Uncertainty in Standardization and Decomposition

Whence uncertainty?

• Unlike regression the Das Gupta method doesn’t really have a concept of an error term - all results are calculated algebraically (Das Gupta 1993)
• But we might still want to quantify uncertainty in the results of a standardization and decomposition
• Bootstrapping is a general purpose way to achieve this (Carsey and Harden 2014)
• It can also show where results might be sensitive to small changes in the data (typically if the absolute difference between the rates you are decomposing is small) - we will see an example of this later
• In this session we will introduce bootstrapping and describe the process to bootstrap a standardization and decomposition analysis

Part One: Bootstrapping

Bootstrapping

• Bootstrapping is a resampling method where you create an empirical distribution of estimates of your quantity of interest through resampling
• It is kind of a dumb strategy - it doesn’t try to be clever and use maths to calculate the appropriate uncertainty with appeals to asymptotics, it just reshuffles your data a bunch and re-runs your analysis on each resampled dataset
• The clever maths way assumes your data comes from a nice smooth distribution; bootstrapping makes an empirical distribution via resampling

An example of sampling with replacement

Bootstrapping

• There are different ways to achieve this
• Here we adopt a non-parametric bootstrap (Carsey and Harden 2014) where you sample with replacement from the original data and re-run the analysis on each bootstrap draw
• This is also what Li (2017) does in the Stata rdecompose command
• We will (briefly) introduce bootstrapping in general, and there is a worked example in the practical

Sampling with replacement example

Bootstrapping example

• In this example we want to calculate the mean and the standard error of the mean for a toy dataset

set.seed(387) # for reproducibility

results <- c(1, 3, 2.7, 10, 1)

results

[1]  1.0  3.0  2.7 10.0  1.0

mean(results)

[1] 3.5

Bootstrapping example

• Now we sample with replacement from the results to create a new set of results the same size as the original results

boot_1 <- sample(results, size = length(results), replace = TRUE)

data.frame(results = results, boot_1 = boot_1)

  results boot_1
1     1.0    1.0
2     3.0    1.0
3     2.7    3.0
4    10.0    1.0
5     1.0    2.7

data.frame(results = mean(results), boot_1 = mean(boot_1))

  results boot_1
1     3.5    1.7

Bootstrapping example

• And do it again…

boot_2 <- sample(results, size = length(results), replace = TRUE)

data.frame(results = results, boot_1 = boot_1, boot_2 = boot_2)

  results boot_1 boot_2
1     1.0    1.0      1
2     3.0    1.0      3
3     2.7    3.0      1
4    10.0    1.0      3
5     1.0    2.7      1

data.frame(results = mean(results), boot_1 = mean(boot_1), boot_2 = mean(boot_2))

  results boot_1 boot_2
1     3.5    1.7    1.8

Bootstrapping example

• … and again …

boot_3 <- sample(results, size = length(results), replace = TRUE)

data.frame(results = results, boot_1 = boot_1, boot_2 = boot_2, boot_3 = boot_3)

  results boot_1 boot_2 boot_3
1     1.0    1.0      1   10.0
2     3.0    1.0      3   10.0
3     2.7    3.0      1    1.0
4    10.0    1.0      3    1.0
5     1.0    2.7      1    2.7

data.frame(results = mean(results), boot_1 = mean(boot_1), boot_2 = mean(boot_2), boot_3 = mean(boot_3))

  results boot_1 boot_2 boot_3
1     3.5    1.7    1.8    4.9

Bootstrapping example

• … like 1000 times. Then we can see the range of results from the resampling.

boot_1000 <- sample(results, size = length(results), replace = TRUE)

data.frame(results = results, boot_1 = boot_1, boot_2 = boot_2, boot_3 = boot_3, boot_1000 = boot_1000)

  results boot_1 boot_2 boot_3 boot_1000
1     1.0    1.0      1   10.0         1
2     3.0    1.0      3   10.0         1
3     2.7    3.0      1    1.0         1
4    10.0    1.0      3    1.0        10
5     1.0    2.7      1    2.7        10

data.frame(results = mean(results), boot_1 = mean(boot_1), boot_2 = mean(boot_2), boot_3 = mean(boot_3), boot_1000 = mean(boot_1000))

  results boot_1 boot_2 boot_3 boot_1000
1     3.5    1.7    1.8    4.9       4.6

Bootstrapping example

Example

Try it for yourself

• Navigate to this website
• (Website made by AI, sorry)
• Play around with the different options

Boostrapping example

• With non-parametric bootstrapping you resample the data and then perform whatever analysis you like on each resample
• You then get an empirical distribution of results (remember the histogram!)
• Pros: this will (almost) always work, makes no assumptions about distributions or anything
• Cons: the amount of memory required scales with the number of units, your computer might run out of memory/catch on fire - if you are analysing population level data from India this approach will take a lot longer than if you are analysing population data from Scotland
• If you are planning to bootstrap some large data you will have to think about the practicalities (feel free to ask us if you are planning this!)

Bootstrapping standardization and decomposition

• Standardization and Decomposition often uses aggregate data
• To use the non-parametric bootstrap here you need¹ to expand the data out to unit-level data, then take the re-samples, then aggregate each re-sample back up to perform the standardization and decomposition

1. At least, I don’t know of any alternatives! And this is also what Li (2017) does. If there are other clever ways to do this let me know!

Bootstrapping in practice

• To do this in practice you need to create a function (in R; a ‘program’ in Stata) to essentially “uncount”/explode/expand the data into individual level data (where the number of rows = size of total population), then re-sample the unit-level data with replacement, then aggregate back up to the level of the original data
• Then you standardize and decompose each resampled, aggregated dataset

Bootstrapping in practice

1. ‘Explode’ the aggregate data
2. Resample the exploded data say 200 times¹
3. Aggregate each of the 200 datasets
4. Standardize and decompose 200 times
5. Summarise the results

1. There isn’t really a good answer for how many is ‘enough’ other than ‘it depends’. As a rule of thumb, the further into the tails you go the more draws you need to get a reliable estimate (see example on slide twenty here). People often use 1,000 but that’s just by convention

Exploding data

Aggregate counts

# A tibble: 2 × 3
  population n_people n_died
  <chr>         <dbl>  <dbl>
1 Scotland        500     20
2 England        6000    200

Individual People

     population died
1       England    0
2       England    0
3       England    0
4       England    0
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8       England    0
9       England    0
10     Scotland    0
11      England    1
12      England    0
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5894    England    0
5895    England    0
5896    England    0
5897    England    0
5898    England    0
5899   Scotland    0
5900    England    0
5901    England    0
5902    England    0
5903    England    0
5904    England    0
5905    England    0
5906    England    0
5907   Scotland    0
5908    England    0
5909    England    0
5910    England    0
5911    England    0
5912    England    0
5913    England    1
5914    England    0
5915    England    0
5916    England    0
5917    England    0
5918    England    0
5919    England    1
5920    England    0
5921   Scotland    1
5922    England    0
5923    England    0
5924    England    0
5925    England    0
5926   Scotland    0
5927    England    0
5928   Scotland    0
5929    England    0
5930    England    1
5931    England    0
5932    England    0
5933    England    0
5934    England    0
5935    England    0
5936    England    0
5937    England    0
5938   Scotland    0
5939    England    0
5940    England    0
5941    England    0
5942    England    0
5943    England    0
5944    England    0
5945    England    0
5946    England    0
5947    England    0
5948    England    0
5949   Scotland    0
5950    England    0
5951    England    0
5952    England    0
5953    England    0
5954    England    0
5955    England    0
5956    England    0
5957    England    0
5958    England    1
5959    England    0
5960    England    0
5961    England    0
5962    England    0
5963    England    0
5964    England    0
5965    England    0
5966    England    0
5967    England    0
5968    England    0
5969    England    1
5970    England    0
5971    England    0
5972    England    0
5973    England    0
5974    England    0
5975   Scotland    0
5976   Scotland    0
5977    England    1
5978   Scotland    0
5979    England    0
5980    England    0
5981    England    0
5982    England    0
5983    England    0
5984   Scotland    0
5985    England    0
5986    England    0
5987    England    0
5988    England    0
5989    England    0
5990    England    0
5991    England    0
5992    England    0
5993    England    0
5994    England    0
5995    England    0
5996    England    0
5997    England    0
5998    England    0
5999    England    0
6000    England    0
6001    England    0
6002   Scotland    0
6003    England    0
6004   Scotland    0
6005    England    0
6006    England    0
6007    England    0
6008    England    0
6009    England    0
6010    England    0
6011    England    0
6012    England    0
6013    England    0
6014    England    0
6015    England    1
6016    England    0
6017    England    0
6018    England    0
6019   Scotland    0
6020    England    0
6021    England    0
6022    England    0
6023    England    0
6024   Scotland    1
6025    England    0
6026    England    0
6027    England    0
6028    England    0
6029    England    0
6030    England    0
6031    England    0
6032    England    0
6033    England    0
6034    England    0
6035    England    0
6036    England    0
6037    England    0
6038    England    0
6039    England    0
6040    England    0
6041   Scotland    0
6042    England    0
6043    England    0
6044    England    0
6045   Scotland    0
6046    England    0
6047    England    0
6048    England    0
6049    England    0
6050    England    0
6051    England    0
6052    England    0
6053    England    0
6054    England    0
6055    England    0
6056    England    0
6057    England    0
6058    England    0
6059    England    0
6060    England    0
6061    England    0
6062    England    0
6063    England    0
6064    England    0
6065    England    0
6066    England    0
6067   Scotland    0
6068    England    0
6069    England    0
6070    England    0
6071   Scotland    0
6072    England    0
6073    England    0
6074    England    0
6075    England    0
6076    England    0
6077    England    0
6078    England    0
6079    England    0
6080    England    1
6081    England    0
6082    England    0
6083    England    0
6084    England    0
6085   Scotland    0
6086    England    0
6087   Scotland    0
6088    England    0
6089    England    0
6090   Scotland    0
6091    England    0
6092    England    0
6093    England    1
6094    England    0
6095    England    0
6096    England    0
6097    England    0
6098    England    1
6099    England    0
6100    England    0
6101    England    0
6102    England    0
6103    England    0
6104    England    0
6105    England    0
6106    England    0
6107    England    0
6108    England    0
6109    England    0
6110    England    0
6111    England    0
6112    England    0
6113    England    0
6114    England    0
6115    England    0
6116    England    0
6117    England    0
6118    England    0
6119    England    0
6120    England    0
6121    England    0
6122    England    0
6123    England    0
6124    England    0
6125    England    0
6126    England    0
6127    England    1
6128    England    0
6129   Scotland    0
6130   Scotland    0
6131    England    0
6132    England    0
6133    England    0
6134    England    0
6135    England    0
6136    England    0
6137   Scotland    0
6138    England    0
6139    England    0
6140    England    0
6141    England    0
6142    England    0
6143    England    0
6144    England    0
6145   Scotland    0
6146    England    0
6147    England    0
6148    England    0
6149   Scotland    1
6150    England    0
6151    England    1
6152    England    0
6153    England    0
6154    England    0
6155    England    0
6156    England    0
6157   Scotland    0
6158    England    1
6159    England    0
6160   Scotland    0
6161   Scotland    0
6162    England    0
6163   Scotland    0
6164    England    0
6165    England    0
6166    England    0
6167    England    0
6168    England    0
6169    England    0
6170    England    0
6171    England    0
6172    England    0
6173    England    0
6174    England    0
6175    England    0
6176    England    0
6177    England    0
6178    England    0
6179    England    0
6180    England    0
6181    England    0
6182   Scotland    0
6183   Scotland    0
6184    England    0
6185    England    0
6186    England    0
6187    England    0
6188    England    0
6189    England    0
6190    England    0
6191    England    0
6192    England    0
6193    England    0
6194    England    0
6195    England    0
6196    England    0
6197    England    0
6198    England    0
6199    England    0
6200    England    0
6201    England    0
6202    England    0
6203    England    0
6204    England    0
6205    England    0
6206    England    0
6207    England    0
6208   Scotland    0
6209    England    0
6210    England    0
6211    England    0
6212    England    0
6213    England    0
6214   Scotland    0
6215    England    0
6216    England    0
6217    England    0
6218    England    0
6219    England    0
6220    England    0
6221    England    0
6222    England    0
6223    England    0
6224   Scotland    0
6225    England    0
6226    England    0
6227   Scotland    1
6228    England    0
6229    England    0
6230    England    0
6231    England    0
6232    England    0
6233    England    0
6234    England    0
6235    England    0
6236    England    0
6237    England    0
6238    England    0
6239    England    0
6240    England    0
6241    England    0
6242    England    0
6243    England    0
6244    England    0
6245    England    0
6246    England    1
6247    England    0
6248    England    0
6249    England    0
6250    England    0
6251    England    0
6252    England    0
6253    England    0
6254    England    0
6255    England    0
6256    England    0
6257    England    0
6258    England    0
6259    England    0
6260    England    0
6261    England    0
6262   Scotland    0
6263   Scotland    0
6264    England    0
6265    England    0
6266    England    0
6267   Scotland    0
6268    England    0
6269    England    0
6270    England    0
6271    England    0
6272    England    0
6273    England    1
6274    England    0
6275    England    0
6276    England    0
6277    England    0
6278    England    0
6279   Scotland    0
6280   Scotland    0
6281    England    0
6282   Scotland    0
6283    England    0
6284    England    0
6285    England    0
6286    England    0
6287    England    0
6288    England    0
6289    England    0
6290    England    0
6291    England    0
6292    England    0
6293    England    0
6294   Scotland    0
6295    England    0
6296    England    0
6297    England    0
6298    England    0
6299    England    0
6300    England    0
6301    England    0
6302    England    0
6303    England    0
6304    England    0
6305   Scotland    0
6306    England    0
6307    England    0
6308    England    0
6309    England    0
6310    England    0
6311    England    0
6312    England    0
6313    England    0
6314   Scotland    0
6315    England    0
6316    England    0
6317    England    0
6318    England    0
6319    England    0
6320    England    0
6321    England    0
6322    England    1
6323    England    0
6324    England    0
6325    England    0
6326    England    0
6327    England    1
6328    England    0
6329    England    0
6330   Scotland    0
6331    England    0
6332    England    0
6333    England    0
6334    England    0
6335    England    0
6336    England    0
6337    England    0
6338    England    0
6339    England    0
6340    England    0
6341    England    0
6342    England    0
6343    England    0
6344    England    0
6345    England    0
6346    England    0
6347    England    0
6348   Scotland    0
6349    England    0
6350    England    0
6351    England    0
6352    England    0
6353    England    1
6354    England    0
6355    England    0
6356    England    0
6357    England    0
6358    England    0
6359    England    0
6360    England    0
6361    England    0
6362    England    0
6363    England    0
6364    England    0
6365    England    0
6366    England    0
6367    England    1
6368    England    0
6369    England    0
6370    England    1
6371    England    0
6372    England    0
6373    England    0
6374    England    0
6375    England    0
6376    England    0
6377    England    0
6378    England    0
6379    England    0
6380    England    0
6381    England    0
6382    England    0
6383    England    0
6384    England    0
6385    England    0
6386    England    0
6387    England    0
6388    England    0
6389    England    0
6390    England    0
6391    England    0
6392    England    0
6393    England    0
6394    England    0
6395    England    0
6396    England    0
6397    England    0
6398    England    0
6399    England    0
6400    England    0
6401    England    0
6402    England    0
6403    England    0
6404    England    0
6405    England    0
6406    England    0
6407    England    0
6408    England    0
6409    England    0
6410    England    0
6411    England    0
6412    England    0
6413    England    0
6414    England    0
6415    England    0
6416    England    0
6417    England    0
6418    England    0
6419    England    0
6420    England    0
6421    England    0
6422    England    0
6423    England    0
6424   Scotland    0
6425    England    0
6426    England    0
6427    England    0
6428    England    0
6429    England    0
6430    England    0
6431    England    0
6432    England    0
6433    England    0
6434    England    0
6435    England    0
6436    England    0
6437    England    0
6438    England    0
6439    England    0
6440    England    0
6441    England    0
6442   Scotland    0
6443    England    0
6444    England    0
6445    England    0
6446    England    0
6447    England    0
6448    England    0
6449    England    0
6450   Scotland    0
6451    England    0
6452    England    0
6453    England    0
6454    England    0
6455    England    0
6456   Scotland    0
6457    England    0
6458    England    0
6459    England    0
6460    England    0
6461   Scotland    0
6462   Scotland    0
6463    England    0
6464    England    0
6465   Scotland    1
6466    England    0
6467    England    0
6468    England    0
6469    England    0
6470   Scotland    1
6471    England    0
6472    England    0
6473   Scotland    0
6474    England    0
6475    England    0
6476    England    0
6477    England    0
6478    England    0
6479    England    0
6480    England    0
6481    England    0
6482    England    0
6483    England    0
6484    England    0
6485    England    0
6486    England    0
6487    England    0
6488    England    0
6489    England    0
6490    England    0
6491    England    0
6492    England    0
6493    England    1
6494    England    0
6495    England    0
6496    England    0
6497    England    0
6498    England    0
6499    England    0
6500    England    0

Resampling the exploded data

• We just run the same resampling process

     population died population.1 died.1
1       England    0      England      0
2       England    0      England      0
3       England    0      England      0
4       England    0      England      0
5       England    0      England      0
6       England    0      England      0
7       England    0      England      0
8       England    0      England      0
9       England    0      England      0
10     Scotland    0      England      0
11      England    1      England      0
12      England    0      England      0
13      England    0      England      0
14      England    0      England      0
15     Scotland    0      England      0
16      England    0      England      0
17      England    0      England      0
18      England    0      England      0
19      England    0      England      0
20      England    0      England      0
21      England    0      England      0
22     Scotland    0      England      0
23      England    0      England      0
24      England    0      England      0
25      England    0      England      0
26     Scotland    0      England      0
27      England    0      England      0
28      England    0      England      0
29      England    0     Scotland      0
30      England    0      England      0
31     Scotland    0      England      0
32     Scotland    0      England      0
33     Scotland    0      England      0
34      England    0      England      0
35      England    0      England      0
36      England    0      England      0
37      England    0      England      0
38      England    0      England      0
39      England    0      England      0
40      England    0      England      0
41      England    0      England      0
42      England    0     Scotland      0
43      England    0      England      0
44      England    0      England      0
45      England    0     Scotland      0
46      England    0      England      0
47      England    0      England      0
48      England    0      England      0
49      England    0      England      0
50      England    0      England      0
51      England    1      England      0
52      England    0      England      0
53      England    0      England      0
54      England    0      England      0
55      England    0      England      0
56      England    0      England      0
57      England    0      England      0
58      England    0      England      0
59      England    0      England      0
60      England    0      England      0
61      England    0      England      0
62      England    0      England      0
63      England    0      England      0
64      England    0      England      0
65      England    0      England      1
66      England    0      England      0
67      England    0      England      1
68      England    0      England      0
69      England    0      England      0
70      England    0      England      0
71     Scotland    0     Scotland      0
72      England    0      England      0
73      England    0      England      0
74      England    0      England      0
75      England    0      England      0
76      England    0      England      0
77      England    0      England      0
78      England    0      England      0
79     Scotland    0      England      1
80      England    0      England      0
81      England    0      England      0
82      England    0      England      0
83      England    0      England      0
84      England    0      England      0
85      England    0      England      0
86      England    0      England      0
87      England    0      England      0
88     Scotland    0      England      0
89      England    0      England      1
90      England    0      England      0
91      England    0      England      0
92      England    0     Scotland      0
93      England    0      England      0
94      England    0      England      0
95      England    0      England      0
96      England    0      England      0
97      England    0      England      0
98     Scotland    0      England      0
99      England    0      England      0
100     England    0      England      0
101     England    0      England      0
102     England    0      England      0
103    Scotland    0     Scotland      0
104     England    0      England      0
105     England    0      England      0
106     England    1      England      0
107    Scotland    0      England      0
108     England    0      England      0
109    Scotland    0      England      0
110     England    0      England      0
111     England    0      England      0
112     England    0      England      0
113     England    0      England      0
114     England    1      England      0
115     England    0      England      0
116     England    0      England      0
117     England    0      England      0
118     England    0      England      0
119     England    0      England      0
120     England    0      England      0
121     England    0      England      0
122     England    0      England      0
123     England    0      England      0
124     England    0      England      1
125     England    0      England      0
126     England    0      England      0
127     England    0      England      0
128     England    0      England      1
129     England    0      England      0
130     England    0      England      0
131     England    0      England      0
132     England    0      England      0
133     England    0      England      0
134     England    0      England      0
135     England    0      England      0
136     England    0      England      0
137     England    0      England      0
138    Scotland    0      England      0
139     England    0      England      0
140     England    0      England      0
141     England    0      England      0
142     England    0      England      0
143     England    0      England      0
144     England    0      England      0
145     England    0      England      0
146     England    0      England      0
147     England    1      England      0
148     England    0      England      0
149     England    0      England      0
150     England    0      England      0
151     England    0      England      0
152     England    0      England      0
153     England    0      England      0
154     England    0      England      1
155     England    0      England      0
156    Scotland    0      England      0
157     England    0      England      0
158    Scotland    0      England      0
159     England    1      England      0
160     England    0     Scotland      0
161     England    0      England      0
162     England    0      England      0
163     England    0      England      0
164     England    0      England      0
165     England    0      England      0
166     England    0      England      0
167     England    0      England      0
168     England    0      England      0
169     England    0      England      1
170     England    0      England      0
171     England    0      England      0
172     England    0      England      1
173     England    0      England      0
174     England    0      England      0
175     England    0      England      0
176     England    1      England      0
177     England    0      England      0
178     England    0      England      0
179     England    0      England      0
180     England    0      England      0
181     England    0      England      0
182     England    0      England      0
183     England    0      England      0
184     England    0      England      0
185     England    0      England      0
186     England    0      England      0
187     England    0      England      0
188     England    0      England      0
189    Scotland    0      England      0
190     England    0      England      0
191     England    0      England      0
192     England    0      England      0
193     England    0      England      0
194     England    0      England      0
195    Scotland    0      England      0
196    Scotland    0      England      0
197     England    0      England      0
198     England    0      England      0
199     England    0      England      0
200     England    0      England      0
201     England    0      England      0
202     England    0      England      0
203     England    0      England      0
204     England    0      England      0
205     England    0      England      0
206     England    0      England      0
207     England    0      England      0
208    Scotland    0      England      0
209     England    0      England      0
210     England    0      England      0
211     England    0      England      0
212    Scotland    0      England      0
213     England    0      England      0
214     England    0      England      0
215     England    0      England      0
216     England    0      England      0
217     England    0      England      0
218     England    0      England      0
219    Scotland    0      England      0
220     England    0      England      0
221     England    1      England      0
222     England    0      England      0
223     England    0      England      0
224     England    0      England      0
225    Scotland    0      England      0
226     England    0      England      0
227     England    0      England      1
228     England    0      England      1
229     England    0      England      0
230     England    0      England      0
231     England    0      England      0
232     England    0      England      0
233     England    0     Scotland      0
234    Scotland    0      England      0
235     England    0      England      0
236     England    0      England      0
237     England    0      England      0
238     England    0      England      0
239     England    0      England      0
240     England    0      England      0
241     England    0      England      0
242     England    0      England      0
243     England    0      England      0
244     England    0      England      0
245     England    0      England      0
246     England    0      England      0
247     England    0      England      0
248     England    0      England      0
249    Scotland    0      England      0
250     England    0      England      0
251     England    0      England      0
252     England    0      England      0
253    Scotland    0      England      0
254     England    0      England      0
255     England    0      England      0
256     England    0      England      0
257     England    0      England      0
258     England    0      England      0
259     England    0      England      1
260     England    0      England      0
261     England    0      England      0
262     England    0      England      0
263     England    0      England      0
264     England    0      England      0
265     England    0      England      0
266     England    0      England      0
267     England    0      England      0
268    Scotland    0      England      0
269     England    0      England      0
270     England    0      England      0
271     England    0      England      0
272    Scotland    0      England      0
273     England    0     Scotland      0
274     England    0      England      0
275    Scotland    0      England      0
276     England    0      England      0
277     England    0      England      1
278     England    0      England      1
279     England    0     Scotland      0
280     England    0      England      0
281    Scotland    0      England      0
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6307    England    0      England      0
6308    England    0      England      0
6309    England    0      England      0
6310    England    0     Scotland      0
6311    England    0      England      0
6312    England    0     Scotland      0
6313    England    0     Scotland      0
6314   Scotland    0      England      0
6315    England    0      England      0
6316    England    0      England      0
6317    England    0      England      0
6318    England    0      England      0
6319    England    0      England      0
6320    England    0      England      0
6321    England    0      England      0
6322    England    1      England      0
6323    England    0      England      0
6324    England    0     Scotland      0
6325    England    0      England      0
6326    England    0      England      0
6327    England    1      England      0
6328    England    0      England      0
6329    England    0      England      0
6330   Scotland    0      England      0
6331    England    0      England      0
6332    England    0      England      0
6333    England    0      England      0
6334    England    0      England      0
6335    England    0      England      0
6336    England    0     Scotland      0
6337    England    0      England      0
6338    England    0      England      0
6339    England    0      England      0
6340    England    0     Scotland      0
6341    England    0      England      0
6342    England    0      England      0
6343    England    0      England      0
6344    England    0     Scotland      0
6345    England    0      England      0
6346    England    0      England      0
6347    England    0      England      0
6348   Scotland    0      England      0
6349    England    0     Scotland      0
6350    England    0      England      0
6351    England    0      England      0
6352    England    0      England      0
6353    England    1      England      0
6354    England    0      England      0
6355    England    0      England      0
6356    England    0      England      1
6357    England    0      England      0
6358    England    0      England      0
6359    England    0      England      0
6360    England    0      England      0
6361    England    0      England      0
6362    England    0      England      0
6363    England    0      England      0
6364    England    0      England      0
6365    England    0      England      0
6366    England    0      England      0
6367    England    1      England      0
6368    England    0      England      0
6369    England    0      England      0
6370    England    1      England      0
6371    England    0      England      0
6372    England    0      England      0
6373    England    0      England      0
6374    England    0      England      0
6375    England    0      England      0
6376    England    0      England      0
6377    England    0      England      0
6378    England    0      England      0
6379    England    0      England      0
6380    England    0      England      0
6381    England    0      England      0
6382    England    0      England      0
6383    England    0      England      0
6384    England    0      England      0
6385    England    0     Scotland      0
6386    England    0      England      0
6387    England    0      England      0
6388    England    0      England      0
6389    England    0     Scotland      0
6390    England    0      England      0
6391    England    0      England      0
6392    England    0      England      1
6393    England    0      England      0
6394    England    0      England      0
6395    England    0      England      0
6396    England    0      England      0
6397    England    0      England      0
6398    England    0      England      0
6399    England    0      England      0
6400    England    0      England      0
6401    England    0      England      0
6402    England    0      England      0
6403    England    0      England      0
6404    England    0      England      1
6405    England    0      England      0
6406    England    0      England      0
6407    England    0      England      0
6408    England    0      England      0
6409    England    0      England      0
6410    England    0      England      0
6411    England    0      England      0
6412    England    0      England      0
6413    England    0      England      0
6414    England    0      England      0
6415    England    0      England      0
6416    England    0      England      0
6417    England    0      England      0
6418    England    0      England      0
6419    England    0      England      0
6420    England    0      England      0
6421    England    0      England      0
6422    England    0      England      0
6423    England    0      England      0
6424   Scotland    0      England      0
6425    England    0      England      0
6426    England    0      England      0
6427    England    0      England      0
6428    England    0      England      0
6429    England    0      England      0
6430    England    0      England      0
6431    England    0      England      0
6432    England    0     Scotland      0
6433    England    0      England      0
6434    England    0      England      0
6435    England    0      England      0
6436    England    0      England      0
6437    England    0      England      0
6438    England    0      England      0
6439    England    0     Scotland      0
6440    England    0      England      0
6441    England    0      England      0
6442   Scotland    0      England      0
6443    England    0      England      0
6444    England    0      England      0
6445    England    0     Scotland      0
6446    England    0      England      0
6447    England    0      England      0
6448    England    0      England      0
6449    England    0      England      0
6450   Scotland    0      England      0
6451    England    0      England      0
6452    England    0      England      0
6453    England    0      England      0
6454    England    0      England      0
6455    England    0      England      0
6456   Scotland    0     Scotland      0
6457    England    0      England      0
6458    England    0      England      0
6459    England    0      England      0
6460    England    0      England      0
6461   Scotland    0      England      0
6462   Scotland    0     Scotland      0
6463    England    0      England      0
6464    England    0      England      0
6465   Scotland    1      England      0
6466    England    0      England      0
6467    England    0      England      0
6468    England    0      England      0
6469    England    0      England      0
6470   Scotland    1      England      0
6471    England    0      England      0
6472    England    0      England      0
6473   Scotland    0      England      0
6474    England    0      England      0
6475    England    0     Scotland      0
6476    England    0     Scotland      1
6477    England    0      England      0
6478    England    0      England      0
6479    England    0      England      0
6480    England    0      England      0
6481    England    0      England      0
6482    England    0      England      0
6483    England    0      England      1
6484    England    0      England      0
6485    England    0      England      0
6486    England    0      England      0
6487    England    0      England      0
6488    England    0      England      0
6489    England    0     Scotland      0
6490    England    0      England      0
6491    England    0     Scotland      0
6492    England    0      England      0
6493    England    1     Scotland      0
6494    England    0      England      0
6495    England    0      England      0
6496    England    0      England      0
6497    England    0      England      0
6498    England    0      England      0
6499    England    0      England      0
6500    England    0      England      0

Aggregating

Aggregated original data

  population died    n
1    England    0 5794
2    England    1  206
3   Scotland    0  486
4   Scotland    1   14

Aggregated bootstrap 1

  population died    n
1    England    0 5815
2    England    1  197
3   Scotland    0  476
4   Scotland    1   12

Summarising the results

• After all this you can calculate a standard error or confidence interval around the decomposition results (see Li (2017))
• There are lots of ways to do this (Efron and Hastie 2016), but often people just take percentiles of the bootstrap distribution (the worst option)
• Or you can visualize the empirical distribution of the results as we did before

A nice example

fine-fig

Any questions

Part Two: Issues bootstrapping standardization and decomposition

When things go wrong

• Decomposition will decompose whatever difference there is - even if it is very small
• The sampling distribution of underlying rates is usually quite nice (a Normal approximation is probably fine)
• But a decomposition is essentially a ratio (remember the session on rates as products) - a transformation of the standardized rates

When things go wrong

• The sampling distribution of a transformed quantity will not necessarily be the same as the sampling distribution of the two underlying quantities
• In particular the sampling distribution of ratios can be nice or really not nice (Franz 2007)
• (If you are not finding this very intuitive don’t worry! I didn’t either!)

The sampling distribution of decomposition

• If there isn’t much difference in the underlying crude rate you are decomposing (in a hand-wavy way, if the difference is not statistically significant) then things can go haywire
• If the standardized rates are very close together then the decomposition can involve dividing by very small numbers
• Uh oh

Dividing by small numbers

1 / 10

[1] 0.1

1 / 1

[1] 1

1 / 0.1

[1] 10

1 / 0.01

[1] 100

1 / 0.001

[1] 1000

Dividing by small numbers

Ratios of normals

• If the crude rates are close together and you add some uncertainty in these rates via bootstrapping they may overlap
• Because the calculations for decomposition involve division the resulting distribution of the decomposition can go haywire
• Ben navigate to the ratios example
• (AI again sorry)

An example

reconviciton-rate

When things work fine

• Before we looked at a decomposition of rates between 2004/05 and 2026/17

A not-so-nice example

• But if we compare 2015 and 2016…

A not-so-nice example

• The scale runs from -10,000% to +10,000%!
• So lets zoom in for a closer look

A not-so-nice example

haywire-fig

A not-so-nice example

• The average bootstrap replication (the blue dot) does not match the observed data (the red dot)!
• This isn’t a sample size problem - more bootstrap replications will give the same results
• Would you be happy to present those results in a paper?

How can this be?

• I was very confused by the bad decomposition - why would the bootstrapping lead to these wild estimates (huge confidence intervals not centered on the true values!)
• The thing I was bootstrapping was the underlying data - and the bootstrapped distribution of the crude rates is fine
• But the decomposition is a transformation of these rates, and the properties of the rates do not necessarily transfer downstream to the transformation

Do I need to worry about this?

• Maybe you could argue that you don’t need to factor in uncertainty in the underlying rates - in this case we are working with adminisrative data , the reconviction cohorts are what they are etc
• This is an actual philosophical debate (see Verlaan and Langton 2024)
• But maybe you are working with survey data when you have a rate and a standard error
• I (Ben) also think it’s useful to know how sensitive your results are to small changes in the input data, as much for pragmatic reasons as philosophical reasons
• Bootstrapping the raw data (nonparametric simulation (see Carsey and Harden 2014)) can show when your results are fragile to small permutations in the data (see also Giordano, Meager, and Broderick 2026)

Two practical suggestions

• Check whether the difference in crude rates between comparator is statistically significant before you do your standardization and decomposition (just a standard prop.test() in R or prtest in Stata)
• You can also bootstrap your results a few times anyway as a matter of course just to see if something breaks

Bonus slides: A worked example of bootstrapping in practice

Bootstrapping in practice: Exploding the data

library(tidyverse)
library(DasGuptR)

# just the first and last year
dat_2004_2016 <- 
  DasGuptR::reconv |> 
  filter(year == "2004" | year == "2016")

# we need to turn the dataset into a 'tidy' dataframe with columns for
# year, Age, Sex, whether a person was reconvicted or not, n

dat_2004_2016_long <- 
  dat_2004_2016 |> 
  mutate(not_reconvicted = offenders - reconvicted) |> 
  select(year, Age, Sex, not_reconvicted, reconvicted) |> 
  pivot_longer(not_reconvicted:reconvicted, names_to = "reconvicted", values_to = "n") |> 
  mutate(n = as.integer(n)) |>
  uncount(n) # this takes the n column and explodes the data using the counts
# of people convicted and not reconvicted

Bootstrapping: 2. Resampling

• Now we have an ‘individual’ level dataset for our reconviction cohorts we can resample from them with replacement
• We use functions from the {rsample} package to do this in a memory efficient way
• Because bootstrapping uses random number generation we need to set the seed before starting so our results are reproducible
• We also need to set the number of draws. We’ll use 200 for speed; but more is always better

library(rsample)

set.seed(12345)

n_draws <- 200

draws_2004_2016 <- 
dat_2004_2016_long |> 
  group_by(year) |> # we bootstrap within years - this keeps the sizes of the reconviction cohorts the same
  rsample::bootstraps(times = n_draws)

Boostrapping: 2. Resampling

• We now have 200 of… something

draws_2004_2016

# Bootstrap sampling 
# A tibble: 200 × 2
   splits                id          
   <list>                <chr>       
 1 <split [89957/32972]> Bootstrap001
 2 <split [89957/32992]> Bootstrap002
 3 <split [89957/33239]> Bootstrap003
 4 <split [89957/33113]> Bootstrap004
 5 <split [89957/32932]> Bootstrap005
 6 <split [89957/33149]> Bootstrap006
 7 <split [89957/33307]> Bootstrap007
 8 <split [89957/33076]> Bootstrap008
 9 <split [89957/32863]> Bootstrap009
10 <split [89957/33139]> Bootstrap010
# ℹ 190 more rows

Bootstrapping: 3. Aggregating

• Now we can convert these memory-efficient, individual level resamples from our data to aggregate data ready to standardize and decompose
• To do this we need to use R’s programming functions to iterate over each bootstrap resample and aggregate them
• We will write a helper function to do this

Writing an R function

• Functions in R let you easily re-run code
• If you have not written an R function before you might want to check out the Functions chapter in R for Data Science (Wickham, Çetinkaya-Rundel, and Grolemund 2023)

Bootstrapping: 3. Aggregating

# create a helper function to aggregate the results o

uncount_split <- function(splits){
  
  rsample::analysis(splits) |> 
    count(year, Age, Sex, reconvicted) |> # this is the opposite of uncount()
    pivot_wider(names_from = "reconvicted", # this gets the data back in the
                values_from = "n") |> # right shape for dgnpop
    mutate(offenders = not_reconvicted + reconvicted,
           prev_rate = reconvicted / offenders)
  
}

Bootstrapping: 3. Aggregating

• Now we have our helper function we can apply it to each bootstrap draw
• We use the map function from the {purrr} package
• Jenny Bryan has written the definitive guide to iteration with {purrr}
• The basic intuition is each bootstrap draw is a row of our draws_2004_2016 dataframe, and we want to aggregate each one
• In other words, we want to map the aggregation to each of the bootstrap draws

Bootstrapping: 3. Aggregating

draws_2004_2016 <- 
draws_2004_2016 |> 
 mutate(agg_data = map(splits, uncount_split))

Bootstrapping: 3. Aggregating

# Bootstrap sampling 
# A tibble: 200 × 3
   splits                id           agg_data           
   <list>                <chr>        <list>             
 1 <split [89957/32972]> Bootstrap001 <gropd_df [20 × 7]>
 2 <split [89957/32992]> Bootstrap002 <gropd_df [20 × 7]>
 3 <split [89957/33239]> Bootstrap003 <gropd_df [20 × 7]>
 4 <split [89957/33113]> Bootstrap004 <gropd_df [20 × 7]>
 5 <split [89957/32932]> Bootstrap005 <gropd_df [20 × 7]>
 6 <split [89957/33149]> Bootstrap006 <gropd_df [20 × 7]>
 7 <split [89957/33307]> Bootstrap007 <gropd_df [20 × 7]>
 8 <split [89957/33076]> Bootstrap008 <gropd_df [20 × 7]>
 9 <split [89957/32863]> Bootstrap009 <gropd_df [20 × 7]>
10 <split [89957/33139]> Bootstrap010 <gropd_df [20 × 7]>
# ℹ 190 more rows

Bootstrapping: 3. Aggregating

• Look at the first aggregated draw

draws_2004_2016$agg_data[[1]]

# A tibble: 20 × 7
# Groups:   year [2]
    year Age      Sex    not_reconvicted reconvicted offenders prev_rate
   <int> <chr>    <chr>            <int>       <int>     <int>     <dbl>
 1  2004 21 to 25 Female            1082         554      1636     0.339
 2  2004 21 to 25 Male              5580        3311      8891     0.372
 3  2004 26 to 30 Female             845         408      1253     0.326
 4  2004 26 to 30 Male              4077        2214      6291     0.352
 5  2004 31 to 40 Female            1730         532      2262     0.235
 6  2004 31 to 40 Male              6895        2871      9766     0.294
 7  2004 over 40  Female            1052         197      1249     0.158
 8  2004 over 40  Male              5151        1232      6383     0.193
 9  2004 under 21 Female            1095         396      1491     0.266
10  2004 under 21 Male              5996        4255     10251     0.415
11  2016 21 to 25 Female             807         227      1034     0.220
12  2016 21 to 25 Male              4093        1862      5955     0.313
13  2016 26 to 30 Female             820         309      1129     0.274
14  2016 26 to 30 Male              4091        1686      5777     0.292
15  2016 31 to 40 Female            1553         659      2212     0.298
16  2016 31 to 40 Male              6146        2702      8848     0.305
17  2016 over 40  Female            1740         356      2096     0.170
18  2016 over 40  Male              7236        1798      9034     0.199
19  2016 under 21 Female             521         178       699     0.255
20  2016 under 21 Male              2473        1227      3700     0.332

Bootstrapping: 3. Aggregating

• Look at the second aggregated draw

draws_2004_2016$agg_data[[2]]

# A tibble: 20 × 7
# Groups:   year [2]
    year Age      Sex    not_reconvicted reconvicted offenders prev_rate
   <int> <chr>    <chr>            <int>       <int>     <int>     <dbl>
 1  2004 21 to 25 Female            1113         576      1689     0.341
 2  2004 21 to 25 Male              5524        3348      8872     0.377
 3  2004 26 to 30 Female             930         387      1317     0.294
 4  2004 26 to 30 Male              4152        2148      6300     0.341
 5  2004 31 to 40 Female            1706         583      2289     0.255
 6  2004 31 to 40 Male              6840        2898      9738     0.298
 7  2004 over 40  Female             946         223      1169     0.191
 8  2004 over 40  Male              5187        1229      6416     0.192
 9  2004 under 21 Female            1077         418      1495     0.280
10  2004 under 21 Male              6053        4164     10217     0.408
11  2016 21 to 25 Female             762         236       998     0.236
12  2016 21 to 25 Male              4137        1796      5933     0.303
13  2016 26 to 30 Female             854         246      1100     0.224
14  2016 26 to 30 Male              3954        1739      5693     0.305
15  2016 31 to 40 Female            1607         626      2233     0.280
16  2016 31 to 40 Male              6376        2682      9058     0.296
17  2016 over 40  Female            1726         356      2082     0.171
18  2016 over 40  Male              7251        1865      9116     0.205
19  2016 under 21 Female             483         146       629     0.232
20  2016 under 21 Male              2368        1245      3613     0.345

Bootstrapping: 4. Standardize and decompose

• We can use the same pattern of creating a helper function and then applying it to each row of draws_2004_2016 to standardize and decompose 200 times

Bootstrapping: 4. Helper function

calc_dg <- function(d){
  # This just wraps the standard dgnpop call for the reconvictions data
  d |> 
    dgnpop(pop="year", 
           factors=c("prev_rate"),id_vars=c("Sex","Age"),
           crossclassified="offenders",
           agg = TRUE) |> 
    dg_table() |> 
    rownames_to_column() # this helps keep track of the results
  
}

Bootstrapping: 4. Standardize and decompose

• NB this might take a while!

draws_2004_2016 <- 
draws_2004_2016 |> 
  mutate(results = map(agg_data, calc_dg), # apply dgnpop to each aggregated dataset
         draw = 1:n_draws) |> # add an index which says which draw number the results are from
  select(draw, results) # only keep the draw number and results column

Bootstrapping: 4. Standardize and decompose

• We now need to expand out the results with unnest

decomp_draws <- 
draws_2004_2016 |> 
    unnest(results) |> 
     pivot_longer(`2004`:`2016`,
               names_to = "year",
               values_to = "rate") |> 
     filter(year == "2016") # the decomposition results are the same for both years, so we'll just keep one set

Bootstrapping: 5. Visualize

Bootstrapping: 5. Visualize

Bootstrapping in practice

• These are the original results

     rowname 2004 2016  diff decomp
1 Age_struct 0.31 0.28 -0.02   41.3
2  prev_rate 0.31 0.28 -0.03   57.2
3 Sex_struct 0.29 0.29  0.00    1.5
4      crude 0.32 0.27 -0.05  100.0

Bootstrapping in practice

Bootstrapping in practice

• Now we can calculate standard errors and confidence intervals for these estimates
• (Actually it’s bad practice to use the mean of the bootstrap draws instead of the observed results (Rainey 2024))

# A tibble: 4 × 5
  rowname    mean_decomp se_est lower_ci upper_ci
  <chr>            <dbl>  <dbl>    <dbl>    <dbl>
1 Age_struct       41.3   2.74    35.9      46.7 
2 Sex_struct        1.48  0.290    0.910     2.05
3 crude           100     0      100       100   
4 prev_rate        57.2   2.82    51.7      62.7 

Bootstrapping in practice

• Or directly calculate confidence intervals by taking percentiles

# A tibble: 4 × 4
  rowname    median lower_ci upper_ci
  <chr>       <dbl>    <dbl>    <dbl>
1 Age_struct  41.2      36.4    46.6 
2 Sex_struct   1.47      1       2.19
3 crude      100       100     100   
4 prev_rate   57.2      51.9    62.6 

References

Carsey, Thomas M., and Jeffrey J. Harden. 2014. Monte Carlo Simulation and Resampling Methods for Social Science. SAGE Publications, Inc. https://doi.org/10.4135/9781483319605.

Das Gupta, Prithwis. 1993. Standardization and Decomposition of Rates: A User’s Manual.

Efron, Bradley, and Trevor Hastie. 2016. “Bootstrap Confidence Intervals.” In Computer Age Statistical Inference: Algorithms, Evidence, and Data Science, 181–207. Institute of Mathematical Statistics Monographs. Cambridge: Cambridge University Press.

Franz, Volker H. 2007. “Ratios: A Short Guide to Confidence Limits and Proper Use.” arXiv. https://doi.org/10.48550/arXiv.0710.2024.

Giordano, Ryan, Rachael Meager, and Tamara Broderick. 2026. “An Automatic Finite-Sample Robustness Metric: When Can Dropping a Little Data Change Conclusions? Part I: Definitions and Experiments.” Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 384 (2321): 20250001. https://doi.org/10.1098/rsta.2025.0001.

Li, Jinjing. 2017. “Rate Decomposition for Aggregate Data Using Das Gupta’s Method.” The Stata Journal 17 (2): 490–502.

Rainey, Carlisle. 2024. “A Careful Consideration of CLARIFY: Simulation-Induced Bias in Point Estimates of Quantities of Interest.” Political Science Research and Methods 12 (3): 614–23. https://doi.org/10.1017/psrm.2023.8.

Verlaan, Tim, and Samuel Langton. 2024. “On the Use of Inferential Statistics on Administrative Police Data.” In The Crime Data Handbook, 197–210. Bristol University Press.

Wickham, Hadley, Mine Çetinkaya-Rundel, and Garrett Grolemund. 2023. R for Data Science (2e). Second.