# The zero-sum-power issue in the `hyper2` package

#### Robin K. S. Hankin

To cite the `hyper2` package in publications, please use Hankin (2017).

The powers of a `hyper2` object generally sum to zero. This is implicit in Bradley-Terry type likelihood functions in which probabilities are generally ratios of linear combinations of strengths. Here I discuss this observation in the context of `hyper2` package idiom. Usually, powers not summing to zero is an indication of a programming bug. For example, consider the following likelihood function, drawn from the `chess` dataset:

\[ \frac{p_1^{30}p_2^{36}p_3^{22}}{ \left(p_1+p_2\right)^{35} \left(p_2+p_3\right)^{35} \left(p_1+p_3\right)^{18} } \]

For a time, one package documentation file contained the following `hyper2` idiom purporting to create the `chess` likelihood function:

    chess <- hyper2()
    chess["Topalov"] <- 30
    chess["Anand"  ] <- 36
    chess["Karpov" ] <- 22
    chess[c("Topalov","Anand" )] <- 35  # bug!  should  be -35
    chess[c("Anand","Karpov"  )] <- 35  # bug!  should  be -35
    chess[c("Karpov","Topalov")] <- 18  # bug!  should  be -18

However, the above commands include an error [the sign of the denominator is incorrect]. The package includes two mechanisms to detect this type of error. Firstly, the print method (by default) detects such unbalanced likelihood functions and gives a warning:

    chess

    ## Warning in print.hyper2(x): powers have nonzero sum

    ## log(Anand^36 * (Anand + Karpov)^35 * (Anand +
    ## Topalov)^35 * Karpov^22 * (Karpov + Topalov)^18 *
    ## Topalov^30)

and secondly, function `loglik()` traps nonzero power sums:

    loglik(equalp(chess),chess)

    ## Error in loglik_single(p, H, log = log): sum(powers(H)) == 0 is not TRUE

The correct idiom would be

    chess[c("Topalov","Anand" )] <- -35
    chess[c("Anand","Karpov"  )] <- -35
    chess[c("Karpov","Topalov")] <- -18
    chess

    ## log(Anand^36 * (Anand + Karpov)^-35 * (Anand +
    ## Topalov)^-35 * Karpov^22 * (Karpov + Topalov)^-18 *
    ## Topalov^30)

    loglik(equalp(chess),chess)

    ## [1] -60.9969519

See how the print method gives immediate assurance that its argument is indeed balanced (and besides, unbalanced likelihood functions cannot even be evaluated with `loglik()`). It is natural to suggest including a check in the creation method—`hyper2()`—which would prevent the creation of a hyper2 object with unbalanced powers. However, this approach is not consistent with the package; consider the following situation. Suppose we wish to incorporate a new observation into `chess`, specifically that Anand played Karpov once, with Anand winning. We might proceed as follows:

    chess["Anand"] %<>% inc
    chess[c("Anand, Karpov")] %<>% dec(2)
    chess

    ## Warning in print.hyper2(x): powers have nonzero sum

    ## log(Anand^37 * (Anand + Karpov)^-35 * (Anand +
    ## Topalov)^-35 * Anand, Karpov^-2 * Karpov^22 * (Karpov
    ## + Topalov)^-18 * Topalov^30)

However, after the increment, but before the decrement, `chess` has a nonzero power sum, pending addition of another term. At this point, `chess` is unbalanced; its nonzero power sum is an indicator that it is a temporary object. That’s OK as long as we remember to add the denominator (as carried out in the next line) which would mean dividing by `(Anand+Karpov)^2`, thereby restoring the zero power sum. If we forget to do this, the print method gives us a warning, and indeed `loglik()` returns an error, which should prompt us to check the coding.

## The print method

The `hyper2` print method is sensitive to option `give_warning_on_nonzero_power_sum`. If `TRUE` (the default), a warning is issued if the powers have nonzero sum. This is usually appropriate. If the option is `FALSE`, the warning is suppressed. Note that the intermediate likelihood functions in the `chess` example are not printed (or indeed evaluated), so unbalanced likelihood functions are permitted, but only ephemerally.

## Function `balance()`

Sometimes it is convenient to accommodate the numerator terms all together and enforce the zero power sum at the end. This is accomplished by `balance()`:

    H <- hyper2()
    H['a'] %<>% inc(5)
    H['b'] %<>% inc(2)
    H['c'] %<>% inc(9)
    H %<>% balance
    H

    ## log(a^5 * (a + b + c)^-16 * b^2 * c^9)

## References

Hankin, R. K. S. 2017. “Partial Rank Data with the `hyper2` Package: Likelihood Functions for Generalized Bradley-Terry Models.” The R Journal 9 (2): 429–39.