Multinomial Regression With Sample-Level Covariates

Why Include Sample-Level Covariates?

Donors may differ in age, batch, disease severity, or other measured characteristics that are associated with cell-type composition. A model with sample-level covariates describes how the composition varies with condition while holding those covariates fixed. The observations for this analysis are donor-level cell counts, not individual cells treated as independent replicates.

Multinomial logistic regression models all cell types jointly. Its fitted probabilities are nonnegative and sum to one for each covariate profile, so the model respects the fact that the cell-type proportions form a composition. Its coefficients describe changes in relative probabilities, not changes in the total number of cells or absolute abundance in tissue.

Baseline-Category Model and Its Assumptions

Choose one cell type as the reference category; here it is cell type \(K\). Let \(x_i\) be the vector of donor-level covariates, including an intercept. For every non-reference type \(j\), the model specifies a linear predictor \(\eta_{ij}=x_i^\mathsf{T}\beta_j\), where \(\beta_j\) is the coefficient vector for that cell type relative to the reference.

The equivalent probability expressions are

The model is used under several substantive and statistical conditions. Cell types should be mutually exclusive and collectively cover the counts being modeled. Donors are the independent observational units; conditional on each donor's total and covariates, the model uses a multinomial sampling distribution for that donor's counts. The mean log ratios are assumed to be linear in the specified covariates unless transformations or interactions are included. The design must contain enough information to estimate the coefficients: severe collinearity, very sparse categories, or separation can make estimates unstable or infinite. Finally, the multinomial sampling assumption should be plausible. Extra donor-to-donor variation beyond that assumption can make model-based uncertainty too small.

The likelihood makes the role of each donor explicit. Conditional on \(n_i\), donor \(i\)'s count vector has probability proportional to \(\prod_j p_{ij}^{Y_{ij}}\), with the multinomial coefficient accounting for the number of cell label arrangements. Assuming donors are independent, the log likelihood is

The coefficient score has a useful interpretation. For a non-reference category \(j\), differentiating the donor's log likelihood with respect to a coefficient multiplying covariate \(x_{im}\) gives \(x_{im}(Y_{ij}-n_i p_{ij})\). Summing over donors gives

Thus estimation seeks coefficients for which covariate-weighted observed counts and model-expected counts are balanced. This also clarifies why a donor with more cells contributes more to the count likelihood, while remaining one independent donor rather than many independent experimental units.

Fitted Probabilities From a Model

Interpreting Covariate Coefficients and Contrasts

For a one-unit increase in a covariate, holding the other modeled covariates fixed, \(\beta_{jm}\) is the change in the log ratio \(p_{ij}/p_{iK}\). Exponentiating gives the multiplicative change in that probability ratio. It is not, in general, a multiplicative change in \(p_{ij}\) alone: changing one linear predictor changes the normalized probabilities for all cell types.

For two non-reference cell types \(j\) and \(r\), subtract their reference-category equations:

Consequently, a condition contrast for type \(j\) versus type \(r\) is the difference between their condition coefficients. This is often more informative than interpreting either coefficient in isolation. The selected reference category changes the coefficient representation, but fitted probabilities and contrasts between cell types do not depend on that choice when the model is reparameterized correctly.

Inference and the Distinction From One-Versus-Rest Models

A condition effect can be assessed for one cell type relative to the reference by testing its condition coefficient, or for a pairwise cell-type contrast by testing the relevant difference of coefficients. A global condition test asks whether condition contributes to any of the non-reference log ratios. For a binary condition and \(K\) cell types, the null hypothesis sets all \(K-1\) condition coefficients to zero. A likelihood-ratio test compares the fitted model with condition to a nested model without condition, using twice the difference in maximized log likelihoods; under standard regularity conditions, its reference distribution is chi-squared with \(K-1\) degrees of freedom. This is an association test conditional on the included covariates, not a test of a causal effect.

A separate one-versus-rest regression instead treats each selected cell type as a binary outcome: its count versus the combined count of all other types. This can be useful for a focused component-specific question, but it is not the same as fitting a joint multinomial model. Separate fits do not constrain their predicted probabilities to sum to one, and their implied odds refer to different denominators. They also do not generally reproduce the multinomial model's linear log-ratio structure.

Diagnostics, Limitations, and Common Pitfalls

Before interpreting coefficients, check that each donor has a valid count vector, with nonnegative counts and a total equal to the sum across the included cell types. Confirm that the covariates are defined at the donor level and that the condition and covariate design is estimable. Examine category counts across covariate profiles: a type absent from a condition or represented by very few donors can lead to unstable estimates or separation. Very large standard errors, coefficients that grow dramatically under small model changes, or fitted probabilities essentially equal to zero or one are warning signs.

Check whether the chosen linear form is credible for continuous covariates, and whether important interactions are needed. Compare observed donor-level counts with model-implied counts across relevant covariate ranges, rather than relying on one aggregate fit summary. If donors show more variability than the multinomial model allows, standard errors and tests based on that sampling model may be unreliable; the Dirichlet-multinomial and logistic-normal models introduced in earlier material allow additional forms of variation.

• Mistake: interpreting \(\beta_{jC}\) as a change in the proportion of type \(j\). It is a change in the log ratio of type \(j\) to the reference type. Convert predictors to probabilities for a specified covariate profile when proportion-scale summaries are needed.
• Mistake: treating every cell as an independent replicate. The model's independent units are donors, with counts grouped within donors. More cells can make a donor's composition more precisely measured, but do not create more independent donors.
• Mistake: assuming separate one-versus-rest fits are a joint composition model. Their predictions need not sum to one and their odds use distinct denominators. Use the multinomial model when the target is a coherent joint distribution over cell types.

The multinomial model is most useful when the cell categories are defined consistently, donor-level counts are adequate, and the conditional multinomial assumption is a reasonable working model. It does not by itself resolve annotation uncertainty, establish absolute tissue abundance, or remove unmeasured confounding. Its adjusted contrasts describe conditional associations in the sampled donors.