Complete 2×2 Crossover Analysis for AUC and Cmax

From Subject-Level Exposure to a Bioequivalence Decision

A 2×2 crossover gives each participant both treatments, in one of two orders: test followed by reference (TR), or reference followed by test (RT). Because each participant serves as their own control, treatment comparisons use within-subject information. The analysis nevertheless has to account for period: a difference between the first and second period can influence the treatment contrast in opposite directions across the two sequences.

This tutorial follows a hypothetical, balanced study through log-transformed AUC and Cmax, the usual primary exposure measures in this setting. The worked data are invented for teaching. Assume that AUC and Cmax have already been calculated for each subject and treatment using the study’s prespecified pharmacokinetic methods. The analysis below concerns the crossover comparison of those measurements, not how to derive them from concentration-time observations.

Definition: For subject \(i\), define the treatment contrast \(d_i=\log(T_i)-\log(R_i)\), where \(T_i\) and \(R_i\) are that subject’s test and reference exposure measurements. The treatment estimate on the log scale is the sequence-adjusted estimate of the mean test-minus-reference contrast. Exponentiating it gives the geometric mean ratio (GMR), test divided by reference.

Model, Assumptions, and Estimation

For either endpoint, a standard model for the logged observation is

Y_{ijk}=\mu+\text{Sequence}_i+\text{Subject(Sequence)}_{ij} +\text{Period}_k+\text{Treatment}_{ijk}+\varepsilon_{ijk}.

Here \(Y\) is log AUC or log Cmax; sequence, period, and treatment are fixed effects; and subject is nested within sequence. The treatment contrast of interest is test minus reference. In a complete, balanced 2×2 design with \(n_1\) and \(n_2\) subjects in the TR and RT sequences, respectively, the residual degrees of freedom for the treatment comparison are \(df=n_1+n_2-2\). The estimate can be calculated directly from within-subject contrasts, as shown below.

The interpretation relies on the randomized crossover being implemented as planned, with each subject receiving both treatments and the required washout being adequate. The model assumes independent subjects, normally distributed within-sequence contrast errors, and a common variance for those errors across sequences. It also assumes that a treatment effect and a period effect can be represented additively, without a treatment-by-period interaction that changes the treatment contrast. Missing periods or a materially unbalanced analysis require a model fitted to the available data rather than the simple balanced calculations here.

Write \(\bar d_{TR}\) and \(\bar d_{RT}\) for the mean test-minus-reference contrasts within each sequence. In the TR sequence, the contrast combines treatment effect with the negative of the period-2 effect; in RT, it combines treatment effect with the positive of that effect. Thus their average estimates treatment, while their half-difference estimates period:

\hat{\tau}=\frac{\bar d_{TR}+\bar d_{RT}}{2}, \qquad \hat{\pi}=\frac{\bar d_{RT}-\bar d_{TR}}{2}.

This is also why simply treating all individual contrasts as though they were interchangeable can be misleading: their sequence matters for the period adjustment. For equal sequence sizes \(n_1=n_2=m\), estimate the common contrast variance by pooling the within-sequence sample variances:

s_p^2= \frac{(m-1)s_{TR}^2+(m-1)s_{RT}^2}{2m-2}, \qquad SE(\hat{\tau})=\sqrt{\frac{s_p^2}{2m}}.

The standard 90% confidence interval on the log scale is \(\hat{\tau}\pm t_{0.95,df}SE(\hat{\tau})\). Exponentiating both endpoints gives the confidence interval for the GMR. For the conventional 80.00%–125.00% limits, bioequivalence is concluded for an endpoint only if the entire 90% interval lies within those bounds. Both AUC and Cmax must meet the criterion for a conclusion of bioequivalence on both endpoints.

Worked Crossover Data

The table gives paired subject-level logged measurements. The entries are natural logarithms of exposure in a consistent unit; the contrast column is log test minus log reference. For each subject, the sequence identifies the order in which the two treatments were administered. Rounding in the table is for display; the calculations below use the displayed contrasts.

Sequence	Subject	Log AUC R	Log AUC T	AUC contrast	Log Cmax R	Log Cmax T	Cmax contrast
TR	01	4.00	4.08	0.08	2.10	2.02	−0.08
TR	02	4.12	4.21	0.09	2.18	2.12	−0.06
TR	03	4.05	4.15	0.10	2.05	2.01	−0.04
TR	04	4.20	4.31	0.11	2.21	2.19	−0.02
TR	05	4.15	4.27	0.12	2.13	2.13	0.00
TR	06	4.08	4.21	0.13	2.16	2.18	0.02
RT	07	4.11	4.20	0.09	2.12	2.05	−0.07
RT	08	4.06	4.16	0.10	2.20	2.15	−0.05
RT	09	4.18	4.29	0.11	2.09	2.06	−0.03
RT	10	4.09	4.21	0.12	2.15	2.14	−0.01
RT	11	4.22	4.35	0.13	2.23	2.24	0.01
RT	12	4.14	4.28	0.14	2.17	2.20	0.03

Worked Example: AUC Treatment Estimate and Confidence Interval

There are six subjects in each sequence, so \(df=6+6-2=10\). The AUC contrasts in TR sum to \(0.63\), giving \(\bar d_{TR}=0.63/6=0.105\). Those in RT sum to \(0.69\), giving \(\bar d_{RT}=0.69/6=0.115\). Therefore,

\hat{\tau}_{AUC}=(0.105+0.115)/2=0.110.

For each sequence, the deviations from its mean are \(-0.025,-0.015,-0.005,0.005,0.015,0.025\). Their squared sum is \(0.00175\), so each sequence has sample variance \(0.00175/5=0.0003500\). Pooling gives \(s_p^2=0.0003500\), and

SE=\sqrt{0.0003500/12}=0.005401.

Using \(t_{0.95,10}=1.81246\), the margin is \(1.81246(0.00540062)=0.009788\). Thus the 90% log interval is approximately \((0.10021,0.11979)\). Exponentiation gives a GMR of \(e^{0.110}=1.116\), or 111.6%, with a 90% interval of approximately \((1.105,1.127)\), or 110.5%–112.7%. The whole interval is within 80.00%–125.00%, so this hypothetical AUC result meets the bioequivalence criterion.

Worked Example: Cmax Treatment Estimate and Confidence Interval

For Cmax, the TR contrasts sum to \(-0.18\), so their mean is \(-0.030\). The RT contrasts sum to \(-0.12\), so their mean is \(-0.020\). The treatment estimate is

\hat{\tau}_{Cmax}=\frac{-0.030+(-0.020)}{2}=-0.025.

Within either sequence, deviations from that sequence’s mean are \(-0.05,-0.03,-0.01,0.01,0.03,0.05\). Their squared sum is \(0.00700\), giving a sample variance of \(0.00700/5=0.001400\); the pooled variance is also \(0.001400\). Hence \(SE=\sqrt{0.001400/12}=0.010801\). With 10 degrees of freedom, the 90% log interval is

-0.025\pm1.81246(0.01080123)=(-0.04458,-0.00542).

The GMR is \(e^{-0.025}=0.9753\), or 97.53%; the exponentiated interval is approximately \((0.9564,0.9946)\), or 95.64%–99.46%. This whole interval is within the equivalence limits, so Cmax also meets the criterion. Taken together, these hypothetical AUC and Cmax results support bioequivalence on both endpoints.

Worked Example: Separating Treatment and Period Effects

The sequence means also show how period adjustment works. For AUC, the TR mean is 0.105 and the RT mean is 0.115. The treatment estimate is their average, \(0.110\), while the estimated period-2 minus period-1 effect is

\hat{\pi}_{AUC}=(0.115-0.105)/2=0.005.

For Cmax, the corresponding sequence means are \(-0.030\) and \(-0.020\), so \(\hat{\tau}_{Cmax}=(-0.030-0.020)/2=-0.025\), and \(\hat{\pi}_{Cmax}=(-0.020-(-0.030))/2=0.005\). The treatment estimates are not either sequence mean alone: averaging cancels the opposite period contributions. This derivation depends on the balanced two-sequence design and the additive model stated above.

Diagnostics, Alternatives, and Limitations

Before interpreting a confidence interval, verify the analysis population, treatment labels, sequence assignments, and paired records. Then examine the within-sequence contrast distributions and model residuals for strong skewness, unusual spread differences, or observations that dominate the variance. With only six subjects per sequence, diagnostic plots have limited power to identify departures from the model, so they supplement rather than replace the design-based assumptions. A data error or a missing period can also invalidate the simple paired-contrast arithmetic.

A balanced crossover model and a simple paired comparison can give the same treatment estimate when sequence and period are handled correctly. The model makes those design terms explicit and supplies the residual variance and degrees of freedom for inference. In contrast, pooling all TR and RT contrasts as if they shared the same period contribution, or analyzing each period as an independent parallel group, discards the crossover structure. A parallel-group analysis is appropriate for a different design, not a substitute for accounting for within-subject pairing in this one.

The 90% confidence interval criterion is equivalent to the usual two one-sided tests at \(\alpha=0.05\): the interval must exclude values below the lower equivalence limit and above the upper limit. This does not mean there is a 90% probability that the fixed true ratio lies inside the computed interval. It means that the procedure, under its assumptions and repeated sampling, has 90% coverage. Nor does a point estimate inside the limits suffice: uncertainty matters, and an interval extending beyond either bound fails the criterion.

A major limitation is that a favorable result cannot repair a poorly conducted or unsuitable study, and these calculations do not establish equivalence for endpoints that were not analyzed. The standard limits and analysis plan should be prespecified and appropriate to the product and applicable guidance. The conclusion here is conditional on the hypothetical design, data, assumptions, and specified endpoints.

Common pitfall: Concluding bioequivalence because the GMR is between 80.00% and 125.00%. The correction is to check the entire 90% confidence interval for each required endpoint. For example, a GMR of 94% with a confidence interval of 79%–110% does not meet the criterion, because its lower endpoint is below 80%.

Check Your Understanding

Answer each question, then compare with the solutions below.

For the AUC data, what is the estimated period-2 minus period-1 effect on the log scale?
Why is a Cmax GMR of 97.53% not by itself enough to decide bioequivalence?
A hypothetical endpoint has a 90% GMR interval of 79%–110%. Does it meet the standard limits, even if its point estimate is 94%?
For a balanced study with six subjects in each sequence, what are the residual degrees of freedom for this treatment comparison?

Solutions. (1) \((0.115-0.105)/2=0.005\) log units for AUC. (2) The point estimate does not show the uncertainty; the entire 90% interval must lie within 80.00%–125.00%. Here the Cmax interval is approximately 95.64%–99.46%, which does satisfy the limits. (3) No. Although 94% is inside the limits, the lower confidence limit of 79% is below 80%. (4) \(df=6+6-2=10\).