TWO-WAY ANOVA

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If a set of treatments is being compared in an experiment involving several replicates, an appropriate design can greatly improve the experiment. If the experimental units can be placed into groups (referred to as blocks) into which the variation is less than over the entire experiment, and treatments are assigned within groups, comparisons between treatments will be more accurate. Ensure the allocation of treatments to units within the blocks is random to guard against effects of patterns.

Read in the silage data. There are 21 plots arranged in three blocks of seven plots. The seven conditioning treatments were allocated randomly to the plots within each block.

To perform the ANOVA analysis the dryrate data must first be unstacked into 7 columns, representing treatment, and 3 rows representing the replicate. This can be done either by copying and pasting into new worksheet cells, or by using the PivotTable Wizard to create the table on the right.
Now select Tools > Data Analysis > ANOVA: two-way without replication

NB: Excel views replication in a special way. In this case 'two-way ANOVA without replication' must be selected even though replication is used in the experimental model. For further information see the Help documentation for the two-way ANOVA.

Enter the input range, including the row and column labels but not the Grand Totals and the output range (on the same sheet) . Click OK to obtain the following output:

The treatment effect is clear and the treatment means are more accurate than if a blocking structure had not been used.

To identify how real are the treatment differences, construct t-tests, using the mean square error in the ANOVA as the estimate of the variance. Calculate the standard error of the differences between pairs of treatments by selecting an empty cell and typing = SQRT(MSError*((1/n1) + (1/n2))) where MSError is the cell reference of the mean square error and n1 and n2 are the number in each treatment.
Use this value to test for differences between pairs of treatments by calculating a p-value from t using the following steps:

  1. Activate the FunctionWizard on the toolbar.
  2. Select TDIST from the list of statistical functions.
  3. Enter the absolute t-value as x, the number of degrees of freedom of the MSE as df, and 2 as the number of tails.
The answer given (known as the p-value) is the probability that t could be greater than the absolute value of the number you fed in.

For example, for a comparison between treatments 1 and 2, s.e.d = 0.209367, df =12, t = 0.11145, and p = TDIST(0.11145,12,2) =0.913104. i.e. There is no evidence of a difference between treatments 1 and 2.

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Basic statistics in Excel   23.2.99   Page: 22 of 25