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## 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**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 • 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**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 …?**F ratioFormula**• .**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