7  BrDU+ cells

❖ Data

❖ Description

Variable Description
Mouse

Mouse unique identifier

Layer

Cerebellar layer

Stage

Developmental stage

Condition

Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)

Dens_BrDU

Density of BrDU^(+) cells (10^(-5) cells/μm^(3))

Variable Description
Mouse

Mouse unique identifier

Layer

Cerebellar layer

Stage

Developmental stage

Condition

Hypoxia condition: Normoxia (N) vs Intermittent Hypoxia (IH)

Dens_BrDU

Density of BrDU^(+) cells (10^(-5) cells/μm^(3))

❖ Correlations

7.1 Density of BrDU+ cells

7.1.1 Data Exploration

❖ Distribution:

Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 3.266 2.028 4.112 0.621 3.961 0.288 6.074 0.133 −1.554 15
IH 7.008 2.52 6.351 0.36 2.927 4.162 12.744 1.129 0.283 18
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 3.266 2.028 4.112 0.621 3.961 0.288 6.074 0.133 −1.554 15
IH 7.008 2.52 6.351 0.36 2.927 4.162 12.744 1.129 0.283 18
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 4.178 2.201 4.843 0.527 1.787 1.807 9.038 1.501 1.716 15
IH 4.843 2.216 4.911 0.458 1.931 2.088 11.005 1.583 2.632 18
Condition Mean SD Variance CoV IQR Min Max Skewness Kurtosis n
N 4.178 2.201 4.843 0.527 1.787 1.807 9.038 1.501 1.716 15
IH 4.843 2.216 4.911 0.458 1.931 2.088 11.005 1.583 2.632 18

7.1.2 Models & Diagnostics

Exploring some Generalized Linear (Mixed) model candidates:

❖ Model call:

```{r}
glmmTMB(formula = Dens_BrDU ~ Condition * Layer + (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
305.46 306.88 318.60 0.42 0.24 0.24 1.86 0.45
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
305.46 306.88 318.60 0.42 0.24 0.24 1.86 0.45

❖ 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_BrDU ~ Condition * Layer + (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
308.23 309.65 321.37 0.03 0.02 0.01 1.86 1.99
AIC AICc BIC R2_conditional R2_marginal ICC RMSE Sigma
308.23 309.65 321.37 0.03 0.02 0.01 1.86 1.99

❖ 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:

7.1.3 Effects Analysis

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

7.1.3.1 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) 3.504 0.534 (2.60, 4.72) 8.220 < .001
Condition2 1.588 0.325 (1.06, 2.37) 2.259 0.024 *
Layer1 0.869 0.073 (0.74, 1.03) -1.664 0.096
Condition2 * Layer1 1.394 0.158 (1.12, 1.74) 2.930 0.003 **
Model: Dens_BrDU ~ Condition * Layer (66 Observations)
Parameter Coefficient SE 95% CI z p
(Intercept) 3.504 0.534 (2.60, 4.72) 8.220 < .001
Condition2 1.588 0.325 (1.06, 2.37) 2.259 0.024 *
Layer1 0.869 0.073 (0.74, 1.03) -1.664 0.096
Condition2 * Layer1 1.394 0.158 (1.12, 1.74) 2.930 0.003 **
Model: Dens_BrDU ~ Condition * Layer (66 Observations)

❖ Main effects (Wald Chi-Square):

car::Anova(mod, type = 3)
term statistic df p.value
Condition 5.10 1 0.020 *
Layer 2.77 1 0.100
Condition:Layer 8.59 1 0.003 **
term statistic df p.value
Condition 5.10 1 0.020 *
Layer 2.77 1 0.100
Condition:Layer 8.59 1 0.003 **

❖ Main effects (Likelihood Ratio Test):

LRT(mod, pred = "Condition")
model df aic bic log_lik deviance chisq chi_df pr_chisq
mod_reduced 4 303.79 312.55 -147.89 295.79
mod_full 6 294.64 307.77 -141.32 282.64 13.15 2 0.001 **
model df aic bic log_lik deviance chisq chi_df pr_chisq
mod_reduced 4 303.79 312.55 -147.89 295.79
mod_full 6 294.64 307.77 -141.32 282.64 13.15 2 0.001 **
Important

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

7.1.3.2 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 3.504 0.534 64 2.583 4.752
IH 5.563 0.763 64 4.23 7.317
Condition response SE df lower.CL upper.CL
N 3.504 0.534 64 2.583 4.752
IH 5.563 0.763 64 4.23 7.317 
- Results are averaged over the levels of: Layer
- 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 0.63 0.129 64 0.418 0.948 1 −2.259 0.027 *
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.63 0.129 64 0.418 0.948 1 −2.259 0.027 * 
- Results are averaged over the levels of: Layer
- 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")
Layer response SE df lower.CL upper.CL
EGL 4.529 0.532 64 3.581 5.728
IGL 4.304 0.504 64 3.405 5.439
Layer response SE df lower.CL upper.CL
EGL 4.529 0.532 64 3.581 5.728
IGL 4.304 0.504 64 3.405 5.439 
- 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
EGL / IGL 1.052 0.119 64 0.839 1.32 1 0.451 0.654
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
EGL / IGL 1.052 0.119 64 0.839 1.32 1 0.451 0.654 
- 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 3.044 0.532 64 2.146 4.317
IH 6.738 1.054 64 4.93 9.209
Condition response SE df lower.CL upper.CL
N 3.044 0.532 64 2.146 4.317
IH 6.738 1.054 64 4.93 9.209
Condition response SE df lower.CL upper.CL
N 4.032 0.701 64 2.85 5.706
IH 4.593 0.72 64 3.358 6.282
Condition response SE df lower.CL upper.CL
N 4.032 0.701 64 2.85 5.706
IH 4.593 0.72 64 3.358 6.282 
- 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 0.452 0.106 64 0.283 0.721 1 −3.393 0.001 **
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.452 0.106 64 0.283 0.721 1 −3.393 0.001 **
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.878 0.205 64 0.55 1.4 1 −0.557 0.580
contrast ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.878 0.205 64 0.55 1.4 1 −0.557 0.580 
- 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 0.515 0.117 64 0.327 0.809 1 −2.93 0.005 **
Condition ratio SE df lower.CL upper.CL null t.ratio p.value
N / IH 0.515 0.117 64 0.327 0.809 1 −2.93 0.005 ** 
- Confidence level used: 0.95
- Intervals are back-transformed from the log scale
- Tests are performed on the log scale

❖ Boxplot: