ANOVA Factorial Designs

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					ANOVA: Factorial Designs
Experimental Design

 Choosing the appropriate statistic or
  design involves an understanding of
   The number of independent variables and levels
   The nature of assignment of subjects to
    treatment levels
 http://psych.athabascau.ca/html/Validity/in
  dex.shtml
      Review of a Few Designs
 Subjects are completely randomly assigned to
  treatments
  Completely Randomized ANOVA: One treatment
   with two or more levels
  Completely Randomized Factorial ANOVA: Two or
   more treatments each with two or more levels
 Treatments assigned to homogenous blocks of
  experimental units or repeated measures used
  to control for nuisance variation
  Randomized Block ANOVA: One treatment with two
   or more levels
The secret to mastering two-factor
analysis of variance
 In a two-factor analysis of variance, we look at
  interactions along with main effects.
   Interactions are the effect that one factor has on
    another factor.
   The main effect of a factor looks at the mean
    response obtained by averaging each factor (IV) over
    all levels.
 A two-factor ANOVA should begin with an
  examination of the interactions. Interpretation of the
  main effects changes according to whether
  interactions are present.
Two-factor ANOVA
 Consists of three significance tests:
   Each of the main effects (A and B)
   Interaction of the two factors (AB)
 There is an F-test for each of the hypotheses: the
  mean square for each main effect and the
  interaction effect divided by the within-variance
  (MSE).
Hypothesis testing
 The first hypothesis (main effect of factor A):
   looks at the mean response for each level of A--that
    is, the mean obtained by averaging over all levels of
    B--and asks whether they are the same. This is the
    case whether or not there is an interaction in the
    underlying model.
 The second hypothesis (main effect of factor B):
   looks at the mean response for each level of B--that
    is, the mean obtained by averaging over all levels of
    A--and asks whether they are the same. This is the
    case whether or not there is an interaction in the
    underlying model.
 The third hypothesis (interaction effect):
   asks whether or not factor A has an effect on factor B.
A demonstration of a two-factor ANOVA

http://www.ruf.rice.edu/~lane/stat_sim/two
 _way/index.html
Let’s use an example we’ve seen before

 In this study, interviewers telephone calls to
  adults in randomly selected households to ask
  opinions about the next election.
 Treatment A: introduction
  A1: Gave name
  A2: Identified the university he/she was representing
  A3: Gave name and identified the university.
 Treatment B: copy of survey?
  B1: Interviewer offered to send a copy of final results
  B2: Interviewer did not offer to send a copy of the final
   results
For default there is an effect of B but not
A. Make up data where…
…there is an effect of A but not B and no
 interaction.
…there is an effect of B and an interaction,
 but no effect of A.
…there is a cross-over interaction.
What part of the ANOVA summary table is
 unaffected by group size?
Is it necessary for lines to cross to have a
 significant interaction?
Conclusions
 If there is no significant interaction, the means for the
  levels of factor A will behave like the expected values
  from any of the individual levels of B.

 If there is a significant interaction, the hypotheses for the
  main effects are the same (we still test whether the
  means obtained from each level of A by averaging over
  all levels of B are the same.) However, if the model
  includes an interaction, this hypothesis might not be
  useful.

 Look first at the interaction; never analyze or interpret
  main effects in the presence of an interaction.
    Never interpret main effects in the
    presence of interactions
Survey response rate based on no compensation vs. $5.
Suppose B1 (circles) were males and B2 (squares) were females.
 Graph 1: Since the means of the two levels of A are equal,
    the main effect of A is 0. Yet, it would be a huge mistake to
    say that A doesn't matter. Who benefits from what?
   Graph 2: How do males compare to females in terms of
    response rate based on compensation? What would you do
    with this information if you were advising how to reward
    completed surveys?
   Graph 3: How do males compare to females in terms of
    response rate based on compensation? What would you do
    with this information if you were advising how to reward for
    completed surveys?
   Graph 4: Do males and females differ in terms of response
    rate? Explain.