Class 24OutlinePath AnalysisNotations and TermsEquation SystemAdvantages of Path DiagramSlide 7Factor AnalysisExploratory and Confirmatory Factor AnalysisExample of Verbal and Math AbilitiesA Two-Factor SolutionTime Series AnalysisThe Birthday ExampleCohort Analysis: Pooling Information from Repeated Cross-Sectional DataTime Series EffectsEasterlin Hypothesis (Cohort Effect)Linear Dependency among Age, Period, and CohortSlide 18Class 24Multivariate Techniques and Demographic MethodsOutline•Path Analysis•Factor Analysis•Time Series Analysis•Age-Cohort-Period EffectsPath AnalysisBackgroundindex X1School SESrating X2Ambition X3IQ X4Class valueindex X5RuRvRw.27.90.92.87.46.12.15.11.23.12.19.06.37Causal model from Turner (1964)Notations and Terms•One-way straight arrow: causation•Two-way curved arrow: correlation•Path coefficient p (standardized coefficient) •Correlation coefficient r (zero-order)•Direct effect, indirect effect and total effect3223pp 3223rr Equation System The path coefficients in the previous diagram are calculated from the following equation system.X3=p32X2+p31X1+p3uRuX4=p43X3+p42X2+p41X1+p4vRvX5=p54X4+p53X3+p52X2+p51X1+P5wRwAdvantages of Path Diagram•It makes explicit the assumptions for the causal interpretation of data.–Assumptions about ordering of the variables–Assumptions about residuals •It presents the complete information about a causal scheme in a single diagram.Advantages of Path Diagram•Path analysis provides a way to calculate indirect effects. –Example: indirect effect of X2 on X5 isr52-p52=.28-.15=.13•Correlation between any pair of variables can be written in terms of the paths leading from common antecedent variables and correlation between exogenous variables.–Example:r32=p32+p31r12=.23+.27*.46=.35Factor Analysis•Origins of factor analysis date back to Spearman’s work on human intelligence in 1905.•Purpose – to describe the relationships between many variables by a few underlying, unobserved (latent) factors. Data reduction.Exploratory and Confirmatory Factor Analysis•Confirmatory factor analysis: theory suggests the number of factors and FA is the statistical procedure to test the theory.•Exploratory factor analysis: seeks to uncover the underlying structure of a relatively large set of variables. There is no prior theory and the number of factors is unknown in advance.Example of Verbal and Math AbilitiesFrench English History Arithmetic Algebra GeometryFrench 1English 0.44 1History 0.41 0.35 1Arithmetic 0.29 0.35 0.16 1Algebra 0.33 0.32 0.19 0.59 1Geometry 0.25 0.33 0.18 0.47 0.46 1Correlation Matrix of Test Scores in Six SubjectsA Two-Factor SolutionFactor 1 (math) Factor 2 (verbal)French 0.236 0.660English 0.321 0.550History 0.591Arithmetic 0.771 0.169Algebra 0.716 0.217Geometry 0.571 0.213Factor loadingsTime Series Analysis•Times series - a series of measurements taken at different times. For example, unemployment rates; population growth.•Panel studies – the same individuals are interviewed multiple times.•Cohort analysis – the same cohort (often not the same individuals) are interviewed at different times. Cohort: a set of individuals who experienced the same event at about the same time.The Birthday Example7000 8000 9000 10000 11000freq0 100 200 300 400dayyearNumber of Births by Day of Year7000 7500 8000 8500 9000 9500freq90 100 110 120dayyearNumber of Births by Day in April7,000 8,000 9,000 10,000 11,0001 2 3 4 5 6 7 8 9 10 11 12Number of Births by Month7,000 8,000 9,000 10,000 11,0000 1 2 3 4 5 6Number of Births by Day of WeekCohort Analysis: Pooling Information from Repeated Cross-Sectional DataAge 4 c1986 c1987 c1988 c1989 c19903 c1987 c1988 c1989 c1990 c19912 c1988 c1989 c1990 c1991 c19921 c1989 c1990 c1991 c1992 c19930 c1990 c1991 c1992 c1993 c19941990 1991 1992 1993 1994yearTime Series Effects•Age effects are effects related to aging or the life-cycle. E.g., individuals tend to become less healthy as they age.•Period effects are effects affecting all cohorts in a given historical period. E.g., effects of famine on mortality.•Cohort effects are effects which reflect the unique reaction of a cohort to an historical event, or which were experienced uniquely by the cohort. E.g., Post-WWII cohort who reached draft age during the Vietnam War experienced unique issues.Easterlin Hypothesis (Cohort Effect) •Richard Easterlin conjectured that people born in large cohorts (i.e., baby boomers) are socially and economically disadvantaged, due to crowding. •Economic disadvantage leads to low fertility. •Individuals growing up in small cohorts (such as children of the Great Depression) are economically advantaged and thus have higher fertility. Easterlin Cycle.Linear Dependency among Age, Period, and Cohort•Cohort + Age = Period–E.g., birth cohort = 1985, age = 20, period (year) = 1985 + 20 = 2005.•Therefore, age, period, and cohort are linearly dependent, meaning that if we know two of them, we know the value of the third.•The effects of linearly-dependent independent variables cannot be estimated in a multiple regression equation. That is, we cannot estimate the following regression equation: Y = a + b*age +c*cohort + d*year + e.•The above model involving age, period, and cohort is said to be “not identified.”Cells contain:-Mean occ. prestige score-N of casesage119-20221-30331-40441-50551-60661-90ROWTOTALcoho rt1 1901-1910.000.000.000.000.00036.971,79836.971,7982 1911-1920.000.000.000.00037.6795937.902,07637.833,0353 1921-1930.000.000.00038.7899638.901,77539.5985439.033,6254 1931-1940.000.00040.371,08240.901,69140.11752.00040.573,5255 1941-1950.00041.141,42242.212,51542.291,162.000.00041.935,0996 1951-196030.2531138.612,91341.491,595.000.000.00039.034,8197 1961-197029.6130837.741,250.000.000.000.00036.131,5588 1971-198027.569.000.000.000.000.00027.569COL TOTAL29.8962839.065,58541.615,19240.773,84938.823,48637.854,72839.3823,4 68Does occupational status increase across cohort (with
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