Overall and group means

We first need to compute the mean age by class (referred as the group means):

• class A: \(\frac{24 + 31 + 26 + 23}{4} = 26\)
• class B: \(\frac{24 + 21 + 19 + 24}{4} = 22\)
• class C: \(\frac{15 + 21 + 18 + 18}{4} = 18\)

and the mean age for the whole sample (referred as the overall mean):

\[\begin{equation} \begin{split} & \frac{24 + 31 + 26 + 23 + 24 + 21 + 19 }{12} \\ &\frac{+ 24 + 15 + 21 + 18 + 18}{12} = 22 \end{split} \end{equation} \]

SSR and SSE

We then need to compute the sum of squares regression (SSR), and the sum of squares error (SSE).

The SSR is computed by taking the square of the difference between the mean group and the overall mean, multiplied by the number of observations in the group:

and then taking the sum of all cells:

\[64+0+64 = 128 = SSR\]

The SSE is computed by taking the square of the difference between each observation and its group mean:

and then taking the sum of all cells:

\[\begin{equation} \begin{split} & 4+25+0+9+4+1+9+4 \\ & +9+9+0+0 = 74 = SSE \end{split} \end{equation} \]

For those interested in computing the sum of square total (SST), it is simply the sum of SSR and SSE, that is,

\[\begin{equation} \begin{split} SST &= SSR + SSE\\ &= 128 + 74 \\ & =202 \end{split} \end{equation} \]

ANOVA table

The ANOVA table looks as follows (we leave it empty and we are going to fill it in step by step):

We start to build the ANOVA table by plugging the SSR and SSE values found above into the table (in the “Sum.of.Sq.” column):

The “Df” column corresponds to the degrees of freedom, and is computed as follows:

• for the line regression: number of groups – 1 = 3 – 1 = 2
• for the line error: number of observations – number of groups = 12 – 3 = 9

With this information, the ANOVA table becomes:

The “Mean.Sq.” column corresponds to the Mean Square, and is equal to the sum of square divided by the degrees of freedom, so the “Sum.of.Sq.” column divided by the “Df” column:

Finally, the F-value corresponds to the ratio between the two mean squares, so \(\frac{64}{8.222} = 7.78\):

This F-value gives the test statistic (also referred as \(F_{obs}\)), which needs to be compared with the critical value found in the Fisher table to conclude the test.

We find the critical value in the Fisher table based on the degrees of freedom (those used in the ANOVA table) and based on the significance level. Suppose we take a significance level \(\alpha = 0.05\), the critical value can be found in the Fisher table as follows:

So we have

\[F_{2; 9; 0.05} = 4.26\]

If you are interested to find this value with R, it can be found with the qf() function, where 0.95 corresponds to \(1 – \alpha\):

qf(0.95, 2, 9)
## [1] 4.256495

Conclusion of the test

The rejection rule says that, if:

• \(F_{obs} > F_{2; 9; 0.05} \Rightarrow\) we reject the null hypothesis
• \(F_{obs} \le F_{2; 9; 0.05} \Rightarrow\) we do not reject the null hypothesis

In our case,

\[F_{obs} = 7.78 > F_{2; 9; 0.05} = 4.26\]

\(\Rightarrow\) We reject the null hypothesis that all means are equal. In other words, it means that at least one class is different than the other two in terms of age.²

To verify our results, here is the ANOVA table using R:

##             Df Sum Sq Mean Sq F value Pr(>F)  
## class        2    128   64.00   7.784 0.0109 *
## Residuals    9     74    8.22                 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

We found the same results by hand, but note that in R, the \(p\)-value is computed instead of comparing the \(F_{obs}\) with the critical value. The \(p\)-value can easily be found in R based on the \(F_{obs}\) and the degrees of freedom:

pf(7.78, 2, 9,
  lower.tail = FALSE
)
## [1] 0.010916