Aggregate Data and Statistics

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Aggregate Data and Statistics
Wendy Watkins Carleton University
Chuck Humphrey University of Alberta
Statistics Canada Data Liberation Initiative
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Outline

• What are aggregate data?
• Why aggregate?
• How to aggregate?
• Computing exercise

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What are aggregate data?

• Lets start with the relationship between
  statistics and data.

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Statistics and Data

• Data
• numeric files created and organized for analysis
• requires processing
• not ready for display

• Statistics
• numeric facts/figures
• created from data, i.e, already processed
• presentation-ready

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Statistics and Data
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Statistics and Data
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Statistics and Data

• In short, statistics are created from data and
  represent summaries of the detail observed in the
  data.

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What is aggregation?

• Building on this example, lets explore
  aggregation.
• We see a table with the number of smokers
  summarized over categories for age, education,
  sex, geography, and different time points.

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Age and Education are in the background and
display totals.
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What is aggregation?

• Aggregation involves tabulating a summary
  statistic across all of the categories or levels
  of a set of variables.

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The summary statistic

• The summary statistic in this example is the
  total number of smokers.

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Variables and categories

• The variables and their categories are
• Region (11) Canada and the ten provinces
• Age (5) Total, 15-19, 20-44, 45-64, 65
• Sex (3) Total, Female, Male
• Education (4) Total, Some secondary or less,
  Secondary graduate or more, Not stated
• Periods (5) 1985, 1989, 1991, 1994-95, 1996-97

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Variables and categories

• The tabulation consists of determining the
  combinations of all categories across variables
  and then counting the number of smokers within
  each of these combinations.
• 11 x 5 x 3 x 4 x 5 3300 category

combinations
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Tabulating or aggregating

• One might be wondering if there is a difference
  between tabulating and aggregating.
• Usually, they are the same thing.

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Tabulating aggregating

• In creating tables from data, the variables are
  arranged in various combinations along the
  columns and the rows.

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Tabulating aggregating

• Placing multiple variables along the columns or
  rows is called nesting.
• Tables may have variables nested on both the
  columns and rows.

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Categories of Sex nested within Periods
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Categories of Education nested within Sex
Categories of Sex nested within Region
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A quick summary

• Up to this point, we have noted that
• statistics are created from data
• aggregations consist of tabulating statistics
  within the categories of selected variables
• variables may be nested within columns and rows
  to display these tabulations

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What are aggregate data?

• Q What is the difference between an aggregation
  or tabulation and aggregate data?
• A The display of the aggregation (that is, the
  structure of the tabulated output).

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Statistical data structure

• A statistical data structure is a fixed,
  two-dimensional matrix with the variables in the
  columns and cases in the rows.

V1
V2
V3
V4
V5
V6
V7
Case 1
Case 2
Case 3
Case 4
Case 5
Case 6
Case 7
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Statistical data structure

• Aggregate data require the same type of
  statistical data structure.
• With aggregate data, the variables are the cells
  of a tabulation while the cases are the
  categories of one or more of the tables
  dimensions.

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What are aggregate data?

• Here is an example of a tabulation of three
  variables. One variable consists of five levels
  and will be used to represent the cases in an
  aggregate data file. The other two variables
  make up a two-way table and the cells in this
  table will be the variables in the aggregate data
  file.

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From tabulation to data
Dimension 1
5
4
Dimension 3
3
2
1
Dimension 2
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From tabulation to data

• Start with dimension 3 and convert this dimension
  into the rows or cases of an aggregate data file.

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From tabulation to data
Dimension 1
5
4
Dimension 3
3
2
1
Dimension 2
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From tabulation to data
Dimension 1
Dimension 2
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From tabulation to data
Dimension 3
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From tabulation to data

• Working with the six cells from the tabulation
  for level 1 of dimension 3, locate these six
  cells as six variables in the new data structure.

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From tabulation to data
R1C1
Dimension 1
Dimension 2
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From tabulation to data
R1C1
Dimension 3
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From tabulation to data
R2C1
Dimension 1
Dimension 2
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From tabulation to data
R1C1
R2C1
Dimension 3
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From tabulation to data
Dimension 1
R3C1
Dimension 2
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From tabulation to data
R1C1
R2C1
R3C1
Dimension 3
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From tabulation to data
R1C2
Dimension 1
Dimension 2
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From tabulation to data
R1C2
R1C1
R2C1
R3C1
Dimension 3
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From tabulation to data
R1C2
Dimension 1
Dimension 2
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From tabulation to data
R1C2
R2C2
R1C1
R2C1
R3C1
Dimension 3
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From tabulation to data
Dimension 1
R3C2
Dimension 2
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From tabulation to data
R1C2
R2C2
R3C2
R1C1
R2C1
R3C1
Dimension 3
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From tabulation to data
Dimension 1
Dimension 2
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From tabulation to data

• Repeat this for level 2 of dimension 3.

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From tabulation to data
Dimension 1
Dimension 2
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From tabulation to data
Dimension 3
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From tabulation to data
Dimension 1
Dimension 2
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From tabulation to data

• Now for level 3 of dimension 3, etc.

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From tabulation to data
Dimension 1
Dimension 2
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From tabulation to data
Dimension 3
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From tabulation to data
Dimension 1
Dimension 2
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From tabulation to data
Dimension 3
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From tabulation to data
Dimension 1
Dimension 2
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From tabulation to data
Dimension 3
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Aggregate data structure

• The data structure just shown is the one used in
  the aggregate data files for 1991 and earlier
  censuses.

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Aggregate data structure

• Another data structure for aggregate files
  consists of records built using nested variables
  along the rows of a table. The categories of
  these nested variables serve as index values.

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Aggregate data structure

• This is the data structure for aggregate data
  from the 1996 Census.

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(11)
(5)
(3)
(4)
(5)
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Aggregate data structure

• age by sex by education by periods
• 5 x 3 x 4 x 5 300 cells
• For each level of geography (region), there are
  300 cells.

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Aggregate data structure

• With the previous data structure, there would be
  one case for each level of region with 300
  variables on each line.

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Aggregate data structure

• With a fully indexed aggregate data structure,
  there are 300 lines for each level of region with
  each line representing a combination of the
  values of the remaining nested or index
  variables, i.e., age, sex, education, period.

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Aggregate data structure

• In Beyond 20/20, the smokers statistics are
  arranged as the column dimension while region,
  age, sex, education and period nested along the
  rows.

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(11)
(5)
(3)
(4)
(5)
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Aggregate data structure

• Because the original table was not created using
  numeric codes for the categories of the index
  variables, writing the data to a file produces
  the following format. Notice that the categories
  are in string or text format.

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Aggregate data structure

• A recode of these string values to numeric
  codes displays the index structure for this type
  of aggregate data file.

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Aggregate Data Structure

• This is a fully indexed data structure. One row
  in the file represents one cell from the table
  while the index values define the combination of
  categories that define the cell.

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Aggregate Data Structure

• A partial index data structure is used by
  Statistics Canada in distributing the 1996
  aggregate data. This kind of structure keeps one
  variable in the column of the table.

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Another example

• Lets look at a table that consists of the
  average length of stay in a hospital by sex, age,
  diagnostic chapter, region, and time period.

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Variables and categories

• diagnostic chapter 19 levels
• sex 3 levels
• age 6 levels
• region 13 levels
• period 28 levels

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Aggregate data structure

• The number of category combinations is equal to
• 13 x 28 x 3 x 6 x 19 124,488 category

combinations
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Aggregate data structure

• Lets convert this into a partially indexed
  aggregate data file structure with the categories
  of region making up the cases the column
  variable consisting of the three latest time
  periods and the summary statistic representing
  the average number of days in the hospital.

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Aggregate average length of hospital stay in days

• The aggregate structure consists of three index
  variables (sex, age, and chapter) with 342
  indexed levels or records per category of region.
• 3 x 6 x 19 342

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What are aggregate data?

• Definition Statistical summaries over
  categorical variables representing social
  phenomena, geography, and time that are organized
  in a statistical data structure.

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Time series aggregate data

• When the data structure of the summaries is
  organized around time, these aggregate statistics
  are called a time series.

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Time Series aggregate data structure
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Annual Time Series
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Geo-spatial aggregate data

• When the data structure of the summaries is
  organized around geography, we recognize these
  aggregations as geo-referenced data.

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Geo-spatial aggregate data structure
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Province
Census Divisions
Census Sub-divisions
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Why aggregate?

• Statistics Canada creates aggregate statistics
  from its major surveys, including the Census, to
  publish select findings.

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Why aggregate?

• The release of aggregate statistics instead of
  the raw data is a way of protecting the
  confidentiality of the respondents and
  safeguarding against disclosure.

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Why aggregate?

• Furthermore, the geographic distribution of
  statistics in Canada is important. As a result,
  aggregate statistics are released by Statistics
  Canada for different levels of geography from
  the nation to small areas.

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Why aggregate?

• Statistics organized into time series is another
  way in which Statistics Canada publishes a large
  amount of statistical information. These time
  series reflect summaries of data that are
  repeatedly collected over time and permit studies
  about trends and change.

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Why aggregate?

• To publish findings
• To safeguard against disclosure
• To provide geographic distributions of statistics
  
• To present statistics over time

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Why aggregate?

• DLI reasons to aggregate
• To modify geo-referenced statistics for GIS
  applications
• for example, finding postal codes within their
  corresponding EA and then aggregating data from
  the postal code level up to the EA level

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Why aggregate?

• Other reasons to aggregate
• To change the unit of analysis
• for the purposes of a specific research question
• to create a common, higher-level unit of analysis
  that can be used in merging files

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How does one aggregate?

• 1. Identify the grouping structure that
  represents all of the variables and their
  categories over which the aggregation is to be
  conducted.
• This group structure defines a new unit of
  analysis and
• with an indexed structure, the levels of indexing.

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How does one aggregate?

• 2. Establish the sort-order for the grouping
  variables, i.e., decide which variable increments
  the slowest, the next slowest, until you reach
  the variable that changes the fastest.

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How does one aggregate?

• 3. Select the summary statistics, such as sums,
  averages, minimums, maximums, etc.

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How does one aggregate?

• As an example, here are the summary statistics
  available in the SPSS Aggregate procedure.

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How does one aggregate?

• The actual aggregation is performed using
  statistical software such as SAS or SPSS.
• SAS offers the Data step and a few procedures
  that can be used to aggregate data, including
  Proc Summary, Proc Tabulate, and Proc Means.

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How does one aggregate?

• SPSS, as already shown, provides the Aggregate
  procedure.

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Time for an exercise
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