## Zinc absorption

• 4 steer
• 3 periods (diets) /steer
• 2 washes (order of buffers)/periods

In each period, the difference is in the diet.

In each wash,

1. Insert a random buffer (9 ppm zinc vs 0 ppm zinc) into rumen
2. measure zinc at times 0, 5, 20, 35, 50, and 65 minutes
3. wash out period
4. Insert the other buffer
5. measure zinc

### Response

• zinc measurement or
• proportion zinc remaining relative to amount measured at 5 minutes

Zinc generally appears to have an exponential decay pattern across time. So responses for the purpose of modeling include parameters from an estimated curve with formula \$B+(X0-B)e^{kt}\$ where \$X0\$ is the measurement at time 5 and \$B\$ and \$k\$ are the estimated parameters.

## Model

The model will be according to the multiple cross-over design.

## Thoughts/Suggestions

• Using the proportion is very sensitive to the amount measured at 5 minutes which could cause a positive correlation amongst the other measurements. One suggestion is to model the actual measurement (rather than the proportion remaining) and use the 5 minute measurement as an offset.
• Consider monotone function estimation to estimate the means at time 5 through 65. Then use the ratio of the estimated mean at time 5 minus the estimated mean at time 15 divided by the estimated mean at time 5 minus the estimated mean at time 65.
• Try to plot the data all on one page with faceting for buffer, wash, diet and period with one line per steer.