Advanced Statistical Methods: Continuous Variables http://statisticalmethods.wordpress.com

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Advanced Statistical Methods Continuous
Variables http//statisticalmethods.wordpress.com

• ANOVA
• tomescu.1_at_sociology.osu.edu

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• test uses 2 estimates of variance variability of
  the sample means ?i about the overall ?
  (between-groups estimate) to the variability of
  the sample observations about their separate
  group means (within-group estimate).
• Total variability Between-group Within-group
  Variability
• F Between estimate/Within estimate
• BSS/(g-1)
• WSS/(N-g)
• where g no. of groups N overall sample size
• WSS also error sum of squares

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• Assumptions
• Normal population distributions on the response
  variable for the g groups
• Equal st. dev. of the population distribution for
  the g groups
• Independent random samples from the g populations
  (one-way ANOVA)

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• Confidence Intervals comparing Means
• No. of pairwise comparisons g(g-1)/2
• - If g is large (i.e. many groups), some pairs of
  of means may appear to be different even if all
  the population means are equal
• For 95 CI, error probability of 0.05 prob.
  that any particular CI does not contain the true
  difference in population means if large no. of
  Cis ? prob. that at least 1 CI in error is much
  larger than the error prob. for any particular
  interval (multiple comparison error rate)
• Simultaneous CI Bonferroni CI
• controls the prob. that all CIs contain the true
  difference
• the prob. at least one set of events occurs
  cannot be greater than the sum of the separate
  probabilities of the events
• Bonferroni 95 CI are wider than the separate 95
  CI

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• Factorial Between-Subjects ANOVA
• 1 response variable (continuous) 2/more
  qualitative explanatory variables
• compare mean of response variable btw. categories
  of any of the IVs, controlling for the other(s)
• with/without interaction effects
• Ex mean income for gender by social classes
  without interaction

Gender Social Class (collapsed) Social Class (collapsed) Social Class (collapsed)
Gender Privileged Disadvantaged Neutral
Male (1) µ11 µ12 µ13
Female (0) µ01 µ02 µ03
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• anova ln_sinc2008 sex2008 class_3grps_2008
• corr sex class -0.136 N 1295
• H0 no diffr. in mean income btw. males and
  females, controlling for social class b10 F
  test Sex mean square/Means square
  error/residual 0.627/0.021 with df11 and df2
  (errorresidual) 739
• Ho no diffr. in mean income for the 3 social
  classes, controlling for gender b2b30 F
  Social Class mean square/Mean square
  error/residual 2.232/0.021 106.7 with df1
  2 df2 (errorresidual) 739

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• Ex mean income for gender by social classes
  with interaction
• Tests 3 types of H0
• a) are means for men women likely to have come
  from same sampling distribution of means income
  is averaged across social classes to eliminate
  that source of variation
• ? compare mean income for males for females,
  controlling for social class
• b) are means for the 3 social classes likely to
  have come from same sampling distribution of
  means, averaged across men and women?
• ? compare means in income for the 3 social
  classes, controlling for gender
• c) are cell means (the means for women the
  means for men within each social class) likely to
  have come from the same sampling distribution of
  differences btw. means?

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• anova ln_sinc2008 sex2008 class_3grps_2008
  int_cls3gr_ssex

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• Two-way ANOVA and Regression
• DV mean income
• IVs dummy for gender (male1 female0)
• set of dummies for social class Privileged
  (yes1 0else)
• Disadvantaged (yes1 0else)
• Neutral reference group
• Y a b1sex b2Privileged b3Disadvantaged
• - no interaction
• Next table based on Agresti Finlay 1999, p. 457

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• µ13 a b1 ?? µ13 µ03 b1 b1 µ13 - µ03
• H0, no difference between males a females in
  mean income, controlling for social class b10
• b2 difference btw. Privileged and Neutral,
  controlling for gender
• b3 diffr. btw. Disadvantaged Neutral,
  controlling for gender
• H0 no difference among social classes in mean
  income, controlling for gender b2b30

Gender Social Class Dummy Variables Dummy Variables Dummy Variables Mean of Income
Gender Social Class Sex Priv Disad ab1sexb2Prb3Dis
Male Privileged 1 1 0 µ11 a b1 b2
Disadvantaged 1 0 1 µ12 a b1 b3
Neutral 1 0 0 µ13 a b1
Female Privileged 0 1 0 µ01 a b2
Disadvantaged 0 0 1 µ02 a b3
Neutral 0 0 0 µ03 a
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• Ho no diffr. in mean income for the 3 social
  classes, controlling for gender b2b30
• H0 no diffr. In mean income btw. Males and
  females, controlling for social class b10

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• Model with Interaction
• Y a b1Sex b2Privileged b3Disadvantaged
  b4(PrivSex) b5(DisadvSex)
• H0 b4 b50 ? F Interaction mean square/Mean
  square errorresidual
• 0.049/0.020 2.45 w df1 2 df2 737

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• Repeated Measures ANOVA
• - when groups are not independent (i.e. have same
  subjects)
• Ex Opinion of Subjects about 3 Influences on
  Children (Agresti Finlay 1999, p. 463)

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• Mixed Models Random Fixed Effects
• Y a b1Movies b2TV b3Subject1 b4Subject2
  b14Subject11
• Y a bj gi where bj effect of influence
  j
• gi effect for subject I
• - expresses the expected response in the cell in
  row i and column j additively in terms of a row
  main effect and a column main effect
• parameter for the last category of each variable
  0 (comparison group)
• main interest estimating influence parameters
  (bs) not subject parameters (gs)
• gs random effects the categories of this
  factor (i.e. subjects) are a random sample of all
  the possible ones
• bs fixed effects analyses refer to all the
  categories of interest (rather than a random
  sample of them) and provide inferences about
  differences among means for those categories

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• Often, in mixed models - more than 1 fixed effect
  
• Ex groups to be compared on the repeated
  responses
• generally, groups have independent samples
• each time (wave), however, has the same subjects
  ? at this level, samples are dependent
• Next table from Agresti Finlay 1999, p. 467

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• treatment time are fixed effects
• the repeated measurements on time creates a 3rd
  effect, a random effect for subjects
• subjects are crossed with the within-subjects
  factor (time)
• subjects are nested within the between-subjects
  factor (treatment)
• Can test for each main effect for interaction
  btw. them. Attn error term has 2 parts (Agresti
  Finaly, 1999, p.468)