Worked Log-Ratio Contrast for Three Cell Types

Why Prespecify a Three-Type Contrast?

A composition records relative shares: if three cell types exhaust the categories being analyzed, their proportions sum to one. An increase in one share therefore changes the shares available to the others, even when no absolute cell count has been measured. A condition comparison is easier to interpret when it specifies in advance which relative feature of the composition is the target.

Here the target is the balance of cell type A against cell types B and C together, with B and C weighted equally on the logarithmic scale. This is a single, prespecified contrast, not three separate tests of proportions. The estimate describes a relative change in the composition; it does not by itself establish a change in absolute cell numbers.

Definition: For positive proportions $p_A,p_B,p_C$ with $p_A+p_B+p_C=1$, define the log-ratio coordinate $$ z=\log\left(\frac{p_A}{\sqrt{p_Bp_C}}\right) =\log(p_A)-\frac{1}{2}\log(p_B)-\frac{1}{2}\log(p_C). $$ This is a contrast because the log-proportion weights $1,-1/2,-1/2$ sum to zero. A larger $z$ means that A is larger relative to the geometric mean of B and C.

Model and Interpretation

Let $Y_i$ indicate the condition for biological sample $i$, with $Y_i=0$ for the reference condition and $Y_i=1$ for the comparison condition. Let $z_i$ be the prespecified log-ratio coordinate calculated for that sample. A simple condition model is

z_i=\beta_0+\beta_1Y_i+\varepsilon_i.

In this model, $\beta_0$ is the mean coordinate in the reference condition and $\beta_1$ is the comparison-minus-reference condition effect. Thus, a positive $\beta_1$ indicates a shift toward A relative to the geometric mean of B and C. Exponentiating gives $\exp(\beta_1)$, the multiplicative change in the ratio $p_A/\sqrt{p_Bp_C}$ between conditions. It is not generally the fold change in $p_A$ alone.

A confidence interval for $\beta_1$ is on the log-ratio scale. Exponentiating both endpoints gives an interval for the multiplicative change in the balance. This transformation preserves endpoint order because the exponential function is increasing. The usual model-based interval requires appropriate independent biological sampling units, a suitable mean model, and a standard error that reflects the sampling variability of those units. For a small-sample $t$ interval, the relevant degrees of freedom and $t$ critical value replace the normal critical value used below.

Assumptions to check: The three categories must be defined consistently across conditions, and the proportions used to calculate $z_i$ must be positive. The condition effect must be estimable rather than confounded with another design feature. The model-based interval also relies on the chosen model and standard error being adequate for the sample-level variation; treating individual cells as independent replicates would generally understate uncertainty.

Worked Example: Calculating the Contrast From One Composition

Suppose a hypothetical reference-condition sample has proportions $p_A=0.20$, $p_B=0.30$, and $p_C=0.50$. They are positive and sum to one, so the log ratio is defined.

Substitute into the definition:

z=\log\left(\frac{0.20}{\sqrt{0.30(0.50)}}\right) =\log\left(\frac{0.20}{\sqrt{0.15}}\right) =\log(0.5164) \approx -0.6609.

The negative coordinate says that A is below the geometric mean of B and C on this relative scale. It does not say that A is absent or that either B or C has increased in absolute number. The calculation also illustrates why the contrast is not the ordinary log odds of A versus all remaining types: its denominator is the geometric mean of B and C.

Worked Condition Effect and Uncertainty Interval

The coefficient and standard error can come from a fitted sample-level model. The following hypothetical summary is used to show the interval calculation; it is not a report from a real study. Suppose the estimated condition effect is $\hat{\beta}_1=0.5272$, with $SE=0.1800$. Use a two-sided 95% normal-based interval for this illustration. This choice assumes the sampling distribution of the estimator is adequately approximated by a normal distribution and that the reported standard error is valid.

Worked Example: Estimating the Relative Balance Change

The 95% interval on the log-ratio scale is the estimate plus or minus 1.96 standard errors:

0.5272\ \pm\ 1.96(0.1800) =0.5272\ \pm\ 0.3528 =[0.1744,\ 0.8800].

The estimated change is positive. Exponentiating the estimate gives the estimated multiplicative change in the A-versus-geometric-mean balance:

\exp(0.5272)\approx1.694.

Thus, under the model, the comparison condition has an estimated balance about 1.694 times the reference-condition balance, or about 69.4% higher. Exponentiating the interval endpoints gives

[\exp(0.1744),\exp(0.8800)] \approx[1.191,\ 2.411].

The interval is compatible with a multiplicative balance increase between about 1.191-fold and 2.411-fold. In repeated use of this interval procedure under its assumptions, 95% of intervals constructed in this way would cover the true condition effect. This is not a 95% probability statement about the fixed effect in this particular interval.

Translating the Coordinate Into Predicted Proportions

A log-ratio effect alone does not determine all three proportions. One additional feature of the composition must be specified. For a transparent illustration, hold the ratio $r=p_B/p_C$ fixed at its reference value, and use the reference composition $(0.20,0.30,0.50)$. This means $r=0.30/0.50=0.6$. The resulting proportions are conditional predictions under this fixed-split assumption, not a general consequence of the estimated contrast.

Write $q=p_A$. When $p_B/p_C=r$ and the proportions sum to one, $p_B=(1-q)r/(1+r)$ and $p_C=(1-q)/(1+r)$. Substitution into the contrast gives an inverse formula:

z=\log\left(\frac{q(1+r)}{(1-q)\sqrt{r}}\right), \qquad q=\frac{\exp(z)\sqrt{r}}{1+r+\exp(z)\sqrt{r}}.

Once $q$ is obtained, the other proportions follow from the fixed split: $p_B=(1-q)r/(1+r)$ and $p_C=(1-q)/(1+r)$. The derivation below shows why that inverse is valid.

Justification of the inverse formula: Since $p_B=(1-q)r/(1+r)$ and $p_C=(1-q)/(1+r)$, their geometric mean is $(1-q)\sqrt{r}/(1+r)$. Therefore $\exp(z)=q(1+r)/((1-q)\sqrt{r})$. Rearranging gives $q/(1-q)=\exp(z)\sqrt{r}/(1+r)$. Solving this equation for $q$ yields the stated formula. The remaining proportions sum to $1-q$ and have ratio $r$, so the reconstructed composition sums to one.

Worked Example: Predicting Proportions at the Estimated Effect

The reference coordinate for $(0.20,0.30,0.50)$ is $z_0\approx-0.6609$. The fitted effect is $0.5272$, so the predicted comparison-condition coordinate is

z_1=z_0+\hat{\beta}_1=-0.6609+0.5272=-0.1337.

Keep $r=0.6$ fixed. Using the inverse formula, the predicted A proportion is

q=\frac{\exp(-0.1337)\sqrt{0.6}}{1.6+\exp(-0.1337)\sqrt{0.6}} \approx0.2975.

The remaining share is $1-0.2975=0.7025$. Dividing it according to the fixed B-to-C ratio gives

p_B=0.7025\left(\frac{0.6}{1.6}\right)\approx0.2634, \qquad p_C=0.7025\left(\frac{1}{1.6}\right)\approx0.4391.

The predicted composition is approximately $(29.75\%,26.34\%,43.91\%)$, which sums to 100.00% after rounding. Relative to the reference composition, A's predicted share rises from 20% to about 29.75% under this particular fixed-split scenario. That share change is not the same as the estimated 69.4% increase in the log-ratio balance.

Worked Example: Mapping the Confidence Limits to Compositions

For the lower effect limit, the corresponding coordinate is $z=-0.6609+0.1744=-0.4865$. Applying the inverse formula with $r=0.6$ gives $p_A\approx0.2294$, then $p_B=0.7706(0.6/1.6)\approx0.2890$ and $p_C=0.7706(1/1.6)\approx0.4816$.

For the upper effect limit, the coordinate is $z=-0.6609+0.8800=0.2191$. Then

p_A=\frac{\exp(0.2191)\sqrt{0.6}}{1.6+\exp(0.2191)\sqrt{0.6}} \approx0.3761.

The other two shares are $p_B=0.6239(0.6/1.6)\approx0.2340$ and $p_C=0.6239(1/1.6)\approx0.3899$. Thus, under the fixed B:C ratio assumption, the endpoint compositions are approximately $(22.94\%,28.90\%,48.16\%)$ and $(37.61\%,23.40\%,38.99\%)$. They are transformed confidence limits under this specified mapping, not a joint confidence region for all three proportions.

Diagnostics, Alternatives, and Limits

Check the analysis at both the model and interpretation stages. Verify that each sample's proportions sum to one, that the category definitions are stable, and that zero proportions have not been inserted into a logarithm. Inspect sample-level coordinates by condition for outliers, unequal spread, and patterns suggesting that a simple condition-only mean model is inadequate. Assess whether the standard error reflects biological-sample variation and whether the condition effect can be separated from batch or other design features. A narrow interval based on many cells but very few biological samples is not reassuring if the cells were treated as independent replicates.

The prespecified contrast has an advantage over examining many component-wise tests: it gives one explicit, interpretable target and avoids selecting the contrast after seeing which result looks strongest. It is not the only possible comparison. A pairwise log ratio, such as $\log(p_A/p_B)$, focuses directly on A relative to B but ignores C in its definition. A contrast of A against the arithmetic sum $p_B+p_C$ answers a different question. Component-wise abundance models target changes in particular proportions and require attention to the dependence induced by the sum-to-one constraint. These quantities should not be treated as interchangeable.

Predicted proportions require extra care. The log-ratio estimate specifies a change in one compositional coordinate, but infinitely many compositions can share that coordinate. Holding B:C fixed is one interpretable choice; another model could estimate changes in the other balance as well. Report the assumption used for any back-transformation, and do not present one conditional reconstruction as uniquely implied by the contrast. Finally, as emphasized in the earlier material on relative abundance versus absolute cell number, a relative shift alone cannot establish that the absolute number of A cells increased or decreased.

Common pitfall: “The balance increased 69.4%, so the proportion of A increased 69.4%.” This confuses a multiplicative change in $p_A/\sqrt{p_Bp_C}$ with a change in $p_A$ itself. The balance depends on all three components. To give predicted proportions, state an additional constraint—such as holding B:C fixed—and apply the inverse formula; otherwise report the effect on the log-ratio scale.

Check Your Understanding

Answer each question, then compare with the brief solutions below.

If the condition coefficient for $z$ is negative, what does that say about A relative to B and C?
For an estimate of $0.30$ with $SE=0.10$, calculate a 95% normal-based interval on the log-ratio scale.
Why does a log-ratio effect not by itself identify a unique three-part predicted composition?
With a fixed $p_B/p_C=1$, what is the predicted composition if $z=0$?
Can an increase in the relative balance establish an increase in the absolute number of A cells? Explain.

Solutions. (1) A is lower relative to the geometric mean of B and C in the comparison condition. (2) $0.30\pm1.96(0.10)=[0.104,0.496]$. (3) It constrains one relative coordinate, while the composition has two independent dimensions; an additional condition on the B:C split or another balance is needed. (4) With equal B and C, $r=1$; $q=\exp(0)\sqrt{1}/(2+\exp(0)\sqrt{1})=1/3$, so the composition is $(1/3,1/3,1/3)$. (5) No. Relative proportions can shift because of changes in any component or in the total; absolute cell counts require information beyond the composition.