.895	0.048	166	0.804	0.995
Med	0.907	0.049	166	0.815	1.009
Post	1.015	0.057	166	0.908	1.135

- Results are averaged over the levels of: Condition
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = pred, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.987	0.056	166	0.882	1.105	1	−0.233	0.816
Ant / Post	0.882	0.052	166	0.784	0.991	1	−2.126	0.035 *
Med / Post	0.893	0.053	166	0.794	1.005	1	−1.895	0.060

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.987	0.056	166	0.882	1.105	1	−0.233	0.816
Ant / Post	0.882	0.052	166	0.784	0.991	1	−2.126	0.035 *
Med / Post	0.893	0.053	166	0.794	1.005	1	−1.895	0.060

- Results are averaged over the levels of: Condition
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

❖ Marginal Means:

emmeans(mod, specs = emmeans_formula, type = "response")

Condition	response	SE	df	lower.CL	upper.CL
N	0.941	0.072	166	0.81	1.093
IH	0.851	0.065	166	0.731	0.99

Condition	response	SE	df	lower.CL	upper.CL
N	0.941	0.072	166	0.81	1.093
IH	0.851	0.065	166	0.731	0.99

Condition	response	SE	df	lower.CL	upper.CL
N	0.849	0.065	166	0.73	0.988
IH	0.968	0.074	166	0.831	1.126

Condition	response	SE	df	lower.CL	upper.CL
N	0.849	0.065	166	0.73	0.988
IH	0.968	0.074	166	0.831	1.126

Condition	response	SE	df	lower.CL	upper.CL
N	0.975	0.079	166	0.831	1.145
IH	1.056	0.083	166	0.904	1.233

Condition	response	SE	df	lower.CL	upper.CL
N	0.975	0.079	166	0.831	1.145
IH	1.056	0.083	166	0.904	1.233

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = emmeans_formula, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.105	0.12	166	0.893	1.369	1	0.927	0.355

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.105	0.12	166	0.893	1.369	1	0.927	0.355

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.878	0.095	166	0.708	1.088	1	−1.199	0.232

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.878	0.095	166	0.708	1.088	1	−1.199	0.232

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.924	0.104	166	0.739	1.155	1	−0.702	0.484

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.924	0.104	166	0.739	1.155	1	−0.702	0.484

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

emmeans(mod, specs = emmeans_formula, type = "response") |>
  contrast(interaction = "pairwise", by = NULL, adjust = "none", infer = T)

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.259	0.144	166	1.004	1.579	1	2.011	0.046 *

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.259	0.144	166	1.004	1.579	1	2.011	0.046 *

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.197	0.142	166	0.947	1.512	1	1.514	0.132

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.197	0.142	166	0.947	1.512	1	1.514	0.132

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.95	0.113	166	0.751	1.201	1	−0.43	0.668

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.95	0.113	166	0.751	1.201	1	−0.43	0.668

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

22.2 Gluδ2-labelled Parallel fiber Area in the ML

22.2.1 Data Exploration

❖ Distribution:

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.979	0.531	0.282	0.268	0.84	1.09	3.24	0.425	−0.348	30
IH	1.811	0.52	0.27	0.287	0.815	0.621	2.92	−0.191	−0.183	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.979	0.531	0.282	0.268	0.84	1.09	3.24	0.425	−0.348	30
IH	1.811	0.52	0.27	0.287	0.815	0.621	2.92	−0.191	−0.183	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.019	0.508	0.258	0.252	0.775	0.98	3.11	−0.061	−0.113	29
IH	1.942	0.464	0.215	0.239	0.695	1.05	2.83	0.047	−0.665	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.019	0.508	0.258	0.252	0.775	0.98	3.11	−0.061	−0.113	29
IH	1.942	0.464	0.215	0.239	0.695	1.05	2.83	0.047	−0.665	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.201	0.59	0.348	0.268	0.745	0.991	3.41	−0.229	−0.118	24
IH	1.844	0.509	0.259	0.276	0.44	0.941	3.25	0.538	1.11	27

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.201	0.59	0.348	0.268	0.745	0.991	3.41	−0.229	−0.118	24
IH	1.844	0.509	0.259	0.276	0.44	0.941	3.25	0.538	1.11	27

22.2.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)
Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_GLUD2 ~ Condition * Z + (1 | Mouse), data = data, 
    family = Gamma("log"), REML = TRUE, ziformula = ~0, dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
293.41	294.32	318.40	0.19	0.06	0.14	0.48	0.26

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
293.41	294.32	318.40	0.19	0.06	0.14	0.48	0.26

❖ Residuals:

performance::check_model(
  mod, panel = FALSE,
  check = c("pp_check", "qq", "reqq", "linearity", "homogeneity")
)

❖ Predictions:

Simulating data from the model for pseudo “Posterior Predictive” plots.

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

❖ Model call:

```{r}
glmmTMB(formula = A_GLUD2 ~ Condition * Z + (1 | Mouse), data = data, 
    family = gaussian("log"), REML = TRUE, ziformula = ~0, dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
290.19	291.10	315.18	0.06	0.02	0.04	0.48	0.49

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
290.19	291.10	315.18	0.06	0.02	0.04	0.48	0.49

❖ Residuals:

performance::check_model(
  mod, panel = FALSE,
  check = c("pp_check", "qq", "reqq", "linearity", "homogeneity")
)

❖ Predictions:

Simulating data from the model for pseudo “Posterior Predictive” plots.

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

22.2.3 Effects Analysis

Chosen model: Gamma (‘log’)

```{r}
glmmTMB(formula = A_GLUD2 ~ Condition * Z + (1 | Mouse), data = data, 
    family = Gamma("log"), REML = TRUE, ziformula = ~0, dispformula = ~1)
```

22.2.3.1 Coefficients

❖ All effects (Wald):

parameters::parameters(
  mod, component = "conditional", effects = "fixed",
  exponentiate = should_exp(mod), p_adjust = "none", summary = TRUE, digits = 3
)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	1.957	0.094	(1.78, 2.15)	14.015	< .001
Condition1	1.054	0.051	(0.96, 1.16)	1.107	0.268
Z1	0.949	0.027	(0.90, 1.00)	-1.801	0.072
Z2	1.008	0.029	(0.95, 1.07)	0.296	0.767
Condition1 * Z1	1.001	0.029	(0.95, 1.06)	0.048	0.961
Condition1 * Z2	0.968	0.028	(0.92, 1.02)	-1.154	0.248
Model: A_GLUD2 ~ Condition * Z (168 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	1.957	0.094	(1.78, 2.15)	14.015	< .001
Condition1	1.054	0.051	(0.96, 1.16)	1.107	0.268
Z1	0.949	0.027	(0.90, 1.00)	-1.801	0.072
Z2	1.008	0.029	(0.95, 1.07)	0.296	0.767
Condition1 * Z1	1.001	0.029	(0.95, 1.06)	0.048	0.961
Condition1 * Z2	0.968	0.028	(0.92, 1.02)	-1.154	0.248
Model: A_GLUD2 ~ Condition * Z (168 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	1.23	1	0.270
Z	3.61	2	0.160
Condition:Z	1.64	2	0.440

term	statistic	df	p.value
Condition	1.23	1	0.270
Z	3.61	2	0.160
Condition:Z	1.64	2	0.440

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	260.08	275.70	-125.04	250.08
mod_full	8	262.86	287.85	-123.43	246.86	3.22	3	0.360

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	260.08	275.70	-125.04	250.08
mod_full	8	262.86	287.85	-123.43	246.86	3.22	3	0.360

Important

Our LRT() method removes the predictor plus all its interactions

22.2.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

❖ Marginal Means:

emmeans(mod, specs = pred, type = "response")

Condition	response	SE	df	lower.CL	upper.CL
N	2.063	0.14	166	1.805	2.359
IH	1.856	0.126	166	1.624	2.121

Condition	response	SE	df	lower.CL	upper.CL
N	2.063	0.14	166	1.805	2.359
IH	1.856	0.126	166	1.624	2.121

- Results are averaged over the levels of: Z
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = pred, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.112	0.107	166	0.92	1.343	1	1.107	0.270

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.112	0.107	166	0.92	1.343	1	1.107	0.270

- Results are averaged over the levels of: Z
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

❖ Marginal Means:

emmeans(mod, specs = pred, type = "response")

Z	response	SE	df	lower.CL	upper.CL
Ant	1.858	0.103	166	1.665	2.073
Med	1.973	0.109	166	1.769	2.201
Post	2.043	0.117	166	1.826	2.287

Z	response	SE	df	lower.CL	upper.CL
Ant	1.858	0.103	166	1.665	2.073
Med	1.973	0.109	166	1.769	2.201
Post	2.043	0.117	166	1.826	2.287

- Results are averaged over the levels of: Condition
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = pred, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.942	0.046	166	0.855	1.037	1	−1.23	0.220
Ant / Post	0.909	0.046	166	0.822	1.006	1	−1.865	0.064
Med / Post	0.966	0.049	166	0.874	1.067	1	−0.692	0.490

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.942	0.046	166	0.855	1.037	1	−1.23	0.220
Ant / Post	0.909	0.046	166	0.822	1.006	1	−1.865	0.064
Med / Post	0.966	0.049	166	0.874	1.067	1	−0.692	0.490

- Results are averaged over the levels of: Condition
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

❖ Marginal Means:

emmeans(mod, specs = emmeans_formula, type = "response")

Condition	response	SE	df	lower.CL	upper.CL
N	1.962	0.153	166	1.682	2.288
IH	1.759	0.139	166	1.506	2.056

Condition	response	SE	df	lower.CL	upper.CL
N	1.962	0.153	166	1.682	2.288
IH	1.759	0.139	166	1.506	2.056

Condition	response	SE	df	lower.CL	upper.CL
N	2.013	0.158	166	1.725	2.35
IH	1.934	0.152	166	1.656	2.258

Condition	response	SE	df	lower.CL	upper.CL
N	2.013	0.158	166	1.725	2.35
IH	1.934	0.152	166	1.656	2.258

Condition	response	SE	df	lower.CL	upper.CL
N	2.224	0.181	166	1.893	2.612
IH	1.878	0.15	166	1.604	2.198

Condition	response	SE	df	lower.CL	upper.CL
N	2.224	0.181	166	1.893	2.612
IH	1.878	0.15	166	1.604	2.198

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = emmeans_formula, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.115	0.124	166	0.896	1.388	1	0.982	0.328

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.115	0.124	166	0.896	1.388	1	0.982	0.328

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.041	0.115	166	0.836	1.296	1	0.363	0.717

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.041	0.115	166	0.836	1.296	1	0.363	0.717

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.184	0.135	166	0.945	1.483	1	1.483	0.140

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.184	0.135	166	0.945	1.483	1	1.483	0.140

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

emmeans(mod, specs = emmeans_formula, type = "response") |>
  contrast(interaction = "pairwise", by = NULL, adjust = "none", infer = T)

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.071	0.105	166	0.883	1.299	1	0.7	0.485

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.071	0.105	166	0.883	1.299	1	0.7	0.485

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.942	0.096	166	0.77	1.151	1	−0.592	0.555

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.942	0.096	166	0.77	1.151	1	−0.592	0.555

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.879	0.089	166	0.72	1.073	1	−1.275	0.204

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.879	0.089	166	0.72	1.073	1	−1.275	0.204

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

22.3 Gluδ2-labelled Parallel fiber Volume in the ML

22.3.1 Data Exploration

❖ Distribution:

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.083	0.28	0.079	0.135	0.39	1.4	2.81	0.187	0.957	30
IH	2.077	0.466	0.218	0.225	0.55	0.27	2.64	−2.125	7.201	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.083	0.28	0.079	0.135	0.39	1.4	2.81	0.187	0.957	30
IH	2.077	0.466	0.218	0.225	0.55	0.27	2.64	−2.125	7.201	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.944	0.317	0.1	0.163	0.545	1.36	2.43	−0.077	−1.138	29
IH	2.133	0.335	0.112	0.157	0.605	1.53	2.61	−0.476	−1.122	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.944	0.317	0.1	0.163	0.545	1.36	2.43	−0.077	−1.138	29
IH	2.133	0.335	0.112	0.157	0.605	1.53	2.61	−0.476	−1.122	29

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.094	0.409	0.167	0.195	0.525	0.963	2.9	−0.622	1.505	24
IH	1.983	0.332	0.11	0.168	0.51	1.34	2.59	−0.12	−0.717	27

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.094	0.409	0.167	0.195	0.525	0.963	2.9	−0.622	1.505	24
IH	1.983	0.332	0.11	0.168	0.51	1.34	2.59	−0.12	−0.717	27

22.3.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)
Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = Vol_GLUD2 ~ Condition * Z + (1 | Mouse), data = data, 
    family = Gamma("log"), REML = TRUE, ziformula = ~0, dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
227.08	227.98	252.07	0.03	0.03	1.94e-03	0.35	0.20

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
227.08	227.98	252.07	0.03	0.03	1.94e-03	0.35	0.20

❖ Residuals:

performance::check_model(
  mod, panel = FALSE,
  check = c("pp_check", "qq", "reqq", "linearity", "homogeneity")
)

❖ Predictions:

Simulating data from the model for pseudo “Posterior Predictive” plots.

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

❖ Model call:

```{r}
glmmTMB(formula = Vol_GLUD2 ~ Condition * Z + (1 | Mouse), data = data, 
    family = gaussian("log"), REML = TRUE, ziformula = ~0, dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
181.96	182.87	206.95	0.01	8.27e-03	2.77e-03	0.35	0.36

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
181.96	182.87	206.95	0.01	8.27e-03	2.77e-03	0.35	0.36

❖ Residuals:

performance::check_model(
  mod, panel = FALSE,
  check = c("pp_check", "qq", "reqq", "linearity", "homogeneity")
)

❖ Predictions:

Simulating data from the model for pseudo “Posterior Predictive” plots.

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

22.3.3 Effects Analysis

Chosen model: Gamma (‘log’)

```{r}
glmmTMB(formula = Vol_GLUD2 ~ Condition * Z + (1 | Mouse), data = data, 
    family = Gamma("log"), REML = TRUE, ziformula = ~0, dispformula = ~1)
```

22.3.3.1 Coefficients

❖ All effects (Wald):

parameters::parameters(
  mod, component = "conditional", effects = "fixed",
  exponentiate = should_exp(mod), p_adjust = "none", summary = TRUE, digits = 3
)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	2.051	0.033	(1.99, 2.12)	44.091	< .001
Condition1	0.994	0.016	(0.96, 1.03)	-0.360	0.719
Z1	1.014	0.023	(0.97, 1.06)	0.634	0.526
Z2	0.993	0.022	(0.95, 1.04)	-0.324	0.746
Condition1 * Z1	1.007	0.022	(0.96, 1.05)	0.329	0.742
Condition1 * Z2	0.960	0.021	(0.92, 1.00)	-1.822	0.068
Model: Vol_GLUD2 ~ Condition * Z (168 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	2.051	0.033	(1.99, 2.12)	44.091	< .001
Condition1	0.994	0.016	(0.96, 1.03)	-0.360	0.719
Z1	1.014	0.023	(0.97, 1.06)	0.634	0.526
Z2	0.993	0.022	(0.95, 1.04)	-0.324	0.746
Condition1 * Z1	1.007	0.022	(0.96, 1.05)	0.329	0.742
Condition1 * Z2	0.960	0.021	(0.92, 1.00)	-1.822	0.068
Model: Vol_GLUD2 ~ Condition * Z (168 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	0.13	1	0.720
Z	0.40	2	0.820
Condition:Z	3.66	2	0.160

term	statistic	df	p.value
Condition	0.13	1	0.720
Z	0.40	2	0.820
Condition:Z	3.66	2	0.160

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	188.39	204.01	-89.20	178.39
mod_full	8	190.43	215.42	-87.22	174.43	3.96	3	0.270

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	188.39	204.01	-89.20	178.39
mod_full	8	190.43	215.42	-87.22	174.43	3.96	3	0.270

Important

Our LRT() method removes the predictor plus all its interactions

22.3.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

❖ Marginal Means:

emmeans(mod, specs = pred, type = "response")

Condition	response	SE	df	lower.CL	upper.CL
N	2.039	0.047	166	1.948	2.135
IH	2.063	0.047	166	1.972	2.158

Condition	response	SE	df	lower.CL	upper.CL
N	2.039	0.047	166	1.948	2.135
IH	2.063	0.047	166	1.972	2.158

- Results are averaged over the levels of: Z
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = pred, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.988	0.032	166	0.927	1.054	1	−0.36	0.719

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.988	0.032	166	0.927	1.054	1	−0.36	0.719

- Results are averaged over the levels of: Z
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

❖ Marginal Means:

emmeans(mod, specs = pred, type = "response")

Z	response	SE	df	lower.CL	upper.CL
Ant	2.08	0.056	166	1.972	2.194
Med	2.036	0.055	166	1.93	2.148
Post	2.037	0.06	166	1.922	2.158

Z	response	SE	df	lower.CL	upper.CL
Ant	2.08	0.056	166	1.972	2.194
Med	2.036	0.055	166	1.93	2.148
Post	2.037	0.06	166	1.922	2.158

- Results are averaged over the levels of: Condition
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = pred, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.022	0.039	166	0.948	1.101	1	0.563	0.575
Ant / Post	1.021	0.041	166	0.944	1.105	1	0.531	0.596
Med / Post	1	0.04	166	0.925	1.081	1	−0.005	0.996

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.022	0.039	166	0.948	1.101	1	0.563	0.575
Ant / Post	1.021	0.041	166	0.944	1.105	1	0.531	0.596
Med / Post	1	0.04	166	0.925	1.081	1	−0.005	0.996

- Results are averaged over the levels of: Condition
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

❖ Marginal Means:

emmeans(mod, specs = emmeans_formula, type = "response")

Condition	response	SE	df	lower.CL	upper.CL
N	2.083	0.079	166	1.934	2.244
IH	2.077	0.08	166	1.925	2.241

Condition	response	SE	df	lower.CL	upper.CL
N	2.083	0.079	166	1.934	2.244
IH	2.077	0.08	166	1.925	2.241

Condition	response	SE	df	lower.CL	upper.CL
N	1.944	0.075	166	1.803	2.097
IH	2.133	0.082	166	1.977	2.3

Condition	response	SE	df	lower.CL	upper.CL
N	1.944	0.075	166	1.803	2.097
IH	2.133	0.082	166	1.977	2.3

Condition	response	SE	df	lower.CL	upper.CL
N	2.093	0.089	166	1.925	2.276
IH	1.982	0.079	166	1.831	2.145

Condition	response	SE	df	lower.CL	upper.CL
N	2.093	0.089	166	1.925	2.276
IH	1.982	0.079	166	1.831	2.145

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale

❖ Contrasts:

emmeans(mod, specs = emmeans_formula, type = "response") |> 
  contrast(method = "pairwise", adjust = "none", infer = TRUE)

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.003	0.054	166	0.902	1.115	1	0.052	0.958

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.003	0.054	166	0.902	1.115	1	0.052	0.958

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.912	0.049	166	0.819	1.015	1	−1.705	0.090

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.912	0.049	166	0.819	1.015	1	−1.705	0.090

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.056	0.061	166	0.942	1.184	1	0.942	0.348

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.056	0.061	166	0.942	1.184	1	0.942	0.348

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

emmeans(mod, specs = emmeans_formula, type = "response") |>
  contrast(interaction = "pairwise", by = NULL, adjust = "none", infer = T)

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.1	0.083	166	0.947	1.277	1	1.259	0.210

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.1	0.083	166	0.947	1.277	1	1.259	0.210

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.95	0.074	166	0.814	1.108	1	−0.66	0.510

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.95	0.074	166	0.814	1.108	1	−0.66	0.510

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.863	0.068	166	0.739	1.008	1	−1.87	0.063

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.863	0.068	166	0.739	1.008	1	−1.87	0.063

- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot: