[PHP] 5 Star Rating - Lower bound of Wilson score confidence interval for a Bernoulli parameter

wilson.php

<?php

/*
|--------------------------------------------------------------------------
| 5 Star Rating
|--------------------------------------------------------------------------
|
| Lower bound of Wilson score confidence interval for a Bernoulli parameter (0.9604)
|
| See: 
|  * http://www.evanmiller.org/how-not-to-sort-by-average-rating.html
|  * https://gist.github.com/richardkundl/2950196
|  * https://onextrapixel.com/how-to-build-a-5-star-rating-system-with-wilson-interval-in-mysql/
|
*/


function score($positive, $negative) {
    return ((($positive + 1.9208) / ($positive + $negative) - 1.96 * sqrt((($positive * $negative) / ($positive + $negative)) + 0.9604) / ($positive + $negative)) / (1 + 3.8416 / ($positive + $negative)));
}

function fiveStarRating($one, $two, $three, $four, $five) {
    $positive = $two * 0.25 + $three * 0.5 + $four * 0.75 + $five;
    $negative = $one + $two * 0.75 + $three * 0.5 + $four * 0.25;

    return score($positive, $negative);
}

function fiveStarRatingAverage($avg, $total)
{
    $positive = ($avg * $total - $total) / 4;
    $negative = $total - $positive;

    return score($positive, $negative);
}

// Examples

echo fiveStarRating(10, 1, 2, 6, 90);     // 0.80390178246001
echo fiveStarRating(80, 1, 2, 6, 90);     // 0.46188074417216
echo fiveStarRating( 0, 1, 2, 6, 0 );     // 0.33136280289755
echo fiveStarRating(10, 1, 2, 0, 2 );     // 0.079648861762752

echo fiveStarRatingAverage(4.8000001907349, 10);        // 0.65545605272928

How do fiveStarRating and fiveStarRatingAverage compare in terms of the quality of the sorting? Do you have any data or examples?
Do you have a reference for the fiveStarRating algo, or is it rather arbitrary?
Thanks, and thank you for sharing this snippet.

I don't know how different the result would be. I'd expect fiveStarRating to be "better" and I'd use fiveStarRatingAverage only if I lost the details and only stored avg and count but I don't really know.

My feeling is that this should only be one criteria and that it's hard to say that 0.46188074417216 is definitely better than 0.46188074417301`.

I'd expect fiveStarRating to be "better"

Actually I proved they're equal. Let's say that I want to implement fiveStarRatingAverage with fiveStarRating's parameters:

function fiveStarRatingAverage2($one, $two, $three, $four, $five)
{
    // Doesn't compile, $avg, $total not defined
    $positive = ($avg * $total - $total) / 4;
    $negative = $total - $positive;

    return score($positive, $negative);
}

Let's complete the implementation:

function fiveStarRatingAverage2($one, $two, $three, $four, $five)
{
    $total = $one + $two + $three + $four + $five;
    $avg = ($one * 1 + $two * 2 + $three * 3 + $four * 4 + $five * 5) / $total;
    $positive = ($avg * $total - $total) / 4;
    $negative = $total - $positive;

    return score($positive, $negative);
}

Now let's inline $total and $avg into $positive:

function fiveStarRatingAverage2($one, $two, $three, $four, $five)
{
    $total = $one + $two + $three + $four + $five;
    $positive = (($one * 1 + $two * 2 + $three * 3 + $four * 4 + $five * 5) - ($one + $two + $three + $four + $five)) / 4;
    $negative = $total - $positive;

    return score($positive, $negative);
}

And simplify:

function fiveStarRatingAverage2($one, $two, $three, $four, $five)
{
    $total = $one + $two + $three + $four + $five;
    $positive = $two * 0.25 + $three * 0.5 + $four * 0.75 + $five;
    $negative = $total - $positive;

    return score($positive, $negative);
}

Voila! We got the same calculation for $positive. It's not difficult to figure out that $negative is also equal for both functions.