Exam

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Exam 2 Results
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Exam 2 Results
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Exam 2 Results

• All t-scores refer to Repeated-Measures t-tests
• For the exam as a whole
• t(30) 3.31, p .002, d 1.21
• Cohens conventions for d .2 small, .5
  medium, .8 large

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One-Way Analysis of Variance (ANOVA)
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One-Way ANOVA

• One-Way Analysis of Variance
• aka One-Way ANOVA
• Most widely used statistical technique in all of
  statistics
• One-Way refers to the fact that only one IV and
  one DV are being analyzed (like the t-test)
• i.e. An independent-samples t-test with treatment
  and control groups where the treatment (present
  in the Tx grp and absent in the control grp) is
  the IV

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One-Way ANOVA

• Unlike the t-test, the ANOVA can look at levels
  or subgroups of IVs
• The t-test can only test if an IV is there or
  not, not differences between subgroups of the IV
• I.e. our experiment is to test the effect of hair
  color (our IV) on intelligence
• One t-test can only test if brunettes are smarter
  than blondes, any other comparison would involve
  doing another t-test
• A one-way ANOVA can test many subgroups or levels
  of our IV hair color, for instance blondes,
  brunettes, and redheads are all subtypes of hair
  color, can so can be tested with one one-way ANOVA

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One-Way ANOVA

• Other examples of subgroups
• If race is your IV, then caucasian,
  african-american, asian-american, hispanic, etc.
  are all subgroups/levels
• If gender is your IV, than male and female are
  you levels
• If treatment is your IV, then some treatment, a
  little treatment, and a lot of treatment can be
  your levels

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One-Way ANOVA

• OK, so why not just do a lot of t-tests and keep
  things simple?
• Many t-tests will inflate our Type I Error rate!
• This is an example of using many statistical
  tests to evaluate one hypothesis see the
  Bonferroni Correction
• It is less time consuming
• There is a simple way to do the same thing in
  ANOVA, they are called post-hoc tests, and we
  will go over them later on
• However, with only one DV and one IV (with only
  two levels), the ANOVA and t-test are
  mathematically identical, since they are
  essentially derived from the same source

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One-Way ANOVA

• Therefore, the ANOVA and the t-test have similar
  assumptions
• Assumption of Normality
• Like the t-test you can place fast and loose with
  this one, especially with large enough sample
  size see the Central Limit Theorem
• Assumption of Homogeneity of Variance
• Like the t-test this isnt problematic unless one
  levels variance is much larger than one the
  others (4 times as large) the one-way ANOVA is
  robust to small violations of this assumption, so
  long as group size is roughly equal

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One-Way ANOVA

• Independence of Observations
• Like the t-test, the ANOVA is very sensitive to
  violations of this assumption if violated it is
  more appropriate to use a Repeated-Measures ANOVA
• The basic logic behind the ANOVA
• The ANOVA yields and F-statistic (just like the
  t-test gave us a t-statistic)
• The basic form of the F-statistic is
  MSgroups/MSerror

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One-Way ANOVA

• The basic logic behind the ANOVA
• MS mean square or the mean of squares (why it
  is called this will be more obvious later)
• MSbetween or MSgroups average variability
  (variance) between the levels of our IV/groups
• Ideally we want to maximize MSgroups, because
  were predicting that our IV will differentially
  effect our groups
• i.e. if our IV is treatment, and the levels are
  no treatment vs. a lot of treatment, we would
  expect the treatment group mean to be very
  different than the no treatment mean this
  results in lots of variability between these
  groups

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One-Way ANOVA

• The basic logic behind the ANOVA
• MSwithin or MSerror average variance among
  subjects in the same group
• Ideally we want to minimize MSerror, because
  ideally our IV (treatment) influences everyone
  equally everyone improves, and does so at the
  same rate (i.e. variability is low)
• If F MSgroups/ MSerror, then making MSgroups
  large and MSerror small will result in a large
  value of F
• Like t, a large value corresponds to small
  p-values, which makes it more likely to reject Ho

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One-Way ANOVA

• However, before we calculate MS, we need to
  calculate what are called sums of squares, or SS
• SS the sum of squared deviations around the
  mean
• Does this sound familiar? What does this sound
  like?
• Just like MS, we have SSerror and SSgroups
• Unlike MS, we also have SStotal SSerror
  SSgroups

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One-Way ANOVA

• SStotal S(Xij - )2
• Its the formula for our old friend variance,
  minus the n-1 denominator!
• Note N the number of subjects in all of the
  groups added together

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One-Way ANOVA

• SSgroups
• This means we
• Subtract the grand mean, or the mean of all of
  the individual data points, from each group mean
• Square these numbers
• Multiply them by the number of subjects from that
  particular group
• Sum them
• Note n number of subjects per group
• Hint The number of numbers that you sum should
  equal the number of groups

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One-Way ANOVA

• That leaves us with SSerror SStotal SSgroups
• Remember SStotal SSerror SSgroups
• Degrees of freedom
• Just as we have SStotal,SSerror, and SSgroups, we
  also have dftotal, dferror, and dfgroups
• dftotal N 1 OR the total number of subjects
  in all groups minus 1
• dfgroups k 1 OR the number of levels of our
  IV (aka groups) minus 1
• dferror N k OR the total number of subjects
  minus the number of groups OR dftotal - dfgroups

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One-Way ANOVA

• Now that we have our SS and df, we can calculate
  MS
• MSgroups SSgroups/dfgroups
• MSerror SSerror/dferror
• Remember
• MSbetween or MSgroups average variability
  (variance) between the levels of our IV/groups
• MSwithin or MSerror average variance among
  subjects in the same group

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One-Way ANOVA

• We then use this to calculate our F-statistic
• F MSgroups/ MSerror
• Then we compare this to the F-Table (Table E.3
  and E.4, page 516 517 in your text)
• There are actually two tables, one if you set
  your a .05 (Table E.3, pg. 516), and one if
  your a .01 (Table E.4, pg. 517)

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One-Way ANOVA

• Degrees of Freedom for Numerator dfgroup
• Degrees of Freedom for Denominator dferror

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One-Way ANOVA

• This value our critical F
• Like the critical t, if our observed F is larger
  than the critical F, then we reject Ho
• Hypothesis testing in ANOVA
• Since ANOVA tests for differences between means
  for multiple groups or levels of our IV, then H1
  is that there is a difference somewhere between
  these group means
• H1 µ1 ? µ2 ? µ3 ? µ4, etc
• Ho µ1 µ2 µ3 µ4, etc

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One-Way ANOVA

• However, our F-statistic does not tell us where
  this difference lies
• If we have 4 groups, group 1 could differ from
  groups 2-4, groups 2 and 4 could differ from
  groups 1 and 3, group 1 and 2 could differ from
  3, but not 4, etc.
• Since our hypothesis should be as precise as
  possible (presuming youre researching something
  that isnt completely new), you will want to
  determine the precise nature of these differences
• You can do this using multiple comparison
  techniques

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One-Way ANOVA

• But before we go into that, an example
• Example 1
• An experimenter wanted to examine how depth of
  processing material and age of the subjects
  affected recall of the material after a delay.
  One group consisted of Younger subjects who were
  presented the words to be recalled in a condition
  that elicited a Low level of processing. A second
  group consisted of Younger subjects who were
  given a task requiring the Highest level of
  processing. The two other groups were Older
  subjects who were given tasks requiring either
  Low or High levels of processing. The data
  follow
• Younger/Low 8 6 4 6 7 6 5
  7 9 7
• Younger/High 21 19 17 15 22 16 22 22 18
  21
• Older/Low 9 8 6 8 10 4 6
  5 7 7
• Older/High 10 19 14 5 10 11 14 15
  11 11

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One-Way ANOVA

• Example 1
• DV memory performance, IV age/depth of
  processing (4 levels)
• H1 That at least one of the four groups will be
  different from the other three
• µ1 ? µ2 ? µ3 ? µ4
• Ho That none of the four groups will differ from
  one another
• µ1 µ2 µ3 µ4
• dftotal 40-1 39 dfgroup 4-1 3 dferror
  40-4 36
• Critical Fa.05 between 2.92 and 2.84
  (2.922.84)/2 2.88

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One-Way ANOVA

• Grand Mean (6519370110)/40 10.95
• MeanY/L 65/10 6.5
• MeanY/H 193/10 19.3
• MeanO/L 70/10 7
• MeanO/H 110/10 11

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One-Way ANOVA

• SStotal
• SX2441 3789 520 1386 6,136
• (SX)2 (65 19370110)2 191,844
• SStotal 1,339.9

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One-Way ANOVA

• SSgroup 1,051.3
• SSerror SStotal SSgroups 1,339.9 1,051.3
  288.6
• MSgroups 1051.3/3 350.4333
• MSerror 288.6/36 8.0166666
• F 350.4333/8.0166 43.71

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One-Way ANOVA

• Example 1
• Since our observed F critical F, we would
  reject Ho and conclude that one of our four
  groups is significantly different from one of our
  other groups

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One-Way ANOVA

• Example 2
• What effect does smoking have on performance?
  Spilich, June, and Renner (1992) asked nonsmokers
  (NS), smokers who had delayed smoking for three
  hours (DS), and smokers who were actively smoking
  (AS) to perform a pattern recognition task in
  which they had to locate a target on a screen.
  The data follow

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One-Way ANOVA

• Example 2
• Get into groups of 2 or more
• Identify the IV, number of levels, and the DV
• Identify H1 and Ho
• Identify your dftotal, dfgroups, and dferror, and
  your critical F
• Calculate your observed F
• Would you reject Ho? State your conclusion in
  words.

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One-Way ANOVA
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One-Way ANOVA

• Multiple Comparison Techniques
• The Bonferroni Method
• You could always run 2-sample t-tests on all
  possible 2-group combinations of your groups,
  although with our 4 group example this is 6
  different tests
• Running 6 tests _at_ (a .05) (a .3) ?
• Running 6 tests _at_ (a .05/6 .083) (a .05)
  ?
• This controls what is called the familywise error
  rate in our previous example, all of the 6
  tests that we run are considered a family of
  tests, and the familywise error rate is the a for
  all 6 tests combined we want to keep this at .05

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One-Way ANOVA

• Multiple Comparison Techniques
• Fishers Least Significant Difference (LSD) Test
• Test requires a significant F although the
  Bonferroni method didnt require a significant F,
  you shouldnt use it unless you have one
• Why would you look for a difference between two
  groups when your F said there isnt one?

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One-Way ANOVA

• Multiple Comparison Techniques
• This is what is called statistical fishing and is
  very bad you should not be conducting
  statistical tests willy-nilly without just cause
  or a theoretical reason for doing so
• Think of someone fishing in a lake, you dont
  know if anything is there, but youll keep trying
  until you find something the idea is that if
  your hypothesis is true, you shouldnt have to
  look to hard to find it, because if you look for
  anything hard enough you tend to find it

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One-Way ANOVA

• Multiple Comparison Techniques
• Fishers LSD
• We replace in our 2-sample t-test formula
  with MSerror, and we get
• We then test this using a critical t, using our
  t-table and dferror as our df
• You can use either a one-tailed or two-tailed
  test, depending on whether or not you think one
  mean is higher or lower (one-tailed) or possibly
  either (two-tailed) than the other

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One-Way ANOVA

• Multiple Comparison Techniques
• Fishers LSD
• However, with more than 3 groups, using Fishers
  LSD results in an inflation of a (i.e. with 4
  groups a .1)
• You could use the Bonferroni method to correct
  for this, but then why not just use it in the
  first place?
• This is why Fishers LSD is no longer widely used
  and other methods are preferred

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One-Way ANOVA

• Multiple Comparison Techniques
• 3. Tukeys Honestly Significant Difference (HSD)
  test
• Very popular, but too conservative in that it
  result in a low degree of Type I Error but too
  high Type II Error (incorrectly rejects H1)
• Scheffes test
• Preferred by most statisticians, as it minimizes
  both Type I and Type II Error but not will not be
  covered in detail, just something to keep in mind

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One-Way ANOVA

• Estimates of Effect Size in ANOVA
• ?2 (eta squared) SSgroup/SStotal
• Unfortunately, this is what most statistical
  computer packages give you, because it is simple
  to calculate, but seriously overestimates the
  size of effect
• ?2 (omega squared)
• Less biased than ?2, but still not ideal

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One-Way ANOVA

• Estimates of Effect Size in ANOVA
• Cohens d
• Remember for d, .2 small effect, .5 medium,
  and .8 large

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One-Way ANOVA

• Reporting and Interpreting Results in ANOVA
• We report our ANOVA as
• F(dfgroups, dftotal) x.xx, p .xx, d .xx
• i.e. for F(4, 299) 1.5, p .01, d .01 We
  have 5 groups, 300 subjects total in all of our
  groups put together We can reject Ho, however
  our small effect size statistic informs us that
  it may be our large sample size that resulted in
  us doing so rather than a large effect of our IV