Matched t test Experimental Designs

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# Matched t test Experimental Designs - PowerPoint PPT Presentation

Matched t test Experimental Designs . Repeated measures Simultaneous Successive Before and after Counterbalanced Matched pairs Experimental Natural. Chapter 12. The one-way independent ANOVA An. O. Va. = Analysis of Variance. More than two groups .

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

### Chapter 12

The one-way independent ANOVA

An. O. Va. = Analysis of Variance

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.
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?
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.
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.
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
So how do we develop this ANOVA ?
• We want to develop a test statistic (like z or t) for the ANOVA.
• We don’t know anything about it
• Like, the t test
• We will start by rewriting the t test, then generalize it to more than two levels
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.
Generalizing the t test
• Next, for no particular reason, square both sides.
• Then rearrange.
• So, how do we compute MSW?
Generalizing the t test
• For equal sized groups, MSW is simply the average of each group’s variance.
Generalizing the t test
• Next generalize the numerator
• It is a measure of the difference between the means
• However, we can’t 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…?
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
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

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 can’t be 0.
• Symmetry
• Tails
• Is a one tailed in the graph.
• Like two tailed test in terms of directionality.

0

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).
F ratioInterpretaion
• What happens if the null hypothesis is not true?
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.
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 …?
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.
ExampleComputing the F ratio
• Plug and chug.
• But we are still missing the mean of the group means.
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.
F ratioDegrees of freedom
• Like the t test, the shape of the F distribution depends on df
• There are two
• dfbet = k-1=3-1=2
• dfW = NT-k = 150 - 3 = 147
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
• Let’s use  =.05
• Table A7 gives ______
• Compare Fcrit to F.
Exercises
• Page 334
• 1,2,3,5,6,7