MI.test: Test for Marginal Independence

Description

The MI.test function offers three approaches for testing multiple marginal independence (MMI) between one SRCV and one MRCV, or simultaneous pairwise marginal independence (SPMI) between two MRCVs.

Usage

MI.test(data, I, J, type = "all", B = 1999, B.max = B, summary.data = 
    FALSE, add.constant = 0.5, plot.hist = FALSE, print.status = TRUE)
    
MMI.test(data, I, J, type, B, B.max, summary.data, add.constant, 
    plot.hist, print.status)
    
SPMI.test(data, I, J, type, B, B.max, summary.data, add.constant, 
    plot.hist, print.status)

MI.stat(data, I, J, summary.data = FALSE, add.constant = 0.5)

MMI.stat(data, I, J, summary.data, add.constant)

SPMI.stat(data, I, J, summary.data, add.constant)

Arguments

data

For summary.data = FALSE: a data frame containing the raw data where rows correspond to the individual item response vectors, and columns correspond to the items, W1, ..., WI and Y1, ..., YJ (in this order). For summary.data = TRUE

I

The number of items corresponding to row variable W. I = 1 for the one MRCV case.

J

The number of items corresponding to column variable Y.

type

A character string specifying one of the following approaches for testing for MI: "boot" specifies a nonparametric bootstrap procedure; "rs2" specifies a Rao-Scott second-order adjustment; "bon" specifies a Bonferron

B

The desired number of bootstrap resamples.

B.max

The maximum number of bootstrap resamples.  A resample is thrown out if at least one of the J (one MRCV case) or IxJ (two MRCV case) contingency tables does not have the correct dimension; MI.test uses the first B valid resamples

summary.data

A logical value indicating whether data is a summary file containing the item response data instead of the raw data.  Only type = "bon" is available for summary.data = TRUE.

add.constant

A positive constant to be added to all zero marginal cell counts.

plot.hist

A logical value indicating whether plots of the emprical bootstrap sampling distributions should be provided.

print.status

A logical value indicating whether bootstrap progress updates should be provided.

`Value`

--- For MI.test, MMI.test, and SPMI.test, a list containing at least general, a list containing the following objects:
data:
 {The original data frame supplied to the data argument.}
I:
{The original value supplied to the I argument.}
J:
{The original value supplied to the J argument.}
summary.data:
{The original value supplied to the summary.data argument.}
X.sq.S:
{The modified Pearson statistic; NA if at least one of the J (one MRCV case) or IxJ (two MRCV case) contingency tables does not have the correct dimension.}
X.sq.S.ij:
{A matrix containing the individual Pearson statistics.}

`bold`

geepack

`code`

SPMI.stat

`itemize`

X.sq.S:

`item`

B.discard:
p.value.boot:
p.combo.min.boot:
p.combo.prod.boot:
X.sq.S.star:
X.sq.S.ij.star:
p.combo.min.star:
p.combo.prod.star:
df.rs2:
p.value.rs2:
X.sq.S.ij.p.bon:
X.sq.S.ij:
valid.margins:

`Details`

The MI.test function is a wrapper function for MMI.test and SPMI.test which perform the testing of MMI or SPMI.  Three sets of testing methods are implemented:
The nonparametric bootstrap resamples under the null hypothesis by independently sampling the W and Y vectors with replacement from the observed data.  Fixed row counts (i.e., fixed counts for each level of the SRCV) are maintained for the one MRCV case.  A modified Pearson statistic (see Bilder, Loughin, and Nettleton (2000) and Bilder and Loughin (2004)) is calculated for each resample.  In addition, bootstrap p-value combination methods are available to take advantage of the modified Pearson statistic's decomposition into J (one MRCV case) or IxJ (two MRCV case) individual Pearson statistics.  The minimum or the product of p-values is the combination for each resample. {}
The Rao-Scott approach applies a second-order adjustment to the modified Pearson statistic and its sampling distribution.  Formulas are provided in Appendix A of Bilder, Loughin, and Nettleton (2000) and Bilder and Loughin (2004).  Note that this test can be conservative at times. {}
The Bonferroni adjustment multiplies each p-value (using a standard chi-square approximation) from the individual Pearson statistics by J (one MRCV case) or IxJ (two MRCV case).  If a resulting p-value is greater than 1 after the multiplication, it is set to a value of 1.  The overall p-value for the test then is the minimum of these adjusted p-values.  Note that the Bonferroni adjustment tends to produce an overly conservative test when the number of individual Pearson statistics is large. {}

`References`

Agresti, A. and Liu, I.-M. (1999)  Modeling a categorical variable allowing arbitrarily many category choices.  Biometrics, 55, 936--943.

Bilder, C. and Loughin, T. (2004)  Testing for marginal independence between two categorical variables with multiple responses.  Biometrics, 36, 433--451.

Bilder, C., Loughin, T., and Nettleton, D. (2000)  Multiple marginal independence testing for pick any/c variables.  Communications in Statistics--Theory and Methods, 29, 1285--1316.

`See Also`

The genloglin function offers a generalized loglinear modeling approach for testing the relationship among two or three MRCVs.

`Examples`

Run this code# Test for MMI using the second-order Rao-Scott adjustment
test.mmi.rs2 <- MI.test(data = farmer1, I = 1, J = 5, type = "rs2")
test.mmi.rs2

# Test for MMI using all three approaches
# A small B is used for demonstration purposes; normally, a larger B should be used
test.mmi.all <- MI.test(data = farmer1, I = 1, J = 5, type = "all", B = 99, 
    plot.hist = TRUE)
test.mmi.all

# Use MI.test() with summary data
# Convert raw data file to summary file for this example 
farmer1.irdframe <- item.response.table(data = farmer1, I = 1, J = 5, create.dataframe = 
    TRUE)
# Test for MMI using the Bonferroni adjustment
test.mmi.bon <- MI.test(data = farmer1.irdframe, I = 1, J = 5, type = "bon", 
    summary.data = TRUE)
test.mmi.bon

# Test for SPMI using the second-order Rao-Scott adjustment
test.spmi.rs2 <- MI.test(data = farmer2, I = 3, J = 4, type = "rs2")
test.spmi.rs2

# Test for MMI using the marginal logit model approach
library(geepack)
n<-nrow(farmer1)
farmer1.id<-cbind(case=1:n, farmer1)
# Reshape raw data into long format as required by geeglm() function
# Assumes 3:ncol(farmer1.id) corresponds to MRCV items
farmer1.gee<-reshape(data = farmer1.id, 
                 varying = names(farmer1.id)[3:ncol(farmer1.id)], 
                 v.names = "response", timevar = "item", idvar = "case", 
                 direction = "long") 
row.names(farmer1.gee)<-NULL
farmer1.gee[,2:3]<-lapply(farmer1.gee[,2:3], factor)
# Data frame must be ordered by case
farmer1.gee<-farmer1.gee[order(farmer1.gee$case),]
head(farmer1.gee)
tail(farmer1.gee)
mod.fit.H0<-geeglm(formula = response ~ item, family = binomial(link = logit), 
                  data = farmer1.gee, na.action = na.omit, id = case, 
                  corstr = "unstructured")
mod.fit.HA<-geeglm(formula = response ~ Ed*item, family = binomial(link = logit), 
                   data = farmer1.gee, na.action = na.omit, id = case, 
                   corstr = "unstructured")
# Compute Wald test
anova(mod.fit.HA, mod.fit.H0)
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