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## Example

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**Example**• Freshman, Sophomore, Junior, Senior • Measure Happiness (1-100)**ANOVA**• Traditional F test just tells you not all the means are equal • Does not tell you which means are different from other means**Why not**• Do t-tests for all pairs • Fresh vs. Sophomore • Fresh vs. Junior • Fresh vs. Senior • Sophomore vs. Junior • Sophomore vs. Senior • Junior vs. Senior**Problem**• What if there were more than four groups? • Probability of a Type 1 error increases. • Maximum value = comparisons (.05) • 6 (.05) = .30**Chapter 12**• A Priori and Post Hoc Comparisons • Multiple t-tests • Linear Contrasts • Orthogonal Contrasts • Trend Analysis • Bonferroni t • Fisher Least Significance Difference • Studentized Range Statistic • Dunnett’s Test**Multiple t-tests**• Good if you have just a couple of planned comparisons • Do a normal t-test, but use the other groups to help estimate your error term • Helps increase you df**Proof**Candy Gender 5.00 1.00 4.00 1.00 7.00 1.00 6.00 1.00 4.00 1.00 5.00 1.00 1.00 2.00 2.00 2.00 3.00 2.00 4.00 2.00 3.00 2.00 2.00 2.00**Within Variability**• Within variability of all the groups represents “error” • You can therefore get a better estimate of error by using all of the groups in your ANOVA when computing a t-value**Hyp 1: Juniors and Seniors will have different levels of**happiness Hyp 2: Seniors and Freshman will have different levels of happiness**Hyp 1: Juniors and Seniors will have different levels of**happiness**Hyp 1: Juniors and Seniors will have different levels of**happiness**Hyp 1: Juniors and Seniors will have different levels of**happiness**Hyp 1: Juniors and Seniors will have different levels of**happiness t crit (20 df) = 2.086**Hyp 1: Juniors and Seniors will have different levels of**happiness t crit (20 df) = 2.086 Juniors and seniors do have significantly different levels of happiness**Hyp 2: Seniors and Freshman will have different levels of**happiness**Hyp 2: Seniors and Freshman will have different levels of**happiness**Hyp 2: Seniors and Freshman will have different levels of**happiness**Hyp 2: Seniors and Freshman will have different levels of**happiness t crit (20 df) = 2.086**Hyp 2: Seniors and Freshman will have different levels of**happiness t crit (20 df) = 2.086 Freshman and seniors do not have significantly different levels of happiness**Hyp 1: Juniors and Sophomores will have different levels of**happiness Hyp 2: Seniors and Sophomores will have different levels of happiness PRACTICE!**Chapter 12**• A Priori and Post Hoc Comparisons • Multiple t-tests • Linear Contrasts • Orthogonal Contrasts • Trend Analysis • Bonferroni t • Fisher Least Significance Difference • Studentized Range Statistic • Dunnett’s Test**Linear Contrasts**• You think that Freshman and Seniors will have different levels of happiness than Sophomores and Juniors**Linear Contrasts**• Allows for the comparison of one group or set of groups with another group or set of groups**Linear Contrasts**a = weight given to a group**Linear Contrasts**1 2 3 4 a1 = 0, a2 = 0, a3 = 1, a4 = -1 L = -23 a1 = 1, a2 = 0, a3 = 0, a4 = -1 L = -9 a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5**SS Contrast**• You can use the linear contrast to compute a SS contrast • SS contrast is like SS between • SS contrast has 1 df • SS contrast is like MS between**SS Contrasts**a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5**SS Contrasts**a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5 n = 6 L = 13.5 Sum a2 = .52+-.52+ -.52 + .52 = 1**SS Contrasts**a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 13.5 n = 6 L = 13.5 Sum a2 = .52+-.52+ -.52 + .52 = 1**SS Contrasts**a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27**SS Contrasts**a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27 n = 6 L = 27 Sum a2 = 12+-12+ -12 + 12 = 4**SS Contrasts**a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27 n = 6 L = 27 Sum a2 = 12+-12+ -12 + 12 = 4**F Test**Note: MS contrast = SS contrast**F Test**Fresh & Senior vs. Sophomore & Junior**F Test**Fresh & Senior vs. Sophomore & Junior**F Test**Fresh & Senior vs. Sophomore & Junior F crit (1, 20) = 4.35