Developed to perform the estimation and inference for regression coefficient parameters in longitudinal marginal models using the method of quadratic inference functions. Like generalized estimating equations, this method is also a quasi-likelihood inference method. It has been showed that the method gives consistent estimators of the regression coefficients even if the correlation structure is misspecified, and it is more efficient than GEE when the correlation structure is misspecified. Based on Qu, A., Lindsay, B.G. and Li, B. (2000) DOI: 10.1093/biomet/87.4.823.

Like generalized estimating equations (GEE), QIF is also a quasilikelihood inference method. It has been showed that

QIF gives consistent estimators of the regression coefficients even if the correlation structure is misspecified. GEE has the same property.

QIF estimators are of the same efficiency as GEE estimators when the correlation structure is correctly specified, but more efficient when the correlation structure is misspecified.

QIF gives a goodness-of-fit test for the validity of the first moment assumption pertaining to the unbiasedness of inference function. This assumption is crucial to ensure the consistency in estimation. GEE cannot provide this test.

QIF is robust against a small portion of outliers/contaminated data; refer to Qu, A. and Song, P. (2004), “Assessing robustness of generalised estimating equations and quadratic inference functions”, Biometrika, 91, 447-459.

QIF is analogous to -2*log-likelihood, so it enables naturally to define some model selection criteria, such as Akaike Information Criterion (AIC) and Bayes Information Criterion (BIC).

You can install the released version of qif from CRAN with:

`install.packages("qif")`

Or install the development version from Github with:

```
install.packages("devtools") # you need devtools to install packages from Github
::install_github("umich-biostatistics/qif") devtools
```

```
## Marginal log-linear model for the epileptic seizures count data
## (Diggle et al., 2002, Analysis of Longitudinal Data, 2nd Ed., Oxford Press).
# Read in the epilepsy data set:
data(epil)
# Fit the QIF model:
<- qif(y ~ base + trt + lage + V4, id=subject, data=epil,
fit family=poisson, corstr="AR-1")
# Alternately, use ginv() from package MASS
<- qif(y ~ base + trt + lage + V4, id=subject, data=epil,
fit family=poisson, corstr="AR-1", invfun = "ginv")
# Print summary of QIF fit:
summary(fit)
## Second example: MS study
data(exacerb)
<-qif(exacerbation ~ treatment + time + duration + time2, id=id,
qif_BIN_INDdata=exacerb, family=binomial, corstr="independence")
<-qif(exacerbation ~ treatment + time + duration + time2, id=id,
qif_BIN_AR1data=exacerb, family=binomial, corstr="AR-1")
<-qif(exacerbation ~ treatment + time + duration + time2, id=id,
qif_BIN_CSdata=exacerb, family=binomial, corstr="exchangeable")
<-qif(exacerbation ~ treatment + time + duration + time2, id=id,
qif_BIN_UNdata=exacerb, family=binomial, corstr="unstructured")
summary(qif_BIN_CS)
$statistics
qif_BIN_CS
$covariance qif_BIN_CS
```