As described in section ref(SCENARIO) the Scenario parameter tt(spread) may be
set to tt(true) resulting in random variations of several parameters. Handling
spreading is illlustrated in figure ref(SPREADFIG).

    figure(spread/spread)
        (Spread used by Simrisc components)
        (SPREADFIG)
 
When an analysis starts tt(Loop::initialize) passes tt(Scenario::spread) on to
the member tt(Loop::spread). Various components support random fluctuations of
their parameters, and if tt(Scenario::spread) is true these random
fluctiations are applied to these components. Figure ref(SPREADFIG)
illustrates where random fluctuations may be applied:
    itemization(
    it() the tt(Mammo) modality's tt(systematicError);
    it() the tt(Screening's attendanceRate);
    it() various tt(Tumor) components: 
        itemization(
        it() tt(Survival);
        it() tt(Growth);
        it() tt(TumorInfo):
            itemization(
            it() tt(Beir7);
            it() tt(Incidence's) carrier specifications.
            )
        )
    )

Random fluctuations can be applied to means, standard deviations and
proportions. 

For means the normal distribution is used by default. For example, the
specification of the tt(meanAge) distribution, specifified for carrier
tt(Normal) in the tt(Incidence) component of the tt(Tumor) specification may
look like this:
        verb(
    #                   value   spread  distr.                    
    meanAge:            72.9    .552    Normal 
        )
    95% of the mean-values which are randomly drawn from this distribution
are located in the interval 71.8 to 74.0. When spreading is used, thus
computed mean values are used as mean values of the tt(Normal) carrier during
the actual simulations.

For standard deviations the chi+sups(2) distribution is used, ensuring that
the standard deviation that is used for the simulations exceeds 0 (see the
next section for a description of the theory and implementation in
tt(simrisc)).  For example, the standard deviations of the tt(Normal) carrier
distribution may look like this:
        verb(
    #                   value   spread  distr.                    
    stdDev:             21.1    .048    Normal  
        )
    The number of degrees of freedom of the chi+sups(2) distribution to use
tt(N - 1), where tt(N) is the number of cases that are used in a simulation.

COMMENT(
For proportions the binomial distrubution is used, ensuring that randomly 
selected proportions lie within the interval 0..1 (see section ref(BINOMIAL) 
for a description of the theory and implementation in
tt(simrisc)).  For example, the tt(Screening attendanceRate) specification may
be specified as:
        verb(
    attendanceRate:     .8     
        ) 
    Likewise, the tt(systematicError) (Modality, Mammo) and risks (e.g.,
tt(Normal Incidence Tomor) carrier are proportions).
)

