Question

An experiment is conducted to determine the effects of four different nitrogen levels (0, 20, 40 and 60 pounds per acre) on three different varieties of oats (Golden, Marvellous and Victory). There are six pieces of land (of equal fertility) available for planting. Each piece of land is divided into three plots, and an oat variety is randomly assigned to one of the plots. Each of the plots is then divided into four subplots, and a level of nitrogen is randomly assigned to the subplot. The response variable is the yield of the oats in quarter pounds per subplot.

1. Write the appropriate model for the analysis of this experiment.

Now suppose the six pieces of land are not necessarily equal in fertility.

2. Write the appropriate model for the analysis of this experiment.

Answer 1

1.The design that will appropriate for this problem would be the Split Plot design. This is done because for each land we split the land into 3 plots i.e., according to 3 types of seeds variety and then further split each plot into 4 subplots i.e., according to 4 types of nitrogen doses.

The model is given by:

$y_{ijk}=\mu +b_{i}+\tau _{j}+\gamma _{k}+\delta _{jk}+e_{ijk}$ , i=1(1)6 , j=1(1)3 , k=1(1)4

where

$y_{ijk}$ = the yield /output due to the k^th nitrogen level, j^th variety of seeds, i^th land /block

$\mu$ = fixed general effect

$b_{i}$ = fixed effect of the ith land

$\tau _{j}$ =fixed effect of the jth seed variety

$\gamma _{k}$ = fixed effect of the kth dose of nitrogen

$\delta _{jk}$ = fixed interaction effect of the jth seed variety and kth nitrogen dose

= random error due to the k^th nitrogen level, j^th variety of seeds, i^th land /block

eAn

and the assumptions:

$\sum_{j}\tau _{j}=0$

$\sum_{k}\gamma _{k}=0$

$\sum_{k}\delta _{jk}=0=\sum_{j}\delta _{jk}$

$b_{i}\sim N(0,\sigma _{b}^{2})$

$e_{ijk}\sim N(0,\sigma _{e}^{2})$

2. When we do not have the 6 lands, we remove the effect of the land from the model:

The model is given by:

$y_{jk}=\mu +\tau _{j}+\gamma _{k}+\delta _{jk}+e_{jk}$ , j=1(1)3 , k=1(1)4

where

$y_{jk}$ = the yield /output due to the k^th nitrogen level, j^th variety of seeds

$\mu$ = fixed general effect

$\tau _{j}$ =fixed effect of the jth seed variety

$\gamma _{k}$ = fixed effect of the kth dose of nitrogen

$\delta _{jk}$ = fixed interaction effect of the jth seed variety and kth nitrogen dose

$e_{jk}$ = random error due to the k^th nitrogen level, j^th variety of seeds

and the assumptions:

$\sum_{j}\tau _{j}=0$

$\sum_{k}\gamma _{k}=0$

$\sum_{k}\delta _{jk}=0=\sum_{j}\delta _{jk}$

$e_{jk}\sim N(0,\sigma _{e}^{2})$