IT233: Applied Statistics TIHE 2005 Lecture 27

Example: With reference to the above examples, check which of the following 3 contrasts (if any) are orthogonal:

(i)

(ii)

(iii)

Solution:We have = 4, = 5, = 6, = 5

Treatments / 1 / 2 / 3 / 4
Coefficients of (i) / 1 / -1 / 1 / -1
Coefficients of (ii) / 1 / -1 / 0 / 0
Coefficients of (iii) / 0 / 0 / 1 / -1

From Contrast (i) & (ii)

From Contrast (i) & (iii)

From Contrast (ii) & (iii)

Conclusion:

The contrasts (ii) and (iii) are orthogonal to each other.

Notes:

  • Orthogonal contrasts result in independent Contrast Sums of Squares ().
  • Therefore, since Treatment Sum of Squares has d.f. , then theoretically, it is possible to partition into orthogonal contrast sums of squares, i.e.

if the contrasts are orthogonal to each other.

t-Statistics Approach:

Sometimes, it is of interest to make several (perhaps all possible) paired comparisons among the treatments. Let

be the contrast for comparison. To test the hypothesis (i.e. ), one may use the following - statistic (as mentioned earlier),

instead of single-degree-of-freedom test.

Notes:

  • The above test statistics can also be used to obtain a confidence interval for .
  • We can use confidence interval approach for the test.
  • We should note that i.e. “if has the -distribution, then has the -distribution.”

.05 / 5 / / 6.6100 /
.05 / 18 / / 4.4142 /
.01 / 12 / / 9.3330 /
.01 / 20 / / 8.0940 /

Multiple Comparisons:

In many situations, analysts do not know in advance how to construct appropriate orthogonal contrasts, or they may wish to test more than comparisons using same data at same time. For example, suppose we have 4 treatments with means and and we wish to test all possible pairs on means i.e.

,,

,,

The null hypotheses would then be . If we test all these pairs of means using t-tests, probability of committing type I error for the entire set of comparisons can be increased. Multiple comparisons are such tests that avoid this problem and compare all possible means at the same time instead of making separate comparisons for all pairs. There are several methods for making paired comparisons. We shall discuss two of them here:

  1. Tukey’s Test
  2. Duncan’s Multiple Range Test

Note: These tests require that all the sample sizes are equal.

Tukey’s Test:

The Tukey’s Test is based on the Studentized range distribution. The Studentized range statistic is

where is the largest sample mean and is the smallest sample mean out of p means, is the mean square error and n is the common sample size.

Let represent the upper percentile point of q, where is level of significance, k is the number of treatment and is the degrees of freedom for . A list of values of is given in Table A.22. Using the Tukey’s Test two means, and (), are considered significantly different if

where

The test procedure is summarized below:

Step 1.Arrange the sample means in ascending order.

Step 2.Find the value of .

Step 3.Find the absolute difference for all possible pairs (i, j).

Step 4.Make conclusion as below:

Two means, and (), are significantly different, if .

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