Statistical Analysis

1
Chapter 16

• Statistical Analysis

2
Chapter Outline

• Introduction
• Descriptive Statistics
• Inferential Statistics
• Other Multivariate Techniques

3
Descriptive Statistics

• Statistical computations that describe the
  characteristics of a sample or the relationship
  among variables in a sample.
• Inferential statistics make inferences about the
  larger population from which the sample
  observations were drawn.

4
Partial Raw Data Matrix
5
Hypothetical Raw Data on Education and Prejudice
6
Measures of Association

• Descriptive statistics summarizing the
  relationships between variables.
• Many measures of association are based on a
  proportionate reduction of error (PRE) model.

7
Proportionate Reduction of Error (PRE)

• A logical model for assessing the strength of a
  relationship by asking how much knowing values on
  one variable would reduce our errors in guessing
  values on the other.
• Example
• If we know how much education people have, we can
  improve our ability to estimate how much they
  earn, thus indicating there is a relationship
  between the two variables.

8
Proportionate Reduction of Error (PRE)

• Based on a comparison of
• The number of errors we would make in attempting
  to guess the attributes of a given variable for
  each of the cases under study - if we knew
  nothing but the distribution of attributes on
  that variable.
• The number of errors we would make if we knew the
  joint distribution overall and were told for each
  case the attribute of one variable each time we
  were asked to guess the attribute of the other.

9
Question

• ______________ is a logical model for assessing
  the strength of a relationship.
• MRI
• DVU
• OMG
• PRE

10
Answer D

• PRE is a logical model for assessing the strength
  of a relationship.

11
Nominal Variables

• If two variables consist of nominal data (gender,
  religious affiliation, race), lambda (l) would be
  one appropriate measure.
• Lambda is based on your ability to guess values
  on one of the variables the PRE achieved through
  knowledge of values on the other variable.

12
Ordinal Variables

• If the variables being related are ordinal
  (social class, religiosity, alienation), gamma
  (g) is one appropriate measure of association.
• Lambda is based on guessing exact values, gamma
  is based on guessing the ordinal arrangement of
  values.

13
Gamma

• Computed from two quantities
• The number of pairs having the same ranking on
  the two variables.
• The number of pairs having the opposite ranking
  on the two variables.
• For pairs having the same ranking
• The frequency of each cell in the table is
  multiplied by the sum of all cells below and to
  the right of it, with all products summed.

14
Question

• If the variables being related are ordinal,
  _________ is an appropriate measure of
  association.
• Gamma
• Lambda
• Rho
• chi square

15
Answer A

• If the variables being related are ordinal, gamma
  is an appropriate measure of association.

16
Interval or Ratio Variables

• If interval or ratio variables (age, income,
  grade point average, and so forth) are being
  associated, one appropriate measure of
  association is Pearsons product-moment
  correlation (r).
• r reflects how closely you can guess the value of
  one variable through your knowledge of the value
  of another.

17
Regression Analysis

• A method of data analysis in which the
  relationships among variables are represented in
  the form of an equation, called a regression
  equation.
• Linear regression analysis
• A form of statistical analysis that seeks the
  equation for the straight line that best
  describes the relationship between two ratio
  variables.

18
Question

• If interval or ratio variables are being
  associated, one appropriate measure is
  ____________.
• gamma
• lambda
• Pearsons product
• none of these choices

19
Answer C

• If interval or ratio variables are being
  associated, one appropriate measure is Pearsons
  product.

20
Simple Scattergram of Values of X and Y
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A Scattergram of the Values of Two Variables with
Regression Line Added
22
Multiple Regression Analysis

• A form of statistical analysis that seeks the
  equation representing the impact of two or more
  independent variables on a single dependent
  variable.

23
Partial Regression Analysis

• A form of regression analysis in which the
  effects of one or more variables are held
  constant, similar to the logic of the elaboration
  model.

24
Curvilinear Regression Analysis

• A form of regression analysis that allows
  relationships among variables to be expressed
  with curved geometric lines instead of straight
  ones.

25
Inferential Statistics

• The body of statistical computations relevant to
  making inferences from findings based on sample
  observations to some larger population.

26
Nonsampling Error

• Those imperfections of data quality that are a
  result of factors other than sampling error.
• Examples misunderstandings of questions by
  respondents, erroneous recordings by interviewers
  and coders, and keypunch errors.

27
Statistical Significance

• A general term referring to the likelihood that
  relationships observed in a sample could be
  attributed to sampling error alone.
• Tests of Statistical Significance
• A class of statistical computations that indicate
  the likelihood that the relationship observed
  between variables in a sample can be attributed
  to sampling error only.

28
The Logic of Statistical Significance

• Assumptions regarding the independence of two
  variables in the population study.
• Assumptions regarding the representativeness of
  samples selected through conventional
  probability-sampling procedures.
• The observed joint distribution of sample
  elements in terms of the two variables.

29
Question

• _________________ indicate the likelihood that
  the relationship observed between variables in a
  sample can be attributed to sampling error only.
• ex post facto hypothesizing
• tests of statistical significance
• disconfirmation
• all of these choices

30
Answer B

• Tests of statistical significance indicate the
  likelihood that the relationship observed between
  variables in a sample can be attributed to
  sampling error only.

31
Hypothetical Population of Men and Women Who
Favor or Oppose Sexual Equality
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A Representative Sample
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An Unrepresentative Sample
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Level of Significance

• In the context of tests of statistical
  significance, the degree of likelihood that an
  observed, empirical relationship could be
  attributable to sampling error.
• A relationship is significant at the .05 level if
  the likelihood of its being only a function of
  sampling error is no greater than 5 out of 100.

35
Representative Sample from a Population in Which
Variables Are Related
36
Chi Square

• Computed as follows.
• For each cell in the tables, the researcher
• Subtracts the expected frequency for that cell
  from the observed frequency.
• Squares this quantity.
• Divides the squared difference by the expected
  frequency.
• This procedure is carried out for each cell in
  the tables.

37
Path Analysis

• A form of multivariate analysis in which the
  causal relationships among variables are
  presented in a graphical format.

38
Diagramming the Religious Sources of Anti-Semitism
39
The Larceny Rates over Time in a Hypothetical City
40
Factor Analysis

• An analytical method of discovering the general
  dimensions represented by a collection of actual
  variables.
• These factors are calculated hypothetical
  dimensions that are not perfectly represented by
  any of the empirical variables under study but
  are highly associated with groups of empirical
  variables.

41
Analysis of Variance (ANOVA)

• Based on comparing variations between and within
  groups and determining whether between-group
  differences could reasonably have occurred in
  simple random sampling or whether they likely
  represent a genuine relationship between the
  variables involved.

42
Discriminant Analysis

• Seeks to account for variation in some dependent
  variable by finding a hypothetical, composite
  dimension that separates categories of the
  dependent variable.
• Results in an equation that scores people on the
  basis of that hypothetical dimension and allows
  us to predict their values on the dependent
  variable.

43
Question

• _________________ is a causal model for
  understanding relationships between variables.
• ex post facto hypothesizing
• tests of statistical significance
• path analysis
• all of these choices

44
Answer C

• Path analysis is a causal model for understanding
  relationships between variables.

45
Two Distribution Patterns of the Incomes of
Republicans and Democrats
46
Six Writers Three Who Write by Hand and Three
Who Use Computers
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Plotting the Six Writers in Terms of Age and
Income
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Income Alone Is Sufficient to Predict Writing
Method
49
A Slightly More Complicated Pattern
50
Separating the Pens from the Computers
51
Log-linear Models

• Offer a method for analyzing complex
  relationships among several nominal variables
  having more than two attributes each.

52
Geographic Information Systems (GIS)

• Map quantitative data that describe geographic
  units for a graphical display.

53
Quick Quiz
54

• 1. _____________ is the applied branch of
  mathematics especially appropriate to a variety
  of research analyses.
• calculus
• probability
• statistics
• none of these choices

55
Answer C

• Statistics is the applied branch of mathematics
  especially appropriate to a variety of research
  analyses.

56

• 2. Gamma is composed of
• the number of pairs having the same ranking on
  two variables
• the number of pairs having the opposite ranking
  on the two variables
• both a and b
• none of these choices

57
Answer C

• 2. Gamma is composed of the number of pairs
  having the same ranking on two variables, and the
  number of pairs having the opposite ranking on
  the two variables.

58

• 3. A __________permits the estimation of values
  on one variable from values on the other.
• multivariate analysis
• indirect analysis
• regression line
• exploratory study

59
Answer C

• A regression line permits the estimation of
  values on one variable from values on the other.

60

• 4. ______________ are statistical measures used
  for making inferences from findings based on
  sample observations to a larger population.
• descriptive statistics
• inferential statistics
• ex post facto statistics
• none of these choices

61
Answer B

• Inferential statistics are statistical measures
  used for making inferences from findings based on
  sample observations to a larger population.

62

• 5. A____________ analysis represents changes in a
  variable over time.
• regression
• bivariate
• time-series analysis
• all of these choices

63
Answer C

• A time-series analysis represents changes in a
  variable over time.

64

• 6. Which type of statistics assists researchers
  in drawing conclusions from their observations.
• descriptive statistics
• inferential statistics
• ordinal statistics
• interval statistics

65
Answer B

• Inferential statistics assists researchers in
  drawing conclusions from their observations.