21  VGLUT

Vesicular-Glutamate Transporter 2

❖ Data

❖ Description

Variable	Description
Mouse	Mouse unique identifier
Condition	Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)
Z	Bregma coordinates (Ant, Med, Post)
A_VGLUT_MF	Vglut2-labelled Mossy fiber area in the IGL (10^-4 μm^(2))
A_VGLUT_CF	Vglut2-labelled Climbing fiber area in the ML (10^-4 μm^(2))
A_DD_per_cell	Purkinje dendrite area per Purkinje cell body (10^-4 μm^(2))
Vol_DD_per_cell	Purkinje dendrite volume per Purkinje cell body (10^-4 μm^(3))
A_ML	Molecular layer area (10^-4 μm^(2))
Thick_ML	Molecular layer thickness (μm)
N_CC	Number of Purkinje cell bodies (per 413x10^3 μm^(3))

Variable	Description
Mouse	Mouse unique identifier
Condition	Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)
Z	Bregma coordinates (Ant, Med, Post)
A_VGLUT_MF	Vglut2-labelled Mossy fiber area in the IGL (10^-4 μm^(2))
A_VGLUT_CF	Vglut2-labelled Climbing fiber area in the ML (10^-4 μm^(2))
A_DD_per_cell	Purkinje dendrite area per Purkinje cell body (10^-4 μm^(2))
Vol_DD_per_cell	Purkinje dendrite volume per Purkinje cell body (10^-4 μm^(3))
A_ML	Molecular layer area (10^-4 μm^(2))
Thick_ML	Molecular layer thickness (μm)
N_CC	Number of Purkinje cell bodies (per 413x10^3 μm^(3))

❖ Correlations

21.1 Vglut2-labelled Mossy fiber area (IGL)

21.1.1 Data Exploration

❖ Distribution:

Ant

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	7.939	2.242	5.026	0.282	2.767	2.61	13.123	0.191	0.184	41
IH	8.377	1.752	3.071	0.209	1.946	4.587	13.023	0.322	1.371	32

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	7.939	2.242	5.026	0.282	2.767	2.61	13.123	0.191	0.184	41
IH	8.377	1.752	3.071	0.209	1.946	4.587	13.023	0.322	1.371	32

Med

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	7.893	1.856	3.446	0.235	1.582	4.754	13.249	0.923	1.303	43
IH	9.658	3.054	9.329	0.316	3.93	4.399	17.734	0.963	1.082	32

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	7.893	1.856	3.446	0.235	1.582	4.754	13.249	0.923	1.303	43
IH	9.658	3.054	9.329	0.316	3.93	4.399	17.734	0.963	1.082	32

Post

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	8.094	1.875	3.517	0.232	2.661	4.435	12.632	−0.078	−0.124	36
IH	10.712	2.254	5.081	0.21	2.943	4.988	13.967	−0.709	0.216	26

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	8.094	1.875	3.517	0.232	2.661	4.435	12.632	−0.078	−0.124	36
IH	10.712	2.254	5.081	0.21	2.943	4.988	13.967	−0.709	0.216	26

21.1.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_VGLUT_MF ~ 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
961.52	962.24	988.30	0.16	0.15	8.46e-03	2.15	0.25

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
961.52	962.24	988.30	0.16	0.15	8.46e-03	2.15	0.25

❖ 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:

Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_VGLUT_MF ~ 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
970.77	971.49	997.55	2.50e-03	2.37e-03	1.32e-04	2.15	2.18

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
970.77	971.49	997.55	2.50e-03	2.37e-03	1.32e-04	2.15	2.18

❖ 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:

21.1.3 Effects Analysis

Chosen model: Gamma (‘log’)

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

21.1.3.1 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)	8.716	0.174	(8.38, 9.06)	108.287	< .001
Condition1	0.915	0.018	(0.88, 0.95)	-4.448	< .001
Z1	0.937	0.024	(0.89, 0.98)	-2.572	0.010 **
Z2	1.001	0.025	(0.95, 1.05)	0.024	0.981
Condition1 * Z1	1.065	0.027	(1.01, 1.12)	2.501	0.012 *
Condition1 * Z2	0.988	0.025	(0.94, 1.04)	-0.479	0.632
Model: A_VGLUT_MF ~ Condition * Z (210 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	8.716	0.174	(8.38, 9.06)	108.287	< .001
Condition1	0.915	0.018	(0.88, 0.95)	-4.448	< .001
Z1	0.937	0.024	(0.89, 0.98)	-2.572	0.010 **
Z2	1.001	0.025	(0.95, 1.05)	0.024	0.981
Condition1 * Z1	1.065	0.027	(1.01, 1.12)	2.501	0.012 *
Condition1 * Z2	0.988	0.025	(0.94, 1.04)	-0.479	0.632
Model: A_VGLUT_MF ~ Condition * Z (210 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	19.79	1	<0.001 ***
Z	8.29	2	0.020 *
Condition:Z	6.79	2	0.030 *

term	statistic	df	p.value
Condition	19.79	1	<0.001 ***
Z	8.29	2	0.020 *
Condition:Z	6.79	2	0.030 *

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	937.90	954.64	-463.95	927.90
mod_full	8	926.42	953.20	-455.21	910.42	17.49	3	<0.001 ***

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	937.90	954.64	-463.95	927.90
mod_full	8	926.42	953.20	-455.21	910.42	17.49	3	<0.001 ***

Important

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

21.1.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

Condition

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	7.975	0.208	208	7.574	8.396
IH	9.527	0.288	208	8.975	10.113

Condition	response	SE	df	lower.CL	upper.CL
N	7.975	0.208	208	7.574	8.396
IH	9.527	0.288	208	8.975	10.113

- 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	0.837	0.033	208	0.774	0.906	1	−4.448	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.837	0.033	208	0.774	0.906	1	−4.448	<0.001 ***

- 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:

Z

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	8.166	0.258	208	7.672	8.691
Med	8.721	0.272	208	8.201	9.275
Post	9.298	0.318	208	8.692	9.947

Z	response	SE	df	lower.CL	upper.CL
Ant	8.166	0.258	208	7.672	8.691
Med	8.721	0.272	208	8.201	9.275
Post	9.298	0.318	208	8.692	9.947

- 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	0.936	0.04	208	0.86	1.019	1	−1.535	0.126
Ant / Post	0.878	0.04	208	0.803	0.96	1	−2.877	0.004 **
Med / Post	0.938	0.042	208	0.86	1.024	1	−1.446	0.150

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.936	0.04	208	0.86	1.019	1	−1.535	0.126
Ant / Post	0.878	0.04	208	0.803	0.96	1	−2.877	0.004 **
Med / Post	0.938	0.042	208	0.86	1.024	1	−1.446	0.150

- 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:

Condition:Z

❖ Marginal Means:

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

Ant

Condition	response	SE	df	lower.CL	upper.CL
N	7.953	0.332	208	7.324	8.636
IH	8.384	0.396	208	7.638	9.204

Condition	response	SE	df	lower.CL	upper.CL
N	7.953	0.332	208	7.324	8.636
IH	8.384	0.396	208	7.638	9.204

Med

Condition	response	SE	df	lower.CL	upper.CL
N	7.885	0.321	208	7.277	8.544
IH	9.647	0.455	208	8.79	10.587

Condition	response	SE	df	lower.CL	upper.CL
N	7.885	0.321	208	7.277	8.544
IH	9.647	0.455	208	8.79	10.587

Post

Condition	response	SE	df	lower.CL	upper.CL
N	8.087	0.357	208	7.413	8.822
IH	10.691	0.557	208	9.648	11.848

Condition	response	SE	df	lower.CL	upper.CL
N	8.087	0.357	208	7.413	8.822
IH	10.691	0.557	208	9.648	11.848

- 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)

Ant

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.949	0.06	208	0.838	1.074	1	−0.84	0.402

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.949	0.06	208	0.838	1.074	1	−0.84	0.402

Med

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.817	0.051	208	0.723	0.924	1	−3.242	0.001 **

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.817	0.051	208	0.723	0.924	1	−3.242	0.001 **

Post

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.756	0.052	208	0.661	0.865	1	−4.097	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.756	0.052	208	0.661	0.865	1	−4.097	<0.001 ***

- 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)

Ant / Med

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.16	0.099	208	0.982	1.372	1	1.753	0.081

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.16	0.099	208	0.982	1.372	1	1.753	0.081

Ant / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.254	0.112	208	1.052	1.495	1	2.536	0.012 *

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.254	0.112	208	1.052	1.495	1	2.536	0.012 *

Med / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.081	0.096	208	0.907	1.287	1	0.875	0.383

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.081	0.096	208	0.907	1.287	1	0.875	0.383

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

❖ Boxplot:

---

21.2 Vglut2-labelled Climbing fiber area (ML)

21.2.1 Data Exploration

❖ Distribution:

Ant

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	3.379	2.888	8.341	0.855	2.461	0.211	12.796	1.847	3.639	38
IH	6.929	5.357	28.696	0.773	5.334	0.702	29.153	2.319	7.987	35

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	3.379	2.888	8.341	0.855	2.461	0.211	12.796	1.847	3.639	38
IH	6.929	5.357	28.696	0.773	5.334	0.702	29.153	2.319	7.987	35

Med

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	3.08	1.942	3.772	0.631	2.328	0.752	11.91	2.211	9.116	43
IH	6.574	6.98	48.718	1.062	5.179	0.204	31.623	2.629	7.592	32

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	3.08	1.942	3.772	0.631	2.328	0.752	11.91	2.211	9.116	43
IH	6.574	6.98	48.718	1.062	5.179	0.204	31.623	2.629	7.592	32

Post

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	3.077	2.15	4.624	0.699	2.483	0.368	10.502	1.586	3.518	35
IH	8.758	6.616	43.775	0.755	8.047	0.387	26.488	1.194	1.177	26

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	3.077	2.15	4.624	0.699	2.483	0.368	10.502	1.586	3.518	35
IH	8.758	6.616	43.775	0.755	8.047	0.387	26.488	1.194	1.177	26

21.2.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_VGLUT_CF ~ 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
1051.68	1052.40	1078.42	0.29	0.23	0.07	4.41	0.74

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
1051.68	1052.40	1078.42	0.29	0.23	0.07	4.41	0.74

❖ 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:

Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_VGLUT_CF ~ 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
1255.13	1255.85	1281.87	9.14e-03	8.42e-03	7.26e-04	4.43	4.52

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
1255.13	1255.85	1281.87	9.14e-03	8.42e-03	7.26e-04	4.43	4.52

❖ 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:

21.2.3 Effects Analysis

Chosen model: Gamma (‘log’)

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

21.2.3.1 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)	4.764	0.445	(3.97, 5.72)	16.727	< .001
Condition1	0.657	0.061	(0.55, 0.79)	-4.507	< .001
Z1	1.024	0.077	(0.88, 1.19)	0.315	0.753
Z2	0.931	0.068	(0.81, 1.07)	-0.980	0.327
Condition1 * Z1	1.039	0.078	(0.90, 1.21)	0.512	0.609
Condition1 * Z2	1.066	0.078	(0.92, 1.23)	0.872	0.383
Model: A_VGLUT_CF ~ Condition * Z (209 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	4.764	0.445	(3.97, 5.72)	16.727	< .001
Condition1	0.657	0.061	(0.55, 0.79)	-4.507	< .001
Z1	1.024	0.077	(0.88, 1.19)	0.315	0.753
Z2	0.931	0.068	(0.81, 1.07)	-0.980	0.327
Condition1 * Z1	1.039	0.078	(0.90, 1.21)	0.512	0.609
Condition1 * Z2	1.066	0.078	(0.92, 1.23)	0.872	0.383
Model: A_VGLUT_CF ~ Condition * Z (209 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	20.32	1	<0.001 ***
Z	0.98	2	0.610
Condition:Z	1.86	2	0.400

term	statistic	df	p.value
Condition	20.32	1	<0.001 ***
Z	0.98	2	0.610
Condition:Z	1.86	2	0.400

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	1038.53	1055.24	-514.26	1028.53
mod_full	8	1031.37	1058.11	-507.69	1015.37	13.15	3	0.004 **

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	1038.53	1055.24	-514.26	1028.53
mod_full	8	1031.37	1058.11	-507.69	1015.37	13.15	3	0.004 **

Important

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

21.2.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

Condition

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	3.128	0.384	207	2.456	3.984
IH	7.254	1.02	207	5.498	9.572

Condition	response	SE	df	lower.CL	upper.CL
N	3.128	0.384	207	2.456	3.984
IH	7.254	1.02	207	5.498	9.572

- 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	0.431	0.08	207	0.298	0.623	1	−4.507	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.431	0.08	207	0.298	0.623	1	−4.507	<0.001 ***

- 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:

Z

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	4.878	0.575	207	3.866	6.156
Med	4.436	0.518	207	3.524	5.584
Post	4.995	0.622	207	3.908	6.384

Z	response	SE	df	lower.CL	upper.CL
Ant	4.878	0.575	207	3.866	6.156
Med	4.436	0.518	207	3.524	5.584
Post	4.995	0.622	207	3.908	6.384

- 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	1.1	0.139	207	0.856	1.412	1	0.75	0.454
Ant / Post	0.977	0.131	207	0.75	1.272	1	−0.177	0.860
Med / Post	0.888	0.115	207	0.688	1.146	1	−0.919	0.359

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.1	0.139	207	0.856	1.412	1	0.75	0.454
Ant / Post	0.977	0.131	207	0.75	1.272	1	−0.177	0.860
Med / Post	0.888	0.115	207	0.688	1.146	1	−0.919	0.359

- 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:

Condition:Z

❖ Marginal Means:

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

Ant

Condition	response	SE	df	lower.CL	upper.CL
N	3.33	0.53	207	2.434	4.556
IH	7.147	1.246	207	5.069	10.078

Condition	response	SE	df	lower.CL	upper.CL
N	3.33	0.53	207	2.434	4.556
IH	7.147	1.246	207	5.069	10.078

Med

Condition	response	SE	df	lower.CL	upper.CL
N	3.105	0.473	207	2.299	4.192
IH	6.338	1.125	207	4.467	8.993

Condition	response	SE	df	lower.CL	upper.CL
N	3.105	0.473	207	2.299	4.192
IH	6.338	1.125	207	4.467	8.993

Post

Condition	response	SE	df	lower.CL	upper.CL
N	2.961	0.477	207	2.154	4.068
IH	8.427	1.593	207	5.806	12.233

Condition	response	SE	df	lower.CL	upper.CL
N	2.961	0.477	207	2.154	4.068
IH	8.427	1.593	207	5.806	12.233

- 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)

Ant

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.466	0.11	207	0.293	0.742	1	−3.237	0.001 **

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.466	0.11	207	0.293	0.742	1	−3.237	0.001 **

Med

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.49	0.115	207	0.309	0.777	1	−3.047	0.003 **

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.49	0.115	207	0.309	0.777	1	−3.047	0.003 **

Post

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.351	0.087	207	0.215	0.573	1	−4.218	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.351	0.087	207	0.215	0.573	1	−4.218	<0.001 ***

- 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)

Ant / Med

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.951	0.243	207	0.575	1.572	1	−0.197	0.844

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.951	0.243	207	0.575	1.572	1	−0.197	0.844

Ant / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.326	0.353	207	0.784	2.242	1	1.059	0.291

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.326	0.353	207	0.784	2.242	1	1.059	0.291

Med / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.394	0.36	207	0.838	2.32	1	1.287	0.199

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.394	0.36	207	0.838	2.32	1	1.287	0.199

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

❖ Boxplot:

---

21.3 Purkinje dendrite area (per Purkinje cell body)

21.3.1 Data Exploration

❖ Distribution:

Ant

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.143	0.389	0.151	0.34	0.445	0.109	2.006	−0.105	0.408	40
IH	1.333	0.549	0.302	0.412	0.639	0.562	3.305	1.492	3.586	35

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.143	0.389	0.151	0.34	0.445	0.109	2.006	−0.105	0.408	40
IH	1.333	0.549	0.302	0.412	0.639	0.562	3.305	1.492	3.586	35

Med

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.347	0.612	0.374	0.454	0.578	0.369	3.386	1.49	2.888	41
IH	1.155	0.405	0.164	0.35	0.499	0.479	2.445	0.898	2.178	31

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.347	0.612	0.374	0.454	0.578	0.369	3.386	1.49	2.888	41
IH	1.155	0.405	0.164	0.35	0.499	0.479	2.445	0.898	2.178	31

Post

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.449	0.732	0.535	0.505	0.593	0.265	4.611	2.508	9.047	39
IH	1.204	0.541	0.292	0.449	0.715	0.497	2.45	0.999	−0.104	27

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.449	0.732	0.535	0.505	0.593	0.265	4.611	2.508	9.047	39
IH	1.204	0.541	0.292	0.449	0.715	0.497	2.45	0.999	−0.104	27

21.3.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘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
337.15	337.86	364.04	0.10	0.04	0.06	0.53	0.40

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
337.15	337.86	364.04	0.10	0.04	0.06	0.53	0.40

❖ 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:

Gaussian (‘log’)

❖ 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
389.93	390.63	416.82	0.05	0.02	0.03	0.53	0.55

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
389.93	390.63	416.82	0.05	0.02	0.03	0.53	0.55

❖ 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:

21.3.3 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)
```

21.3.3.1 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)	1.264	0.060	(1.15, 1.39)	4.920	< .001
Condition1	1.030	0.049	(0.94, 1.13)	0.626	0.531
Z1	0.969	0.038	(0.90, 1.05)	-0.819	0.413
Z2	0.988	0.039	(0.92, 1.07)	-0.299	0.765
Condition1 * Z1	0.912	0.036	(0.84, 0.99)	-2.338	0.019 *
Condition1 * Z2	1.045	0.041	(0.97, 1.13)	1.123	0.261
Model: A_DD_per_cell ~ Condition * Z (213 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	1.264	0.060	(1.15, 1.39)	4.920	< .001
Condition1	1.030	0.049	(0.94, 1.13)	0.626	0.531
Z1	0.969	0.038	(0.90, 1.05)	-0.819	0.413
Z2	0.988	0.039	(0.92, 1.07)	-0.299	0.765
Condition1 * Z1	0.912	0.036	(0.84, 0.99)	-2.338	0.019 *
Condition1 * Z2	1.045	0.041	(0.97, 1.13)	1.123	0.261
Model: A_DD_per_cell ~ Condition * Z (213 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	0.39	1	0.530
Z	1.27	2	0.530
Condition:Z	5.47	2	0.060

term	statistic	df	p.value
Condition	0.39	1	0.530
Z	1.27	2	0.530
Condition:Z	5.47	2	0.060

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	309.34	326.14	-149.67	299.34
mod_full	8	309.09	335.98	-146.55	293.09	6.24	3	0.100

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	309.34	326.14	-149.67	299.34
mod_full	8	309.09	335.98	-146.55	293.09	6.24	3	0.100

Important

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

21.3.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

Condition

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	1.303	0.081	211	1.151	1.473
IH	1.227	0.088	211	1.065	1.414

Condition	response	SE	df	lower.CL	upper.CL
N	1.303	0.081	211	1.151	1.473
IH	1.227	0.088	211	1.065	1.414

- 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	1.062	0.101	211	0.88	1.281	1	0.626	0.532

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.062	0.101	211	0.88	1.281	1	0.626	0.532

- 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:

Z

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	1.224	0.074	211	1.086	1.38
Med	1.25	0.077	211	1.107	1.41
Post	1.321	0.084	211	1.165	1.497

Z	response	SE	df	lower.CL	upper.CL
Ant	1.224	0.074	211	1.086	1.38
Med	1.25	0.077	211	1.107	1.41
Post	1.321	0.084	211	1.165	1.497

- 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	0.98	0.066	211	0.859	1.118	1	−0.302	0.763
Ant / Post	0.927	0.064	211	0.809	1.062	1	−1.096	0.275
Med / Post	0.946	0.066	211	0.825	1.085	1	−0.799	0.425

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.98	0.066	211	0.859	1.118	1	−0.302	0.763
Ant / Post	0.927	0.064	211	0.809	1.062	1	−1.096	0.275
Med / Post	0.946	0.066	211	0.825	1.085	1	−0.799	0.425

- 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:

Condition:Z

❖ Marginal Means:

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

Ant

Condition	response	SE	df	lower.CL	upper.CL
N	1.151	0.093	211	0.98	1.351
IH	1.303	0.118	211	1.09	1.557

Condition	response	SE	df	lower.CL	upper.CL
N	1.151	0.093	211	0.98	1.351
IH	1.303	0.118	211	1.09	1.557

Med

Condition	response	SE	df	lower.CL	upper.CL
N	1.345	0.109	211	1.148	1.577
IH	1.161	0.108	211	0.967	1.393

Condition	response	SE	df	lower.CL	upper.CL
N	1.345	0.109	211	1.148	1.577
IH	1.161	0.108	211	0.967	1.393

Post

Condition	response	SE	df	lower.CL	upper.CL
N	1.427	0.117	211	1.214	1.678
IH	1.222	0.119	211	1.009	1.48

Condition	response	SE	df	lower.CL	upper.CL
N	1.427	0.117	211	1.214	1.678
IH	1.222	0.119	211	1.009	1.48

- 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)

Ant

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.883	0.108	211	0.695	1.123	1	−1.02	0.309

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.883	0.108	211	0.695	1.123	1	−1.02	0.309

Med

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.159	0.142	211	0.91	1.477	1	1.202	0.231

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.159	0.142	211	0.91	1.477	1	1.202	0.231

Post

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.168	0.149	211	0.909	1.502	1	1.22	0.224

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.168	0.149	211	0.909	1.502	1	1.22	0.224

- 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)

Ant / Med

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.762	0.102	211	0.585	0.993	1	−2.024	0.044 *

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.762	0.102	211	0.585	0.993	1	−2.024	0.044 *

Ant / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.756	0.105	211	0.575	0.995	1	−2.007	0.046 *

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.756	0.105	211	0.575	0.995	1	−2.007	0.046 *

Med / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.992	0.138	211	0.755	1.305	1	−0.056	0.955

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.992	0.138	211	0.755	1.305	1	−0.056	0.955

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

❖ Boxplot:

---

21.4 Purkinje dendrite volume (per Purkinje cell body)

21.4.1 Data Exploration

❖ Distribution:

Ant

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.759	0.25	0.063	0.329	0.285	0.321	1.637	1.025	2.557	41
IH	1.196	0.6	0.36	0.501	1.012	0.447	2.5	0.763	−0.592	35

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.759	0.25	0.063	0.329	0.285	0.321	1.637	1.025	2.557	41
IH	1.196	0.6	0.36	0.501	1.012	0.447	2.5	0.763	−0.592	35

Med

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.757	0.371	0.138	0.491	0.452	0.126	1.921	1.26	2.163	43
IH	1.015	0.515	0.265	0.507	0.393	0.45	2.375	1.342	0.951	31

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.757	0.371	0.138	0.491	0.452	0.126	1.921	1.26	2.163	43
IH	1.015	0.515	0.265	0.507	0.393	0.45	2.375	1.342	0.951	31

Post

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.761	0.323	0.105	0.425	0.414	0.147	1.689	0.723	0.775	39
IH	1.234	0.516	0.266	0.418	0.884	0.44	2.087	0.089	−1.109	28

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.761	0.323	0.105	0.425	0.414	0.147	1.689	0.723	0.775	39
IH	1.234	0.516	0.266	0.418	0.884	0.44	2.087	0.089	−1.109	28

21.4.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = Vol_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	RMSE	Sigma
227.18	227.87	254.22		0.19	0.43	0.44

AIC	AICc	BIC	R2_conditional	R2_marginal	RMSE	Sigma
227.18	227.87	254.22		0.19	0.43	0.44

❖ 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:

Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = Vol_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	RMSE	Sigma
291.16	291.86	318.20		0.19	0.43	0.43

AIC	AICc	BIC	R2_conditional	R2_marginal	RMSE	Sigma
291.16	291.86	318.20		0.19	0.43	0.43

❖ 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:

21.4.3 Effects Analysis

Chosen model: Gamma (‘log’)

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

21.4.3.1 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)	0.932	0.028	(0.88, 0.99)	-2.337	0.019 *
Condition1	0.814	0.025	(0.77, 0.86)	-6.795	< .001
Z1	1.023	0.043	(0.94, 1.11)	0.531	0.595
Z2	0.941	0.040	(0.87, 1.02)	-1.438	0.150
Condition1 * Z1	0.978	0.041	(0.90, 1.06)	-0.532	0.595
Condition1 * Z2	1.061	0.045	(0.98, 1.15)	1.384	0.166
Model: Vol_DD_per_cell ~ Condition * Z (217 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	0.932	0.028	(0.88, 0.99)	-2.337	0.019 *
Condition1	0.814	0.025	(0.77, 0.86)	-6.795	< .001
Z1	1.023	0.043	(0.94, 1.11)	0.531	0.595
Z2	0.941	0.040	(0.87, 1.02)	-1.438	0.150
Condition1 * Z1	0.978	0.041	(0.90, 1.06)	-0.532	0.595
Condition1 * Z2	1.061	0.045	(0.98, 1.15)	1.384	0.166
Model: Vol_DD_per_cell ~ Condition * Z (217 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	46.18	1	<0.001 ***
Z	2.09	2	0.350
Condition:Z	1.93	2	0.380

term	statistic	df	p.value
Condition	46.18	1	<0.001 ***
Z	2.09	2	0.350
Condition:Z	1.93	2	0.380

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	212.36	229.26	-101.18	202.36
mod_full	8	198.25	225.29	-91.13	182.25	20.11	3	<0.001 ***

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	212.36	229.26	-101.18	202.36
mod_full	8	198.25	225.29	-91.13	182.25	20.11	3	<0.001 ***

Important

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

21.4.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

Condition

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	0.759	0.03	215	0.702	0.821
IH	1.144	0.052	215	1.046	1.252

Condition	response	SE	df	lower.CL	upper.CL
N	0.759	0.03	215	0.702	0.821
IH	1.144	0.052	215	1.046	1.252

- 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	0.663	0.04	215	0.589	0.747	1	−6.795	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.663	0.04	215	0.589	0.747	1	−6.795	<0.001 ***

- 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:

Z

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	0.953	0.048	215	0.862	1.053
Med	0.877	0.045	215	0.792	0.971
Post	0.969	0.053	215	0.87	1.078

Z	response	SE	df	lower.CL	upper.CL
Ant	0.953	0.048	215	0.862	1.053
Med	0.877	0.045	215	0.792	0.971
Post	0.969	0.053	215	0.87	1.078

- 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	1.087	0.079	215	0.942	1.254	1	1.152	0.250
Ant / Post	0.984	0.073	215	0.85	1.139	1	−0.222	0.825
Med / Post	0.905	0.068	215	0.78	1.049	1	−1.33	0.185

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.087	0.079	215	0.942	1.254	1	1.152	0.250
Ant / Post	0.984	0.073	215	0.85	1.139	1	−0.222	0.825
Med / Post	0.905	0.068	215	0.78	1.049	1	−1.33	0.185

- 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:

Condition:Z

❖ Marginal Means:

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

Ant

Condition	response	SE	df	lower.CL	upper.CL
N	0.759	0.052	215	0.663	0.869
IH	1.196	0.089	215	1.033	1.385

Condition	response	SE	df	lower.CL	upper.CL
N	0.759	0.052	215	0.663	0.869
IH	1.196	0.089	215	1.033	1.385

Med

Condition	response	SE	df	lower.CL	upper.CL
N	0.757	0.051	215	0.664	0.864
IH	1.015	0.08	215	0.869	1.186

Condition	response	SE	df	lower.CL	upper.CL
N	0.757	0.051	215	0.664	0.864
IH	1.015	0.08	215	0.869	1.186

Post

Condition	response	SE	df	lower.CL	upper.CL
N	0.761	0.054	215	0.662	0.874
IH	1.234	0.102	215	1.047	1.453

Condition	response	SE	df	lower.CL	upper.CL
N	0.761	0.054	215	0.662	0.874
IH	1.234	0.102	215	1.047	1.453

- 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)

Ant

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.634	0.064	215	0.52	0.774	1	−4.498	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.634	0.064	215	0.52	0.774	1	−4.498	<0.001 ***

Med

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.746	0.077	215	0.608	0.915	1	−2.826	0.005 **

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.746	0.077	215	0.608	0.915	1	−2.826	0.005 **

Post

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.617	0.067	215	0.498	0.764	1	−4.438	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.617	0.067	215	0.498	0.764	1	−4.438	<0.001 ***

- 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)

Ant / Med

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.85	0.123	215	0.639	1.131	1	−1.121	0.263

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.85	0.123	215	0.639	1.131	1	−1.121	0.263

Ant / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.029	0.153	215	0.767	1.379	1	0.19	0.849

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.029	0.153	215	0.767	1.379	1	0.19	0.849

Med / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.21	0.182	215	0.9	1.627	1	1.268	0.206

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.21	0.182	215	0.9	1.627	1	1.268	0.206

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

❖ Boxplot:

---

21.5 Molecular layer area

21.5.1 Data Exploration

❖ Distribution:

Ant

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.616	0.367	0.135	0.14	0.456	1.695	3.167	−1.055	0.297	41
IH	2.733	0.256	0.065	0.094	0.37	2.148	3.149	−0.669	−0.155	36

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.616	0.367	0.135	0.14	0.456	1.695	3.167	−1.055	0.297	41
IH	2.733	0.256	0.065	0.094	0.37	2.148	3.149	−0.669	−0.155	36

Med

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.697	0.297	0.088	0.11	0.306	1.653	3.079	−1.385	2.465	43
IH	2.63	0.333	0.111	0.127	0.417	1.745	3.024	−1.008	0.409	32

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.697	0.297	0.088	0.11	0.306	1.653	3.079	−1.385	2.465	43
IH	2.63	0.333	0.111	0.127	0.417	1.745	3.024	−1.008	0.409	32

Post

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.749	0.296	0.088	0.108	0.39	1.802	3.12	−1.321	2.033	39
IH	2.701	0.321	0.103	0.119	0.494	1.943	3.096	−1	−0.046	28

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.749	0.296	0.088	0.108	0.39	1.802	3.12	−1.321	2.033	39
IH	2.701	0.321	0.103	0.119	0.494	1.943	3.096	−1	−0.046	28

21.5.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_ML ~ 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
190.47	191.15	217.58	0.03	0.02	8.78e-03	0.31	0.12

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
190.47	191.15	217.58	0.03	0.02	8.78e-03	0.31	0.12

❖ 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:

Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_ML ~ 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
167.92	168.61	195.03	5.32e-03	3.28e-03	2.05e-03	0.31	0.31

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
167.92	168.61	195.03	5.32e-03	3.28e-03	2.05e-03	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:

21.5.3 Effects Analysis

Chosen model: Gamma (‘log’)

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

21.5.3.1 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)	2.687	0.026	(2.64, 2.74)	103.920	< .001
Condition1	1.000	0.010	(0.98, 1.02)	-0.030	0.976
Z1	0.995	0.012	(0.97, 1.02)	-0.454	0.650
Z2	0.991	0.012	(0.97, 1.01)	-0.737	0.461
Condition1 * Z1	0.980	0.012	(0.96, 1.00)	-1.723	0.085
Condition1 * Z2	1.012	0.012	(0.99, 1.04)	1.043	0.297
Model: A_ML ~ Condition * Z (219 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	2.687	0.026	(2.64, 2.74)	103.920	< .001
Condition1	1.000	0.010	(0.98, 1.02)	-0.030	0.976
Z1	0.995	0.012	(0.97, 1.02)	-0.454	0.650
Z2	0.991	0.012	(0.97, 1.01)	-0.737	0.461
Condition1 * Z1	0.980	0.012	(0.96, 1.00)	-1.723	0.085
Condition1 * Z2	1.012	0.012	(0.99, 1.04)	1.043	0.297
Model: A_ML ~ Condition * Z (219 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	9.22e-04	1	0.980
Z	1.36	2	0.510
Condition:Z	3.04	2	0.220

term	statistic	df	p.value
Condition	9.22e-04	1	0.980
Z	1.36	2	0.510
Condition:Z	3.04	2	0.220

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	143.92	160.87	-66.96	133.92
mod_full	8	146.40	173.51	-65.20	130.40	3.52	3	0.320

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	143.92	160.87	-66.96	133.92
mod_full	8	146.40	173.51	-65.20	130.40	3.52	3	0.320

Important

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

21.5.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

Condition

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	2.686	0.034	217	2.621	2.753
IH	2.688	0.038	217	2.613	2.765

Condition	response	SE	df	lower.CL	upper.CL
N	2.686	0.034	217	2.621	2.753
IH	2.688	0.038	217	2.613	2.765

- 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	0.999	0.019	217	0.963	1.038	1	−0.03	0.976

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.999	0.019	217	0.963	1.038	1	−0.03	0.976

- 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:

Z

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	2.673	0.04	217	2.596	2.752
Med	2.664	0.04	217	2.586	2.744
Post	2.725	0.043	217	2.641	2.812

Z	response	SE	df	lower.CL	upper.CL
Ant	2.673	0.04	217	2.596	2.752
Med	2.664	0.04	217	2.586	2.744
Post	2.725	0.043	217	2.641	2.812

- 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	1.003	0.02	217	0.964	1.044	1	0.168	0.867
Ant / Post	0.981	0.02	217	0.941	1.022	1	−0.932	0.352
Med / Post	0.977	0.021	217	0.938	1.019	1	−1.088	0.278

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.003	0.02	217	0.964	1.044	1	0.168	0.867
Ant / Post	0.981	0.02	217	0.941	1.022	1	−0.932	0.352
Med / Post	0.977	0.021	217	0.938	1.019	1	−1.088	0.278

- 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:

Condition:Z

❖ Marginal Means:

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

Ant

Condition	response	SE	df	lower.CL	upper.CL
N	2.617	0.053	217	2.515	2.723
IH	2.729	0.06	217	2.614	2.85

Condition	response	SE	df	lower.CL	upper.CL
N	2.617	0.053	217	2.515	2.723
IH	2.729	0.06	217	2.614	2.85

Med

Condition	response	SE	df	lower.CL	upper.CL
N	2.696	0.053	217	2.593	2.803
IH	2.632	0.06	217	2.516	2.753

Condition	response	SE	df	lower.CL	upper.CL
N	2.696	0.053	217	2.593	2.803
IH	2.632	0.06	217	2.516	2.753

Post

Condition	response	SE	df	lower.CL	upper.CL
N	2.747	0.057	217	2.638	2.861
IH	2.703	0.066	217	2.577	2.836

Condition	response	SE	df	lower.CL	upper.CL
N	2.747	0.057	217	2.638	2.861
IH	2.703	0.066	217	2.577	2.836

- 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)

Ant

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.959	0.029	217	0.904	1.017	1	−1.4	0.163

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.959	0.029	217	0.904	1.017	1	−1.4	0.163

Med

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.025	0.031	217	0.965	1.087	1	0.802	0.423

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.025	0.031	217	0.965	1.087	1	0.802	0.423

Post

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.016	0.033	217	0.954	1.082	1	0.501	0.617

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.016	0.033	217	0.954	1.082	1	0.501	0.617

- 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)

Ant / Med

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.936	0.038	217	0.863	1.015	1	−1.616	0.108

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.936	0.038	217	0.863	1.015	1	−1.616	0.108

Ant / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.944	0.04	217	0.868	1.026	1	−1.366	0.173

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.944	0.04	217	0.868	1.026	1	−1.366	0.173

Med / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.008	0.042	217	0.928	1.095	1	0.195	0.846

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.008	0.042	217	0.928	1.095	1	0.195	0.846

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

❖ Boxplot:

---

21.6 Molecular layer thickness

21.6.1 Data Exploration

❖ Distribution:

Ant

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	134.994	18.948	359.011	0.14	23.557	87.509	163.477	−1.055	0.297	41
IH	141.071	13.205	174.359	0.094	19.086	110.876	162.531	−0.669	−0.155	36

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	134.994	18.948	359.011	0.14	23.557	87.509	163.477	−1.055	0.297	41
IH	141.071	13.205	174.359	0.094	19.086	110.876	162.531	−0.669	−0.155	36

Med

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	139.183	15.339	235.29	0.11	15.799	85.333	158.936	−1.385	2.465	43
IH	135.74	17.201	295.878	0.127	21.499	90.063	156.097	−1.008	0.409	32

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	139.183	15.339	235.29	0.11	15.799	85.333	158.936	−1.385	2.465	43
IH	135.74	17.201	295.878	0.127	21.499	90.063	156.097	−1.008	0.409	32

Post

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	141.88	15.27	233.172	0.108	20.151	92.996	161.017	−1.321	2.033	39
IH	139.42	16.558	274.177	0.119	25.52	100.281	159.787	−1	−0.046	28

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	141.88	15.27	233.172	0.108	20.151	92.996	161.017	−1.321	2.033	39
IH	139.42	16.558	274.177	0.119	25.52	100.281	159.787	−1	−0.046	28

21.6.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = Thick_ML ~ 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
1917.84	1918.52	1944.95	0.03	0.02	8.78e-03	15.86	0.12

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
1917.84	1918.52	1944.95	0.03	0.02	8.78e-03	15.86	0.12

❖ 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:

Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = Thick_ML ~ 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
1895.29	1895.98	1922.40	2.01e-06	1.24e-06	7.70e-07	15.82	16.10

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
1895.29	1895.98	1922.40	2.01e-06	1.24e-06	7.70e-07	15.82	16.10

❖ 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:

21.6.3 Effects Analysis

Chosen model: Gamma (‘log’)

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

21.6.3.1 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)	138.686	1.319	(136.12, 141.30)	518.550	< .001
Condition1	1.000	0.010	(0.98, 1.02)	-0.030	0.976
Z1	0.995	0.012	(0.97, 1.02)	-0.454	0.650
Z2	0.991	0.012	(0.97, 1.01)	-0.737	0.461
Condition1 * Z1	0.980	0.012	(0.96, 1.00)	-1.723	0.085
Condition1 * Z2	1.012	0.012	(0.99, 1.04)	1.043	0.297
Model: Thick_ML ~ Condition * Z (219 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	138.686	1.319	(136.12, 141.30)	518.550	< .001
Condition1	1.000	0.010	(0.98, 1.02)	-0.030	0.976
Z1	0.995	0.012	(0.97, 1.02)	-0.454	0.650
Z2	0.991	0.012	(0.97, 1.01)	-0.737	0.461
Condition1 * Z1	0.980	0.012	(0.96, 1.00)	-1.723	0.085
Condition1 * Z2	1.012	0.012	(0.99, 1.04)	1.043	0.297
Model: Thick_ML ~ Condition * Z (219 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	9.22e-04	1	0.980
Z	1.36	2	0.510
Condition:Z	3.04	2	0.220

term	statistic	df	p.value
Condition	9.22e-04	1	0.980
Z	1.36	2	0.510
Condition:Z	3.04	2	0.220

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	1871.29	1888.24	-930.65	1861.29
mod_full	8	1873.77	1900.89	-928.89	1857.77	3.52	3	0.320

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	1871.29	1888.24	-930.65	1861.29
mod_full	8	1873.77	1900.89	-928.89	1857.77	3.52	3	0.320

Important

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

21.6.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

Condition

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	138.646	1.739	217	135.26	142.116
IH	138.726	1.984	217	134.87	142.692

Condition	response	SE	df	lower.CL	upper.CL
N	138.646	1.739	217	135.26	142.116
IH	138.726	1.984	217	134.87	142.692

- 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	0.999	0.019	217	0.963	1.038	1	−0.03	0.976

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.999	0.019	217	0.963	1.038	1	−0.03	0.976

- 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:

Z

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	137.946	2.044	217	133.977	142.034
Med	137.478	2.072	217	133.455	141.623
Post	140.653	2.237	217	136.313	145.132

Z	response	SE	df	lower.CL	upper.CL
Ant	137.946	2.044	217	133.977	142.034
Med	137.478	2.072	217	133.455	141.623
Post	140.653	2.237	217	136.313	145.132

- 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	1.003	0.02	217	0.964	1.044	1	0.168	0.867
Ant / Post	0.981	0.02	217	0.941	1.022	1	−0.932	0.352
Med / Post	0.977	0.021	217	0.938	1.019	1	−1.088	0.278

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.003	0.02	217	0.964	1.044	1	0.168	0.867
Ant / Post	0.981	0.02	217	0.941	1.022	1	−0.932	0.352
Med / Post	0.977	0.021	217	0.938	1.019	1	−1.088	0.278

- 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:

Condition:Z

❖ Marginal Means:

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

Ant

Condition	response	SE	df	lower.CL	upper.CL
N	135.083	2.728	217	129.812	140.568
IH	140.871	3.087	217	134.916	147.089

Condition	response	SE	df	lower.CL	upper.CL
N	135.083	2.728	217	129.812	140.568
IH	140.871	3.087	217	134.916	147.089

Med

Condition	response	SE	df	lower.CL	upper.CL
N	139.153	2.74	217	133.855	144.66
IH	135.824	3.104	217	129.843	142.081

Condition	response	SE	df	lower.CL	upper.CL
N	139.153	2.74	217	133.855	144.66
IH	135.824	3.104	217	129.843	142.081

Post

Condition	response	SE	df	lower.CL	upper.CL
N	141.784	2.927	217	136.132	147.671
IH	139.531	3.393	217	133.001	146.382

Condition	response	SE	df	lower.CL	upper.CL
N	141.784	2.927	217	136.132	147.671
IH	139.531	3.393	217	133.001	146.382

- 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)

Ant

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.959	0.029	217	0.904	1.017	1	−1.4	0.163

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.959	0.029	217	0.904	1.017	1	−1.4	0.163

Med

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.025	0.031	217	0.965	1.087	1	0.802	0.423

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.025	0.031	217	0.965	1.087	1	0.802	0.423

Post

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.016	0.033	217	0.954	1.082	1	0.501	0.617

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.016	0.033	217	0.954	1.082	1	0.501	0.617

- 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)

Ant / Med

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.936	0.038	217	0.863	1.015	1	−1.616	0.108

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.936	0.038	217	0.863	1.015	1	−1.616	0.108

Ant / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.944	0.04	217	0.868	1.026	1	−1.366	0.173

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.944	0.04	217	0.868	1.026	1	−1.366	0.173

Med / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.008	0.042	217	0.928	1.095	1	0.195	0.846

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.008	0.042	217	0.928	1.095	1	0.195	0.846

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

❖ Boxplot:

---

21.7 Number of Purkinje cell bodies

21.7.1 Data Exploration

❖ Distribution:

Ant

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	9.805	1.913	3.661	0.195	2.5	7	17	1.373	3.766	41
IH	9.457	1.975	3.903	0.209	3	5	14	0.262	0.001	35

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	9.805	1.913	3.661	0.195	2.5	7	17	1.373	3.766	41
IH	9.457	1.975	3.903	0.209	3	5	14	0.262	0.001	35

Med

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	10.349	2.654	7.042	0.256	4	5	16	0.31	−0.288	43
IH	9.968	1.853	3.432	0.186	2	6	15	0.286	0.66	31

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	10.349	2.654	7.042	0.256	4	5	16	0.31	−0.288	43
IH	9.968	1.853	3.432	0.186	2	6	15	0.286	0.66	31

Post

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	9.41	3.168	10.038	0.337	2	4	18	0.968	1.003	39
IH	9.786	3.071	9.434	0.314	3	5	21	1.845	5.682	28

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	9.41	3.168	10.038	0.337	2	4	18	0.968	1.003	39
IH	9.786	3.071	9.434	0.314	3	5	21	1.845	5.682	28

21.7.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Genpois (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = N_CC ~ Condition * Z + (1 | Mouse), data = data, 
    family = genpois("log"), control = glmmTMBControl(optimizer = optim, 
        optArgs = list(method = "BFGS")), REML = TRUE, ziformula = ~0, 
    dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma	Score_log	Score_spherical
1052.49	1053.18	1079.53	0.02	0.02	4.61e-04	2.46	0.67	-2.35	0.07

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma	Score_log	Score_spherical
1052.49	1053.18	1079.53	0.02	0.02	4.61e-04	2.46	0.67	-2.35	0.07

❖ Residuals:

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

performance::check_overdispersion(mod)

# Overdispersion test

       dispersion ratio =   0.930
  Pearson's Chi-Squared = 200.051
                p-value =    0.76

❖ Predictions:

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

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

Nbinom2 (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = N_CC ~ Condition * Z + (1 | Mouse), data = data, 
    family = nbinom2("log"), control = glmmTMBControl(optimizer = optim, 
        optArgs = list(method = "BFGS")), REML = TRUE, ziformula = ~0, 
    dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	RMSE	Sigma	Score_log	Score_spherical
1069.24	1069.94	1096.28		0.01	2.46	1.20e+09	-3.04	0.07

AIC	AICc	BIC	R2_conditional	R2_marginal	RMSE	Sigma	Score_log	Score_spherical
1069.24	1069.94	1096.28		0.01	2.46	1.20e+09	-3.04	0.07

❖ Residuals:

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

performance::check_overdispersion(mod)

# Overdispersion test

       dispersion ratio =   0.625
  Pearson's Chi-Squared = 134.424
                p-value =       1

❖ Predictions:

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

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

Poisson (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = N_CC ~ Condition * Z + (1 | Mouse), data = data, 
    family = poisson("log"), control = glmmTMBControl(optimizer = optim, 
        optArgs = list(method = "BFGS")), REML = TRUE, ziformula = ~0, 
    dispformula = ~1)
```

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	RMSE	Sigma	Score_log	Score_spherical
1067.24	1067.78	1090.90		0.01	2.46	1	-2.35	0.07

AIC	AICc	BIC	R2_conditional	R2_marginal	RMSE	Sigma	Score_log	Score_spherical
1067.24	1067.78	1090.90		0.01	2.46	1	-2.35	0.07

❖ Residuals:

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

performance::check_overdispersion(mod)

# Overdispersion test

       dispersion ratio =   0.622
  Pearson's Chi-Squared = 134.433
                p-value =       1

❖ Predictions:

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

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

21.7.3 Effects Analysis

Chosen model: Genpois (‘log’)

```{r}
glmmTMB(formula = N_CC ~ Condition * Z + (1 | Mouse), data = data, 
    family = genpois("log"), control = glmmTMBControl(optimizer = optim, 
        optArgs = list(method = "BFGS")), REML = TRUE, ziformula = ~0, 
    dispformula = ~1)
```

21.7.3.1 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)	9.788	0.177	(9.45, 10.14)	126.010	< .001
Condition1	1.007	0.018	(0.97, 1.04)	0.400	0.689
Z1	0.977	0.025	(0.93, 1.03)	-0.906	0.365
Z2	1.034	0.026	(0.98, 1.09)	1.324	0.186
Condition1 * Z1	1.011	0.025	(0.96, 1.06)	0.428	0.668
Condition1 * Z2	1.015	0.025	(0.97, 1.07)	0.609	0.543
Model: N_CC ~ Condition * Z (217 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	9.788	0.177	(9.45, 10.14)	126.010	< .001
Condition1	1.007	0.018	(0.97, 1.04)	0.400	0.689
Z1	0.977	0.025	(0.93, 1.03)	-0.906	0.365
Z2	1.034	0.026	(0.98, 1.09)	1.324	0.186
Condition1 * Z1	1.011	0.025	(0.96, 1.06)	0.428	0.668
Condition1 * Z2	1.015	0.025	(0.97, 1.07)	0.609	0.543
Model: N_CC ~ Condition * Z (217 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	0.16	1	0.690
Z	1.86	2	0.390
Condition:Z	1.03	2	0.600

term	statistic	df	p.value
Condition	0.16	1	0.690
Z	1.86	2	0.390
Condition:Z	1.03	2	0.600

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	1012.57	1029.47	-501.28	1002.57
mod_full	8	1017.28	1044.32	-500.64	1001.28	1.28	3	0.730

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	1012.57	1029.47	-501.28	1002.57
mod_full	8	1017.28	1044.32	-500.64	1001.28	1.28	3	0.730

Important

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

21.7.3.2 Marginal Effects

Marginal means & Contrasts for each predictor:

Condition

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	9.859	0.233	215	9.409	10.33
IH	9.717	0.265	215	9.208	10.255

Condition	response	SE	df	lower.CL	upper.CL
N	9.859	0.233	215	9.409	10.33
IH	9.717	0.265	215	9.208	10.255

- 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	1.015	0.037	215	0.945	1.089	1	0.4	0.689

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.015	0.037	215	0.945	1.089	1	0.4	0.689

- 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:

Z

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	9.567	0.293	215	9.007	10.161
Med	10.118	0.308	215	9.529	10.744
Post	9.687	0.315	215	9.085	10.328

Z	response	SE	df	lower.CL	upper.CL
Ant	9.567	0.293	215	9.007	10.161
Med	10.118	0.308	215	9.529	10.744
Post	9.687	0.315	215	9.085	10.328

- 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	0.945	0.041	215	0.869	1.029	1	−1.302	0.194
Ant / Post	0.988	0.044	215	0.905	1.078	1	−0.279	0.780
Med / Post	1.045	0.046	215	0.957	1.14	1	0.982	0.327

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.945	0.041	215	0.869	1.029	1	−1.302	0.194
Ant / Post	0.988	0.044	215	0.905	1.078	1	−0.279	0.780
Med / Post	1.045	0.046	215	0.957	1.14	1	0.982	0.327

- 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:

Condition:Z

❖ Marginal Means:

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

Ant

Condition	response	SE	df	lower.CL	upper.CL
N	9.74	0.4	215	8.982	10.562
IH	9.396	0.426	215	8.594	10.274

Condition	response	SE	df	lower.CL	upper.CL
N	9.74	0.4	215	8.982	10.562
IH	9.396	0.426	215	8.594	10.274

Med

Condition	response	SE	df	lower.CL	upper.CL
N	10.348	0.402	215	9.586	11.171
IH	9.893	0.464	215	9.019	10.852

Condition	response	SE	df	lower.CL	upper.CL
N	10.348	0.402	215	9.586	11.171
IH	9.893	0.464	215	9.019	10.852

Post

Condition	response	SE	df	lower.CL	upper.CL
N	9.506	0.403	215	8.744	10.334
IH	9.87	0.484	215	8.961	10.872

Condition	response	SE	df	lower.CL	upper.CL
N	9.506	0.403	215	8.744	10.334
IH	9.87	0.484	215	8.961	10.872

- 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)

Ant

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.037	0.063	215	0.919	1.169	1	0.588	0.557

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.037	0.063	215	0.919	1.169	1	0.588	0.557

Med

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.046	0.064	215	0.928	1.179	1	0.738	0.461

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.046	0.064	215	0.928	1.179	1	0.738	0.461

Post

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.963	0.062	215	0.848	1.094	1	−0.583	0.560

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.963	0.062	215	0.848	1.094	1	−0.583	0.560

- 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)

Ant / Med

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.991	0.085	215	0.836	1.174	1	−0.105	0.917

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.991	0.085	215	0.836	1.174	1	−0.105	0.917

Ant / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.076	0.095	215	0.904	1.282	1	0.83	0.408

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.076	0.095	215	0.904	1.282	1	0.83	0.408

Med / Post

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.086	0.096	215	0.912	1.293	1	0.933	0.352

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.086	0.096	215	0.912	1.293	1	0.933	0.352

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

❖ Boxplot:

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