1 Weighted Jonckheere Test for Ordered Alternatives in Repeated Measures of Randomized Blocks and a Comparison Hatice Kübra ŞENOL AKDUR Gazi Üniversity Fikri GÖKPINAR Hülya BAYRAK Esra GÖKPINAR Gazi University

2 Repeated Measurements Design of the study

3 RM Designs We can use same persons or subjects and test them on 2 or more seperate occasions.  Suppose, for example, we want to find out if people react more quickly to an auditory stimulus (like a bell) or to a visual stimulus (like a light).   We can use the same participants and try them out with both types of stimulus.  This is called a repeated measures design and is often more accurate than the independent measures design. 

4 Ordered Alternative Hypothesis in RM Designs.In the study, we focus on ordered alternative hypothesis of RM designs. The main idea of this kind of experiment is the study of change or development over time/treatments. For instance, based on an educational psychology experiment [18], aggression level of students tends to increase from primary to middle school .

5 Simulated Experiment ResultTime Periods Time1 Time2 Time3 Time4 Time5 Time6 Time7 Time8 Time9 Subjects of Experiment Subject1 -1,291 -0,394 -2,023 -0,672 -0,284 -0,786 0,653 -1,164 1,551 Subject2 0,679 1,222 0,690 0,681 0,505 0,682 0,200 0,024 0,366 Subject3 0,407 0,392 0,584 0,719 0,203 0,477 0,883 -1,228 -1,565 Subject4 0,287 -1,054 -0,807 0,941 1,354 2,478 0,470 0,650 -1,182 Subject5 1,729 1,142 1,519 1,855 0,611 0,294 -1,167 -0,600 -0,239

6 Model of the study

7 Model of the error terms

8 Model of the error terms

9 Test Statistics of the study

10 Test Statistics of the study

11 Hypotheses of the study

12 The Jonckheere Test

13 The Jonckheere Test

14 The Modified Jonckheere Test

15 Independent and Identically Distributed and Circular Bootstrap Method

16 Independent and Identically Distributed and Circular Bootstrap Method

17 Simulation Study

18 Simulation Study

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21 Table1 Result As seen in Table1, the lowest results of type 1 error rate are generated by mostly MJI for the block size,5. For lower positive values of rho, we can prefer MJI, whereas for the higher positive values of rho (>0.3) ,it would be reasonable to prefer MJC. After moderate values of rho, circular based tests(MJC, JC) should be preferred.

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24 Table2 Result For the negative values of rho, MJI and JI give very low type 1 error rates, this result shows very conservative test. Hence, choosing circular-based test would be more logical for negative values of rho. After moderate values of rho(>0.5), circular based should be preferred.

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27 Table3 Result As given in Table 3, the lowest type1 error rates are generated by MJI when correlation terms are negative for the block size 5 and 10. It can be seen that that MJC generates very close type1 error rates to the nominal level for the negative values of rho. When the observation terms are independent, MJI mostly gives the closest type 1 error rates to the nominal alpha. MJI gives lower type1 error rates when correlation terms is 0.1. For the higher positive ( 0.3,0.5, 0.7, 0.9) correlation terms, MJC generates more reasonable type1 error rates.

28 Discussion We modify Jonckheere test to decrease type 1 error rates when blocks have dependent observations. In the literature, most authors investigate large time interval(25,50,100)[5]. Due to this attitude, they miss out IID bootstrap importance for small time intervals.Therefore, we take into considerations small time intervals(9,12,24) in this study. The pattern of results above stay almost similar with block size 5, as the block size is increased to 10, 20, and 30. However, type1 error rates are decreased about %2.

29 Discussion As seen from tables and graphs, the strong correlation increase the type 1 error rates. For the negative values of rho, type 1 error rates of IID bootstrap based tests are very smaller than nominal level, which shows a very conservative test. Whereas, the type 1 error rates for large positive values of rho are highly larger than nominal level that indicates a very liberal test. These problems moderately can be solved for larger block sizes. Also, nominal level of tests can be reduced to 0.01. We also should note that our tests will mostly be liberal for small block sizes. Our tests should be applied with caution when both n and b are small.

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