Intermittent hypoxia and cerebellar development

Variable	Description
Stage	Developmental stage
Mouse	Mouse unique identifier
Condition	Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)
Z	Bregma coordinates (Ant, Med, Post)
A_ML_PCL	Area of ML and PC (μm²)
Dens_ML_PCL	Density of cleaved caspase 3^(+) cells in the ML and PC areas (Cells/μm²)
Prop_C_ML_PCL	Proportion of cleaved caspase 3^(+) cells marking in the ML and PC areas

Variable	Description
Stage	Developmental stage
Mouse	Mouse unique identifier
Condition	Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)
Z	Bregma coordinates (Ant, Med, Post)
A_ML_PCL	Area of ML and PC (μm²)
Dens_ML_PCL	Density of cleaved caspase 3^(+) cells in the ML and PC areas (Cells/μm²)
Prop_C_ML_PCL	Proportion of cleaved caspase 3^(+) cells marking in the ML and PC areas

4.1 Area of the ML and PC

4.1.1 Data Exploration

❖ Distribution:

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.012	0.375	0.14	0.186	0.722	1.577	2.529	0.342	−1.615	6
IH	1.778	1.191	1.418	0.67	2.225	0.443	2.891	−0.235	−4.333	4

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.012	0.375	0.14	0.186	0.722	1.577	2.529	0.342	−1.615	6
IH	1.778	1.191	1.418	0.67	2.225	0.443	2.891	−0.235	−4.333	4

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.972	1.937	3.752	0.652	2.169	1.434	7.226	1.793	3.401	8
IH	1.184	0.87	0.757	0.735	1.575	0.433	2.426	0.869	−1.497	5

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	2.972	1.937	3.752	0.652	2.169	1.434	7.226	1.793	3.401	8
IH	1.184	0.87	0.757	0.735	1.575	0.433	2.426	0.869	−1.497	5

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.202	0.978	0.957	0.814	2.271	0.245	2.633	0.726	−1.216	11
IH	0.843	0.675	0.456	0.801	0.976	0.13	2.149	1.195	0.817	8

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	1.202	0.978	0.957	0.814	2.271	0.245	2.633	0.726	−1.216	11
IH	0.843	0.675	0.456	0.801	0.976	0.13	2.149	1.195	0.817	8

4.1.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)
Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = A_ML_PCL ~ 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
111.39	115.75	125.29	0.81	0.36	0.70	0.59	0.42

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
111.39	115.75	125.29	0.81	0.36	0.70	0.59	0.42

❖ 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:

No potential outliers detected by the model.

❖ Model call:

```{r}
glmmTMB(formula = A_ML_PCL ~ 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
130.47	134.83	144.37	0.53	0.21	0.40	0.54	0.65

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
130.47	134.83	144.37	0.53	0.21	0.40	0.54	0.65

❖ 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:

No potential outliers detected by the model.

4.1.3 Effects Analysis

Chosen model: Gamma (‘log’)

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

4.1.3.1 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)	1.304	0.266	(0.87, 1.95)	1.300	0.193
Condition1	1.375	0.281	(0.92, 2.05)	1.558	0.119
Z1	1.467	0.158	(1.19, 1.81)	3.559	< .001
Z2	1.234	0.123	(1.01, 1.50)	2.100	0.036 *
Condition1 * Z1	0.926	0.099	(0.75, 1.14)	-0.713	0.476
Condition1 * Z2	1.159	0.116	(0.95, 1.41)	1.477	0.140
Model: A_ML_PCL ~ Condition * Z (42 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	1.304	0.266	(0.87, 1.95)	1.300	0.193
Condition1	1.375	0.281	(0.92, 2.05)	1.558	0.119
Z1	1.467	0.158	(1.19, 1.81)	3.559	< .001
Z2	1.234	0.123	(1.01, 1.50)	2.100	0.036 *
Condition1 * Z1	0.926	0.099	(0.75, 1.14)	-0.713	0.476
Condition1 * Z2	1.159	0.116	(0.95, 1.41)	1.477	0.140
Model: A_ML_PCL ~ Condition * Z (42 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	2.43	1	0.120
Z	37.99	2	<0.001 ***
Condition:Z	2.21	2	0.330

term	statistic	df	p.value
Condition	2.43	1	0.120
Z	37.99	2	<0.001 ***
Condition:Z	2.21	2	0.330

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	95.31	104.00	-42.66	85.31
mod_full	8	96.47	110.38	-40.24	80.47	4.84	3	0.180

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	95.31	104.00	-42.66	85.31
mod_full	8	96.47	110.38	-40.24	80.47	4.84	3	0.180

Important

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

4.1.3.2 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	1.793	0.489	40	1.034	3.11
IH	0.949	0.289	40	0.513	1.755

Condition	response	SE	df	lower.CL	upper.CL
N	1.793	0.489	40	1.034	3.11
IH	0.949	0.289	40	0.513	1.755

- 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	1.89	0.772	40	0.827	4.316	1	1.558	0.127

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.89	0.772	40	0.827	4.316	1	1.558	0.127

- 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	1.913	0.455	40	1.182	3.095
Med	1.609	0.368	40	1.013	2.555
Post	0.721	0.156	40	0.465	1.118

Z	response	SE	df	lower.CL	upper.CL
Ant	1.913	0.455	40	1.182	3.095
Med	1.609	0.368	40	1.013	2.555
Post	0.721	0.156	40	0.465	1.118

- 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	1.189	0.219	40	0.82	1.724	1	0.94	0.353
Ant / Post	2.653	0.473	40	1.851	3.803	1	5.478	<0.001 ***
Med / Post	2.232	0.367	40	1.601	3.111	1	4.885	<0.001 ***

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.189	0.219	40	0.82	1.724	1	0.94	0.353
Ant / Post	2.653	0.473	40	1.851	3.803	1	5.478	<0.001 ***
Med / Post	2.232	0.367	40	1.601	3.111	1	4.885	<0.001 ***

- 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	2.436	0.765	40	1.291	4.596
IH	1.502	0.537	40	0.729	3.095

Condition	response	SE	df	lower.CL	upper.CL
N	2.436	0.765	40	1.291	4.596
IH	1.502	0.537	40	0.729	3.095

Condition	response	SE	df	lower.CL	upper.CL
N	2.564	0.766	40	1.402	4.691
IH	1.01	0.35	40	0.501	2.033

Condition	response	SE	df	lower.CL	upper.CL
N	2.564	0.766	40	1.402	4.691
IH	1.01	0.35	40	0.501	2.033

Condition	response	SE	df	lower.CL	upper.CL
N	0.923	0.266	40	0.515	1.653
IH	0.563	0.182	40	0.293	1.083

Condition	response	SE	df	lower.CL	upper.CL
N	0.923	0.266	40	0.515	1.653
IH	0.563	0.182	40	0.293	1.083

- 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	1.622	0.772	40	0.62	4.243	1	1.016	0.316

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.622	0.772	40	0.62	4.243	1	1.016	0.316

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	2.54	1.162	40	1.008	6.403	1	2.038	0.048 *

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	2.54	1.162	40	1.008	6.403	1	2.038	0.048 *

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.638	0.709	40	0.683	3.93	1	1.14	0.261

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.638	0.709	40	0.683	3.93	1	1.14	0.261

- 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	0.639	0.235	40	0.303	1.344	1	−1.218	0.230

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.639	0.235	40	0.303	1.344	1	−1.218	0.230

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.99	0.349	40	0.485	2.02	1	−0.028	0.977

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.99	0.349	40	0.485	2.02	1	−0.028	0.977

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.55	0.507	40	0.801	3.003	1	1.341	0.188

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.55	0.507	40	0.801	3.003	1	1.341	0.188

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

❖ Boxplot:

4.2 Density of Cleaved Caspase 3+ cells (ML and PC)

4.2.1 Data Exploration

❖ Distribution:

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	118.681	62.476	3,903.221	0.526	110.229	21.181	180.048	−0.921	−0.81	6
IH	381.893	312.177	97,454.741	0.817	593.951	70.506	792.046	0.773	−0.062	4

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	118.681	62.476	3,903.221	0.526	110.229	21.181	180.048	−0.921	−0.81	6
IH	381.893	312.177	97,454.741	0.817	593.951	70.506	792.046	0.773	−0.062	4

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	196.461	116.389	13,546.434	0.592	217.329	33.484	345.084	−0.206	−1.694	8
IH	180.429	188.765	35,632.324	1.046	272.29	20.613	505.988	1.836	3.76	5

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	196.461	116.389	13,546.434	0.592	217.329	33.484	345.084	−0.206	−1.694	8
IH	180.429	188.765	35,632.324	1.046	272.29	20.613	505.988	1.836	3.76	5

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	173.485	115.28	13,289.445	0.664	246.997	16.351	344.711	−0.075	−1.341	11
IH	246.381	234.398	54,942.373	0.951	361.759	10.49	697.585	1.119	0.684	8

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	173.485	115.28	13,289.445	0.664	246.997	16.351	344.711	−0.075	−1.341	11
IH	246.381	234.398	54,942.373	0.951	361.759	10.49	697.585	1.119	0.684	8

4.2.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Gamma (‘log’)
Gaussian (‘log’)

❖ Model call:

```{r}
glmmTMB(formula = Dens_ML_PCL ~ 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
545.01	549.37	558.91	0.45	0.07	0.41	110.71	0.73

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
545.01	549.37	558.91	0.45	0.07	0.41	110.71	0.73

❖ 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:

No potential outliers detected by the model.

❖ Model call:

```{r}
glmmTMB(formula = Dens_ML_PCL ~ 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
561.63	565.99	575.53	3.32e-05	9.40e-06	2.38e-05	100.24	119.21

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
561.63	565.99	575.53	3.32e-05	9.40e-06	2.38e-05	100.24	119.21

❖ 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:

No potential outliers detected by the model.

4.2.3 Effects Analysis

Chosen model: Gamma (‘log’)

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

4.2.3.1 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)	190.524	41.772	(123.97, 292.80)	23.944	< .001
Condition1	0.812	0.178	(0.53, 1.25)	-0.951	0.341
Z1	1.122	0.217	(0.77, 1.64)	0.593	0.553
Z2	1.006	0.187	(0.70, 1.45)	0.035	0.972
Condition1 * Z1	0.754	0.147	(0.51, 1.10)	-1.453	0.146
Condition1 * Z2	1.221	0.227	(0.85, 1.76)	1.073	0.283
Model: Dens_ML_PCL ~ Condition * Z (42 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	190.524	41.772	(123.97, 292.80)	23.944	< .001
Condition1	0.812	0.178	(0.53, 1.25)	-0.951	0.341
Z1	1.122	0.217	(0.77, 1.64)	0.593	0.553
Z2	1.006	0.187	(0.70, 1.45)	0.035	0.972
Condition1 * Z1	0.754	0.147	(0.51, 1.10)	-1.453	0.146
Condition1 * Z2	1.221	0.227	(0.85, 1.76)	1.073	0.283
Model: Dens_ML_PCL ~ Condition * Z (42 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	0.91	1	0.340
Z	0.55	2	0.760
Condition:Z	2.18	2	0.340

term	statistic	df	p.value
Condition	0.91	1	0.340
Z	0.55	2	0.760
Condition:Z	2.18	2	0.340

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	532.33	541.02	-261.17	522.33
mod_full	8	534.98	548.88	-259.49	518.98	3.35	3	0.340

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	532.33	541.02	-261.17	522.33
mod_full	8	534.98	548.88	-259.49	518.98	3.35	3	0.340

Important

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

4.2.3.2 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	154.683	44.769	40	86.18	277.64
IH	234.669	77.233	40	120.664	456.388

Condition	response	SE	df	lower.CL	upper.CL
N	154.683	44.769	40	86.18	277.64
IH	234.669	77.233	40	120.664	456.388

- 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	0.659	0.289	40	0.272	1.598	1	−0.951	0.347

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.659	0.289	40	0.272	1.598	1	−0.951	0.347

- 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	213.717	66.273	40	114.197	399.966
Med	191.754	56.022	40	106.245	346.085
Post	168.758	43.432	40	100.315	283.899

Z	response	SE	df	lower.CL	upper.CL
Ant	213.717	66.273	40	114.197	399.966
Med	191.754	56.022	40	106.245	346.085
Post	168.758	43.432	40	100.315	283.899

- 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	1.115	0.374	40	0.566	2.196	1	0.323	0.748
Ant / Post	1.266	0.408	40	0.66	2.428	1	0.733	0.468
Med / Post	1.136	0.349	40	0.611	2.114	1	0.416	0.680

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	1.115	0.374	40	0.566	2.196	1	0.323	0.748
Ant / Post	1.266	0.408	40	0.66	2.428	1	0.733	0.468
Med / Post	1.136	0.349	40	0.611	2.114	1	0.416	0.680

- 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	130.791	53.245	40	57.445	297.786
IH	349.221	162.975	40	135.98	896.866

Condition	response	SE	df	lower.CL	upper.CL
N	130.791	53.245	40	57.445	297.786
IH	349.221	162.975	40	135.98	896.866

Condition	response	SE	df	lower.CL	upper.CL
N	190.078	67.775	40	92.461	390.753
IH	193.446	89.442	40	75.984	492.486

Condition	response	SE	df	lower.CL	upper.CL
N	190.078	67.775	40	92.461	390.753
IH	193.446	89.442	40	75.984	492.486

Condition	response	SE	df	lower.CL	upper.CL
N	148.874	50.458	40	75.046	295.335
IH	191.297	73.399	40	88.09	415.424

Condition	response	SE	df	lower.CL	upper.CL
N	148.874	50.458	40	75.046	295.335
IH	191.297	73.399	40	88.09	415.424

- 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	0.375	0.232	40	0.107	1.307	1	−1.588	0.120

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.375	0.232	40	0.107	1.307	1	−1.588	0.120

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.983	0.573	40	0.302	3.195	1	−0.03	0.976

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.983	0.573	40	0.302	3.195	1	−0.03	0.976

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.778	0.396	40	0.278	2.178	1	−0.492	0.625

contrast	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.778	0.396	40	0.278	2.178	1	−0.492	0.625

- 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	0.381	0.259	40	0.096	1.508	1	−1.417	0.164

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.381	0.259	40	0.096	1.508	1	−1.417	0.164

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.481	0.303	40	0.135	1.721	1	−1.16	0.253

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.481	0.303	40	0.135	1.721	1	−1.16	0.253

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.263	0.756	40	0.376	4.236	1	0.389	0.699

Condition	ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.263	0.756	40	0.376	4.236	1	0.389	0.699

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

❖ Boxplot:

4.3 Proportion of Cleaved Caspase 3+ marking (ML and PC)

4.3.1 Data Exploration

❖ Distribution:

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.006	0.004	0	0.583	0.007	0.001	0.011	−0.524	−1.109	6
IH	0.021	0.016	0	0.765	0.031	0.004	0.04	0.259	−1.662	4

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.006	0.004	0	0.583	0.007	0.001	0.011	−0.524	−1.109	6
IH	0.021	0.016	0	0.765	0.031	0.004	0.04	0.259	−1.662	4

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.011	0.008	0	0.698	0.014	0.001	0.023	0.195	−1.202	8
IH	0.01	0.012	0	1.194	0.018	0.001	0.032	1.935	3.946	5

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.011	0.008	0	0.698	0.014	0.001	0.023	0.195	−1.202	8
IH	0.01	0.012	0	1.194	0.018	0.001	0.032	1.935	3.946	5

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.009	0.007	0	0.708	0.012	0.001	0.02	0.064	−1.157	11
IH	0.013	0.014	0	1.014	0.02	0	0.04	1.263	1.185	8

Condition	Mean	SD	Variance	CoV	IQR	Min	Max	Skewness	Kurtosis	n
N	0.009	0.007	0	0.708	0.012	0.001	0.02	0.064	−1.157	11
IH	0.013	0.014	0	1.014	0.02	0	0.04	1.263	1.185	8

4.3.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

Beta (‘logit’)
Gaussian (‘log’)

❖ Model call:

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

❖ Performance:

performance::performance(mod)

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
-280.20	-275.84	-266.30	0.61	0.14	0.55	5.81e-03	223.06

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
-280.20	-275.84	-266.30	0.61	0.14	0.55	5.81e-03	223.06

❖ 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:

No potential outliers detected by the model.

❖ Model call:

```{r}
glmmTMB(formula = Prop_C_ML_PCL ~ 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
-258.73	-254.37	-244.83	1.00	0.27	1.00	5.67e-03	6.75e-03

AIC	AICc	BIC	R2_conditional	R2_marginal	ICC	RMSE	Sigma
-258.73	-254.37	-244.83	1.00	0.27	1.00	5.67e-03	6.75e-03

❖ 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:

No potential outliers detected by the model.

4.3.3 Effects Analysis

Chosen model: Beta (‘logit’)

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

4.3.3.1 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)	0.011	0.003	(7.35e-03, 0.02)	-20.118	< .001
Condition1	0.775	0.166	(0.51, 1.18)	-1.188	0.235
Z1	0.985	0.148	(0.73, 1.32)	-0.102	0.918
Z2	1.047	0.150	(0.79, 1.39)	0.322	0.747
Condition1 * Z1	0.673	0.102	(0.50, 0.90)	-2.622	0.009 **
Condition1 * Z2	1.334	0.190	(1.01, 1.76)	2.017	0.044 *
Model: Prop_C_ML_PCL ~ Condition * Z (42 Observations)

Parameter	Coefficient	SE	95% CI	z	p
(Intercept)	0.011	0.003	(7.35e-03, 0.02)	-20.118	< .001
Condition1	0.775	0.166	(0.51, 1.18)	-1.188	0.235
Z1	0.985	0.148	(0.73, 1.32)	-0.102	0.918
Z2	1.047	0.150	(0.79, 1.39)	0.322	0.747
Condition1 * Z1	0.673	0.102	(0.50, 0.90)	-2.622	0.009 **
Condition1 * Z2	1.334	0.190	(1.01, 1.76)	2.017	0.044 *
Model: Prop_C_ML_PCL ~ Condition * Z (42 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)

term	statistic	df	p.value
Condition	1.41	1	0.230
Z	0.12	2	0.940
Condition:Z	7.14	2	0.030 *

term	statistic	df	p.value
Condition	1.41	1	0.230
Z	0.12	2	0.940
Condition:Z	7.14	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	-290.56	-281.87	150.28	-300.56
mod_full	8	-292.22	-278.32	154.11	-308.22	7.67	3	0.050 *

model	df	aic	bic	log_lik	deviance	chisq	chi_df	pr_chisq
mod_reduced	5	-290.56	-281.87	150.28	-300.56
mod_full	8	-292.22	-278.32	154.11	-308.22	7.67	3	0.050 *

Important

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

4.3.3.2 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	0.009	0.003	40	0.005	0.016
IH	0.014	0.005	40	0.008	0.027

Condition	response	SE	df	lower.CL	upper.CL
N	0.009	0.003	40	0.005	0.016
IH	0.014	0.005	40	0.008	0.027

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

❖ Contrasts:

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

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.6	0.258	40	0.252	1.43	1	−1.188	0.242

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.6	0.258	40	0.252	1.43	1	−1.188	0.242

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

❖ Boxplot:

❖ Marginal Means:

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

Z	response	SE	df	lower.CL	upper.CL
Ant	0.011	0.003	40	0.006	0.019
Med	0.012	0.003	40	0.007	0.02
Post	0.011	0.003	40	0.007	0.018

Z	response	SE	df	lower.CL	upper.CL
Ant	0.011	0.003	40	0.006	0.019
Med	0.012	0.003	40	0.007	0.02
Post	0.011	0.003	40	0.007	0.018

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

❖ Contrasts:

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

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.94	0.248	40	0.552	1.601	1	−0.233	0.817
Ant / Post	1.015	0.244	40	0.625	1.65	1	0.064	0.950
Med / Post	1.08	0.245	40	0.683	1.708	1	0.338	0.737

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
Ant / Med	0.94	0.248	40	0.552	1.601	1	−0.233	0.817
Ant / Post	1.015	0.244	40	0.625	1.65	1	0.064	0.950
Med / Post	1.08	0.245	40	0.683	1.708	1	0.338	0.737

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

❖ Boxplot:

❖ Marginal Means:

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

Condition	response	SE	df	lower.CL	upper.CL
N	0.006	0.002	40	0.003	0.013
IH	0.021	0.008	40	0.01	0.044

Condition	response	SE	df	lower.CL	upper.CL
N	0.006	0.002	40	0.003	0.013
IH	0.021	0.008	40	0.01	0.044

Condition	response	SE	df	lower.CL	upper.CL
N	0.012	0.004	40	0.006	0.024
IH	0.011	0.005	40	0.005	0.025

Condition	response	SE	df	lower.CL	upper.CL
N	0.012	0.004	40	0.006	0.024
IH	0.011	0.005	40	0.005	0.025

Condition	response	SE	df	lower.CL	upper.CL
N	0.009	0.003	40	0.005	0.018
IH	0.013	0.004	40	0.006	0.025

Condition	response	SE	df	lower.CL	upper.CL
N	0.009	0.003	40	0.005	0.018
IH	0.013	0.004	40	0.006	0.025

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

❖ Contrasts:

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

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.272	0.15	40	0.089	0.829	1	−2.361	0.023 *

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.272	0.15	40	0.089	0.829	1	−2.361	0.023 *

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.068	0.553	40	0.375	3.04	1	0.127	0.900

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.068	0.553	40	0.375	3.04	1	0.127	0.900

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.744	0.35	40	0.288	1.923	1	−0.629	0.533

contrast	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.744	0.35	40	0.288	1.923	1	−0.629	0.533

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

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

Condition	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.255	0.134	40	0.088	0.74	1	−2.591	0.013 *

Condition	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.255	0.134	40	0.088	0.74	1	−2.591	0.013 *

Condition	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.366	0.177	40	0.138	0.972	1	−2.081	0.044 *

Condition	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	0.366	0.177	40	0.138	0.972	1	−2.081	0.044 *

Condition	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.434	0.65	40	0.574	3.585	1	0.796	0.431

Condition	odds.ratio	SE	df	lower.CL	upper.CL	null	t.ratio	p.value
N / IH	1.434	0.65	40	0.574	3.585	1	0.796	0.431

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

❖ Boxplot: