Recall that \(\text{std.err} = \text{std.dev} / \sqrt{n}\). We can use this to calculate the standard deviation and variance for each case:

ex.df$S = ex.df$std.err * sqrt(ex.df$n)
ex.df$S2 = ex.df$S^2

Now, \(SSE = \sum_i\sum_j(Y_{ij}-\overline{Y}_{i*})^2 = \sum_i(n_i-1)S_i^2\). We can put each contribution in a column:

ex.df$SSE = (ex.df$n-1)*ex.df$S2

Next, \(SST = \sum_i\sum_j(\overline{Y}_{i*}-\overline{Y})^2 = \sum_in_i(\overline{Y}_{i*}-\overline{Y})^2\) where \(\overline{Y} = \frac{1}{n}\sum_i n_i\overline{Y}_{i*}\).

ex.n = sum(ex.df$n)
ex.x = (1/ex.n) * sum(ex.df$n * ex.df$x.i)
ex.df$SST = ex.df$n * (ex.df$x.i - ex.x)^2

Now we have all the ingredients for our ANOVA table!

ex.dfT = nrow(ex.df) - 1
ex.dfE = ex.n - nrow(ex.df)
ex.SST = sum(ex.df$SST)
ex.SSE = sum(ex.df$SSE)
ex.MST = ex.SST / ex.dfT
ex.MSE = ex.SSE / ex.dfE
ex.F = ex.MST / ex.MSE
ex.p = pf(ex.F, ex.dfT, ex.dfE, lower.tail=FALSE)

ex.df = data.frame(
  group = c(rep("A", 4), rep("B",7), rep("C", 6), rep("D", 5), rep("E", 5)),
  data = c(.8,.6,.6,.5, 
           .7,.8,.5,.5,.6,.9,.7, 
           1.2,1.0,.9,1.2,1.3,.8, 
           1.0,.9,.9,1.1,.7,
           .6,.4,.4,.7,.3
           )
)

aov(data ~ group, ex.df)

## Call:
##    aov(formula = data ~ group, data = ex.df)
## 
## Terms:
##                    group Residuals
## Sum of Squares  1.211844  0.571119
## Deg. of Freedom        4        22
## 
## Residual standard error: 0.1611209
## Estimated effects may be unbalanced

ANOVA in R is a special case of linear regression. Using the command aov ensures that summary printouts match the multiple means use case.

anova( aov(data ~ group, ex.df) )

summary( aov(data ~ group, ex.df) )

##             Df Sum Sq Mean Sq F value   Pr(>F)    
## group        4 1.2118 0.30296   11.67 3.09e-05 ***
## Residuals   22 0.5711 0.02596                     
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1