library(knitr) pdf.options(pointsize=12) oldopt <- options(digits=4, formatR.arrow=FALSE, scipen=999, width=70) opts_chunk$set(comment=NA, rmLeadingLF=TRUE, tidy.opts = list(replace.assign=FALSE), fig.align='center', crop=TRUE) knitr::knit_hooks$set(crop = knitr::hook_pdfcrop) read_chunk("discrete-code.R")

if("lme4" %in% (.packages())) detach("package:lme4", unload=TRUE) req_suggested_packages <- c("gamlss") noneMiss <- suppressMessages(requireNamespace('gamlss', quietly = TRUE)) nogamlss <- !noneMiss if (nogamlss) { message("This vignette requires the `gamlss` package, that is not available/installed.") message("Code that requires this package will not be executed.") }

op <- options(width=90) knitr::opts_chunk$set( collapse = TRUE )

Consider first the Bernoulli distribution. For this, there are two possible outcomes, e..g. 0 and 1, or success and failure. For tosses of a fair coin, the probability of a Head is $\frac{1}{2}$. For tosses of an unbiased die, the probability of a six is $\frac{1}{6}$.

The binomial distribution, with size $n$, is the distribution of the total number of 1's (or 'successes') in $n$ independent Bernoulli trials with the same probability $\pi$. On the assumption that heads appear independently between coin tosses, with probability $\pi$ = 0.5 at each toss, the total number of heads in 10 tosses is binomial with size $n$ = 10 and $\pi$ = 0.5. The mean is $n \pi = 5$, while the variance is $n \pi (1-\pi) = 2.5$ .

There is no necessary reason why Bernoulli trials must be independent, or why the probability should be the same for all trials. As an example, if insects are exposed to a fumigant that is not inevitably fatal, it is unlikely that the probability of death will be the same for all insects. This has led to interest in other distributions, able to model a wider variety of types of data. In this vignette, the primary concern is with such alternatives.

Event processes lead to the Poisson distribution and its generalization. For an event process (e.g., radioactive decay events), the number of counts in any time interval will be Poisson if:

- Events occur independently -- the occurrence of one event does not change the probability of a further event, and
- The rate $\lambda$ at which events occur is constant.

Thus, in radioactive decay, atoms appear to decay independently (and emit ionizing radiation), at a rate that is the same for all atoms. The mean is $\lambda$, which is also the variance. If the sample mean and the sample variance differ only by statistical error, data are to this extent consistent with a Poisson distribution.

Note that the Poisson distribution with rate $\lambda$ is the limiting distribution of the number of events (counting 1 or "success" as an event) for a binomial distribution as $n$ goes to infinity and $\pi$ goes to zero, with the binomial mean constant at $n \pi = \lambda$. It is, then, a limiting case of the binomial distribution.

Both for the binomial and for the Poisson, one parameter
determines both the mean and the variance. This limits the
data for which they can provide useful models, so that it
becomes necessary to look for alternatives. The most commonly
implemented alternative to the binomial is the betabinomial.
There are a number of alternatives to the Poisson that have
been widely implemented, with the negative binomial is the
best known. See @rigby2019distributions for details of
distributions that are implemented in the `gamlss`

package.

For each distribution, there are four functions, with names
whose first letter is one of `d`

(**d**ensity or,
for discrete distributions, probability),
`p`

(cumulative **p**robability),
`q`

(**q**uantile), and `r`

(generate a
**r**andom sample). For an example, look ahead to
the discussion of the binomial distribution in the next
subsection. The following, using functions in the `stats`

package that is by default available in R sessions,
demonstrates the `d`

/`p`

/`q`

/`r`

nomenclature for the use
of the binomial distribution to model the probability of
0, 1, 2, or 3 heads, in (`size`

) 3 tosses, with probability
(`prob`

= 0.5) of a head. The functions `dbinom()`

and
`pbinom()`

take number $x$ of `heads`

as their first
argument, and return (for `dbinom()`

) a probability
or (for `pbinom()`

) a cumulative probability:

Observe the different names, in the two cases, for the argument that gives the number of `successes.

The inverse function to `pbinom()`

is `qbinom()`

.
This takes as first argument a cumulative probability `p`

,
and returns the smallest value of $x$ such that
`pbinom(p)`

$\leq$ `p`

.

The result is unchanged if

- 0.125 is replaced by 0, or by any quantity between 0 and 0.125
- 0.5 is replaced by 0.126, or by any quantity $>$0.125 and $\leq$ 0.5
- 0.875 by 0.501, or ...
- 1.0 by 0.876, or ...

The inverse function, which determines quantiles, is defined to be 0 for any $p \leq 0.125$, stepping up to 1 for $0.875 < p \leq 1.0$. Figure \@ref(fig:plot-gph12) shows the (cumulative) distribution function, with the quantile function alongside:

cap1 <- "Panel A shows the (cumulative) distribution function for the binomial. Panel B shows the inverse function, i.e., it plots the quantiles."

plot(gph1, position=c(0,0,0.485,1), more=TRUE) plot(gph2, position=c(.515,0,1,1), more=TRUE)

We would like to compare quantiles of a fitted distribution with quantiles of the data, as a mechanism for checking whether a model is a good fit to the data. For this, we prefer quantiles to be on a continuous dispersion. A complication, for the present data, and for other such discrete data, is that each of the horizontal lines in Figure \@ref(fig:plot-gph12)B corresponds to a range of probabilities, thus:

Functions such as `qbinom()`

return the point that,
moving from the vertical to the horizontal scale, is at
the upper end of the relevant horizontal scale.
Quantiles that incorporate a random point along the
relevant horizontal line are preferable for use in model
diagnostics, preferably repeating any check with several
different sets of such ``randomized quantile residuals.''
Depending on the number $x$ of diseased plants,
choose $u$ to be a uniform random number in the
corresponding interval shown above. If $k=0$, choose
$u$ to be a uniform random number in the interval [0,0.125],
and so on.

Then, if the model is correct, $u$ will be uniformly distributed on the interval $0 <= u <=1$. Given a set of values $u_i$, with $i=1, \ldots m$, these might be plotted against quantiles of the uniform distribution on the unit interval. It is, however, usual to transform to a standard normal distribution quantile scale. Thus 0.2 will translate to -0.84, 0.5 to 0, and 0.8 to 0.84.

Thus transformed, the quantiles can be plotted against quantiles of the normal distribution. Where a model has been fitted, the process is applied at each fitted value, generating "randomized quantile residuals." The "randomized quantile residuals" are residuals, on a normal quantile scale, from the median of the fitted distribution.

The `glm()`

function in the `stats`

package allows
`quasibinomial`

(and `quasipoisson`

)
families. This is not a formally defined distribution.
Instead, they fit just as for the `binomial`

or `poisson`

,
but estimate a `dispersion`

from the fitted model
that allows the variance to be larger (or, possibly, smaller)
than the respective binomial or poisson variance.

The `gamlss`

package implements, in each case, several
alternatives. Parameters follow naming conventions that
are different from those used for the `stats`

package's
binomial family functions. The location parameter `prob`

becomes `mu`

for `gamlss`

functions, while `size`

becomes `bd`

(= 'bound'). The `gamlss.dist`

package has
an accompanying pdf "The gamlss.family distributions"
that describes the distributions that the package makes
available.

Bernoulli trials may not be independent, and/or the
probability may change from one trial to the next.
This is a common situation, which has not had the
attention that it merits in the scientific literature.
In contexts where it is thought plausible that
the variance is a constant multiple $\Phi$
of the binomial variance, use of the `quasibinomial`

model fitting strategy has been common.
Quasibinomial fits proceed by fitting a binomial
distribution, then multiplying the variance by a
'dispersion' factor $\Phi$ that is estimated from
the data. With $\Phi$ thus defined (and in this
context termed the 'dispersion'), the variance
for the number $x$ of `successes' out of $n$ is
$n \pi (1-\pi) \Phi$.

Alternatives to the binomial that are implemented in the
`gamlss`

package are the betabinomial and the double binomial.
These both have the binomial ($\Phi = 1$) as a limiting case.
In the `gamlss`

implementation, the parameters are `mu`

and
`sigma`

, while the `glmmTMB`

betabinomial implementation
has `mu`

and `phi`

, where $\phi$ = $\sigma^{-1}$. Both
`sigma`

and `phi`

are described as "dispersion parameters"
that, together with `mu`

( = $\pi$), determine the variance.
The relationship to the variance is in general different
for different distributional families.

In addition, there are zero-inflated versions of all the
distributions noted, and zero-adjusted versions
of all except the double binomial. These have a further
parameter, named `nu`

in the `gamlss`

implementation, that
is described as a ‘shape’ parameter.

An insightful way to
relate the different parameterizations of the betabinomial
is to express the dispersion parameter as a function of
the intra-class correlation $\rho$. A positive correlation
leads to more homogeneous responses within replicates, and
manifests itself in greater between replicate differences,
leading to a dispersion index $\Phi$ that is greater than
one. Then:
$$
\begin{aligned}
\rho &= \dfrac{\sigma}{\sigma+1} \quad \mbox{(}\sigma
\mbox{ is the dispersion parameter in gamlss)}\
&= \dfrac{1}{\phi+1} \quad \mbox{(}\phi
\mbox{ is the dispersion parameter in glmmTMB)}
\end{aligned}
$$
The dispersion index (multiplier for $n \pi(\pi-1)$) is then
$$
\Phi = 1 + (n-1) \rho = \dfrac{1+ n \sigma}{1 + \sigma}
= \dfrac{\phi + n}{\phi + 1}
$$
I am not aware of any such simple formulae for the double
binomial. The double binomial allows for dispersion indices
that can be less than as well as greater than one.
For values of `sigma`

for the double binomial, that are between 0.1
and 2.8, for $n$ = 10 and $\pi$ = 0.5, `sigma`

never
differs from the multiplier $\Phi$ for the binomial variance
by more than 2%.

The betabinomial can be implemented in a manner
that allows an intra-class correlation $\rho$ that is
somewhat less than 0, and hence a dispersion index
$\Phi$ that is somewhat less than one. See
@prentice1986binary. However, most implementations
require $\Phi >= 1$ or (as for **glmmTMB**) $\Phi > 1$.

For the betabinomial, again taking $\Phi$ to be the multiplier for the binomial variance: [ \sigma = \frac{\Phi-1}{n - \Phi}; \qquad \Phi = 1 + \frac{(n-1)\sigma}{\sigma+1} ]

Figure \@ref(fig:cfDBI-BB) compares the binomial distribution
with `size`

$n$ =10, and probability $\pi$ = 0.5, with the
dispersion parameter in each case chosen so that the dispersion
factor is $\Phi = 2.0$ in Panel A, and $\Phi = 4.75$ in Panel B.

message("Subsequent code that requires `gamlss` will not be executed.")

cap2 <- "Panel A compares the double binomial (DBI) and the betabinomial (BB) distribution with the dispersion parameter (DBI: $\\sigma = 2.0$; BB: $\\sigma = 0.125$) in each case chosen so that the dispersion index is $\\Phi = 2.0$. Panel B repeats the comparison with the dispersion parameters (DBI: $\\sigma = 8.27$; BB: $\\sigma = \\frac{5}{7}$) chosen so that $\\Phi = 4.75$. The binomial distribution ($\\Phi$ = 1), is shown for comparison, in both panels. As $\\Phi$ increases, the change in shape needed to accomodate the increased variance becomes more extreme."

Note also the correlated binomial, implemented in the
`fitODBOD`

package, but with limited functionality
for working with the fitted model. It is noted here in
order to reinforce the point that there are multiple
alternatives to the betabinomial.

The betabinomial is a generalization, allowing continuous
parameter values, of the Pólya urn model. An urn holds
$\alpha$ red balls and $\beta$ blue balls. In $n$
(= `size`

or `bd`

) draws, each ball that is withdrawn is
replaced, prior to the next draw, by two balls of the same
color. The effect is to move probability away from
the mean or mode, and towards the extremes.
See https://en.wikipedia.org/wiki/P%C3%B3lya_urn_model

R's `glm()`

function offers the option of a quasibinomial error.
Specification of a quasibinomial error has the consequence that
the model is fitted as for a binomial distribution, with the
the binomial variance $n \pi (1- \pi)$ then multiplied by a
constant factor $\Phi$ that is usually estimated using the
Pearson chi-squared statistic. For the betabinomial, the multiplier
is $\Phi = 1+(n-1)\rho$, i.e., it increases with $n$. This is an
important difference.

Figure \@ref(fig:cfDBI-BB) highlighted the extent to which the assumption of a distribution that has a binomial-like shape will be seriously wrong, if the dispersion index $\Phi$ is substantially greater than one. Whatever the distribution, if $\Phi >> 1$, the probability is pushed out towards the extremes, in ways that are sufficient to multiply the variance by the relevant factor $\Phi$.

Figure \@ref(fig:plot-binAB) compares observed with fitted binomial counts, for two datasets. The data in Figure \@ref(fig:plot-binAB)A appear close to binomial. That in Figure \@ref(fig:plot-binAB)B shows clear differences from the binomial.

cap3 <- "In both panels, the vertical black lines show fitted values when a binomial distribution is fitted to the data. The data in Panel A has a distribution that is close to binomial. That in Panel B appears to deviate from binomial."

plot(gph1, position=c(0,0,0.485,1), more=TRUE) plot(gph2, position=c(.515,0,1,1), more=FALSE)

The Panel A data are the numbers of heads in 200 sets of ten coin tosses:

htab

The data can be obtained from the `testDriveR`

package, thus

The Panel B data are the numbers of diseased plants, out of 6, in 62 field quadrats.

tastab

The data can be obtained from the `testDriveR`

package, thus

For data such as here, a poor fit to a binomial model is to be expected. The probability of disease is likely to vary from one quadrat to another, with some clustering of diseased plants within quadrats.

The code that now follows was used to obtain binomial, double binomial,
and betabinomial fits for the data, stored in the data frame `diseased`

,
that were used for Figure \@ref(fig:plot-binAB)B.
The fit is in each case handled as a special case of fitting a
regression model. The only parameter is a location parameter
--- hence the `~1`

.

Now examine, and compare, diagnostic plots for the binomial
(Figure \@ref(fig:cfsim)), and for the betabinomial (Figure
\@ref(fig:cfq2)) model. Each plot is based on six sets of
randomized quantile residuals (`howmany=6`

is the default),
while the setting `plot.type="all"`

for Panels A and B has
the effect that all points are shown (in gray), while the
medians are shown as solid black dots.
Panel B (a 'worm plot') is a detrended version of Panel A,
with the dashed curves marking out 95% confidence bounds.

Figure \@ref(fig:cfsim) shows diagnostic plots for a binomial distribution:

cap4 <- "Diagnostic plots of randomized quantile residuals (identified as sample quantiles), for a _binomial model_ fitted to the plant disease data. Panel B (a 'worm plot') is a detrended version of Panel A, with the dotted curves marking out 95% confidence bounds."

Figure \@ref(fig:cfq2) shows diagnostic plots for a betabinomial distribution:

cap5 <- "Diagnostic plots of randomized quantile residuals, for a __betabinomial__ model fitted to the plant disease data."

The worm plots give the clearest picture. Figure \@ref(fig:cfsim)B makes it clear that the data is not binomial, while Figure \@ref(fig:cfq2)B indicates that the data are broadly consistent with a betabinomial distribution.

Code is:

Replace `doBI`

by `doBB`

, for the code for the betabinomial diagnostic
plots.

The AIC statistic can be used for a theoretically based
comparison between model fits. As with other such
'information' statistics, the AIC statistic is not
designed to provide statistical tests. The following
uses the AIC statistic to compare the three models that
have been fitted -- the binomial, the betabinomial,
and the double binomial (for which diagnostic plots
have not been shown.) The values given in the `dAIC`

column are increases from the best fitting model.

The comparison favors the betabinomial, by a smallish margin.

Do the differences matter? That depends on the use that will be made of the results. If the use of the model depends on accurately predicting the proportion of diseased samples, the differences clearly will matter. A confidence interval, e.g., for a difference between two different sample areas, may be acceptably accurate provided that normal approximations are used for the sampling distributions of the two means, and the variances are calculated from the data. For this, we are relying on Central Limit Theorem effects (see ??) to bring the sampling distribution of the mean close to the normal.

Data from large families indicates that variation in the
proportion of males is greater than would be expected for a
binomial distribution. The dataset `qra::malesINfirst12`

,
from hospital records in Saxony in the nineteenth century, gives
the number of males among the first 12 children of family size 13
in 6115 families. The probability that a child will be male
varies, within and/or between families.
(The 13th child is ignored to counter the effect of families
non-randomly stopping when a desired gender is reached.) Data is:

The following fits the model with binomial errors.

For the betabinomial (`doBB`

), replace `family=BI`

by `family=BB`

.
For the double binomial (`doDBI`

), replace `family=BI`

by `family=DBI`

.

Fitted probabilities for the betabinomial can, if required, be
calculated and added to the data frame `maleFit`

thus:

The code can be modified in the obvious way to add fitted values for the binomial and/or the double binomial.

cap6 <- "Data, from hospital records in Saxony in the nineteenth century, gave the number of males among the first 12 children in families of size 13, in 6115 families. Panel A shows a worm plot for the model that fitted a binomial distribution. Panel B repeats the worm plot, now for the model that fitted a betabinomial distribution."

Figure \@ref(fig:rqmales) shows worm plots for the binomial and betabinomial models:

The Panel A plot, with points all lying close to a line, indicates that data follow a binomial-like pattern of variation. The upward slope of the line indicates that points are more dispersed than for a binomial distribution. The Panel B plot indicates that data are consistent with the betabinomial fit from which this plot was generated.

\noindent Code for the plots is:

AIC statistics for the three models are:

The double binomial is by a small and inconsequential margin the preferred model.

Note that for data with a 0/1 response, neither the
`quasibinomial`

nor the `betabinomial`

or other such
model can be fitted, unless the 0/1 responses can be
grouped to give repeated $x$ out of $n$ sets of results.
With a 0/1 response, the residual deviance is a function
of the fitted parameters, and gives no information on
the variance. See @McCullagh, Section 4.5.1.

We now move to examine models for count data, initially fitting Poisson distributions.

Figure \@ref(fig:plot-poisAB) compares observed counts with fitted poisson counts, for two datasets.

The first set of counts are the @rutherford1908electrical polonium radioactive decay counts that give the number of scintillations in 2608 1/8 minute intervals.

The second set of counts are numbers of accidents among 414 machinists from a three months study conducted around the end of WWI

The following code calculates the means, for the two datasets, then calculating fitted values for fits to the Poisson distribution

cap7 <- "In both panels, the vertical black lines show fitted values when a Poisson distribution is fitted to the data. The data in Panel A has a distribution that appears close to poisson. That in Panel B appears to deviate from Poisson. Notice that the variance for the poisson fit (`r round(mumach,2)`) is smaller than the variance that is estimated from the data (1.01) by a factor of a little more than 2."

plot(gph1, position=c(0,0,0.485,1), more=TRUE) plot(gph2, position=c(.515,0,1,1), more=FALSE)

In Panel A of Figure \@ref(fig:plot-poisAB), the data appear consistent with a Poisson distribution. In Panel B, the Poisson fit underestimates the number of zeros, and overestimates the number of ones. In principle, the underestimation of the number of zeros can be fixed by fitting a zero-inflated Poisson distribution. It turns out, that while this improves the fit, the fit is still less than satisfactory. The accident risk is likely to vary between machinists, invalidating the constant rate assumption required for a Poisson distribution.

Among alternatives to the Poisson that allow for departures
from the Poisson assumptions of constant rate $\lambda$ and
independence between the events that are counted, the
negative binomial has been the most widely used, in part
because it was for a time the only alternative that was
widely implemented. The `gamlss`

and other R packages
now implement a number of alternatives that can be used
with count data.

Modify `family=poisson`

as required for `doNBI`

(`family=NBI`

),
`doPIG`

(`family=PIG`

), and `doZIP`

(`family=ZIP`

)

Now compare the AIC statistic between these three models:

The zero-inflated Poisson does do a better job than the Poisson, but is much less satisfactory than a negative binomial or a Poisson inverse gamma model. As there is little to choose between the negative binomial and the Poisson inverse gamma model, the more widely implemented negative binomial fit is likely to be preferred.

cap8 <- "Worm plots for the Poisson, for the zero-inflated Poisson, and for the negative binomial."

The following demonstrates the calculation of the fitted frequencies for the negative binomial, Poisson inverse gamma, and zero inflated Poisson fits. The zero inflated Poisson ensures that the estimated number of zeros exactly equals the observed number, but gets the relative numbers of frequencies 1, 2 and 3 badly wrong.

options(oldopt) knitr::knit_exit()

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