Quantitative Data Analysis

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Quantitative Data Analysis

• Descriptive statistics
• Bivariate relationships

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Bivariate statistics

• Describe the relationship between two variables
• e.g. water pollution and health
• education and health promoting behaviour
• gender and liberalism
• The statistical analysis of two variables tells
  us about the relationship between these variables

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Covariance and independence

• Two important concepts for understanding
  bivariate statistical relationships
• Covariance means that two things go together
  they are associated with each other.
• changes in one variable are reflected in changes
  in the other variable (they covary).
• higher incomes higher life expectancy
• higher education higher job prestige

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• Independence means that the variables are not
  related there is no association
• opposite of covariation
• The two variables are independent/unrelated.
• changes in one variable are not associated with
  changes in the other variable.
• number of siblings does not influence life
  expectancy

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Causal statements/hypotheses

• In inferential statistics (discussed later),
  independence and covariance are reflected in the
  null hypothesis and the hypothesis.
• The null hypothesis the vars are not related
• independence
• A hypothesis there is a causal relationship
• covariance
• you predict the variables are related - covary

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Bivariate techniques

• Scattergram
• a graph of the relationship
• cross-tabulation or contingency table
• display the relationship on a table
• measures of association
• statistical measures (the amount of
  covariation is expressed in terms of a value)
• also called a correlation coefficient

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Bivariate Contingency Table Cross-tabulation

• A bivariate table cross-tabulates or
  cross-classifies two (or more) variables
• used for categorical or grouped data (re to
  condense interval or ratio level data)
• Called a contingency table the distribution of
  cases in one category of a variable are
  contingent upon the categories of the second
  variable

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Percentage Table (see Table 11.9)

• Contingency tables report (or count) the number
  of cases in each cell reports raw data
• Researchers convert raw counts tables into
  percentage tables. Why?
• Constructing Cross-tabulation tables (See Rules
  on p.371)
• - run percentages toward the independent
  variable

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Bivariate Tables Comparing Means

• Cross-tabulation is used when variables are
  categorical (nominal/ordinal) or when interval or
  ratio level data are grouped.
• When the independent variable is nominal and the
  dependent variable is interval/ratio, we compare
  the means for the two (or more) categories (See
  table 11.19, p. 376)

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Correlation scattergram (see p.383)

• researcher plots all the cases on a graph, with
  each axis representing one variable
• used for interval/ratio level data
• not used for nominal or ordinal data
• place the independent variable on the horizontal
  (or X) axis and the dependent variable on the
  vertical (or Y) axis
• lowest value in lower left corner highest values
  at the top (for Y) and far right (for X)

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1. Form of the relationship independence,
linear, and curvilinear

• Independence no relationship
• Cases form a random scatter no pattern
• Linear relationship
• Cases are located around an imagery straight
  line from one corner to another
• Curvilinear relationship
• Cases form a U curve, and inverted U curve, or
  and S curve

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2. Direction of the relationship positive or
negative

• Positive relationship the higher the value of
  the X variable, the higher the value of the Y
  variable ( visa versa). Your examples?
• higher education higher income
• lower education lower income
• the cases form a diagonal pattern (line) from the
  lower left hand corner to the upper right

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2. Direction of the relationship positive or
negative

• Negative relationship the higher the value on X,
  the lower the value on Y Examples?
• higher education, lower of arrests
• greater social integration, lower depression
• the cases form a diagonal pattern (line) from the
  top left-hand side of the graph to the lower
  right-hand side of the graph
• can have a shallow or steep slope

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3. The degree of precision

• Precision refers to spread of points on the
  graph the amount of spread
• High precision cases hug the line (not spread)
• Low precision considerable dispersion (spread)
  of cases around the line
• Scattergram researchers eyeball precision
• Can also use advanced statistics to measure the
  degree of precision of a relationship

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Measures of Association

• A measure of association is a statistical
  computation which produces a single value or
  number that indicates the strength of the
  relationship between two variables.
• It indicates the degree to which the two
  variables go together or covary
• Is there an association between the variables?
• Are they correlated?
• Is there a strong or weak relationship?

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Measures of Association

• There are many measures of association (lambda,
  gamma, tau, chi-squared, rho)
• the correct choice depends on the level of
  measurement of the variables
• interpretation depends upon the measure used

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Five most commonly used measures of association

• Measure Type of Data High Assn Independence
• Lambda Nominal 1.0 0
• Gamma Ordinal 1.0, -1.0 0
• Tau Ordinal 1.0, -1.0 0
• Rho Interval, ratio 1.0, -1.0 0
• Chi-squared Nominal, ordinal Infinity 0

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Summary of Measures of Association

• Rho (Pearsons r) is the most commonly used
  measure of association
• tells how well the data fit the (regression)
  line on a scattergram (-1 to 1, 0 independence)
• rho measures linear relationships only 0 can
  mean no relationship or curvilinear relationship
• Chi-squared is used as descriptive statistic
  (measure of association) and as an inferential
  statistic

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Interpreting measures of associationproportion
reduction in error

• How much does information about one variable
  reduce the errors that are made when guessing the
  values of the other variable?
• Independence
• The measure of association 0
• knowing about one variable will not help you to
  guess the value of the other variable

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Interpreting measures of associationproportion
reduction in error

• Correlation/Association
• knowing about one variable reduces the error in
  predicting the values of the other variable
• strong association few errors in predicting the
  second variable on the basis of the first
• weak association proportion of errors is larger

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Understanding Rho

• 0 the variables are not correlated at all
• 1 perfect positive correlation
• -1 perfect negative correlation
• Interpret the following
• The correlation between womens full-time
  salaries and mens full-time salaries is .70
• height and intelligence is .02