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Quantitative Reasoning

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Lecture Notes - January 31, 2002

Studying the Social Sciences Scientifically

Characteristics of Scientific Knowledge
1. Empirically Verified: demonstration by means of objective observation that a statement is true.
2. Explanatory: signifies that a conclusion can be derived from a set of general propositions and specific initial considerations.
3. General: applicable to many rather than a few cases
4. Nonnormative: knowledge is not concerned with evaluation or prescription but with factual or objective determinations.
5. Predictive: indicates an ability to correctly anticipate future events. The application of explanation to events in the future forms a prediction.
6. Provisional: subject to revision and change.
7. Transmissible: indicates that the methods used in making scientific discoveries are made explicit.
Explanation
1. Explanation is a systemic, empirically verified understanding of why a phenomenon occurs as it does.
2. Probabilistic explanation: an explanation that does not explain or predict events with 100% accuracy.
3. Theory: a statement or series of statements that organize, explain, and predict knowledge.
Acquiring Scientific Knowledge
1. Deduction: a process of reasoning from a theory to specific observations.
2. Induction: a process of reasoning from specific observations to general principle.
The Research Process
- Research Question
- Theory & Hypothesis
- Operationalization & Measurement
- Research Design
- Data Collection
- Data Analysis
- Data Interpretation
- Implications

Lecture Notes - February 4, 2002

The Research Process: Theory & Hypothesis

Form a question: Why...?
- The question may come from practical studies, studies based on interest, or normative concerns.
Research the question and review past studies
- Come up with a theory, a statement or series of statments that organize, explain, and predict knowledge. It should be general and applicable to many situations. Not all theories are empirically derived, some begin with assumptions.
- Determine the independent variable(s), the measure(s) of the phenomena that influence the dependent variable.
- Determine any antecedent variables, variables from the past that may influence the current dependent variable.
- Determine any intervening variables, variables that occur outside the study that may affect the independent and dependent variables.
Formulate a Hypothesis
- A hypothesis is an explicit statement that indicates how a researcher thinks the phenomena of interest are related.
- Characteristics of a good hypothesis:
  1. Must be empirical, not normative
  2. Must be general
  3. Must be plausible (make sense in respect to the research done)
  4. Must be directional: it must specify the expected relationship between the two variables as either positive or negative
  5. Must be consistent with the data
  6. Must be testable
  7. Must specify the types or levels of actor (unit of analysis).

Lab Notes - February 6, 2002

Concepts lead to a theory. Through operationalization, the variables involved in the concept are described. The variables are then used to form a general, directional, and testable hypothesis.

Lecture Notes - February 6, 2002

Guidelines for Critically Analyzing Social Science Research

What is the research question? and Why is it important?
- What are its theoretical implications? What do we know now that we did not know before?
- What are its methodological implications? What new ways of studying the subject have we learned?
- What are its practical implications? What meaning does it have for various people?
What is the research hypothesis?
- What are the theoretical justifications for this hypothesis?
- What is the dependent variable?
- What is the independent variable?
- Are there any antecedent or intervening variables?
- What is the unit of analysis?
How are the concepts operationalized?
How are the concepts measured?

Lecture Notes - February 7, 2002

Measurement - Systematic observation and representation by scores or numerals of the variables under investigation.
Operational Definition - The rules by which a concept is measured and scores assigned.
Validity - The correspondence between a measure and the concept it is supposed to measure.
- Face validity - Validity asserted by arguing that a measure corresponds closely to the concept it is designed to measure.
- Construct validity - Validity demonstrated for a measure by showing that it is related to measure of another concept.
- Content validity
- Interitem validity
Level of Measurement - The degree of information that we can assign and then extract from a variable.
1. Nominal variables (ie gender, race, religion, marital status) - Set of categories that are exhaustive and mutually exclusive, numerical values have no substantive meaning.
  - The categories of the variable are not numerical and can be compared to each other only in terms of the number of cases classified within them.
  - The categories must be mutually exclusive of each other so that no ambiguity exists concerning classification of any given case. There must exist an "other" or "misc." category. The categories must be relatively homogenous.
  - The only mathematical operation allowed: counting the number of occurences that have been classified into the various categories of the variable.
2. Ordinal variables (ie social class, attitude, opinion) - Set of categories that is exhaustive and mutually exclusive. Numerical values indicate ranking.
  - Allows the categories themselves to be ranked with respect to how much of the trait being measured they possess.
  - In addition to being able to count the number of cases in each category, we can rank the cases with respect to each other.
  - One fault of ordinal variables is that the distance between scores cannot be described in precise terms.
3. Interval-Ratio Variables (ie age, number of children, income)
  - Measured in units that have equal intervals
  - True zero point exists
  - Interval variables have a continuous scale with real, meaningful units and a zero point that means lack of the variable or that none of the variable exists.
  - Ratio variables are like interval variables, but they have a meaningful zero-point (ie temperature - zero degrees does not mean that temperature does not exist).
Lab Notes - February 13, 2002
1. The categories of a variable should be:
  1. Mutually exclusive: A person (etc.) can only belong to one category
  2. Collectively exhaustive: All categories are included, complete range of possiblities is covered.
2. Variables should be valid and reliable (accurate and consistent)
  1. Validity: Does the variable measure the concept that it is intended to measure? (ie using a scale to measure height would be invalid)
  2. Reliability: Would you get the same result if you repeated the question or observation? (Note: using multiple people to measure increases reliability)
  3. Factors that affect reliability: random error (ie sudden sickness of the person under study), ambiguous questions (on a survey)
  4. Factors that affect validity: systematic error, reactive measurement effect (people react to being measured, "the Hawthorne effect"), memory
3. Ways to classify variables
  1. Level of measuremnt (NOIR) - higher levels include characteristics of lower levels
    - Nominal: categories cannot be ordered, ranked, or mathematically analyzed
    - Ordinal: (ie "Likert scale") categories can be ordered/ranked
    - Interval: uniform interval between adjacent categories, can be mathematically analyzed via addition and subtration
    - Ratio: non-arbitrary (meaningful) zero, can be mathematically analyzed via multiplication and division
  2. Continuous vs. Discrete
    - Continuous: Age, time, money --> can be broken into hundredths with many choices
    - Discrete: Religion, gender, number of siblings --> cannot have halves
    - Nominal and Ordinal variables are always discrete
Lecture Notes - February 14, 2002
Research Designs
1. Classic Experimental Design
  - Experimental Group goes through Assignmnet, Pre-test, Treatment, and Post-Test
  - Control Group goes through Assignment, Pre-test, and Post-test
2. Time Series Quasi-Experimental Design
  - In addition to the treatment given to the experimental group, you must take other factors that affect both groups into account.
  - The researcher does not have complete control over the independent variable(s).
3. Non-Experimental Time Series Design
4. Panel-Study Design
  - Assignment to groups is not random
  - There are only two points in time considered (the more, the better)
  - Questionairre about background may improve the study.
5. Case Study Design
  - Allows the researcher to get specific explanations, but limits the researcher in attaining general information (ie - the researcher can get information on an individual student, but not on the entire school)
6. Cross-Sectional Study
  - Assignment is random, "Real World" Treatment, one-time post-test
Introduction to Sampling
1. Probability Sample - Simple random, Systematic, Stratified (horizontal), or Cluster (vertical)
2. Non-Probability Sample
3. Study Population - the specific group we are trying to get information on
4. Elements and Populations
Data Collection
1. Methods: Experimentation, Field Research (participant observation, interviewing, focus groups --> highly valid with questionable reliability), Content Analysis, Existing Sources, Survey Research
Notes from Healey Book
1. Data Reduction - The process of organizing data into a presentable form. This can be done by using numbers, a table, or a graphic device to summarize or stand for a large array of data.
  1. Percentage (%) is equal to 100 times frequency (the number of cases in any category) divided by the number of cases in all categories.
  2. Proportion (p) is equal to frequency divided by the number of cases in all categories.
    - When working with a small number of cases (say, fewer than 20), it is usually preferable to report the actual frequencies rather than percentages or proportions.
    - Always report the number of observations along with proportions and percentages.
    - Proportions and percentages can be calculated for variables at any level of measurement.
  3. Ratio equals the number of cases in the first category divided by the number of cases in the second category.
  4. Rates are defined as the number of actual occurrences of some phenomenon divided by the number of possible occurrences per some unit of time.
  5. Frequency distributions are tables that summarize the distribution of a variable by reporting the number of cases contained in each category of the variable.
    - The categories fo the frequency distribution must be exhaustive and mutually exclusive.
    - The table should have a descriptive title, clearly labeled categories, and a report of the total number of cases at the bottom of the frequency column. These items must be included in the table regardless of the variable or level of measurement.
    - The scores of interval-ratio variables must be grouped into larger categories to heighten clarity and ease of comprehension. Note: No overlapping of categories (class intervals) is allowed.
    - Midpoint is defined as the point exactly halfway between the upper and lower limits.
  6. Cumulative frequency: For each higher interval, the cumulative frequency will be all cases in the interval plus all the cases in the preceeding intervals.
  7. Cumulative percentage: For each higher interval, the cumulative percentage will be the total of the percentage for that interval and the percentages of the intervals preceeding it.
  8. Procedures for contructing frequency distributions for Interval-Ratio Variables.
    - Decide how many class intervals you wish to use.
    - Find the size of the class interval.
    - State the lowest (highest) interval so that its lower limit is equal to or below the lowest (highest score. All intervals must be equal in size.
    - Do not overlap intervals.
    - Count the number of cases in each class interval and report these subtotals in a column labeled "frequency." Next, inspect the frequency distribution carefully.
    - Give a clear, concise title, and number the table if your report contains more than one. All categories and columns must also be clearly labeled.
2. Charts and Graphs
  1. Pie charts: Begin by computing the percentage of all cases that fall into each category of the variable. Then divide a circle (the pie) into segements (slices) proportional to the percentage distribution. Label the chart and all segments clearly.
  2. Bar charts: Conventionally, the categories of the variable are arrayed along the horizontal axis (or abscissa) and frequencies, or percentages if you prefer, along the vertical axis (or ordinate). For each category, construct a rectangle of constant width and with a height that corresponds to the number of cases in the category. Bar charts are particularly effective ways to display the relative frequencies for two or more categories of a variable when you want to emphasize some comparisons.
  3. Histograms: Like bar charts, except the bars are contiguous to each other. Most appropriate for continuous interval-ratio-level variables. To construct a histogram:
    - Arrray the class intervals or scores along the horizontal axis (abscissa) using the class limits.
    - Array frequencies along the vertical axis (ordinate).
    - For each category in the frequency distribution, construct a bar with height corresponding to the number of cases in the category and width corresponding to the limits of the class intervals.
    - Label each axis of the graph.
    - Title the graph.
  4. Line Charts (frequncy polygon): Like histograms, except dots are used at the midpoint of each interval and then connected by a line. Line charts can be used to display trends across time.