Simultaneous Inference for Multiple Marginal Models

Description

Calculation of correlation between test statistics from multiple marginal models using the score decomposition

Usage

mmm(...)
mlf(...)

Arguments

Details

Estimated correlations of the estimated parameters of interest from the multiple marginal models are obtained using a stacked version of the i.i.d. decomposition of parameter estimates by means of score components (first derivatives of the log likelihood). The method is less conservative than the Bonferroni correction. The details are provided by Pipper and Ritz (2012).

The implementation assumes that the model were fitted to the same data, i.e., the rows of the matrices returned by estfun belong to the same observations for each model.

The reference distribution is always multivariate normal, if you want to use the multivariate t, please specify the corresponding degrees of freedom as an additional df argument to glht.

Observations with missing values contribute zero to the score function. Models have to be fitted using na.exclude as na.action argument.

Value

An object of class mmm or mlf, basically a named list of the arguments with a special method for glht being available for the latter. vcov, estfun, and bread methods are available for objects of class mmm.

Author(s)

Code for the computation of the joint covariance and sandwich matrices was contributed by Christian Ritz and Christian B. Pipper.

References

Christian Bressen Pipper, Christian Ritz and Hans Bisgaard (2012), A Versatile Method for Confirmatory Evaluation of the Effects of a Covariate in Multiple Models, Journal of the Royal Statistical Society, Series C (Applied Statistics), 61, 315–326.

Examples

### replicate analysis of Hasler & Hothorn (2011), 
### A Dunnett-Type Procedure for Multiple Endpoints,
### The International Journal of Biostatistics: Vol. 7: Iss. 1, Article 3.
### DOI: 10.2202/1557-4679.1258

### see ?coagulation
if (require("SimComp")) {
    data("coagulation", package = "SimComp")

    ### level "S" is the standard, "H" and "B" are novel procedures
    coagulation$Group <- relevel(coagulation$Group, ref = "S")

    ### fit marginal models
    (m1 <- lm(Thromb.count ~ Group, data = coagulation))
    (m2 <- lm(ADP ~ Group, data = coagulation))
    (m3 <- lm(TRAP ~ Group, data = coagulation))

    ### set-up Dunnett comparisons for H - S and B - S 
    ### for all three models
    g <- glht(mmm(Thromb = m1, ADP = m2, TRAP = m3),
              mlf(mcp(Group = "Dunnett")), alternative = "greater")

    ### joint correlation
    cov2cor(vcov(g))

    ### simultaneous p-values adjusted by taking the correlation
    ### between the score contributions into account
    summary(g)
    ### simultaneous confidence intervals
    confint(g)

    ### compare with
    ## Not run: 
        library("SimComp")
        SimCiDiff(data = coagulation, grp = "Group",
                  resp = c("Thromb.count","ADP","TRAP"), 
                  type = "Dunnett", alternative = "greater",
                  covar.equal = TRUE)
    
## End(Not run)
 
    ### use sandwich variance matrix
    library("sandwich")
    g <- glht(mmm(Thromb = m1, ADP = m2, TRAP = m3),
              mlf(mcp(Group = "Dunnett")), 
              alternative = "greater", vcov = sandwich)
    summary(g)
    confint(g)
}

### attitude towards science data
data("mn6.9", package = "TH.data")

### one model for each item
mn6.9.y1 <- glm(y1 ~ group, family = binomial(), 
                na.action = na.omit, data = mn6.9)
mn6.9.y2 <- glm(y2 ~ group, family = binomial(), 
                na.action = na.omit, data = mn6.9)
mn6.9.y3 <- glm(y3 ~ group, family = binomial(), 
                na.action = na.omit, data = mn6.9)
mn6.9.y4 <- glm(y4 ~ group, family = binomial(), 
                na.action = na.omit, data = mn6.9)

### test all parameters simulaneously
summary(glht(mmm(mn6.9.y1, mn6.9.y2, mn6.9.y3, mn6.9.y4), 
             mlf(diag(2))))
### group differences
summary(glht(mmm(mn6.9.y1, mn6.9.y2, mn6.9.y3, mn6.9.y4), 
             mlf("group2 = 0")))

### alternative analysis of Klingenberg & Satopaa (2013),
### Simultaneous Confidence Intervals for Comparing Margins of
### Multivariate Binary Data, CSDA, 64, 87-98
### http://dx.doi.org/10.1016/j.csda.2013.02.016

### see supplementary material for data description
### NOTE: this is not the real data but only a subsample
influenza <- structure(list(
HEADACHE = c(1L, 0L, 0L, 1L, 0L, 0L, 1L, 1L, 1L,
0L, 0L, 1L, 0L, 1L, 0L, 1L, 1L, 1L, 1L, 1L, 1L, 0L, 0L, 0L, 0L,
1L, 1L), MALAISE = c(0L, 0L, 1L, 1L, 0L, 1L, 1L, 1L, 0L, 1L,
0L, 0L, 1L, 1L, 0L, 0L, 1L, 0L, 1L, 0L, 1L, 0L, 1L, 1L, 0L, 1L,
0L), PYREXIA = c(0L, 0L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 0L, 1L,
1L, 1L, 1L, 0L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 0L, 0L, 0L, 1L, 1L
), ARTHRALGIA = c(0L, 0L, 0L, 0L, 1L, 0L, 1L, 0L, 1L, 1L, 0L,
0L, 1L, 1L, 0L, 1L, 0L, 0L, 1L, 0L, 0L, 0L, 0L, 1L, 1L, 1L, 1L
), group = structure(c(2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L, 2L,
2L, 2L, 2L, 2L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L, 1L,
1L), .Label = c("pla", "trt"), class = "factor"), Freq = c(32L,
165L, 10L, 23L, 3L, 1L, 4L, 2L, 4L, 2L, 1L, 1L, 1L, 1L, 167L,
1L, 11L, 37L, 7L, 7L, 5L, 3L, 3L, 1L, 2L, 4L, 2L)), .Names = c("HEADACHE",
"MALAISE", "PYREXIA", "ARTHRALGIA", "group", "Freq"), row.names = c(1L,
2L, 3L, 5L, 9L, 36L, 43L, 50L, 74L, 83L, 139L, 175L, 183L, 205L,
251L, 254L, 255L, 259L, 279L, 281L, 282L, 286L, 302L, 322L, 323L,
366L, 382L), class = "data.frame")
influenza <- influenza[rep(1:nrow(influenza), influenza$Freq), 1:5]

### Fitting marginal logistic regression models
(head_logreg <- glm(HEADACHE ~ group, data = influenza, 
                    family = binomial()))
(mala_logreg <- glm(MALAISE ~ group, data = influenza, 
                    family = binomial()))
(pyre_logreg <- glm(PYREXIA ~ group, data = influenza, 
                    family = binomial()))
(arth_logreg <- glm(ARTHRALGIA ~ group, data = influenza, 
                    family = binomial()))

### Simultaneous inference for log-odds
xy.sim <- glht(mmm(head = head_logreg,
                   mala = mala_logreg,
                   pyre = pyre_logreg,
                   arth = arth_logreg),
               mlf("grouptrt = 0"))
summary(xy.sim)
confint(xy.sim)

### Artificial examples
### Combining linear regression and logistic regression
set.seed(29)
y1 <- rnorm(100)
y2 <- factor(y1 + rnorm(100, sd = .1) > 0)
x1 <- gl(4, 25) 
x2 <- runif(100, 0, 10)

m1 <- lm(y1 ~ x1 + x2)
m2 <- glm(y2 ~ x1 + x2, family = binomial())
### Note that the same explanatory variables are considered in both models
### but the resulting parameter estimates are on 2 different scales 
### (original and log-odds scales)

### Simultaneous inference for the same parameter in the 2 model fits
summary(glht(mmm(m1 = m1, m2 = m2), mlf("x12 = 0")))

### Simultaneous inference for different parameters in the 2 model fits
summary(glht(mmm(m1 = m1, m2 = m2),
             mlf(m1 = "x12 = 0", m2 = "x13 = 0")))

### Simultaneous inference for different and identical parameters in the 2
### model fits
summary(glht(mmm(m1 = m1, m2 = m2),
             mlf(m1 = c("x12 = 0", "x13 = 0"), m2 = "x13 = 0")))

### Examples for binomial data
### Two independent outcomes
y1.1 <- rbinom(100, 1, 0.5)
y1.2 <- rbinom(100, 1, 0.5)
group <- factor(rep(c("A", "B"), 50))

m1 <- glm(y1.1 ~ group, family = binomial)
m2 <- glm(y1.2 ~ group, family = binomial)

summary(glht(mmm(m1 = m1, m2 = m2), 
             mlf("groupB = 0")))

### Two perfectly correlated outcomes
y2.1 <- rbinom(100, 1, 0.5)
y2.2 <- y2.1
group <- factor(rep(c("A", "B"), 50))

m1 <- glm(y2.1 ~ group, family = binomial)
m2 <- glm(y2.2 ~ group, family = binomial)

summary(glht(mmm(m1 = m1, m2 = m2), 
             mlf("groupB = 0")))

### use sandwich covariance matrix
summary(glht(mmm(m1 = m1, m2 = m2), 
             mlf("groupB = 0"), vcov = sandwich))

Results