Day 28 Crossover Designs II

July 22nd, 2025

28.1 Announcements

  • Project due on Wednesday to send for peer review.
  • The Difference Between “Significant” and “Not Significant” is not Itself Statistically Significant - Gelman and Stern (2012)
  • Homework due tomorrow, 7am: write a question in this Google Docs, or answer a question someone made. Make sure you have your track changes on, or leave your last name in a comment.

28.2 Crossover designs

Baseline:

  • Apply treatments to the same experimental unit sequentially, to eliminate between-experimental unit variation when comparing treatments
  • Fewer EUs (probably living beings) are required than in non-crossover designs
  • Power (?)
  • Between-EU variability is accounted for in the model
  • Sources of variability:
    • Treatment ( \(\geq\) 2)
    • Period ( \(\geq\) 2)
    • Sequence ( \(\geq\) 2)
    • Carryover (residual) effects
    • Between-individuals
  • The EUs are randomized to sequences
  • What is the difference between crossover designs capturing between-individuals variability, and subsampling?

Overall model:

\[y_{ijklm} = \mu + T_i + P_j + S_k +C_{l} + u_m + \varepsilon_{ijklm}, \\ u_{m} \sim N(0, \sigma^2_u), \\ \varepsilon_{ijklm} \sim N(0, \sigma^2_\varepsilon),\] where:

  • \(y_{ijk}\) is the observation for the \(i\)th treatment, \(j\)th period with the \(m\)th previous treatment and the \(m\)th individual that received the treatments in the \(k\)th sequence,
  • \(T_i\) is the effect of the \(i\)th treatment,
  • \(P_j\) is the effect of the \(j\)th period,
  • \(S_k\) is the effect of the \(k\)th sequence,
  • \(C_{l}\) is the carryover effect of the \(l\)th treatment,
  • \(u_{mk}\) is the (random) effect of the \(m\)th individual under the \(k\)th sequence, and \(\varepsilon_{ijkl}\) is the residual.

Considerations:

Carryover effects:

  • Experiment designs may be better or worsely designed to separate carryover effect from treatment effect.
    • Type of treatment, adequate time between treatments (“wash out” period)
    • Some treatments may damage the individual for indefinite time (e.g., damaging the liver)

Repeated measures:

  • When \(P > 2\), the several measurements on the same individual assume a compound symmetry correlation function.
  • Other types of correlation functions assume that the correlation “wears out” with time (e.g., AR(1) correlation function)
Schematic representation of a 2x2 crossover design

Figure 28.1: Schematic representation of a 2x2 crossover design

28.3 Applied example

6 cows were used to test 3 different treatments in a crossover design.

  • Repeated measures. Now we are assuming compound symmetry (a specific type of correlation function).
  • How does carryover affect power?
url <- "https://raw.githubusercontent.com/stat720/summer2025/refs/heads/main/data/cow_prod_cox.csv"
df <- read.csv(url)
df
##    Cow Sequence Period Treatment Prior_trt milk_production
## 1    1      ABC      1         A         O              38
## 2    1      ABC      2         B         A              25
## 3    1      ABC      3         C         B              15
## 4    2      BCA      1         B         O             109
## 5    2      BCA      2         C         B              86
## 6    2      BCA      3         A         C              39
## 7    3      CAB      1         C         O             124
## 8    3      CAB      2         A         C              72
## 9    3      CAB      3         B         A              27
## 10   4      ACB      1         A         O              86
## 11   4      ACB      2         C         A              76
## 12   4      ACB      3         B         C              46
## 13   5      BAC      1         B         O              75
## 14   5      BAC      2         A         B              35
## 15   5      BAC      3         C         A              34
## 16   6      CBA      1         C         O             101
## 17   6      CBA      2         B         C              63
## 18   6      CBA      3         A         B               1
library(lme4)
library(emmeans)
m_milk.nocarryover <- lmer(milk_production ~ Sequence + Period + Treatment + (1|Sequence:Cow),
               data = df)
emmeans(m_milk.nocarryover, ~ Treatment )
##  Treatment emmean   SE   df lower.CL upper.CL
##  A           45.2 4.65 17.6     35.4     54.9
##  B           57.5 4.65 17.6     47.7     67.3
##  C           72.7 4.65 17.6     62.9     82.4
## 
## Results are averaged over the levels of: Sequence 
## Degrees-of-freedom method: kenward-roger 
## Confidence level used: 0.95
m_milk.carryover <- lmer(milk_production ~ Sequence + Period + Treatment + Prior_trt + (1|Sequence:Cow),
               data = df)

car::Anova(m_milk.carryover, test.statistic = "F")
## Analysis of Deviance Table (Type II Wald F tests with Kenward-Roger df)
## 
## Response: milk_production
##                 F Df Df.res    Pr(>F)    
## Sequence  10.8253  5  26.11 1.023e-05 ***
## Period    91.2965  1   6.00 7.504e-05 ***
## Treatment 41.1220  2   6.00 0.0003143 ***
## Prior_trt  6.0141  3   6.00 0.0306394 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
emmeans(m_milk.carryover, ~ Treatment )
##  Treatment emmean   SE   df lower.CL upper.CL
##  A           42.8 3.05 10.6     36.0     49.5
##  B           56.4 3.05 10.6     49.6     63.1
##  C           77.0 3.05 10.6     70.3     83.7
## 
## Results are averaged over the levels of: Sequence, Prior_trt 
## Degrees-of-freedom method: kenward-roger 
## Confidence level used: 0.95

28.4 Tomorrow

  • Wrapup