*2.4.6 Two-way ANOVA*

*2.4.5 Calculation*

• Degrees of freedom

k � 1 for SSG

An F statistic is obtained from ANOVA test or a regression analysis to find out if the means between two populations are significantly different [1]. F statistics is used to decide the acceptance or rejection of null hypothesis. F value is calculated from the data, if calculated is larger than F statistics the null hypothesis is rejected.

The ANOVA table showing F value is given in **Table 5**. SS = Sum of Squares (sum of squared deviations):

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observations about the overall mean.

MSE = ð Þþ n1�<sup>1</sup> ð Þþ n2�<sup>1</sup> ð Þþ n3�<sup>1</sup> … … ð Þ nk�<sup>1</sup> <sup>s</sup><sup>2</sup>

Total SST n-1

Error n � IJ SSE MSE

Total n � 1 SST

ð Þþ n1�1 ð Þþ n2�1 … … *:*þð Þ nk�1

• MS = Mean Square = SS/df :

• The F statistic = MSG/MSE

**Table 5.** *ANOVA table.*

**Table 6.**

**170**

*Two-way ANOVA table.*

degrees of freedom.

SST measures the variation of the data around the overall mean x

SSG measures the variation of the group means around the overall mean x SSE measures the variation of each observation around its group mean xi

n � k for SSE, since it measures the variation of the n observations about k

• This is like a standard deviation. Its numerator was a sum of squared deviations (just like our SS formulas), and it was divided by the appropriate number of

k

If the null hypothesis is true, the F statistic has an F distribution with k-1 and n-k degrees of freedom in the numerator/denominator respectively. If the alternative hypothesis is true, then F tends to be large. We reject Ho in favor of Ha if the F

**Source SS df MS F**

**df SS MS F p-value**

K�1

n�1

MSG MSE

group means. n � 1 for SST, since it measures the variation of all n

It is interesting to note that another formula for MSE is

Model/group SSG k � 1 MSG SSG

Residual/Error SSE n � k MSG SSE

A I � 1 SSA MSA MSA/MSE B J � 1 SSB MSB MSB/MSE AXB (I � 1) (J � 1) SSAB MSAB MSAB/MSE

In the two-way ANOVA model, there are two factors, each with several levels as shown in **Table 6**.
