Statistics bootcamp

1
Statistics bootcamp

• Laine Ruus
• Data Library Service, University of Toronto
• Rev. 2005-04-26

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Outline

• Describing a variable
• Describing relationships among two or more
  variables

3
First, some vocabulary

• Variable In social science research, for each
  unit of analysis, each item of data (e.g., age of
  person, income of family, consumer price index)
  is called a variable.
• Unit of analysis The basic observable entity
  being analyzed by a study and for which data are
  collected in the form of variables. A unit of
  analysis is sometimes referred to as a case or
  "observation"

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The variable Sex can be coded as

• 1male 2 female 3 no response, or
• 1female 2 male, or
• 1male 2 female, or
• Mmale F female, or
• male female, or even
• 1yes 2 no 3 maybe

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The values a variable can take must be

• exhaustive include the characteristics of all
  cases, and
• mutually exclusive each case must have one and
  only one value or code for each variable

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Whats wrong with this coding scheme?

• Under 3,000
• 3,000-7,000
• 8,000-12,000
• 13,000-17,000
• 18,000-22,000
• 23,000-27,000
• 28,000-32,000
• 33,000-37,000
• 38,000 over
• (Source Census of Canada, 1961 user summary
  tapes)

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Variables are normally coded numerically, because

• arithmetic is easier with numbers than with
  letters
• some characteristics are inherently numeric age,
  weight in kilograms or pounds, number of children
  ever borne, income, value of dwelling, years of
  schooling, etc.
• space/size of data sets has, until recently, been
  a major consideration

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Three basic types of variables

• Categorical
• Nominal, aka nonorderable discrete
• Eg gender, ethnicity, immigrant status, province
  of residence, marital status, labour force
  status, etc.
• Ordinal, aka orderable discrete
• Eg highest level of schooling, social class,
  left-right political identification, Likert
  scales, income groups, age groups, etc.
• Continuous, aka interval, numeric
• Eg actual age, income, etc.

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Descriptive statistics summarize the properties
of a sample of observations

• how the units of observation are the same
  (central tendency)
• how they are different (dispersion)
• how representative the sample is of the
  population at large (significance)

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Nominal variable

• Central tendency
• mode
• Dispersion
• frequency distribution
• percentages, proportions, odds
• Index of qualitative variation (IQV)
• Significance
• coefficient of variation (CV)
• Visualization
• bar chart

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Mode the category with the largest number or
percentage of observations in a frequency
distribution
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The frequencies can be visualized as a bar chart
(based on percentages)
13
The same distribution, from one of the Canadian
overview files
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and showing percentages
15
Notice the differences
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Why the differences?

• What is the population in each table?
• What is in the denominator in each table?
• Which one is correct?

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The most important thing to know about any
distribution, whether it is a rate, a
proportion, or a percent, is what is in the
denominator. And it must always be reported.
18
We can also derive the distribution information
from the 2001 individual pumf using a statistical
package
19
If we weight the distribution, both the
frequencies and the percentages will almost match
the distribution from the Profile file
20
Just a few words on weighting

• The weight is the chance that any member of the
  population (universe) had to be selected for the
  sample
• In general
• the weight can be used to produce estimates for
  the total population (population weight)
• and/or the weight can adjust for known
  deficiencies in the sample (sample weight)

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and a few more words on weighting

• The 2001 census public use microdata file of
  individuals is a 2.7 sample of the population
• The weight variable (weightp) ranges from
  35.545777-39.464996
• Knowing who was excluded from the sample is as
  important as knowing who was included

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And some final words on weighting

• When to use the population weight variable
• when you are producing frequencies to reflect the
  frequency in the population
• When you dont need to use the population weight
  variable
• when you are producing percentages, proportions,
  ratios, rates, etc.
• When to use a sample weight variable
• always

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Proportions, percents, and odds

• Percent the percent married in the population
  15 years and over is
• (married/population gt15 years)10049.47
• Proportion the proportion married in the
  population 15 years and over is
• (married/population gt15 years).4947
• Odds the odds on being married are
• (married/not married in the population) or
• proportion married/(1-proportion married)
• .4947/(1-.4947).4947/.5053.9790

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Coefficient of variation

• measures how representative the variable in the
  sample is of the distribution in the population
• computed as ((standard deviation/mean)100)
• we will discuss these measures in the context
  of continuous variables
• see Stats Can guidelines in user guides
• cvlt 16.6 is ok to publish, cvgt33.3 do not
  publish
• SDA reports the cv when generating frequencies

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Ordinal variable

• Central tendency
• median and mode
• Dispersion
• frequency distribution
• range
• percentages/quantiles, proportions, odds
• Index of qualitative variation (IQV)
• Significance
• coefficient of variation (CV)
• Visualization
• histogram

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Example frequency distribution
28
Example relative percentages
29
The Median is the value that divides an orderable
distribution exactly into halves.

• Finding the median is easier if we compute
  cumulative percentages, eg in Excel

30
Cum 0-4 years 5.65 5.65 5-9 years
6.59 12.24 10-14 years 6.84 19.08 15-24
years 13.36 32.44 25-34 years 13.31 45.75 35-4
4 years 17.00 62.75 45-54 years 14.73 77.48 55
-64 years 9.56 87.04 65-74 years
7.14 94.18 75-84 years 4.43 98.61 85 years
and over 1.39 100
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So how can we describe this distribution, using
the vocabulary we have so far?

• What is the mode of this distribution?
• What is the median?
• What is the range?

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A statistical package can also report these
measures
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Percentiles/quantiles

• percentiles/quantiles are the value below which a
  given percentage of the cases fall
• the median the 50th percentile
• quartiles are the values of a distribution broken
  into 4 even intervals each containing 25 of the
  cases

35
Interquartile range

• The interquartile range is the difference between
  the value at 75 of cases, and the value at 25
  of cases
• What is the interquartile range for the following
  distribution?

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Cum 0-4 years 5.65 5.65 5-9 years
6.59 12.24 10-14 years 6.84 19.08 15-24
years 13.36 32.44 25-34 years 13.31 45.75 35-4
4 years 17.00 62.75 45-54 years 14.73 77.48 55
-64 years 9.56 87.04 65-74 years
7.14 94.18 75-84 years 4.43 98.61 85 years
and over 1.39 100
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Continuous variable

• Central tendency
• mean, median and mode
• Dispersion
• range
• variance or standard deviation
• quantiles/percentiles
• interquartile range
• Significance
• standard error
• coefficient of variation
• Visualization
• polygon (line graph)

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Means, variances, and standard deviations

• Mean the average, computed by adding up the
  values of all observations and dividing by the
  number of observations (cases)
• Variance computed by taking each value and
  subtracting the value of the mean from it,
  squaring the results, adding them up, and
  dividing by the number of cases (actually N-1)
• Standard deviation the value that cuts off 68
  of the cases above or below the mean, in a normal
  distribution. Its the square root of the
  variance, in the same metric as the variable.

39
Availability of continuous variables in Stats Can
products

• Stats Can rarely publishes truly continuous
  variables in its aggregate statistics products
• Some exceptions are
• age by single years (census)
• estimates of population by single years of age
  (Annual demographic statistics)

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Statistics Canada generally reports the
distribution of continuous variables as

• Measures of central tendency
• an ordinal (categorical) variable
• an average (mean) and standard error, variance,
  or standard deviation
• a median
• rates (other than of 100), eg. children ever born
  per 1,000 women in the 1991 census 2B profile
• Measures of dispersion
• percentage essentially, a rate per 100
  population. Eg, Incidence of low income as a of
  economic families, in the census profiles. This
  includes employment and unemployment rates
• quantiles (eg quintiles in Income trends in
  Canada)
• Gini coefficients (computed from the coefficient
  of variation) (eg Income trends in Canada)
• Measures of significance
• standard error

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In the following distribution

• What is the median?
• What is the mean?
• What is the range?

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Using the percentages and cumulative percentages

• What is your best estimate of the interquartile
  range?
• What is your best estimate of the standard
  deviation?
• Why is the average (mean) income so much higher
  than the median?
• See pages 18-19 of your handout

44
As a percentage distribution
45
The polygon produced by Beyond 20/20 isnt very
useful
46
Using the standard error to describe more of the
distribution

• standard error of the mean is a measure of how
  likely it is that the mean in the data we are
  looking at is the same as or similar to the mean
  in the population at large
• computed from the standard deviation divided by
  the square root of the N
• the larger the N, the smaller the standard error,
  and the more confidence we can have in the
  distribution in the sample as representative of
  the population

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For example.
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Confidence intervals

• The standard error makes most sense when we use
  it to compute a confidence interval around the
  mean
• For Canada, in the previous example
• 95 upper confidence limit (UCL)
• (mean)1.96(standard error)
• 297691.96(19) 29769 37.2429,806.24
• 95 lower confidence limit (LCL)
• (mean)-1.96(standard error)
• 29769 -1.96(19) 29769 - 37.24 29,731.76
• 1.96 is the Z-statistic that represents 95 of a
  normal distribution
• The handout contains computed confidence
  intervals for each of the provinces (p.22)

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How do we interpret this?

• if we draw repeated random samples from the same
  population, 95 of them will have a mean total
  income between 29,732 and 29,806
• this is not the same as saying that we are 95
  confident that the population mean falls within
  those two limits.

50
Using microdata

• Statistical packages such as SAS, SPSS, etc. will
  compute the mean, the standard deviation,
  standard error, etc. for continuous variables
• SDA will only report these measures for variables
  with less than 8,000 values

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For the following example
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How many values?

• What is the maximum number of values for the
  wagesp variable? For the totincp variable?
• If you were to do a frequency distribution of
  wagesp, how many rows might you have in the
  distribution?
• How would you go about finding out what the modal
  category for wagesp is?

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Some final points about variables

• they are not immutable or un-changeable
• continuous variables can be changed into ordinal,
  or even nominal variables
• ordinal variables can be changed into nominal
  variables, and
• nominal variables can be collapsed still further
• nominal and ordinal variables can be combined to
  create indices or scales

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Describing relationships among two or more
variables

• Objectives of describing multivariate
  relationships
• description
• improved ability to predict the value of a
  variable for a case, by using the value of
  another variable (or variables)
• examine causation, which requires
• covariation
• the causal variable must occur before the outcome
  variable in time (temporal precedence)
• A non-spurious relationship

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Variables are either

• Dependent (aka outcome variables) or
• Independent (eg causal variables)

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The same variable can be an independent variable
in one hypothesis, and a dependent variable in
another hypothesis

• Does gender make a difference in level of
  education?
• level of education is dependant gender is
  independant
• Does level of education make a difference to
  earned income?
• earned income is dependant level of
  education is independent
• Is the effect of education on earned income the
  same for men and women?
• earned income is dependant level of
  education is independent, and gender is the
  control variable

58
Statistical measures describe

• The strength of the relationship between two (or
  more) variables
• The direction of the relationship between two (or
  more) variables
• The significance of the relationship between two
  (or more) variables in the sample vis-à-vis the
  population

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The choice of appropriate statistical technique
is dependant on

• whether the dependant variable is nominal,
  ordinal, or continuous
• whether the dependant variable is a dichotomy or
  a polytomy
• whether the independent variable(s) is a
  dichotomy, a polytomy, or continuous

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Relationships between two or more nominal or
ordinal variables

• cross-tabulations
• a common census output product (all the
  Topic-based tabulations in 2001)
• strength and direction
• odds ratios
• significance
• chi-square statistic

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A cross-tabulation
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Computing the odds ratio
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Graphed in Excel, it looks like this
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The chi-square statistic

• Tells us how likely we are to be wrong if we say
  there is a relationship between these two
  variables
• Significance (probability of being wrong)
• Evaluated based on
• Critical value (found in statistics texts)
• Degrees of freedom (df1)
• Level of significance (a 5 possibility of being
  wrong is normally acceptable in the social
  sciences)

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So we know

• Direction strength from graphing the odds
• Significance from a chi-square statistic
• Can be computed in eg Excel
• For the previous table, 2.97 (see handout page
  30)
• Critical value 3.84 with 1 degree of freedom, at
  .05 significance (ie probability of being wrong)
• This table is not statistically significant,
  therefore a good chance of seeing this
  relationship as a chance of eg measurement error

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Adding a control variable in Beyond 20/20
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Computing the conditional odds
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And graphed in Excel, it looks like this
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The original 1996 2-way table looked like this
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But when we added visible minority as a control
variable

• it ended up looking like this

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Did you notice?

• In the 1996 data, when we looked only at
  immigrants, versus non-immigrants, the
  non-immigrants were more likely to be employed
• When we break it out by visible minority status,
• Non-visible majority immigrants are more likely
  to be employed than non-immigrants
  (11.73/9.161.28)
• Visible minorities immigrants are also more
  likely to be employed than non-immigrants
  (6.30/5.521.14)
• This is an example of Simpsons paradox

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Using a statistical package to examine these
relationships

• Automatically computes a chi-square
• Computes degrees of freedom and probability
• Is sensitive to sample size (so we need to take a
  subsample)

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And produces these results
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In the table on the previous slide

• How many cases are there in this subsample?
• What percentage overall are unemployed?
• What percentage of immigrants are unemployed?
• Is this different from the percentage of
  non-immigrants that are unemployed?
• What percentage are immigrants? How would you
  find out?

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Now, you compute the tables for the visible
minority groups
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And tell me what you found

• How many tables are produced?
• What is the difference between them?
• How many of these tables are statistically
  significant (based on the Chisq)?
• In which two groups is the relationship the
  opposite of the majority of the groups?

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Comparison of means

• Relationship between a continuous dependant
  variable, and a dichotomous independent variable
• Significance
• Students t-statistic (similar to Z-statistic)
• Example average employment income for visible
  minorities versus all others (2001 census
  dimensions series)

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Visible minority status and employment income

• In 1996 census, a Dimensions series table showed
  visible minority status and employment income
• This table not available in 2001 census
• Relationship can only be examined using
  microdata, or requesting a custom tabulation

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And the result is
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Computing 95 confidence intervals
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ANOVA (analysis of variance)

• When the dependent variable is a continuous
  variable, and the independent variable is a
  polytomy
• Significance
• F-ratio
• Eta-squared (amount of explained variance)
• Example mean earned income for individual
  visible minorities

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And the result is
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And the associated statistical measures
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Rerunning with highest level of schooling
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And the associated statistical tests
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Pairwise relationships among continuous and
dichotomous variables

• Correlations
• Strength and direction
• Pearson correlation coefficient
• Significance
• Z-statistics for each pair
• (r-to-Z transformation table)

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The correlation matrix
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Combined effects of one or more independent
variables on a continuous dependant variable

• Regression analysis
• Categorical variables must be recoded to dummy
  variables
• Strength and direction
• t-statistic for each independent variable
• Significance
• R²
• F-ratio

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Techniques weve covered
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What you need to remember

• To examine the relationships between 2 or more
  variables in Beyond 20/20, the variables must be
  in the same file and in different dimensions
• If no table exists with those variables, the
  alternative is to use a relevant microdata file
• To generate the table (or request a custom
  tabulation from Statistics Canada)
• To compute the more complicated measures of
  association and significance

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What you need to remember (contd)

• A user who needs to do correlations or regression
  analysis needs continuous outcome (dependant
  variables)
• Information about statistical techniques is
  readily available on the WWW