5  Caspase (IGL + WM)

❖ Data

❖ Description

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_IGL_WM

Area of IGL and WM (μm²)

Dens_IGL_WM

Density of cleaved caspase 3^(+) cells in the IGL and WM areas (Cells/μm²)

Prop_C_IGL_WM

Proportion of cleaved caspase 3^(+) marking in the IGL and WM 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_IGL_WM

Area of IGL and WM (μm²)

Dens_IGL_WM

Density of cleaved caspase 3^(+) cells in the IGL and WM areas (Cells/μm²)

Prop_C_IGL_WM

Proportion of cleaved caspase 3^(+) marking in the IGL and WM areas

❖ Correlations

5.1 Area of the IGL and WM

5.1.1 Data Exploration

❖ Distribution:

Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 4.603 1.504 2.261 0.327 2.966 2.474 6.231 −0.38 −1.525 6
IH 5.189 3.375 11.39 0.65 6.518 1.502 9.201 0.213 −1.827 4
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 4.603 1.504 2.261 0.327 2.966 2.474 6.231 −0.38 −1.525 6
IH 5.189 3.375 11.39 0.65 6.518 1.502 9.201 0.213 −1.827 4
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 3.332 1.24 1.538 0.372 1.82 1.005 4.836 −0.709 0.591 8
IH 2.402 1.562 2.439 0.65 2.72 0.89 4.871 1.14 1.189 5
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 3.332 1.24 1.538 0.372 1.82 1.005 4.836 −0.709 0.591 8
IH 2.402 1.562 2.439 0.65 2.72 0.89 4.871 1.14 1.189 5
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 1.807 1.382 1.91 0.765 2.331 0.159 4.573 1.012 −0.026 11
IH 1.418 1.191 1.418 0.84 2.331 0.184 3.317 0.775 −0.874 8
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 1.807 1.382 1.91 0.765 2.331 0.159 4.573 1.012 −0.026 11
IH 1.418 1.191 1.418 0.84 2.331 0.184 3.317 0.775 −0.874 8

5.1.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

❖ Model call:

```{r}
glmmTMB(formula = A_IGL_WM ~ 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
172.93 177.30 186.83 0.55 0.37 0.29 1.06 0.59
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
172.93 177.30 186.83 0.55 0.37 0.29 1.06 0.59

❖ Residuals:

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

❖ Predictions:

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

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

❖ Model call:

```{r}
glmmTMB(formula = A_IGL_WM ~ 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
173.40 177.76 187.30 0.23 0.13 0.11 0.94 1.12
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
173.40 177.76 187.30 0.23 0.13 0.11 0.94 1.12

❖ 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:

No potential outliers detected by the model.

5.1.3 Effects Analysis

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

5.1.3.1 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) 2.609 0.400 (1.93, 3.52) 6.246 < .001
Condition1 1.135 0.174 (0.84, 1.53) 0.827 0.408
Z1 1.840 0.281 (1.36, 2.48) 3.988 < .001
Z2 1.048 0.149 (0.79, 1.39) 0.331 0.741
Condition1 * Z1 0.918 0.140 (0.68, 1.24) -0.557 0.577
Condition1 * Z2 1.051 0.149 (0.80, 1.39) 0.350 0.726
Model: A_IGL_WM ~ Condition * Z (42 Observations)
Parameter Coefficient SE 95% CI z p
(Intercept) 2.609 0.400 (1.93, 3.52) 6.246 < .001
Condition1 1.135 0.174 (0.84, 1.53) 0.827 0.408
Z1 1.840 0.281 (1.36, 2.48) 3.988 < .001
Z2 1.048 0.149 (0.79, 1.39) 0.331 0.741
Condition1 * Z1 0.918 0.140 (0.68, 1.24) -0.557 0.577
Condition1 * Z2 1.051 0.149 (0.80, 1.39) 0.350 0.726
Model: A_IGL_WM ~ Condition * Z (42 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)
term statistic df p.value
Condition 0.68 1 0.410
Z 26.07 2 <0.001 ***
Condition:Z 0.31 2 0.860
term statistic df p.value
Condition 0.68 1 0.410
Z 26.07 2 <0.001 ***
Condition:Z 0.31 2 0.860

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")
model df aic bic log_lik deviance chisq chi_df pr_chisq
mod_reduced 5 154.85 163.54 -72.43 144.85
mod_full 8 159.55 173.45 -71.77 143.55 1.31 3 0.730
model df aic bic log_lik deviance chisq chi_df pr_chisq
mod_reduced 5 154.85 163.54 -72.43 144.85
mod_full 8 159.55 173.45 -71.77 143.55 1.31 3 0.730
Important

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

5.1.3.2 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 2.962 0.591 40 1.979 4.432
IH 2.298 0.536 40 1.433 3.683
Condition response SE df lower.CL upper.CL
N 2.962 0.591 40 1.979 4.432
IH 2.298 0.536 40 1.433 3.683 
- 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 1.289 0.396 40 0.693 2.398 1 0.827 0.413
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.289 0.396 40 0.693 2.398 1 0.827 0.413 
- 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 4.799 1.102 40 3.017 7.632
Med 2.734 0.574 40 1.79 4.178
Post 1.353 0.258 40 0.92 1.988
Z response SE df lower.CL upper.CL
Ant 4.799 1.102 40 3.017 7.632
Med 2.734 0.574 40 1.79 4.178
Post 1.353 0.258 40 0.92 1.988 
- 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 1.755 0.459 40 1.034 2.979 1 2.149 0.038 *
Ant / Post 3.547 0.897 40 2.128 5.912 1 5.009 <0.001 ***
Med / Post 2.021 0.472 40 1.261 3.24 1 3.013 0.004 **
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
Ant / Med 1.755 0.459 40 1.034 2.979 1 2.149 0.038 *
Ant / Post 3.547 0.897 40 2.128 5.912 1 5.009 <0.001 ***
Med / Post 2.021 0.472 40 1.261 3.24 1 3.013 0.004 ** 
- 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 5.004 1.476 40 2.756 9.084
IH 4.603 1.628 40 2.252 9.408
Condition response SE df lower.CL upper.CL
N 5.004 1.476 40 2.756 9.084
IH 4.603 1.628 40 2.252 9.408
Condition response SE df lower.CL upper.CL
N 3.263 0.862 40 1.913 5.566
IH 2.291 0.746 40 1.186 4.426
Condition response SE df lower.CL upper.CL
N 3.263 0.862 40 1.913 5.566
IH 2.291 0.746 40 1.186 4.426
Condition response SE df lower.CL upper.CL
N 1.591 0.391 40 0.969 2.614
IH 1.15 0.328 40 0.646 2.048
Condition response SE df lower.CL upper.CL
N 1.591 0.391 40 0.969 2.614
IH 1.15 0.328 40 0.646 2.048 
- 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 1.087 0.502 40 0.427 2.766 1 0.181 0.857
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.087 0.502 40 0.427 2.766 1 0.181 0.857
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.424 0.597 40 0.61 3.324 1 0.843 0.404
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.424 0.597 40 0.61 3.324 1 0.843 0.404
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.384 0.514 40 0.653 2.933 1 0.873 0.388
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.384 0.514 40 0.653 2.933 1 0.873 0.388 
- 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 0.763 0.403 40 0.263 2.217 1 −0.512 0.612
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.763 0.403 40 0.263 2.217 1 −0.512 0.612
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.786 0.39 40 0.288 2.145 1 −0.485 0.630
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.786 0.39 40 0.288 2.145 1 −0.485 0.630
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.029 0.471 40 0.408 2.595 1 0.063 0.950
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.029 0.471 40 0.408 2.595 1 0.063 0.950 
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

5.2 Density of Cleaved Caspase 3+ cells (IGL and WM)

5.2.1 Data Exploration

❖ Distribution:

Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 227.181 172.974 29,919.878 0.761 220.939 80.955 552.678 1.682 3.086 6
IH 325.825 146.93 21,588.571 0.451 272.213 131.967 488.71 −0.62 1.638 4
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 227.181 172.974 29,919.878 0.761 220.939 80.955 552.678 1.682 3.086 6
IH 325.825 146.93 21,588.571 0.451 272.213 131.967 488.71 −0.62 1.638 4
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 390.433 318.464 101,419.043 0.816 487.851 14.586 977.175 0.618 0.423 8
IH 164.156 86.513 7,484.486 0.527 154.148 32.438 254.465 −0.923 0.403 5
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 390.433 318.464 101,419.043 0.816 487.851 14.586 977.175 0.618 0.423 8
IH 164.156 86.513 7,484.486 0.527 154.148 32.438 254.465 −0.923 0.403 5
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 340.84 274.325 75,254.051 0.805 336.526 4.851 976.294 1.141 2.028 11
IH 339.921 290.704 84,509.032 0.855 463.775 38.013 890.515 1.004 0.421 8
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 340.84 274.325 75,254.051 0.805 336.526 4.851 976.294 1.141 2.028 11
IH 339.921 290.704 84,509.032 0.855 463.775 38.013 890.515 1.004 0.421 8

5.2.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

❖ Model call:

```{r}
glmmTMB(formula = Dens_IGL_WM ~ 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
581.95 586.31 595.85 0.49 0.02 0.48 168.86 0.74
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
581.95 586.31 595.85 0.49 0.02 0.48 168.86 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:

❖ Model call:

```{r}
glmmTMB(formula = Dens_IGL_WM ~ 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
597.23 601.60 611.13 1.14e-05 2.39e-06 8.97e-06 158.26 187.50
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
597.23 601.60 611.13 1.14e-05 2.39e-06 8.97e-06 158.26 187.50

❖ 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.

5.2.3 Effects Analysis

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

5.2.3.1 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) 271.126 68.068 (165.76, 443.48) 22.316 < .001
Condition1 0.977 0.245 (0.60, 1.60) -0.091 0.928
Z1 1.104 0.226 (0.74, 1.65) 0.484 0.628
Z2 0.936 0.182 (0.64, 1.37) -0.337 0.736
Condition1 * Z1 0.823 0.167 (0.55, 1.22) -0.960 0.337
Condition1 * Z2 1.209 0.241 (0.82, 1.79) 0.950 0.342
Model: Dens_IGL_WM ~ Condition * Z (42 Observations)
Parameter Coefficient SE 95% CI z p
(Intercept) 271.126 68.068 (165.76, 443.48) 22.316 < .001
Condition1 0.977 0.245 (0.60, 1.60) -0.091 0.928
Z1 1.104 0.226 (0.74, 1.65) 0.484 0.628
Z2 0.936 0.182 (0.64, 1.37) -0.337 0.736
Condition1 * Z1 0.823 0.167 (0.55, 1.22) -0.960 0.337
Condition1 * Z2 1.209 0.241 (0.82, 1.79) 0.950 0.342
Model: Dens_IGL_WM ~ Condition * Z (42 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)
term statistic df p.value
Condition 8.25e-03 1 0.930
Z 0.24 2 0.890
Condition:Z 1.12 2 0.570
term statistic df p.value
Condition 8.25e-03 1 0.930
Z 0.24 2 0.890
Condition:Z 1.12 2 0.570

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")
model df aic bic log_lik deviance chisq chi_df pr_chisq
mod_reduced 5 568.18 576.87 -279.09 558.18
mod_full 8 572.79 586.69 -278.39 556.79 1.40 3 0.710
model df aic bic log_lik deviance chisq chi_df pr_chisq
mod_reduced 5 568.18 576.87 -279.09 558.18
mod_full 8 572.79 586.69 -278.39 556.79 1.40 3 0.710
Important

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

5.2.3.2 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 265.012 87.914 40 135.547 518.135
IH 277.381 104.567 40 129.475 594.248
Condition response SE df lower.CL upper.CL
N 265.012 87.914 40 135.547 518.135
IH 277.381 104.567 40 129.475 594.248 
- 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 0.955 0.48 40 0.346 2.636 1 −0.091 0.928
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.955 0.48 40 0.346 2.636 1 −0.091 0.928 
- 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 299.416 101.619 40 150.792 594.526
Med 253.894 82.139 40 132.035 488.22
Post 262.173 74.904 40 147.167 467.051
Z response SE df lower.CL upper.CL
Ant 299.416 101.619 40 150.792 594.526
Med 253.894 82.139 40 132.035 488.22
Post 262.173 74.904 40 147.167 467.051 
- 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 1.179 0.421 40 0.573 2.426 1 0.462 0.647
Ant / Post 1.142 0.381 40 0.582 2.24 1 0.399 0.692
Med / Post 0.968 0.304 40 0.513 1.828 1 −0.102 0.919
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
Ant / Med 1.179 0.421 40 0.573 2.426 1 0.462 0.647
Ant / Post 1.142 0.381 40 0.582 2.24 1 0.399 0.692
Med / Post 0.968 0.304 40 0.513 1.828 1 −0.102 0.919 
- 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 240.908 107.738 40 97.569 594.828
IH 372.132 187.945 40 134.089 1,032.764
Condition response SE df lower.CL upper.CL
N 240.908 107.738 40 97.569 594.828
IH 372.132 187.945 40 134.089 1,032.764
Condition response SE df lower.CL upper.CL
N 299.99 121.355 40 132.445 679.485
IH 214.88 109.804 40 76.501 603.564
Condition response SE df lower.CL upper.CL
N 299.99 121.355 40 132.445 679.485
IH 214.88 109.804 40 76.501 603.564
Condition response SE df lower.CL upper.CL
N 257.536 97.101 40 120.197 551.8
IH 266.893 113.757 40 112.777 631.617
Condition response SE df lower.CL upper.CL
N 257.536 97.101 40 120.197 551.8
IH 266.893 113.757 40 112.777 631.617 
- 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 0.647 0.434 40 0.167 2.509 1 −0.649 0.520
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.647 0.434 40 0.167 2.509 1 −0.649 0.520
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.396 0.916 40 0.37 5.261 1 0.508 0.614
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.396 0.916 40 0.37 5.261 1 0.508 0.614
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.965 0.547 40 0.307 3.033 1 −0.063 0.950
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.965 0.547 40 0.307 3.033 1 −0.063 0.950 
- 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 0.464 0.337 40 0.107 2.012 1 −1.058 0.296
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.464 0.337 40 0.107 2.012 1 −1.058 0.296
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.671 0.43 40 0.184 2.451 1 −0.623 0.537
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.671 0.43 40 0.184 2.451 1 −0.623 0.537
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.447 0.911 40 0.405 5.164 1 0.587 0.561
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.447 0.911 40 0.405 5.164 1 0.587 0.561 
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot:

5.3 Proportion of Cleaved Caspase 3+ marking (IGL and WM)

5.3.1 Data Exploration

❖ Distribution:

Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 0.012 0.009 0 0.754 0.014 0.004 0.029 1.215 1.225 6
IH 0.019 0.008 0 0.424 0.014 0.007 0.025 −1.648 3.083 4
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 0.012 0.009 0 0.754 0.014 0.004 0.029 1.215 1.225 6
IH 0.019 0.008 0 0.424 0.014 0.007 0.025 −1.648 3.083 4
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 0.021 0.017 0 0.793 0.032 0.001 0.047 0.186 −1.178 8
IH 0.009 0.005 0 0.571 0.008 0.002 0.015 −0.334 0.9 5
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 0.021 0.017 0 0.793 0.032 0.001 0.047 0.186 −1.178 8
IH 0.009 0.005 0 0.571 0.008 0.002 0.015 −0.334 0.9 5
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 0.019 0.018 0 0.948 0.018 0 0.067 1.911 4.988 11
IH 0.019 0.018 0 0.94 0.028 0.001 0.055 1.185 0.88 8
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 0.019 0.018 0 0.948 0.018 0 0.067 1.911 4.988 11
IH 0.019 0.018 0 0.94 0.028 0.001 0.055 1.185 0.88 8

5.3.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

❖ Model call:

```{r}
glmmTMB(formula = Prop_C_IGL_WM ~ 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
-247.33 -242.96 -233.42 0.68 0.08 0.65 9.04e-03 167.66
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
-247.33 -242.96 -233.42 0.68 0.08 0.65 9.04e-03 167.66

❖ Residuals:

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

❖ Predictions:

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

♦ Simulated data vs observed data:

♦ Simulated statistics vs observed ones:

❖ Potential outliers:

❖ Model call:

```{r}
glmmTMB(formula = Prop_C_IGL_WM ~ 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
-221.84 -217.48 -207.94 1.00 0.21 1.00 9.10e-03 0.01
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
-221.84 -217.48 -207.94 1.00 0.21 1.00 9.10e-03 0.01

❖ 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.

5.3.3 Effects Analysis

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

5.3.3.1 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) 0.016 0.004 (0.01, 0.03) -16.517 < .001
Condition1 0.926 0.224 (0.58, 1.49) -0.319 0.750
Z1 0.978 0.143 (0.73, 1.30) -0.155 0.877
Z2 0.959 0.139 (0.72, 1.27) -0.289 0.772
Condition1 * Z1 0.691 0.103 (0.52, 0.93) -2.467 0.014 *
Condition1 * Z2 1.372 0.200 (1.03, 1.82) 2.172 0.030 *
Model: Prop_C_IGL_WM ~ Condition * Z (42 Observations)
Parameter Coefficient SE 95% CI z p
(Intercept) 0.016 0.004 (0.01, 0.03) -16.517 < .001
Condition1 0.926 0.224 (0.58, 1.49) -0.319 0.750
Z1 0.978 0.143 (0.73, 1.30) -0.155 0.877
Z2 0.959 0.139 (0.72, 1.27) -0.289 0.772
Condition1 * Z1 0.691 0.103 (0.52, 0.93) -2.467 0.014 *
Condition1 * Z2 1.372 0.200 (1.03, 1.82) 2.172 0.030 *
Model: Prop_C_IGL_WM ~ Condition * Z (42 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)
term statistic df p.value
Condition 0.10 1 0.750
Z 0.26 2 0.880
Condition:Z 6.73 2 0.030 *
term statistic df p.value
Condition 0.10 1 0.750
Z 0.26 2 0.880
Condition:Z 6.73 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 -259.02 -250.34 134.51 -269.02
mod_full 8 -259.04 -245.14 137.52 -275.04 6.02 3 0.110
model df aic bic log_lik deviance chisq chi_df pr_chisq
mod_reduced 5 -259.02 -250.34 134.51 -269.02
mod_full 8 -259.04 -245.14 137.52 -275.04 6.02 3 0.110
Important

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

5.3.3.2 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 0.015 0.005 40 0.008 0.029
IH 0.017 0.006 40 0.008 0.036
Condition response SE df lower.CL upper.CL
N 0.015 0.005 40 0.008 0.029
IH 0.017 0.006 40 0.008 0.036 
- 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 0.857 0.415 40 0.322 2.281 1 −0.319 0.752
contrast odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.857 0.415 40 0.322 2.281 1 −0.319 0.752 
- 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 0.016 0.005 40 0.009 0.029
Med 0.016 0.004 40 0.009 0.028
Post 0.017 0.004 40 0.01 0.029
Z response SE df lower.CL upper.CL
Ant 0.016 0.005 40 0.009 0.029
Med 0.016 0.004 40 0.009 0.028
Post 0.017 0.004 40 0.01 0.029 
- 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 1.019 0.267 40 0.6 1.732 1 0.073 0.942
Ant / Post 0.916 0.214 40 0.572 1.468 1 −0.374 0.710
Med / Post 0.899 0.206 40 0.566 1.429 1 −0.464 0.645
contrast odds.ratio SE df lower.CL upper.CL null t.ratio p.value
Ant / Med 1.019 0.267 40 0.6 1.732 1 0.073 0.942
Ant / Post 0.916 0.214 40 0.572 1.468 1 −0.374 0.710
Med / Post 0.899 0.206 40 0.566 1.429 1 −0.464 0.645 
- 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 0.01 0.004 40 0.004 0.023
IH 0.025 0.01 40 0.011 0.056
Condition response SE df lower.CL upper.CL
N 0.01 0.004 40 0.004 0.023
IH 0.025 0.01 40 0.011 0.056
Condition response SE df lower.CL upper.CL
N 0.02 0.007 40 0.01 0.039
IH 0.012 0.006 40 0.005 0.031
Condition response SE df lower.CL upper.CL
N 0.02 0.007 40 0.01 0.039
IH 0.012 0.006 40 0.005 0.031
Condition response SE df lower.CL upper.CL
N 0.017 0.006 40 0.008 0.033
IH 0.018 0.007 40 0.008 0.038
Condition response SE df lower.CL upper.CL
N 0.017 0.006 40 0.008 0.033
IH 0.018 0.007 40 0.008 0.038 
- 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 0.41 0.243 40 0.124 1.355 1 −1.508 0.140
contrast odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.41 0.243 40 0.124 1.355 1 −1.508 0.140
contrast odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.613 0.929 40 0.504 5.165 1 0.83 0.411
contrast odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.613 0.929 40 0.504 5.165 1 0.83 0.411
contrast odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.953 0.487 40 0.339 2.675 1 −0.095 0.925
contrast odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.953 0.487 40 0.339 2.675 1 −0.095 0.925 
- 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 0.254 0.135 40 0.087 0.745 1 −2.575 0.014 *
Condition odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.254 0.135 40 0.087 0.745 1 −2.575 0.014 *
Condition odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.43 0.204 40 0.165 1.121 1 −1.781 0.083
Condition odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.43 0.204 40 0.165 1.121 1 −1.781 0.083
Condition odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.693 0.776 40 0.67 4.277 1 1.148 0.258
Condition odds.ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 1.693 0.776 40 0.67 4.277 1 1.148 0.258 
- 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: