The research question and the one-way ANOVA model
The ANOVA model above tells us that the response variable (yield in our example) depends on:
A part we can explain by the treatment effect (τ_i ), the insecticide.
A part that we do not explain (ε_ij), the error.
The biological question we want to answer is whether the treatment (insecticides) make a difference in terms of the response (yield) or in other words whether the means of the different insecticides treatments are equal to each other or not.
Note that the mean of a particular treatment is: μ_i=μ+τ_i.
To answer this question, we need to estimate the model parameters using the data we have collected and perform a hypothesis test.
The estimation of the model parameters is obtained with the method of Least Squares, which means that the sum of the squares of the estimated errors (residuals) with respect to the estimated parameters is minimized.
We can express the SSE as the sum of the difference of each observation y_ij with the corresponding group mean μ+τ_i and take the square of it: