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

Trend Analysis The logic of trend analysis is exactly the same logic we just talked about with contrasts!

Example You collect axon firing rate scores from rats in one of four conditions. Condition 1 = 10 mm of Zeta inhibitor Condition 2 = 20 mm of Zeta inhibitor Condition 3 = 30 mm of Zeta inhibitor Condition 4 = 40 mm of Zeta inhibitor Condition 5 = 50 mm of Zeta inhibitor You think Zeta inhibitor reduces the number of times an axon fires – are you right?

What does this tell you ?

Trend Analysis Contrast Codes! -2 -1 0 1 2

Trend Analysis

a1 = -2, a2 = -1, a3 = 0, a4 = 1, a5 = 2 L = 7.2 F crit (1, 20) = 4.35

Note

Example You place subjects into one of five different conditions of anxiety. 1) Low anxiety 2) Low-Moderate anxiety 3) Moderate anxiety 4) High-Moderate anxiety 5) High anxiety You think subjects will perform best on a test at level 3 (and will do worse at both lower and higher levels of anxiety)

What does this tell you ?

-2 1 2 1 -2 Contrast Codes!

Trend Analysis Create contrast codes that will examine a quadratic trend. -2, 1, 2, 1, -2

a1 = -2, a2 = 1, a3 = 2, a4 = 1, a5 = -2 L = 10 F crit (1, 20) = 4.35

Trend Analysis How do you know which numbers to use? Page 742

Linear (NO BENDS)

Quadratic (ONE BEND)

Cubic (TWO BENDS)

Practice You believe a balance between school and one’s social life is the key to happiness. Therefore you hypothesize that people who focus too much on school (i.e., people who get good grades) and people who focus too much on their social life (i.e., people who get bad grades) will be more depressed. You collect data from 25 subjects 5 subjects = F 5 subjects = D 5 subjects = C 5 subjects = B 5 subjects = A You measured their depression

Practice Below are your findings – interpret!

Trend Analysis Create contrast codes that will examine a quadratic trend. -2, 1, 2, 1, -2

a1 = -2, a2 = 1, a3 = 2, a4 = 1, a5 = -2 L = -12.8 F crit (1, 20) = 4.35

Remember 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

Bonferoni t Controls for Type I error by using a more conservative alpha

Do t-tests for all pairs Fresh vs. Sophomore Fresh vs. Junior Fresh vs. Senior Sophomore vs. Junior Sophomore vs. Senior Junior vs. Senior

Maximum probability of a Type I error 6 (.05) = .30 But what if we use Alpha = .05/C .00833 = .05 / 6 6 (.00855) = .05

t-table Compute the t-value the exact same way Problem: normal t table does not have these p values Test for significance using the Bonferroni t table (page 751)

Practice

Practice Fresh vs. Sophomore t = .69 Fresh vs. Junior t = 2.41 Fresh vs. Senior t = -1.55 Sophomore vs. Junior t = 1.72 Sophomore vs. Senior t = -2.24 Junior vs. Senior t = -3.97* Critical t = 6 comp/ df = 20 = 2.93

Bonferoni t Problem Silly What should you use as the value in C? Increases the chances of the Type II error!

Practice Data from exercise 11.1 1) Use linear contrasts to compare 5 days vs 20 and 35 days 2) Imagine you had no hypotheses and you were concerned about Type 1 error. Compare all conditions to each other using Bonferroni’s t. 5 days vs 20 days 5 days vs 35 days 20 days vs 35 days

Practice 1) Use linear contrasts to compare 5 days vs 20 and 35 days F(1,15) = 4.54

Practice 2) Imagine you had no hypotheses and you were concerned about Type 1 error. Compute all possible t-tests using Bonferroni’s t. 5 vs 20 Bonferroni t critical 3 comp, df = 15 2.69 5 vs 35 20 vs 35

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

Fisher Least Significance Difference Simple 1) Do a normal omnibus ANOVA 2) If there it is significant you know that there is a difference somewhere! 3) Do individual t-test to determine where significance is located

Fisher Least Significance Difference Problem You may have an ANOVA that is not significant and still have results that occur in a manner that you predict! If you used this method you would not have “permission” to look for these effects.

Remember

Remember

Studentized Range Statistic Which groups would you likely select to determine if they are different?

Studentized Range Statistic Which groups would you likely select to determine if they are different? This statistics controls for Type I error if (after looking at the data) you select the two means that are most different.

Studentized Range Statistic Easy! 1) Do a normal t-test

Studentized Range Statistic Easy! 2) Convert the t to a q

Studentized Range Statistic 3) Critical value of q (note: this is a two-tailed test) Figure out df (same as t) Example = 20 Figure out r r = the number of groups

Studentized Range Statistic 3) Critical value of q (note: this is a two-tailed test) Figure out df (same as t) Example = 20 Figure out r r = the number of groups Example = 4

Studentized Range Statistic 3) Critical value of q Page 744 Example q critical = +/- 3.96

Studentized Range Statistic 4) Compare q obs and q critical same way as t values q = -5.61 q critical = +/– 3.96

Practice You collect axon firing rate scores from rates in one of four conditions. Condition 1 = 10 mm of Zeta inhibitor Condition 2 = 20 mm of Zeta inhibitor Condition 3 = 30 mm of Zeta inhibitor Condition 4 = 40 mm of Zeta inhibitor Condition 5 = 50 mm of Zeta inhibitor You are simply interested in determining if any two groups are different from each other – use the Studentized Range Statistic

Studentized Range Statistic Easy! 1) Do a normal t-test

Studentized Range Statistic Easy! 2) Convert the t to a q

Studentized Range Statistic 3) Critical value of qnote: this is a two-tailed test) Figure out df (same as t) Example = 20 Figure out r r = the number of groups Example = 5

Studentized Range Statistic 3) Critical value of q Page 744 Example q critical = +/- 4.23

Studentized Range Statistic 4) Compare q obs and q critical same way as t values q = -4.34 q critical = +/– 4.23

Dunnett’s Test Used when there are several experimental groups and one control group (or one reference group) Example: Effect of psychotherapy on happiness Group 1) Psychoanalytic Group 2) Humanistic Group 3) Behaviorism Group 4) Control (no therapy)

Psyana vs. Control Human vs. Control Behavior vs. Control

Psyana vs. Control = 47.8 – 51.4 = -3.6 Human vs. Control = 50.8 – 51. 4 = -0.6 Behavior vs. Control = 59 – 51.4 = 7.6

Psyana vs. Control = 47.8 – 51.4 = -3.6 Human vs. Control = 50.8 – 51. 4 = -0.6 Behavior vs. Control = 59 – 51.4 = 7.6 How different do these means need to be in order to reach significance?

Dunnett’s t is on page 753 df = Within groups df / k = number of groups

Dunnett’s t is on page 753 df = 16 / k = 4

Dunnett’s t is on page 753 df = 16 / k = 4

Psyana vs. Control = 47.8 – 51.4 = -3.6 Human vs. Control = 50.8 – 51. 4 = -0.6 Behavior vs. Control = 59 – 51.4 = 7.6* How different do these means need to be in order to reach significance?

Practice As a graduate student you wonder what undergraduate students (freshman, sophomore, etc.) have different levels of happiness then you.

Dunnett’s t is on page 753 df = 25 / k = 5

Fresh vs. Grad = -17.5* Soph vs. Grad = -21.5* Jun vs. Grad = -31.5* Senior vs. Grad = -8.5