Statistics

1
Statistics

• ANOVA

2
Summary of inferential statistics so far

• Z-test used to compare a sample mean to the
  population mean
• Large samples are needed
• Population mean must be known
• Population standard deviation must be known
• One-sample t-test used to compare a sample mean
  to the population mean
• Small samples are alright
• Population mean must be known
• Dependent-samples t-test - used to compare a
  sample mean to another sample mean
• Two samples are related to each other

3

• Independent-samples t-test used to compare the
  means of two independent groups.
• Experimental
• Control
• Ex. Cheapest car rental
• Hertz
• Cooterz
• Problem What if there is more than 2 levels of
  the independent variable?
• Hertz
• Cooterz
• Enterprise

4

• Solution Multiple t-tests?
• C1 Hertz vs. Cooterz
• C2 Hertz vs. Enterprise
• C3 Cooterz vs. Enterprise
• Problem a becomes inflated
• a .05 for every test
• C1 .05
• C2 .05
• C3 .05
• Overall probability of committing a type I error
  goes up dramatically.
• p (C1) or p (C2) p (C1) p(C2) p(C1 and C2)
• .05 .05 (.05.05) .10 - .0025 .0975
• With 3 comparisons, a goes up to .1426

5

• Analysis of Variance (ANOVA) allows you to
  determine whether there is a significant
  difference between 3 or more samples without
  increasing the size of alpha.
• In principle, works the same as a t-test
• i.e., testing the null hypothesis
• t-test H0 µ1 µ2
• ANOVA H0 µ1 µ2 µ3 µk
• k the number of levels of the independent
  variable
• ANOVA tests this equality of means

6

• Using variance to evaluate differences may sound
  strange at first.
• Purpose of experiment determine the systematic
  variance and reduce the error variance as much as
  possible
• i.e., there are two types of variance to consider
• ANOVA uses this notion for its test
• Compares the systematic variance to the error
  variance
• Systematic variance variance between groups
• Error variance variance within groups
• One-way ANOVA used when there is one
  independent variable
• Two-way ANOVA used when there is two
  independent variables a factorial design

7

• If the variance between groups is very low
  relative to error variance, we fail to reject the
  H0
• What does this mean?
• The means of the 3 groups were very similar.
• The 3 groups came from the same population.
• If the variance between groups is very high
  relative to error variance, we reject the H0.
  Perhaps the H1 is more fitting.
• What does this mean?
• The means of the 3 groups were significantly
  different.
• The 3 groups did not come from the same
  population.

8
Calculating the two types of variance

• The variance estimate used is not S2
• Mean square (MS) the mean of the squared
  deviation scores used to calculate the variation.

9
k the total number of samples (groups)
10
Computational formulas for MS

• Sum of Squares (SS)

• Three different SS of interest in the ANOVA
• SSbetween
• SSwithin
• SStotal

11
dftotal N - 1 dfbetween k-1 dfwithin N - k
12

• F is the statistic utilized with the ANOVA.
• Note F is a ratio
• The expected value of F with H0 being assumed 1
• The test is still testing where the samples fit
  in.
• The calculation of F and t is actually quite
  similar
• F t2

13
Example sleep deprivation and aggression
14
Creating an ANOVA table
15
Significance testing

• Works similar to testing with t-tests
• i.e., obtained value is compared to a critical
  value
• Critical values are different for F
• Higher numbers
• F distribution is quite a bit different than t
• All tests are one-tailed with F
• Why is this?

16
(No Transcript)
17

• To find critical value, consult the F table
• Slight difference, there are two degrees of
  freedom
• dfbetween
• dfwithin

18
(No Transcript)
19
ANOVA results

• So the F was significant. What does that mean?
• We decide to reject the H0
• H0 All of the groups are equal
• H1 The groups are not all equal
• There is a difference, but what is the source of
  the difference?

20
Example sleep deprivation and aggression
Totals ?
Means ?
21
Further testing

• Multiple comparison procedures (MCPs) Tests
  that seek to identify the source of the
  differences that was found in the omnibus ANOVA
• post-hoc vs. planned comparisons
• Post-hoc After the fact
• Planned comparisons a priori, theoretically
  interesting
• Often called pair-wise comparisons
• Ex. 0 vs. 24 hours
• Ex. 0 vs. 48 hours
• Ex. 24 vs. 48 hours
• MCPs should be used following a significant F
• Source of debate

22

• Many different MCPs exist
• See Toothaker
• Major issue for MCPs controlling for the error
• a for the omnibus ANOVA was .05
• What to make a for the pair-wise comparisons?
• The LSD (Least Significant Difference) does not
  control for a. Pairwise t-tests are carried out,
  each at level a.
• Very liberal test not readily accepted

23
Tukey HSD

• HSD honestly significant difference
• Utilizes the studentized range distribution
• Controls for a inflation by increasing critical
  values depending on how many comparisons are
  being made.

24
Formula
We already have the MSw and n from the ANOVA, we
just need to find q. To find q, look at the
studentized range table. (p. 434 of book)
25
(No Transcript)
26
Example
Any mean difference of 4.8 or more is significant
27
(No Transcript)
28
Another example in-class homework