Matched t test Experimental Designs

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Matched t testExperimental Designs

• Repeated measures
• Simultaneous
• Successive
• Before and after
• Counterbalanced
• Matched pairs
• Experimental
• Natural

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Chapter 12

• The one-way independent ANOVA
• An. O. Va. Analysis of Variance

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More than two groups

• So far we have considered only one or two (sub)
  populations
• What if there are more?
• One could compare A to B, B to C, C to D etc.

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Example

• Three groups of manic-depressive patients are
  compared.
• One group is given only psychotherapy, one group
  is given medication, and one group is given
  psychotherapy and medication.
• After treatment, each patient is given a wellness
  test.
• Are there differences between these treatments?

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How many pairs can be derived from k levels?

• This quickly gets out of hand in cases where
  there are many subpopulations (or groups).
• If there are k populations
• There are
• pairs to compare.
• We need another way.

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One way analysis of variance

• That way is the one way analysis of variance
• One way There is only one independent variable.
• The independent variable is called a factor.
• The factor takes on different values
• Each value is called a level
• Each level denotes a population
• A single test compares k levels.

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Another reason to use ANOVA

• Suppose there is no significant difference
  between the k levels.
• If there are k levels then there are
• different combinations
• So, as k increases so does the number of
  combinations.
• Given a .05 significance level
• One now has many chances to reach it and make a
  type I error.
• Somehow this must be taken into account

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So how do we develop this ANOVA ?

• We want to develop a test statistic (like z or t)
  for the ANOVA.
• We dont know anything about it
• So we must start with something we do know
• Like, the t test
• We will start by rewriting the t test, then
  generalize it to more than two levels

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Generalizing the t test

• sp2 is the weighted average of two variances.
• Since we now have k groups or levels, we could
  just as easily average over k variances.
• Denote as MSW
• Call this generalized sp2 the mean square within
  MSW
• This is a pool of k variances.
• The rationale for the mean square terminology is
  that a variance is an unbiased mean of a sum of
  squares.
• Within refers to the fact that the variances are
  within each group.

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Generalizing the t test

• Next, for no particular reason, square both
  sides.
• Then rearrange.
• So, how do we compute MSW?

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Generalizing the t test

• For equal sized groups, MSW is simply the
  average of each groups variance.

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Generalizing the t test

• Next generalize the numerator
• It is a measure of the difference between the
  means
• However, we cant subtract k things
• But, a difference is also a measure of spread
• We need a measure of spread that can be applied
  to k things
• This might be?

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Generalizing the t test

• We will use the variance
• Replace n times the difference of means with n
  times a variance of means
• Call this variance the mean squares between
  MSbet

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Large or small?

• t2 should it be large or small?
• MSbet should it be large or small? 2 reasons
• MSW should to be large or small? 2 reasons

0
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Give t2 a new name

• t2 will be called the F-ratio.
• F has a distribution that can be used in a way,
  similar to, the way we use the normal
  distribution.
• How does this distribution differ from the normal
  and t distributions?
• Mode cant be 0.
• Symmetry
• Tails
• Is a one tailed in the graph.
• Like two tailed test in terms of directionality.

0
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F ratiothe numerator

• The numerator is the actually the variance of the
  general population!
• Recall the definition of the standard error
  (which assumes that the null hypothesis is true).

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F ratioInterpretaion

• What happens if the null hypothesis is not true?

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F ratioInterpretaion

• Look at the formula for the variance of the mean
• If the null hypothesis is not true, then the
  differences can be arbitrarily large, depending
  on the differences between the subpopulations.

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F ratioInterpretaion

• MSbet and MSW behave differently
• In the bottom figure, spreading of the means has
  an effect (increase) on F numerator, but has no
  effect on the denominator
• So, if the null hypothesis is not true, F is ?

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F ratioFormula

• .

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ExampleComputing the F ratio

• Three groups of manic-depressive patients are
  compared.
• One group is given only psychotherapy, one group
  is given medication, and one group is given
  psychotherapy and medication.
• Each group has 50 individuals.
• After treatment, each patient is given a wellness
  test.

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ExampleComputing the F ratio

• Plug and chug.
• But we are still missing the mean of the group
  means.

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ExampleComputing the F ratio

• The mean of group means is easy to compute.
• Plug it in and do the arithmetic.
• Now we need a Fcrit.

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F ratioDegrees of freedom

• Like the t test, the shape of the F distribution
  depends on df
• There are two
• dfbet k-13-12
• dfW NT-k 150 - 3 147

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F distributionthe tables

• Since there are two df, there will be many
  combinations
• Hence, the tables are limited to a few ? levels
• One per table
• See tables A7, A8, A9
• Lets use ? .05
• Table A7 gives ______
• Compare Fcrit to F.

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Exercises

• Page 334
• 1,2,3,5,6,7