Measuring the Complexity and Importance of Businesses in Order to 

Better Manage our Data Collection Efforts 

Serge Godbout and SungJin Youn 

Serge Godbout, Statistics Canada, serge.godbout@statcan.gc.ca

SungJin Youn, Statistics Canada, sungjin.youn@statcan.gc.ca

1. Introduction

The Unified Enterprise Survey (UES) is an establishment-based program and covers a large portion of the Canadian economy 

(Brodeur et al., 2006). The domains of interest are defined by a two-dimensional grid from industrial (using the North 

American Industrial Classification System, NAICS) and geographical (Canadian provinces and territories) components. To 

provide estimates that meet the users’ requirements, the UES combines data collected from a sample and tax data. 

Both data sources bring their challenges. Data collection is resource consuming, relationships with respondents need to be 

carefully managed and compromises have to be made on the required level of desegregations to reduce response burden. On 

the other hand, tax data are cheaper and reduce response burden but the processing is different and data are available at a 

higher aggregated level so they need to be allocated into the structure using a profiling process. The obstacles related to both 

data sources get more complicated with the complexity of a business structure. This highlights the significance of being able 

to measure the complexity of a business structure and to identify the ones that are the most critical to estimates. 

The focus of the paper is on describing the methodological framework developed to help the business profiling and the 

response management of the UES. In section 2, we describe the challenges coming from the profiling process on the Business 

Register, the UES collection and response management process. In sections 3 and 4, we present the complexity metric 

developed to rank the largest and most complex businesses and the rules applied to identify the businesses that are the most 

important to a survey in order to better manage the data collection efforts. In section 5, we show some results based on 2008 

UES production cycle. Finally, a conclusion is given in section 6. 

2. Profiling and collection challenges

2.1 Business structure and profiling on the Canadian Business Register 

Statistics Canada’s Business Register (BR) contains key information on businesses in Canada, like the activity status, 

industrial and geographical classifications, business structure, many size variables, contact information etc. All entities that 

generate economic activity in all Canadian economic sectors are covered by the BR. For more details, see Business Register 

Division (2009). 

All businesses are classified into simple or complex. A simple business is involved in only one economic activity in only one 

location. The other businesses are called complex; they have economic activities in more than one industry, province or 

location (Beaucage, Hunsberger and Pursey, 2005). In figure 1, business A is simple and businesses B and C are complex, 

although C’s structure looks much more complex than B’s. 
Figure 1: Some examples of simple and complex businesses 

Enterprise C 

Division 1 Division 2 

Estab.1 

Estab.2 

Estab.1 

Enterprise A Enterprise B 

Estab. 1 

Estab.2 

Estab. 3 Adapted from Business Register Division (2007) 

Among the size variables provided by the BR, we are more interested in the total revenues and the number of employees. 

These variables are available from tax data at the business level. For a business with a complex structure, tax data need to be 

allocated into the operating structure in order to meet the survey requirements. This is done by applying a ratio derived from 

profiling information provided by a profiling specialist. A business profile needs to be updated periodically because of 

economic fluctuations, business restructuring etc. Since this process is costly, it is important to prioritize the complex 

businesses to help managers choosing the best updating plan for each complex business. 

In the current BR process, there is a subset of the complex businesses called the BR profiling critical units. This subset 

includes roughly 5000 businesses identified by subject-matter specialists for profiling updates. The identification process is 

done under broad guidelines provided by the BR and it is not consistent between the industrial sectors since each specialist is 

attached to one industry sector. Another weakness is that the importance of the multi-sector businesses might be 

underestimated compared to the large and complex ones under one sector. 

2.2 The Unified Enterprise Survey sampling, collection design and response management 

In the UES sampling design, all the establishments of a given business that are in the same domain of interest are clustered to 

form a sampling unit (SU). First, some SUs are automatically placed in the sample as must-takes (MT). Then, the rest of the 

UES population is stratified in take-all (TA), take-some (TS) and take-none (TN) strata using the Lavallée-Hidiroglou 

algorithm (Lavallée and Hidiroglou, 1988) as shown in figure 2. 

Figure 2: Unified Enterprise Survey stratification 

Take-All

Stratum

Take-None

Stratum

Large

Take-Some

Stratum

Must Take

Units

Size

Small

Take-Some

Stratum

The sample is selected through a network sampling design, i.e. a preliminary simple random simple (SRS) is selected in each 

stratum first, then all non-selected SUs belonging to a business for which at least one of its SUs – within the same industrial 
grouping as the non-selected one – were selected in the SRS sample are brought in the final sample. For more details on the 

UES sampling design, see Simard, Girard, Parent and Smith (2001). 

After the sample selection stage, the selected SUs of a business are reorganized into collection entities (CE) in order to help 

managing the data collection process and reducing the response burden. A UES CE is defined as being one or a group of SUs 

from one business that belongs to the same industrial class, using preset default rules or customized according to the 

respondents’ requirements. 

Figure 3: Sampling units and collection entities for complex businesses 

Geographical Dimension 

Industrial 

Dimension 

1 

2 

3 

4 

A B C D E F 

Business C 

Business A 

Business B 

Business A’s Sampled Establishment 

Business B’s Sampled Establishment 

Business C’s Sampled Establishment 

Legend Sampling Unit 

Collection Entity 

Figure 3 illustrates three examples of businesses. 

– Business A has only one sampled establishment so this entity corresponds to the SU and the CE.

– Business B has 11 sampled establishments divided into 4 domains of interest (4 provinces but only one industrial

class), 4 SUs and only one CE.

– Business C has 9 sampled establishments divided into 6 domains of interest (2 provinces and 3 industrial class), 6

SUs and 4 CEs.

For response management purposes, each business in the UES sample is assigned to one of four tiers depending on its size 

and its significance (Sear, Hughes and Lalande, 2007 ; Sear, Hughes, Vinette and Bozzato, 2007 ; Godbout, 2009). 

– Tier I: The largest, most complex businesses in Canada. Tier I is managed by the Enterprise Portfolio

Management Program (EPMP), which updates the business profiling, manages the data collection and maintains

proactive and ongoing relationships with them (for more details, see Enterprise Statistics Division, 2008).

– Tier II: Other businesses important for the UES but simpler or less complex than Tier I businesses.

– Tier III: Mid-size businesses carrying less individual significance than units in Tiers I and II. Tier III usually

forms the bulk of the sample, and the sampled units (SUs) of Tier III businesses are considered “replaceable”, in that

there are many other SUs that represent the same homogeneous population as well as these ones.

– Tier IV: Small businesses usually excluded from the sample, because they have low impact on the estimates. Tax

information is used for Tier IV businesses.

The Tier I businesses are selected by EPMP managers among the largest and most complex ones. Currently, there are roughly 

350 businesses managed by the EPM program. Once the Tier I is set, the sample remaining portion is split into Tiers II, III 

and IV based on their importance to UES. The proposed methodological framework for the business profiling prioritization 

and the tier definition uses two main tools: a complexity metric and a priority flag, as described in sections 3 and 4. 
3. Identifying the largest and most complex businesses

The proposal includes developing a metric to identify the largest and most complex businesses. This metric considers two 

dimensions: the business size (measured in revenue or in number of employees) and its structure complexity (based on the 

same size variable’s distribution) as shown in figure 4. 

Figure 4: Complexity metric dimensions 

Largest and Most 

Business Complex 

Structure 

Complexity 

Business Size 

The proposed complexity measure for a business i has the general form of κ i = f (yi,η i) where yi and ηi are respectively 

the business size and its structure complexity. A good and simple choice for f (yi

,ηi) is the product of both dimensions.

( )

ii

iii

y

yf

η

ηκ

=

= ,

(1) 

The complexity factor ηi is defined using the Shannon’s entropy theory as shown in section 3.1. 

3.1 Complexity factor and Shannon’s entropy 

The profiling and the data collection problems are very similar since we are in both cases interested in collecting a variable 

yi and in its distribution in the business structure. The business structure can be represented in statistical terms using, for a 

partition Pi

 of a complex business i, a vector pi = (pi1, pi2,..., pin ) with 0 ≤ pik ≤ 1 for every k and ∑ pik =1 so we get

k

for the size variable yi its allocated size vector 

( )

( )

ii

iniii

iniii

y

pppy

yy y

p

y

=

=

=

...,

...,

21

21

(2) 

For a business i, its profiling vector piis not fixed in time because of restructuring, changes in revenues, employment etc. 

The updated or realized allocation p

~

i

of the variable yiis expected to be close to pi without being necessarily equal. The

information theory could help studying the uncertainty related to the profiling vector, specially the Shannon’s entropy theory. 

The entropy (Shannon, 1948 ; Reesor and McLeish, 2002 ; Wikipedia, 2009) is a measure of the uncertainty associated with a 

random variable X having n possible values x1, x2...xnusing a distance between 2 probability distributions. Shannon’s 

entropy measures the minimal amount of information needed to complete a set without any losses. It’s defined as the 

expected value of the logarithm of the inverse probabilities: 

[ ]

∑=

−

−=

= =

n

k

kk

k

pp

XH E xXP

1

1

log

( )(log)

(3)
with pk

log pk ≡ 0 when pk = 0 . We have 0 ≤ H(X ) ≤ log n with H(X ) = 0 when X takes only one value with a 

probability of 1 ( P(X = xi) = 1 ) and H(X ) = log n when all n possible values of X follow a uniform distribution 

( P(X = xi) = 1/ n ). 

If a random variable X is associated to a business partition Pi with n operating entities, and takes the value set yi

respectively with probabilities pi

, H(X ) gives the amount of yik ’s required to have a complete set of information on the 

business total yi so Shannon’s entropy becomes a good candidate for measuring a business structure complexity. 

– For a business with a simple structure (only one establishment, i.e. n =1), its entropy equals 0 ( H(X ) = 0 ). 

– For a given number n of establishments within a business, the most complex structure would be for a uniformly 

distributed business with pi = (1/ n,1/ n,...,1/ n) since every pieces of information we can get from the business let a 

maximum of uncertainty for its remaining portion. Its entropy would be maximized with H (X ) = log n . 

– If a complex business contains many establishments but one represents a very large proportion of its size, the 

entropy would be very low since the uncertainty is highly reduced once we know the information related to the main 

piece of the business. 

We propose to measure the complexity factor ηi for a business i and its partition Pi related to the size variable yi as the 

following: 

( )

∑∈

−=

=

Pk i

ikik

i i

p p

PYH

log

η |

(4) 

The proposed complexity metric κ i for a business i, becomes: 

κ i = yiη i

= yi∑−

(5) pik log pik

k∈Pi

One may consider more than one partition of a given business whether we want to measure the structure complexity from the 

whole business or from its partition by industry, by geography or both or combine them by putting different weights on the 

respective complexity factors. A general complexity factor that combines the J entropy measures for the different partitions 

Pij is proposed as the following: 

( )

∑ ∑

∑

= ∈

=





















= −

=

J

j Pk

j ikik

J

j

i j ij

ij

pp

PYH

1

1

log

|

α

αη

(6) 

with 0 < α j < 1 and ∑α j =1 . The α j’s are selected to balance all complexity factors one want to consider. 

3.2 Example 

To illustrate the proposed complexity metric, the table 4 shows five examples of businesses having the same size but different 

structures. 
Table 4: Examples of the complexity metric 

Size Partition at Establishment Level 

ik p

Business 

i

Business 

Size 

i

y k = 1 k = 2 k = 3 k = 4 k = 5 

Structure 

Complexity 

( ) i PYH i η = |

Complexity 

Metric 

iii = y ηκ

A 500 1.000 0.000 0 

B 500 0.500 0.500 0.301 151 

C 500 0.900 0.025 0.025 0.025 0.025 0.201 101 

D 500 0.500 0.125 0.125 0.125 0.125 0.602 301 

E 500 0.200 0.200 0.200 0.200 0.200 0.699 349 

Here are some observations: 

– Business A has only one establishment so its complexity factor and its complexity metric are 0. 

– Business B has two establishments equally divided. Its complexity factor is ηi = log 2 . 

– Businesses C, D and E have five establishments. However, C is heavily concentrated on one establishment so its 

complexity factor is highly reduced while E is uniformly distributed among its five establishments so η i = log 5 . 

To distinguish the Tiers II, III and IV, we need to identify the units with the highest importance in the response management. 

This is done in section 4. 

4. Assigning a response management priority to units 

The second portion of the methodological framework is to set a unit response management. For that, we proposed to base the 

priority rule on the way each SU represents its population according to the sampling design. 

4.1 Response management priority at sampling unit level 

A priority level is assigned to each SU according to the selection stratum. 

– MT and TA are assigned a high priority, since they are very large or unique, and they are self-representative in 

the sample. 

– TS are assigned a regular priority, since they are in a randomly selected group representing a population that is 

assumed to be homogeneous. Therefore, they do not carry the same individual significance in the representation of 

the population as MT or TA do, especially since the sampling fraction of TS strata usually includes an offset for 

expected ignorable non-response. 

–TN selected through network sampling are assigned a zero priority, since they have been selected for coherence 

purposes and to ease the reporting burden of some complex businesses and not necessarily to represent a part of the 

population. 

The relationship between the UES sampling plan and the SU response management priority is illustrated in figure 5. 

Figure 5: UES sampling plan and the sampling unit response management priority 

Population

TS

TA

MT

HIGH PRIORITY

. Large / Unique

. Self-representing

REGULAR PRIORITY

. Homogeneous

. Many-to-many

relationship

. Oversampled for NR

NO PRIORITY

. TN portion

Sample
4.2 Response management priority at collection entity level 

The respondent completes a single questionnaire for the CE; therefore, assigning a priority to each CE is important for the 

response management. Some CEs include SUs with different priorities. The priority of a CE is defined as the highest priority 

of its SUs. Therefore, a high-priority CE has at least one high-priority SU. A regular-priority CE has at least one regularpriority SU, but no high-priority SUs. A zero-priority CE has zero-priority SUs only (no high or regular-priority SUs). 

At the CEs creation step, a response management priority flag is attached to each, based on the survey design, and are 

available to all response managers through different BR tools. 

5. Results for reference year 2008

The original plans were to implement the proposed methodology for the reference year 2008 as a pilot project but some 

operational constraints forced us to postpone the study. Nevertheless, most of the features from the proposal were tested on 

the 2008 UES data. 

First, using the methodology described above, all the UES sampled businesses have been assigned a tier. The Tier I set was 

associated to the current EPM business list. All the non-Tier I businesses were assigned the remaining tiers II, III and IV 

based on the response management priority variable derived from the proposed model. The model and the distribution of all 

UES sampled businesses are given in table 6. 

Table 6: Tier identification model and business counts 

Tier Identification 

Rule 

Nb of 

Businesses 

I All business part of the EPM program 235 

II All non-Tier I businesses that have at 

least 1 high-priority SU 14,000 

III All non-Tier I & II businesses that 

have at least 1 regular-priority SU 33,000 

IV All the remaining businesses 200 

Total All UES sampled businesses 47,435 

Also, all CEs have received a response management priority based on the highest level of priority among its SUs. Table 7 

shows the distribution of all CEs by their response management priority and their business’ tier. 

Table 7: UES collection entities by response management priority and tier 

CE Response Management 

Tier Priority 

High Regular No

Total

I 1,500 500 50 2,050 

II 16,000 1,500 500 18,000 

III 0 34,000 200 34,200 

IV 0 0 250 250 

Total 17,500 36,000 1,000 54,500 

By definition, the 235 Tier I businesses selected by the UES are very large and complex. In average, each Tier I business has 

10 CEs and 75% of these have a high priority. One can observe that 500 Tier I CEs have a regular priority. Even if they are 

critical for the UES program, the Tier I businesses might have some parts that are less important to surveys. This information 

is critical for a response manager when dealing with a reluctant respondent. 

On the other hand, the number of Tier II businesses is overwhelming. There are 14,000 Tier II businesses with a low average 

of 1.3 CE each. Since they have at least one high priority CE, roughly 90% of their 18,000 CEs have a high priority. The 

response management cannot be approached the same way as the Tier I. 
Second, the complexity metric was calculated on all complex businesses on the BR according to three overlapping partitions 

described in Table 8. 

Table 8: Business structure partitions and weighting factors 

Partition Business Desegregation Level Factor α j

1 Establishment 0.3 

2 4-Digit NAICS x Province/Territory 0.4 

3 4-Digit NAICS 0.3 

These three partitions were selected to assign a minimal complexity on the multi-establishment businesses but to put on top 

the multi-industry ones because they lead to more complex collection challenges. Figure 9 shows the distribution of a random 

subpopulation (to lighten the graph) of all the complex businesses by size and structure complexity according to their total 

revenue. 

Figure 9: Complex businesses by size and structure complexity 

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1 10 100 1,000 10,000 100,000 1,000,000 10,000,00

0

100,000,0

00

1,000,000,

000

10,000,00

0,000

100,000,0

00,000

1,000,000,

000,000 Size

Complexity Factor

Top 350 Top 5000 Others BR Profiling Critical EPM

All the green triangles represent a business from the current EPM program; the pink circles are the current BR profiling 

critical businesses while the blue squares are the other complex businesses. The continuous black line delimits the top 350 

businesses according to the complexity metric and the dotted one delimits the top 5000 businesses. All the points on the 

horizontal axis represent complex businesses having all their revenue concentrated on only one establishment. One can 

observe that in most cases, the businesses from the EPM and the BR profiling critical programs are usually among the largest 

according to the complexity metric but the consistency could be improved. The table 10 gives the proportion of the 

businesses from both programs that are above their respective threshold (top 350 for the EPM businesses, top 5000 from the 

BR profiling critical businesses). 
Table 10: Proportion of the complex businesses from both programs that are above their respective threshold 

Class Current EPM 

Businesses 

Profiling Critical 

Businesses 

Total 334 4669 

Above Threshold - N 201 2498 

Above Threshold - % 60.2 53.5 

Note that in table 10, there are only 334 EPM businesses because some of the current Tier I businesses are no longer 

complex. Table 10 shows that roughly 60% of the EPM complex businesses and 55% of the BR Profiling Critical Businesses 

would respectively fall in the top 350 and the top 5000 complex ones. So, the complexity metric could help updating the lists 

of both groups. 

6. Conclusion 

In order to prioritize the businesses according to their size and their structure complexity, a metric was proposed. In addition, 

a response management priority flag was defined based on the sampling design. These tools can prevent managers from 

relying only on subjective criteria to identify important units for profiling and data collection. 

The future plans are to allow the complexity metric to combine the measures calculated from different size variables (e.g. 

total revenue, number of employees, total assets etc.). Also, another feature is to integrate the response management priority 

and the complexity metric in order to allow measuring the complexity on a subset of the businesses that are important to a 

survey. 

Finally, a study will try to see how the complexity metric might help identifying and managing the Tier I and the BR 

profiling critical businesses. For RY2009, the pilot will continue and the non-Tier I portion of the UES sample will be 

classified into the Tiers II, III and IV using the proposed framework. A study will be conducted after the 2009 survey cycle to 

measure the benefits of the proposed methodology on the response management. 

7. Acknowledgments 

The authors would like to acknowledge Claude Turmelle and Janet Hughes from Statistics Canada for their contribution to 

the project and the revisers for their relevant comments. 

8. References 

Beaucage, Y., Hunsberger, P. and Pursey, S. (2005), The Redesign of the Statistics Canada’s Business Register, Proceedings 

of the 2005 Joint Statistical Meetings, Minneapolis (United States). 

Brodeur, M. (2006) et al, The Integrated Approach to Economic Surveys in Canada, Statistics Canada, Ottawa (Canada). 

Catalogue 68-514. 

Business Register Division (2009), A Brief Guide to the BR, Statistics Canada, Ottawa (Canada). 

http://www.statcan.gc.ca/imdb-bmdi/document/1105_D2_T1_V2-eng.pdf

Business Register Division (2007), An Introduction to Concepts, Statistics Canada, Ottawa (Canada). 

Enterprise Statistics Division (2008), Unified Enterprise Survey Bulletin quarterly, Number 2, Fall 2008, Statistics Canada, 

Ottawa (Canada). 

Godbout, S. (2009), Methodological Aspects of Holistic Response Management in the Unified Enterprise Survey, SIMP II: 

Economic Statistics Project, Document #36, Statistics Canada, Ottawa (Canada). 
Lavallée P. and Hidiroglou M. (1988), On the Stratification of Skewed Populations, Survey Methodology. June 1988. 

Volume 14, no 1, Ottawa, Canada: Statistics Canada 

Reesor, R.M. and McLeish, D.L. (2002), Risk, Entropy amd the Transformation of Distributions, Bank of Canada, Working 

Paper 2002-11. 

Sear, J., Hughes, J. and Lalande, D. (2007), Controlling and Managing Response Burden in Business Surveys at Statistics 

Canada, SIMP II: Economic Statistics Project, Document #16, Statistics Canada, Ottawa (Canada). 

Sear, J., Hughes, J., Vinette, L. and Bozzato, W. (2007), Holistic Response Management of Business Surveys at Statistics 

Canada, ICES-III, Montréal (Canada). 

Shannon, C.E. (1948), A Mathematical Theory of Communication, The Bell System Technical Journal, Volume 27, pp. 379-

423, 623-656, July, October 1948, 

Simard, M., Girard, C., Parent, M.-N. and Smith, J. (2001), Sampling Designs for the Unified Enterprise Surveys: The Early 

Years, Working paper BSMD-2001-003E, Statistics Canada, Ottawa (Canada). 

Wikipedia (2009), Entropy – Information theory, http://en.wikipedia.org/wiki/Information_entropy