• Immune responses vary a lot across participants, even in the same study.
• A biologically interesting difference is not enough; we also need an estimate of uncertainty.
• Immunology studies often compare vaccine formulations, adjuvants, doses, or treatment groups.
• In this example, the question is: Does a TLR-agonist adjuvant increase antibody response at day 28 compared with alum?

• Design: simulated vaccine immunogenicity study with 120 participants.
• Groups: alum control vs. TLR-agonist adjuvant.
• Continuous outcome: log2 fold change in antibody titer from baseline to day 28.
• Binary outcome: seroconversion, defined as a 4-fold or greater increase in titer.
• Covariate: age, because immune response can decrease with age.

• The TLR group shows a higher center and more values above a 4-fold increase threshold.
• There is still substantial subject-to-subject variability, which is why estimation and testing are useful.

We estimate the group effect with the difference in mean log2 fold change:

\[ \hat{\Delta} = \bar{Y}_{\mathrm{TLR}} - \bar{Y}_{\mathrm{Alum}} \]

A 95% confidence interval is

\[ \hat{\Delta} \pm t^* \cdot SE(\hat{\Delta}). \]

• Estimated mean difference: 0.65 log2 units.
• 95% CI: (0.36, 0.93).
• Interpretation: because the interval is entirely above 0, the TLR adjuvant is associated with a stronger antibody response in this study.

We test

\[ H_0: \mu_{\mathrm{TLR}} - \mu_{\mathrm{Alum}} = 0 \qquad \text{vs.} \qquad H_A: \mu_{\mathrm{TLR}} - \mu_{\mathrm{Alum}} \ne 0. \]

Using a linear model, the test statistic is

\[ t = \frac{\hat{\Delta}}{SE(\hat{\Delta})}. \]

• Observed test statistic: 4.52.
• p-value: 1.51^{-5}.
• Since the p-value is very small, we reject the null hypothesis and conclude that the adjuvant groups differ in mean antibody response.
• The p-value answers a probability question under the null model; it is not the probability that the null hypothesis is true.

Now we model the probability of seroconversion while adjusting for age:

\[ \log\left(\frac{P(\text{response}=1)}{1-P(\text{response}=1)}\right) = \beta_0 + \beta_1 I(\text{TLR}) + \beta_2 \cdot \text{Age}. \]

• Estimated odds ratio for TLR vs. alum: 8.52.
• Approximate 95% CI for the odds ratio: (1.01, 71.68).

• This plot helps visualize multivariable structure that is harder to see in a 2D figure.
• In immunology, interactive graphics are useful for exploring response heterogeneity.

ggplot(dat, aes(x = group, y = log2_fc, fill = group)) +
  geom_violin(alpha = 0.28, color = NA, width = 0.9) +
  geom_boxplot(width = 0.18, outlier.shape = NA, alpha = 0.65) +
  geom_jitter(width = 0.08, alpha = 0.55, size = 2) +
  labs(
    title = "Day-28 antibody response is higher in the TLR-adjuvanted group",
    x = "Adjuvant group",
    y = "Log2 fold change in antibody titer"
  ) +
  theme_minimal(base_size = 14) +
  theme(legend.position = "none")

• This code creates the first ggplot shown in the presentation.
• Because the entire analysis is inside one .Rmd file, the report is easy to reproduce and update.

• Statistics helps convert biological variation into quantitative evidence.
• In this simulated immunology study, the TLR adjuvant produced a larger average antibody response than alum.
• Point estimates, confidence intervals, and p-values answer different questions and should be interpreted together.
• A regression model extends the analysis by adjusting for age and estimating seroconversion probability.
• The same workflow can be adapted to cytokine data, flow cytometry summaries, neutralization titers, or survival outcomes in immunology.