Conceptual Understanding Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05.
Conceptual Understanding Fcrit = 3.18 Complete the above table for an ANOVA having 3 levels of the independent variable and n = 20. Test for significant at .05. Fcrit (2, 57) = 3.15
Conceptual Understanding Distinguish between: Between-group variability and within-group variability
Conceptual Understanding Distinguish between: Between-group variability and within-group variability Between concerns the differences between the mean scores in various groups Within concerns the variability of scores within each group
Between and Within Group Variability Between-group variability Within-group variability
Between and Within Group Variability sampling error + effect of variable sampling error
Conceptual Understanding Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00?
Conceptual Understanding Under what circumstance will the F ratio, over the long run, approach 1.00? Under what circumstances will the F ratio be greater than 1.00? F ratio will approach 1.00 when the null hypothesis is true F ratio will be greater than 1.00 when the null hypothesis is not true
Conceptual Understanding Without computing the SS within, what must its value be? Why?
Conceptual Understanding The SS within is 0. All the scores within a group are the same (i.e., there is NO variability within groups)
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
Remember
Note
Proof Candy Gender 5.00 1.00 4.00 1.00 7.00 1.00 6.00 1.00 1.00 2.00 2.00 2.00 3.00 2.00 4.00 2.00
t = 2.667 / .641 = 4.16
t = 2.667 / .641 = 4.16
t = 2.667 / .641 = 4.16
t = 2.667 / .641 = 4.16
Also, when F has 1 df between
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
Note: This formula is for equal n
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 a1 = 0, a2 = 0, a3 = 1, a4 = -1 L = -23
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 Contrast
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 Sum a2 = .52+-.52+ -.52 + .52 = 1
SS Contrasts a1 = .5, a2 = -.5, a3 = -.5, a4 = .5 L = 80.5 – 67 = 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 Sum a2 = 12+-12+ -12 + 12 = 4
SS Contrasts a1 = 1, a2 = -1, a3 = -1, a4 = 1 L = 161 – 134 = 27 n = 6 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
SPSS
Make contrasts to determine If seniors are happier than everyone else? 2) If juniors and sophomores have different levels of happiness?
If seniors are happier than everyone else? a1 = -1, a2 = -1, a3 = -1, a4 = 3 L = 45 F crit (1, 20) = 4.35
2) If juniors and sophomores have different levels of happiness? a1 = 0, a2 = -1, a3 = 1, a4 = 0 L = -10 F crit (1, 20) = 4.35
Practice To investigate the maternal behavior of lab rats, we move the rat pup a fixed distance from the mother and record the time required for the mother to retrieve the pup. We run the study with 5, 20, and 35 day old pups. Figure out if 5 days is different than 35 days. SPSS Homework (Do the ANOVA analysis in SPSS – use output to answer question above) 5 days 15 10 25 20 18 20 days 30 23 35 days 40 35 50 43 45