35.9 Example: Food digestibility

A study evaluated various food for sheep (Moir 1961). One combination of variables assessed is shown in Fig. 33.6.

The RQ is:

Does the digestible energy requirement of feed increase with dry matter digestibility percentage (and if so, what is the relationship)?

In this study, $x$ is the dry matter weight digestibility percentage, and $y$ is the digestible energy. The data are shown in Fig. 35.11. Using the software output (Fig. 35.12 (jamovi); Fig. 35.13 (SPSS)), the value of the slope and $y$ -intercept in the sample are $b_0 = -0.193$ and $b_1 = 0.047$ . The regression equation is

$\hat{y} = -0.193 + 0.047x.$

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	DryMatterDigest	EnergyDigest	Energy
1	30.5	27.8	1.243
2	63	61.5	2.75
3	62.8	60.4	2.701
4	50	49.5	2.213
5	60.3	58.7	2.681
6	64.1	63	2.877
7	63.7	62.8	2.895
8	63.4	62.8	2.895
9	65.4	64.2	2.952
10	68.1	66.5	3.059

Showing 1 to 10 of 36 entries

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FIGURE 35.11: The digestibility data set

The slope means that when the dry matter weight digestibility increases by 1 percentage point, the digestible energy increases, on average, by 0.047 Cal/gram.

Each sample will produce slightly different sample slopes, so we can test to see if the slope in the population is non-zero due to sampling variation, using a hypothesis test:

$H_0$ : $\beta_1 = 0$ ;
$H_1$ : $\beta_1 > 0$ .

The parameter is $\beta$ , the population slope for the regression equation predicting digestible energy from dry matter weight.

The alternative hypothesis is one-tailed, based on the RQ.

From the software output, $t = 39.322$ , which is huge; the two-tailed $P$ -value is $P<0.001$ . Since we have a one-tailed alternative hypothesis, the $P$ -value is less than $0.001/2 = 0.0005$ . There is very strong evidence that the digestible energy increases as the dry matter weight digestibility increases.

The approximate 95% CI for the population slope $\beta_1$ is

$0.047 \pm (2\times 0.001),$ or from 0.045 to 0.049.

FIGURE 35.12: jamovi output for the sheep-feed data

FIGURE 35.13: SPSS output for the sheep-feed data

The results are statistically valid. We write:

The sample presents very strong evidence ( $t = 39.322$ ; one-tailed $P < 0.0005$ ) of a relationship between dry matter weight digestibility and the digestible energy (slope: $0.047$ ; $n= 36$ ; 95% CI from $-0.045$ to $-0.049$ ) in the population.

Click on the hotspots in the following image, to see what the areas under the normal curve mean.

References

Moir RJ. A note on the relationship between the digestible dry matter and the digestable energy content of ruminant diets. Australian Journal of Experimental Agriculture and Animal Husbandry. 1961;1:24–6.