Question:
Which of the following is not a treatment contrast among three treatments?
Which of the following is not a treatment contrast among three treatments?
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A contrast needs its coefficients to add up to zero; check which option's coefficients fail that test.
Updated On: Jul 4, 2026
- \(T_1 + 2T_2 - 3T_3\)
- \(T_1 - T_2\)
- \(T_1 - 2T_2 + T_3\)
- \(T_1 + T_2 + T_3\)
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The Correct Option is D
Solution and Explanation
Step 1: A linear combination of treatment totals or effects \(L = \sum_i c_i T_i\) is called a contrast if and only if the coefficients sum to zero, \(\sum_i c_i = 0\). This condition ensures the combination is estimating a difference between treatment effects and not the overall mean level.
Step 2: Check option A, \(T_1 + 2T_2 - 3T_3\): coefficients are \(1, 2, -3\), and \(1+2-3 = 0\). This is a valid contrast.
Step 3: Check option B, \(T_1 - T_2\): coefficients are \(1, -1, 0\) (treatment 3 not involved), and \(1-1+0 = 0\). This is a valid contrast.
Step 4: Check option C, \(T_1 - 2T_2 + T_3\): coefficients are \(1, -2, 1\), and \(1-2+1 = 0\). This is a valid contrast.
Step 5: Check option D, \(T_1 + T_2 + T_3\): coefficients are \(1, 1, 1\), and \(1+1+1 = 3 \ne 0\). The coefficients do not sum to zero, so this combination is just proportional to the sum (or mean) of all treatment effects, not a contrast between them.
Final answer: \(T_1 + T_2 + T_3\) is not a treatment contrast (Option D).
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