Using Analysis of Variance

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• Using Analysis of Variance
• (ANOVA)
• Lecture 1
• One-way between-subjects ANOVA
• Kevin Paterson
• kbp3_at_le.ac.uk

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One-way between subjects ANOVA

• This is what you should be able to do
• Describe basic motivation behind one-way ANOVA
  technique.
• Conduct and interpret one-way ANOVA, including
  the use of post hoc tests.
• Report ANOVA results appropriately.
• Gravetter Wallnau Chapter 13

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One-way between subjects ANOVA

• Psychologist wants to know if weather condition
  affects problem-solving
• 30 undergrads assigned to 1 of 3 conditions.
• Raining outside
• Snowing outside
• Sunny outside
• Subjects took 30 minutes to solve problems.

• Null hypothesis no significant difference in
  number of problems solved in 3 conditions.

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One-way between subjects ANOVA

• How are you going to examine these data?
• Could conduct t-tests to compare raining
  snowing, raining sunny, snowing sunny.

• More problems solved on sunny than rainy days
  (t(18) 3.54, p lt 0.05).
• More problems solved in sunny than snowy days
  (t(18) 3.54, p lt 0.05).
• No difference in number of problems solved in
  rainy and snowy conditions (t(18) 0.0, p gt
  0.05).

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One-way between subjects ANOVA

• Theres a problem with this.
• Each t-test is accepted with a 5 chance of
  finding a difference when none exists (thats
  what p lt 0.05 means).
• So, if you do 3 t-tests, then thats 3 x 5
  chance of error 15.
• More tests you do, the more likely it is you will
  find a supposed difference when none actually
  exists (this is called a Type 1 error).
• Cant we just do one test on data?

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One-way between subjects ANOVA

• Relationship between ANOVA and t-test
• A t-test and an ANOVA with 1 factor with 2 levels
  are very similar, using similar calculations.
• F t2
• ANOVA and regression are also related tests.
• ANOVA uses factorial method to test for
  differences.
• Regression looks for predictive relationships.

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One-way between subjects ANOVA

• Sampling theory and analysis of variance.
• When you run an experiment, you take samples of
  data. Question is, do these samples belong to the
  same or a different underlying population?
  Samples from same population should have same
  variance mean. ANOVA tests if means are from
  same or different populations.

Raining
Snowing
Sunny
mean 15.0 variance 4.0
mean 15.0 variance 4.0
mean 18.0 variance 4.0
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One-way between subjects ANOVA

• How ANOVA works
• 1. Estimate spread of scores within groups
  (called MSwithin or MSerror).
• 2. Estimate spread of scores between groups
  (called MSbetween or MStreatment).
• If samples belong to same population then the
  ratio of 1 and 2 should be small.
• If samples belong to different populations then
  the ratio of 1 and 2 should be large.

Example 1
Example 2
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One-way between subjects ANOVA

• Source of error
• 1. Individual differences Everyone has
  different characteristics, and may behave
  slightly differently on different occasions, for
  a variety of reasons that are outside of our
  control. These differences may have an effect on
  the variability of scores.
• 2. Experimental error whenever you make a
  measurement there is a potential for error, which
  may influence the variability of scores.

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One-way between subjects ANOVA

• F-ratios
• Anova is based on F-statistics (as compared with
  t-statistics etc.)

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One-way between subjects ANOVA

• Conducting ANOVA
• Entry your data into SPSS just as you would for
  independent samples t-test
• Separate columns for condition codes and scores.
• Only the condition column will have 3 condition
  codes, corresponding to the 3 levels of the
  manipulated variable.

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One-way between subjects ANOVA

• Select descriptive statistics homogeneity of
  variance options.
• To see if test can be used.
• Select post-hoc bonferroni test option.
• To see what conditions cause significant effects
  using ANOVA

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One-way between subjects ANOVA
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One-way between subjects ANOVA

• Homogeneity of variance
• Strictly speaking, you should only use ANOVA if
  samples have similar spread (variance).
• In SPSS you can test this using Levines test for
  Homogeneity of Variance.
• If sample variances do not differ significantly
  then you can use ANOVA.

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One-way between subjects ANOVA

• Sums of squares represent gross variability
  between and within subjects
• df gives the degrees of freedom associated with
  between subjects and within subjects effects.
• Mean square provides MSbetween and Mswithin
• F is F-ratio of MSbetween and Mswithin
• Sig. provides precise p-value for F-ratio.

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One-way between subjects ANOVA

• Post hoc tests
• ANOVA tells you if sample belong to different
  populations (i.e. if experimental manipulation
  had an effect).
• However, it doesnt tell you which conditions are
  different.
• Post hoc tests (such as the bonferroni test) are
  used to compare pairs of samples.
• Therefore, having first established that the
  experimental manipulation had an effect, you can
  compare the different levels of that condition.

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One-way between subjects ANOVA

• Interpreting bonferroni tests
• Table gives pair-wise comparisons, mean
  differences, and associated p-values.

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One-way between subjects ANOVA

• Reporting results
• Say what you did in experiment.
• Present mean results in text or table or graph
  (but not all 3).
• State ANOVA results giving df, F-ratio p-value.
• State results of post hoc tests.

Table 1 mean number of problems solved (with
standard deviations).
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One-way between subjects ANOVA

• We used a between subjects design to examine
  mathematical problem-solving in three external
  weather conditions rain, snow and sun. The mean
  number of problems solved in each weather
  condition is shown in Table 1. There was a
  significant effect of weather condition (F(2, 27)
  7.50, plt0.01). Post hoc bonferroni tests
  indicated that more mathematical problems were
  solved when it was sunny than when it was raining
  (plt0.05), and more mathematical problems were
  solved when it was sunny than when it was snowing
  (plt0.05). However, there was no difference in
  number of mathematical problems solved when it
  was raining or snowing (pgt0.05).

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One-way between subjects ANOVA

• Things I havent told you
• What happens if you have unequal sample sizes.
• Answer is that the method of calculation is
  modified (but not much in the case of one-way
  ANOVA - see Gravetter, p423).
• What happens if sample is not normal?
• Dont worry too much. ANOVA is robust and can
  endure violations of assumptions. However, you
  might consider transforming data -see Howell,
  p309.

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One-way Analysis of Variance
One-way between subjects ANOVA

• What happens if samples do not have same
  variance?
• Again, ANOVA is robust and can deal with this (to
  some extent). If homogeneity of variance is
  seriously violated, then Howell advises using
  Welchs Procedure (Howell, p309). However, I
  cant imagine you ever having to do this.
• What about other post hoc (and a priori) tests?
• There are many others - Gravetter, pp 425-430.