# Test of the Mean Interaction Contrast Performs either an Adjusted Rank Transform or ANOVA test for an interaction at the mean level. ```r mic.test(HH, HL, LH, LL, method=c("art", "anova")) ``` ## Arguments - `HH`: Response times from the High--High condition. - `HL`: Response times from the High--Low condition. - `LH`: Response times from the Low--High condition. - `LL`: Response times from the Low--Low condition. - `method`: If "art", use the adjusted rank transform test. If "anova" use ANOVA. ## Details The mean interaction contrast (MIC) indicates the architecture of a process. Serial processes result in MIC equal to zero. Parallel-OR and Coactive process have a positive MIC. Parallel-AND process have a negative MIC. A test for a significant MIC can be done with a nonparametric adjusted rank transform test (described below) or an ANOVA. The Adjusted Rank Transform is a nonparametric test for an interaction between two discrete variables. The test is carried out by first subtracting the mean effect of the salience level on each channel. Suppose, $m_{H,\cdot} =$ E[RT; Level of Channel 1 is Fast], $m_{L,\cdot} =$ E[RT; Level of Channel 1 is Slow], $m_{\cdot, H} =$ E[RT; Level of Channel 2 is Fast], $m_{\cdot, L} =$ E[RT; Level of Channel 2 is Slow]. Then for each response time from the fast--fast condition, $m_{H, \cdot}$ and $m_{\cdot,H}$ are subtracted. Likewise, for each of the other conditions, the appropriate $m$ is subtracted. Next, each mean subtracted response time is replaced with its rank across all conditions (e.g., the fastest time of all conditions would be replaced with a 1). In this implementation, tied response times are assigned using the average rank. Finally, a standard ANOVA on the ranks is done on the ranks and the $p$-value of that test is returned. This test was recommended by Sawilowsky (1990) based on a survey of a number of nonparametric tests for interactions. He credits Reinach (1965) for first developing the test. ## Returns A list of class "htest" containing: - **statistic**: The value of the MIC. - **p.value**: The $p$-value of the ART or ANOVA test. - **alternative**: A description of the alternative hypothesis. - **method**: A string indicating that the Houpt-Townsend statistic was used. - **data.name**: A string indicating which data were used for which input. ## References Reinach, S.G. (1965). A nonparametric analysis for a multiway classification with one element per cell. **South African Journal of Agricultural Science, 8**, 941--960. Sawilowsky, S.S. (1990). Nonparametric tests of interaction in experimental design. **Review of Educational Research, 60**, 91--126. Houpt, J.W. & Townsend, J.T. (2010). The statistical properties of the survivor interaction contrast. **Journal of Mathematical Psychology, 54**, 446-453. ## Author(s) Joe Houpt ## Examples ```r T1.h <- rweibull(300, shape=2 , scale=400 ) T1.l <- rweibull(300, shape=2 , scale=800 ) T2.h <- rweibull(300, shape=2 , scale=400 ) T2.l <- rweibull(300, shape=2 , scale=800 ) Serial.hh <- T1.h + T2.h Serial.hl <- T1.h + T2.l Serial.lh <- T1.l + T2.h Serial.ll <- T1.l + T2.l mic.test(HH=Serial.hh, HL=Serial.hl, LH=Serial.lh, LL=Serial.ll) Parallel.hh <- pmax(T1.h, T2.h) Parallel.hl <- pmax(T1.h, T2.l) Parallel.lh <- pmax(T1.l, T2.h) Parallel.ll <- pmax(T1.l, T2.l) mic.test(HH=Parallel.hh, HL=Parallel.hl, LH=Parallel.lh, LL=Parallel.ll, method="art") ```

mic_test function