How to determine and correct bias in your data (Motivation Decay Part 6)

We now have a new hypothesis to test. How well does our porposed algorithm fit the data. As a quick reminder – we settled in part 4 on the likely algorithm of

T = 250 – 72 x log10 (CP)

We can compare the calculated to the observed values. Unfortunately we see a clear bias in the calculated values.

CP	Decay	Stdev	Hours	Hours calc	Diff (in seconds)
2744	10.000	0.003	8.00	2.44	na
2355	10.005	0.003	8.00	7.22	na
2320	9.999	0.023	8.00	7.68	1136
2310	9.785	0.013	8.18	7.82	1282
2303	9.675	0.011	8.27	7.91	1275
2143	7.606	0.008	10.52	10.17	1266
2029	6.547	0.003	12.22	11.88	1237
1913	5.684	0.003	14.07	13.72	1287
1827	5.168	0.003	15.48	15.15	1170
1454	3.533	0.004	22.65	22.30	1263
1095	2.543	0.004	31.46	31.16	1080
1019	2.355	0.014	33.96	33.41	1988
971	2.270	0.004	35.24	34.92	1148
969	2.269	0.003	35.26	34.98	976
942	2.202	0.028	36.32	35.87	1636
731	1.811	0.003	44.17	43.80	1325
657	1.684	0.003	47.50	47.14	1299
500	1.438	0.013	55.62	55.67	-201
359	1.198	0.012	66.78	66.03	2690
300	1.110	0.001	72.07	71.65	1522
100	1.127	0.041	70.98	106.00	na
95	1.179	0.051	67.86	107.60	na
10	1.111	na	72.00	178.00	na

On average all predicted times are too small. Taking the average off only the best data (stdev < 0.05) we get a positive bias of 1231 seconds. We can easily correct this in the function by rounding it to 1200 seconds and using 250 1/3 hours. This is also closer to the actual best fit.

Personally I opted to use 1269 seconds instead to get a value of 250.35 hours. So why do I think the final value is 250.35 hours?

Remember the Farfetched from Part 5. I had selected the Farfetched because of the CP it represented – and because I like rounding to close numbers. Mathematically I should have taken the closest CP to 296.6.

Using 72 and 250 I calculate 71.647 hours as expected values for the time it takes for the motivation to decay to the minimal value. But this misses the point from the game developer. What if the programmer was asked to keep it flat until 300 CP and from there onward motivation decays more rapidly.

As game developer I face the following function

72 = k – 72 x log10 (300)

Solving this equation for k results in 250.35273 – with 0.35273 hours being 1269 seconds.

I therefore propose that the final answer to the motivation decay is

T(max) = 250.35 hours – 72 hours x log10 (CP)

Below 300 CP T(max) is fixed to 72 hours. Above 2320 CP t(max) is fixed to 8 hours. We can now use this information to calculate the hourly decay as 80 / T(max)

CP range	Hourly decay in %
10-300	1.11% (or more precice 80/72 %)
300-2322	80 / ( 250.35 hours – 72 hours x log10 (CP) )
2323+	10%