# CUSUM Chart based on logistic regression model

In this example we consider an application to CUSUM charts based on a logistic regression models.

Assume we have \(n\) past in-control data \((Y_{-n},X_{-n}),\ldots,(Y_{-1},X_{-1})\), where \(Y_i\) is a binary response variable and \(X_i\) is a corresponding vector of covariates.

Suppose that in control \(\mbox{logit}(\mbox{P}(Y_i=1|X_i))=X_i\beta\). A maximum likelihood estimate \(\hat\beta\) of the parameters is obtained e.g. by the glm function.

For detecting a change to \(\mbox{logit}(\mbox{P}(Y_i=1|X_i))=\Delta+X_i\beta\), a CUSUM chart based on the cumulative sum of likelihood ratios of the out-of-control versus in-control model can be defined by (Steiner et al., \(Biostatistics\) 2000, pp 441-452) \[S_0=0, \quad S_t=\max(0, S_{t-1}+R_t) \] where \[ \exp(R_t)=\frac{\exp(\Delta+X_t\beta)^{Y_t}/(1+\exp(\Delta+X_t\beta))}{\exp(X_t\beta)^{Y_t}/(1+\exp(X_t\beta))} =\exp(Y_t\Delta)\frac{1+\exp(X_t\beta)}{1+\exp(\Delta+X_t\beta)}. \]

The following generates a data set of past observations (replace this with your observed past data) from the model \(\mbox{logit}(\mbox{P}(Y_i=1|X_i))=-1+x_1+x_2+x_3\) and distribution of the covariate values as specified below.

    n <- 1000
    Xlogreg <- data.frame(x1=rbinom(n,1,0.4), x2=runif(n,0,1), x3=rnorm(n))
    xbeta <- -1+Xlogreg$x1+Xlogreg$x2+Xlogreg$x3
    Xlogreg$y <- rbinom(n,1,exp(xbeta)/(1+exp(xbeta)))

Next, we initialise the chart and compute the estimate needed for running the chart - in this case \(\hat \beta\).

    library(spcadjust)
    chartlogreg <- new("SPCCUSUM",model=SPCModellogregLikRatio(Delta= 1, formula="y~x1+x2+x3"))
    xihat <- xiofdata(chartlogreg,Xlogreg)
    xihat

    ##
    ## Call:  glm(formula = formula, family = binomial("logit"), data = P)
    ##
    ## Coefficients:
    ## (Intercept)           x1           x2           x3
    ##     -1.0768       1.0341       1.0357       0.9918
    ##
    ## Degrees of Freedom: 999 Total (i.e. Null);  996 Residual
    ## Null Deviance:       1385
    ## Residual Deviance: 1142  AIC: 1150

## Calibrating the chart to a given average run length (ARL)

We now compute a threshold that with roughly 90% probability results in an average run length of at least 1000 in control. In this regression model this is computed by non-parametric bootstrapping. The number of bootstrap replications (the argument nrep) shoud be increased for real applications.

    cal <- SPCproperty(data=Xlogreg,
                nrep=100,chart=chartlogreg,
                property="calARL",params=list(target=1000),quiet=TRUE)
    cal

    ## 90 % CI: A threshold of 4.906 gives an in-control ARL of at least
    ##   1000.
    ## Unadjusted result:  4.206
    ## Based on  100 bootstrap repetitions.

## Run the chart

Next, we run the chart with new observations that are in-control.

    n <- 100
    newXlogreg <- data.frame(x1=rbinom(n,1,0.4), x2=runif(n,0,1), x3=rnorm(n))
    newxbeta <- -1+newXlogreg$x1+newXlogreg$x2+newXlogreg$x3
    newXlogreg$y <- rbinom(n,1,exp(newxbeta)/(1+exp(newxbeta)))
    S <- runchart(chartlogreg, newdata=newXlogreg,xi=xihat)

In the next example, the chart is run with data that are out-of-control from time 51 and onwards.

    n <- 100
    newXlogreg <- data.frame(x1=rbinom(n,1,0.4), x2=runif(n,0,1), x3=rnorm(n))
    outind <- c(rep(0,50),rep(1,50))
    newxbeta <- -1+newXlogreg$x1+newXlogreg$x2+newXlogreg$x3+outind
    newXlogreg$y <- rbinom(n,1,exp(newxbeta)/(1+exp(newxbeta)))
    S <- runchart(chartlogreg, newdata=newXlogreg,xi=xihat)