mic_test: Test of the Mean Interaction Contrast
In jhoupt/sft: Functions for Systems Factorial Technology Analysis of Data

Description Usage Arguments Details Value Author(s) References Examples

Performs either an Adjusted Rank Transform or ANOVA test for an interaction at the mean level.

1	mic.test(HH, HL, LH, LL, method=c("art", "anova"))

`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.

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.

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.

Joe Houpt <joseph.houpt@wright.edu>

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.

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")