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

Contrasts Some contrasts are independent Some are not Freshman vs. Sophomore (1, -1, 0, 0) Junior vs. Senior (0, 0, 1, -1) Some are not Freshman vs. Sophomore, Junior, Senior (3, -1, -1, -1) Freshman vs. Sophomore & Junior (2, -1, -1, 0)

Orthogonal Contrasts If you have a complete set of orthogonal contrasts The sum of SScontrast = SSbetween

Orthogonal Contrasts 1) ∑ aj = 0 2) ∑ aj bj = 0 Already talked about 2) ∑ aj bj = 0 Ensures contrasts of independent of one another 3) Number of comparisons = K -1 Ensures enough comparisons are used

Orthogonal Contrasts ∑ aj bj = 0 Fresh, Sophomore, Junior, Senior (3, -1, -1, -1) and (2, -1, -1, 0) (3*2)+(-1*-1)+(-1*-1) = 8

Orthogonal Contrasts ∑ aj bj = 0 Fresh, Sophomore, Junior, Senior (-1, 1, 0, 0) & (0, 0, -1, 1) (-1*0)+(1*0)+(-1*0)+(1*0) = 0 *Note: this is not a complete set of contrasts (rule 3)

Orthogonal Contrasts Lets go to five groups What would the complete set contrasts be that would satisfy the earlier rules?

Orthogonal Contrasts General rule There is more than one right answer

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen 2 limbs are created The elements on different limbs can not be combined with each other Elements on the same limbs can be combined with each other (making new limbs)

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen 3, 3, -2, -2, -2

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen 3, 3, -2, -2, -2 1, -1, 0, 0, 0

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen 3, 3, -2, -2, -2 1, -1, 0, 0, 0 0, 0, 1, 1, -2

Orthogonal Contrasts Fresh, Soph, Jun, Sen, Grad Fresh & Soph vs. Jun, Sen, & Grad Fresh vs. Soph Jun & Sen vs. Grad Jun vs. Sen 3, 3, -2, -2, -2 1, -1, 0, 0, 0 0, 0, 1, 1, -2 0, 0, 1, -1, 0

Orthogonal Contrasts 1) ∑ aj = 0 2) ∑ aj bj = 0 3) Number of comparisons = K -1 3, 3, -2, -2, -2 1, -1, 0, 0, 0 0, 0, 1, 1, -2 0, 0, 1, -1, 0

Orthogonal Contrasts 1) ∑ aj = 0 2) ∑ aj bj = 0 3) Number of comparisons = K -1 3, 3, -2, -2, -2 = 0 1, -1, 0, 0, 0 = 0 0, 0, 1, 1, -2 = 0 0, 0, 1, -1, 0 = 0

Orthogonal Contrasts A) 3, 3, -2, -2, -2 B) 1, -1, 0, 0, 0 D) 0, 0, 1, -1, 0 A, B = 0; A, C = 0; A, D = 0 B, C = 0; B, D = 0 C, D = 0

Orthogonal Contrasts If you have a complete set of orthogonal contrasts The sum of SScontrast = SSbetween

Compute a complete set of orthogonal contrasts for the following data. Test each of the contrasts you create for significance

Orthogonal Contrasts Fresh, Soph, Jun, Sen Fresh & Soph vs. Jun & Sen Fresh vs. Soph Jun vs. Sen 1, 1, -1, -1 1, -1, 0, 0 0, 0, 1, -1

1, 1, -1, -1 L = 1 SScontrast = 1.5; F = .014 1, -1, 0, 0 L = 4 SScontrast = 48; F = .48 0, 0, 1, -1 L = -23 SScontrast = 1587; F = 15.72* F crit (1, 20) = 4.35

SScontrast = 1.5 SScontrast = 48 SScontrast = 1587 1.5 + 48 + 1587 = 1636.50 F crit (1, 20) = 4.35

Orthogonal Contrasts Why use them? People like that they sum together People like that they are independent History I would rather have contrasts based on reason then simply because they are orthogonal!

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