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Two Way Anova
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Two Way Anova



Two-Way ANOVA introduction and example

Two-Way ANOVA introduction and example



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Two Way Anova Two Way Anova Presentation Transcript

  • Two-Way ANOVA TWO Categorical Independent Variables ONE Continuous Dependent Variable
  • Two Types of ANOVA so far...
    • Oneway ANOVA compares the mean and variability between groups
      • One dependent interval/ratio variable
      • One independent nominal variable that defines the groups
    • Assumptions
      • Groups should be independent of each other (no subject in more than one grp)
      • Dependent variable has a normal distribution (symmetric, unimodal)
  • Figure from Chapter 13: Formulas for ANOVA.
  • Repeated Measures ANOVA
    • Repeated Measures ANOVA compares the level of the dependent variable measured at more than 2 points in time, or on more than 2 related subjects (e.g., husband/wife)
    • Assumptions
      • Dependent variable is interval/ratio
      • Dependent variable has normal distribution
      • Dependent variable is measured multiple times on the same or related subjects
  • Figure from Chapter 14 The partitioning of variability for a repeated-measures experiment .
  • Figure from Ch 14: Structure of the analysis for a two-factor analysis of variance.
  • From one-way to two-way designs:
    • Often, we wish to study 2 (or more) factors in a single experiment
      • Compare a new and standard style of noise filter (inside a muffler) on a car
      • The size of the car might also be an important factor in noise level.
    • The baseline experiment will therefore have two factors as Independent Variables
      • Type of noise filter (Octel vs Standard)
      • Size of car (Small, Midsize, Large)
    • First Variable: Filter Type
      • Nominal – Dichotomy
      • Dependent variable is noise level (ratio level)
    • Test: Two-Sample t
      • Compare means (above)
      • View boxplot (at right)
      • t (34)=1.116, p = .272
    • RETAIN H 0
    • Type of filter does not cause a significant difference in noise.
    • Second Variable: Car Size
      • Nominal – 3 groups
      • Dep.Var: noise level (ratio)
    • Test: Oneway ANOVA
      • Compare means (above)
      • View boxplot (at left)
      • F (2,33) =112.44, p < .0005
    • REJECT H 0
    • Size of car is related to a significant difference in noise.
  • ANOVA is significant, so we need Post-hoc Tests. Groups: Same Size so Test: Tukey HSD - Small vs Large = Sig. - Midsize vs Large = Sig. - Small vs Midsize = n.s.
  • Summary of our findings so far
    • Filters – Octel vs Standard
      • Independent sample t -test
      • No significant differences
    • Size of Car – Small, Midsize, Large
      • ANOVA
      • Significant differences
      • Large cars are significantly more quiet
    • BUT – is it possible that the Octel filter might work better with just one of the types of cars?
  • What questions does Two-Way ANOVA answer?
    • Is car size related to noise level, if effect of filter type is controlled?
    • Is filter type related to noise level, if effect of size of car is controlled?
    • Is there a combination of Size of Car and Noise Filter Type that is especially loud, or especially soft?
      • called an INTERACTION effect .
    • Multiple comparison tests
  • What do you see in this graph?
  • Summary: Two-Way ANOVA
    • Factorial (Two or more way) ANOVA
      • One dependent variable
        • interval or ratio
        • normal distribution
      • Two independent variables
        • nominal (define groups)
        • independent of each other
      • Test effect of each I.V. controlling for the effects of the other I.V.
      • Test interaction effect for combinations of categories
  • What do we mean, “controlled”?
    • Not the same as experimental control.
    • Statistical control: we look for the effect of one independent variable within each group of the other dependent variable.
    • This removes the impact of the second independent variable.
    • Sometimes a variable which showed no significant effect in a Oneway ANOVA becomes significant if another effect is controlled.
  • SPSS: Basic Hypothesis Tests
    • SIZE effect is still significant
    • TYPE effect is significant when size is controlled
    • INTERACTION effect is significant
      • There is a combination which shows more than the combined impact of SIZE and TYPE
  • Means of each combination of Size & Type INTERACTION: Whenever lines not parallel
  • Means graphed again – TYPE on axis. CROSSED LINES = Interaction
  • Reporting Results Manufacturers of the new Octel noise filter claim that it reduces noise levels in cars of all sizes. In a Two-Way ANOVA, this claim proved to be true. The Size of Car effect was significant (F 2,36 =199.119, p <.0005). When the impact of size was controlled, the Filter Type effect was also significant (F 1,36 =16.146, p <.0005), with the Octel Filter having lower noise levels than standard filters. The Interaction effect was also significant ( F 2,30 =6.146, p =.006). For Small cars, the noise difference between filter types was 3.33; for Large cars it was 5.000 – but Midsize cars with the Octel filter averaged 24.166 lower on the Noise Level scale. In some situations, one would choose to report the means of each cell. This paragraph could also include effect size.