SPSS SPSS Problem (Part 1)

SPSS Problem (Part 2) Due Wed 12.1

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

Hyp 1: Juniors and Seniors will have different levels of happiness Hyp 2: Seniors and Freshman will have different levels of happiness

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

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!

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

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

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