# Empirical Tower Drop Rate

## Introduction

During the last event (2 Santas), I decided to do a little experiment regarding drop rate. There are much better daemons that affect drop rate these days (remember the days when Konohana Sakuya’s 7% Drop rate boost was the best in game). The experiment aims to compare the drop rate outcomes with and without drop rate daemons, and then with one copy of the event drop helper (Star Atop Christmas Tree) also with and without drop rate daemons on the team.

## Methodology:

**Stage selection**: 4-10 of tower (final stage). The data was taken during the second half of the event. 1 copy of the ranking daemon was acquired. All the chapters were already completed with 3 stars, all the event daemons acquired.

**Play mode**: Auto, 4x speed. This would randomize skill killing, and would make it independent of player skill / style / skill combo.

**Accounts**: In order to get a larger number of data points, I made use of 2 different accounts.

**Farming team** – the teams were largely similar but not identical (unfortunately my accounts have similar but not identical cards). There are some other small variance between the non farming related assists, and team positioning, but as you can see later they are irrelevant to the drop rate.

**Account 1**: Kintaro [Yokozuna], Orpheus, Dead Sea Scrolls, Cintamani + Cintamani Ally + Reserves Lady Yodo and Hermes / Mara with Sherlock

**Account 2**: Kintaro [Yokozuna], Sola, Dead Sea Scrolls, Cintamani + Cintamani Ally + Reserves Lady Yodo and Hermes / Mara with Sherlock

There were some minor differences in the bonded assists but are largely irrelevant. The only bond that I check is Sherlock, since he affects drop rate.

If the team dies during the run I make use of a **revive charm** to continue the run. Back to back runs are recorded. No resetting even when extremely poor drops are encountered.

**1x SE** runs were done to get more data points so that I would not break the bank in terms of item use.

I collected **4 sets of data** for each account, adjusting only the reserve and the ally assist. I check the bonds on the ally Cintamani for Sherlock bonds as needed for each type of run.

- [0 Helper Full XP] No event drop helper; Lady Yodo + Hermes Reserve + Cintamani Ally without Sherlock bonded
- [0 Helper 45% Drop] No event drop helper; Lady Yodo + Mara with Holmes bonded + Cintamani Ally with Sherlock bonded
- [1 Helper Full XP] 1 event drop helper; Lady Yodo + Hermes Reserve + Cintamani Ally without Sherlock bonded
- [1 Helper 45% Drop] 1 event drop helper; Lady Yodo + Mara with Holmes bonded + Cintamani Ally with Sherlock bonded

I recorded the individual drops for each run in excel. I calculated the average and variance and sample size for each data set from each account.

**Statistical Analysis**

The P value comparing the data from the 2 accounts for each data set was calculated to test whether the results from the different accounts were significantly different from each other. (They were not). This is the calculator that I used.

The data from both accounts were combined, and a combined mean, variance, standard deviation and sample size was generated.

The combined data sets were compared against each other and P value calculated to test whether the results were significantly different between data sets.

## Results

Here is the raw data in terms of data points:

Data Set | 0 Helper 0 Drop Rate | 0 Helper 45% Drop Rate | 1 Helper 0 Drop Rate | 1 Helper 45% Drop Rate | ||||
---|---|---|---|---|---|---|---|---|

Account | 1 | 2 | 1 | 2 | 1 | 2 | 1 | 2 |

– | 114 | 117 | 166 | 182 | 133 | 117 | 200 | 242 |

– | 118 | 114 | 155 | 153 | 176 | 169 | 236 | 194 |

– | 97 | 119 | 167 | 152 | 164 | 125 | 211 | 251 |

– | 114 | 96 | 172 | 161 | 166 | 185 | 215 | 241 |

– | 115 | 110 | 154 | 169 | 161 | 130 | 177 | 202 |

– | 110 | 117 | 159 | 149 | 156 | 152 | 167 | 260 |

– | 100 | 102 | 167 | 149 | 165 | 139 | 207 | 205 |

– | 123 | 129 | 157 | 144 | 173 | 114 | 197 | 214 |

– | 95 | 96 | 168 | 163 | 168 | 161 | 186 | 201 |

– | 112 | 105 | 154 | 168 | 200 | 129 | 262 | 268 |

– | 124 | 116 | 148 | 171 | 131 | 144 | 205 | 241 |

– | 119 | 116 | 173 | 154 | 154 | 184 | 243 | 223 |

– | – | 122 | 153 | 162 | 142 | 128 | 188 | 221 |

– | – | – | 153 | 142 | 139 | 121 | 241 | 233 |

– | – | – | 160 | 163 | 145 | 136 | 251 | 238 |

– | – | – | 174 | 150 | 137 | – | 223 | 192 |

– | – | – | 171 | 150 | – | – | 248 | 224 |

– | – | – | 160 | 153 | – | – | 229 | 213 |

– | – | – | 162 | 171 | – | – | 284 | 206 |

– | – | – | 167 | 155 | – | – | 189 | 219 |

– | – | – | 152 | – | – | – | 240 | 200 |

– | – | – | 167 | – | – | – | 180 | 248 |

– | – | – | 163 | – | – | – | 212 | 192 |

– | – | – | – | – | – | – | 201 | 212 |

– | – | – | – | – | – | – | – | 205 |

– | – | – | – | – | – | – | – | – |

Mean | 111.8 | 112 | 162 | 158 | 157 | 142 | 216 | 222 |

Variance | 85.7 | 91.4 | 55.4 | 103 | 343 | 536 | 880 | 482 |

Standard Deviation | 9.25 | 9.56 | 7.45 | 10.2 | 18.5 | 23.1 | 29.7 | 22.0 |

Sample Size | 12 | 13 | 23 | 20 | 16 | 15 | 24 | 25 |

Data Set | 0 Helper 0 Drop Rate | 0 Helper 45% Drop Rate | 1 Helper 0 Drop Rate | 1 Helper 45% Drop Rate |

P value between Account 1 and Account 2 | 0.8996 | 0.8617 | 0.0609 | 0.4665 |

Significant difference between Accounts? | No | No | Somewhat | No |

Looking at the raw data there is a very small difference between the drop rate results between the accounts. The P values for the comparisons between the accounts are all quite high with the exception of the 1 Helper 0 Drop Rate set. I think there were 2 extremely low outliers in the data set for account 2, which caused the numbers to turn out that way. Despite that, I think it is fair to make use of the combined mean, variance and sample size to compare between the different data sets.

CombinedData Set | 0 Helper 0 Drop Rate | 0 Helper 45% Drop Rate | 1 Helper 0 Drop Rate | 1 Helper 45% Drop Rate |

Mean | 112 | 160 | 149.8 | 219.1 |

Variance | 92.4 | 83.1 | 476.1 | 670.7 |

Standard Deviation | 9.61 | 9.17 | 21.8 | 25.9 |

Sample Size | 25 | 43 | 31 | 49 |

I entered the combined data into the calculator and generated P values, comparing the data sets against each other:

- 0 Helper 0 Drop Rate vs 0 Helper 45% Drop Rate: P<0.0001 – Significantly different
- 0 Helper 0 Drop Rate vs 1 Helper 0 Drop Rate: P<0.0001 – Significantly different
- 0 Helper 0 Drop Rate vs 1 Helper 45% Drop Rate: P<0.0001 – Significantly different
- 0 Helper 45% Drop Rate vs 1 Helper 0 Drop Rate: P=0.0074 – Significantly different
- 0 Helper 45% Drop Rate vs 1 Helper 45% Drop Rate: P<0.0001 – Significantly different
- 1 Helper 0 Drop Rate vs 1 Helper 45% Drop Rate: P<0.0001 – Significantly different

The P values of the comparisons between the data sets are all very low, which means that there is a significant difference between the data sets. Between 1 Helper 0 Drop rate and 0 Helper 45% Drop rate the difference is of lower significance, but still below the commonly used limits of 0.1 (90% CI), 0.05 (95% CI) and 0.01 (99% CI).

## Discussion

The statistical analysis show that the difference in the data collected from the 2 accounts is not significant. The combined data set was used so as to be able to make comparison between sets with a larger sample size. When the different data sets were compared to each other they were shown to be significantly different from each other.

#### If we make use of the simple means to calculate the drop rate per SE:

0 Helper 0 Drop Rate = 1.4

0 Helper 45% Drop Rate = 2

1 Helper 0 Drop Rate = 1.87

1 Helper 45% Drop Rate = 2.74

#### Do these numbers match if we treat drop rate multiplicatively? In general, yes.

Example

1.4*1.45=2.03 (which is very close to the 2, the empirical result of 0 helper 45% drop rate)

1.4*1.33=1.86 (which is very close to 1.87, the empirical result of 1 helper 0 drop rate)

And when they are stacked

1.4*1.33*1.45= 2.67 (multiplicatively)

Vs

1.4*(1+0.33+0.45) =2.49 (additively)

Treating drop rate and drop helper effects multiplicatively rather than additively get us closer to 2.74, the empirical result of 1 helper 45% drop rate.

#### Drop Rate and the Event Helper affect drop rate in different ways.

The Drop Rate daemon increases the size of the mini boss’s drop. The event helper increases the rate of double drops from both brutes and mini bosses. The variance of the data sets with the event helper is higher because of this mechanism of increasing drops. Drop rate daemon does not seem to affect the variance as much. When you have both, you get larger drops from your event bosses, and the larger drops also have a chance of being doubled, hence the multiplicative effect of Drop Rate and the Event Helper.

#### Limits of Drop Rate Daemons

I arbitrarily made use of 45% drop rate daemon in my experiment. While it is possible to get more drop rate than the 45% I have used for this experiment, realistically one wants to have plenty of XP boost also, so only 1 slot was given to the best drop rate daemon in the game – Mara with Holmes bonded. To test limits, the current max drop rate would include: 2x Mara with Holmes (30% each), Cosimo de’ Medici (15%), Hymenaios (12%), He Shi Bi (10%), Konohana Sakuya (7%) and Treasure Ship (5%). Total 109% boost to drop rate. Unfortunately such a team is highly unlikely to survive 4-10 on auto without an excess of revival charms, and you sacrifice the use of really great XP boosters. The cap of the drop rate in the game is still unknown, and at this point it would be very difficult to test the limit with the 109% Drop Rate team empirically because of survival issues.

## Conclusion

I believe I conducted a reasonably well constructed and realistic experiment on drop rate daemons and drop rate assist and how they interact with each other. Granted there are limitations in the experiment – the sample size could always be larger, and the teams would ideally be absolutely identical right down to assists, bonds and positioning. However I do not believe that those factors are likely to make a difference in the outcome.

Previously I believed that once the event drop helper was acquired the effect on drop rate daemons would be diminished significantly, and it would be pointless to use drop rate daemons if one has drawn or traded the event helper. This experiment has persuaded me otherwise. The drop rate daemons and the event drop helpers work synergistically to improve drops. I think previous experience with drop helpers were of much lower percentage boosts hence the effect would have been less detectable.

## Tl;dr Learning Points

- Baseline Drop Rate after all event daemons are unlocked is 1.4 drops per SE on 4-10.
- Both Event helper and Drop Rate daemons affect the baseline drop rate multiplicatively.
- The Drop Rate daemons increase drop rate more consistently (has lower variance) than the Event Helper.
- When both Event helper and Drop Rate daemons are present, their effect is multiplicative to each other.

otogi: high math shit

Yay now an actual testing run of what ive been saying all this time 😀 glad it changed your opinion and good work!

(ah also while 109% will sound nice at that point you will have some diminishing returns as it is scripted to never truly reach 100%, for minions its made to be a 0>x<1 value while for the bosses… it should be able to go past this 100% since the fact that they drop is a guarantee so they are affected differently, lowest/highest i recorded so far is 3 and 34 drops, but sadly as im still lacking mara this needs further testing :/ )

Poll results don’t display very well on mobile or desktop versions of Chrome (that’s all I’ve tried). I think the selection names are too long. Thanks for your effort =) validated my assumptions about drop rates. I am fortunate enough to have Mara, Sherlock, and Medici, so I usually use them as well as Cinta and DSS(with Amy and Siren to help survive).

Yes I am aware that the text does not show up in full next to the bar graph. I need to make my options shorter.

Again, thank you for a solid, thorough, well-crafted article backed by hard numbers and data. I felt that putting Mara in seemed to give me a few more socks than not but thank you so much for turning that flimsy feeling into methodical math.

Also, again, I’d like to express undying gratitude for the existence and maintenance of the effective stats spreadsheet.

I stopped doing the stats spreadsheet like months ago? I only do my own personal one (which has MLB stats only), or were you referring to this one?

My use of drop rate and event drop help cards depends on my need of mochi and if any of the cards I can get with the drops look good. Though I guess you can never have enough mochi. I tend to load up with exp boosting cards and someone that can soak up enemy attacks.