Chapter 13, Problem 88SE

To determine

Show that $\text{Total SS}\text{= SST+SSB+SSE}$.

Explanation of Solution

The formulas for sum of squares are as follows:

$\begin{array}{l}\text{TSS =}{{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left(Y_{ij}-\overline{Y}\right)}}}^2\\ \text{SST =}b{{\displaystyle \sum _{i=1}^k\left({\overline{Y}}_{i.}-\overline{Y}\right)}}^2\\ \text{SSB =}k{{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{.j}-\overline{Y}\right)}}^2\\ \text{SSE=}{{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}-\overline{Y}\right)}}}^2\end{array}$.

The total sum of squares is calculated is as follows:

$\begin{array}{c}\text{TSS =}{{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left(Y_{ij}-{\overline{Y}}_{i.}+{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+{\overline{Y}}_{.j}-\overline{Y}+\overline{Y}-\overline{Y}\right)}}}^2\\ ={{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left(\left({\overline{Y}}_{i.}-\overline{Y}\right)+\left({\overline{Y}}_{.j}-\overline{Y}\right)+Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}}}^2\\ =\left[\begin{array}{c}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{i.}-\overline{Y}\right)}^2}}+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{.j}-\overline{Y}\right)}^2+}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}^2+}}\\ 2{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{i.}-\overline{Y}\right)\left({\overline{Y}}_{.j}-\overline{Y}\right)+2{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{i.}-\overline{Y}\right)\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}+}}}\\ 2{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{.j}-\overline{Y}\right)\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}}\end{array}\right]\end{array}$

Consider

$\begin{array}{c}{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{.j}-\overline{Y}\right)}=\left[{\displaystyle \sum _{j=1}^b\frac{1}{k}}\left({\displaystyle \sum _{i=1}^k{Y}_{ij}}\right)-b\times \frac{{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}}{bk}\right]\\ =\left[\frac{1}{k}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}-\frac{1}{k}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}\right]\\ =0\end{array}$

$\begin{array}{c}{\displaystyle \sum _{i=1}^k\left({\overline{Y}}_{.i}-\overline{Y}\right)}=\left[{\displaystyle \sum _{i=1}^k\frac{1}{b}}\left({\displaystyle \sum _{j=1}^b{Y}_{ij}}\right)-k\times \frac{{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}}{bk}\right]\\ =\left[\frac{1}{b}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}-\frac{1}{b}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}\right]\\ =0\end{array}$

$\begin{array}{c}{\displaystyle \sum _{i=1}^k\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}={\displaystyle \sum _{i=1}^k\left(Y_{ij}-\frac{1}{b}{\displaystyle \sum _{j=1}^b{Y}_{ij}}-\frac{1}{k}{\displaystyle \sum _{i=1}^k{Y}_{ij}}-\frac{1}{bk}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}\right)}\\ ={\displaystyle \sum _{i=1}^k{Y}_{ij}}-\frac{1}{b}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}-\frac{k}{k}{\displaystyle \sum _{i=1}^k{Y}_{ij}}+\frac{1}{b}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{Y}_{ij}}}\\ =0\end{array}$

Thus, the required TSS is as follows:

$\begin{array}{c}\text{TSS }=\left[\begin{array}{c}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{i.}-\overline{Y}\right)}^2}}+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{.j}-\overline{Y}\right)}^2+}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}^2+}}\\ 2{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{i.}-\overline{Y}\right)\left({\overline{Y}}_{.j}-\overline{Y}\right)+2{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{i.}-\overline{Y}\right)\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}+}}}\\ 2{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{.j}-\overline{Y}\right)\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}}\end{array}\right]\\ =\left[\begin{array}{l}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{i.}-\overline{Y}\right)}^2}}+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{.j}-\overline{Y}\right)}^2+}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}^2+}}\\ 2{\displaystyle \sum _{i=1}^k\left({\overline{Y}}_{i.}-\overline{Y}\right){\displaystyle \sum _{i=1}^k\left({\overline{Y}}_{.j}-\overline{Y}\right)+}}2{\displaystyle \sum _{j=1}^b\left({\overline{Y}}_{i.}-\overline{Y}\right){\displaystyle \sum _{i=1}^k\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)+}}\\ 2{\displaystyle \sum _{j=1}^k\left({\overline{Y}}_{.j}-\overline{Y}\right){\displaystyle \sum _{i=1}^b\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}}\end{array}\right]\\ ={\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{i.}-\overline{Y}\right)}^2}}+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{.j}-\overline{Y}\right)}^2+}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}^2+}}0+0+0\\ ={\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{i.}-\overline{Y}\right)}^2}}+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left({\overline{Y}}_{.j}-\overline{Y}\right)}^2+}}{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}^2}}\\ =b{\displaystyle \sum _{i=1}^k{\left({\overline{Y}}_{i.}-\overline{Y}\right)}^2}+k{\displaystyle \sum _{j=1}^k{\left({\overline{Y}}_{.j}-\overline{Y}\right)}^2}+{\displaystyle \sum _{i=1}^k{\displaystyle \sum _{j=1}^b{\left(Y_{ij}-{\overline{Y}}_{i.}-{\overline{Y}}_{.j}+\overline{Y}\right)}^2}}\\ =\text{SST+SSB+SSE}\end{array}$