Common models and contrasts. Tuesday, Lecture 5 Jeanette Mumford University of Wisconsin - Madison

Transcription

1 ommon models and contrasts Tuesday, Lecture 5 Jeanette Mumford University of Wisconsin - Madison

2 Let s set up some simple models 1-sample t-test -sample t-test With contrasts!

3 1-sample t-test Y = 0 +

4 1-sample t-test Y Y = Y N 1

5 1-sample t-test Y Y = Y N 1 Multiply out the right hand side

6 1-sample t-test Y Y = Y N 0 Multiply out the right hand side

7 ut why is it the mean and not something else?

8 ut why is it the mean and not something else? ecause we re using least squares!

9 Two-sample t-test There are at least ways I can think of parameterizing this! Start with the easiest a person is either in group 1 or in group Y i = 1 {sub i in group 1?} + {sub i in group?} +

10 Two-sample t-test There are at least ways I can think of parameterizing this! Start with the easiest a person is either in group 1 or in group Y i = 1 G 1i + G i + group indicator variables

11 Two-sample t-test Group 1 Group 6 4 Y 1 Y Y Y 4 Y = apple 1 + Y 6 0 1

12 Two-sample t-test Y 1 Y Y Y 4 Y 5 Y =

13 Two-sample t-test 6 4 Y 1 Y Y Y 4 Y = mean for Group 1 + mean for Group Y 6

14 Two-sample t-test (another way) Now you do it. Unwrap what this means Y i = {subject i in Group 1} +

15 ontrasts ontrasts are vectors that pull out what we d like to test Using the two sample t-test from the first example, we might test Is the mean of G1 larger than 0? Is the mean of G larger than 0? Is the mean of G1 > G?

16 General idea Take your contrast statement and get it to look like something > 0 Figure out the vector, c, such that c = something

17 6 4 Y 1 Y Y Y 4 Y 5 Is group 1 > 0? 7 5 = Y We ve already established the first beta represents group 1 s mean c = [1, 0] pulls out the first beta apple

18 6 4 Y 1 Y Y Y 4 Y 5 Is group > 0? 7 5 = Y We ve already established the second beta represents group s mean c = [0, 1] pulls out the first beta apple

19 6 4 Is group 1 > group? Y First, get something > 0 group 1 group > 0 c = [1, -1] Y 1 Y Y Y 4 Y = apple = 1 1 apple 1 1 +

20 6 4 Is group 1 > group? Y First, get something > 0 group 1 group > 0 c = [1, -1] Y 1 Y Y Y 4 Y = apple = 1 1 apple 1 1 +

21 6 4 Is group 1 > group? Y First, get something > 0 group 1 group > 0 c = [1, -1] Y 1 Y Y Y 4 Y = apple = 1 1 apple 1 1 +

22 6 4 Is group 1 > group? Y First, get something > 0 group 1 group > 0 c = [1, -1] Y 1 Y Y Y 4 Y = apple = 1 1 apple 1 1 +

23 an you do this for the second setup of the -sample t-test?

24 Take away Did you feel pretty confident with the last example? Yes = Yay! No = Ask questions!

25 What you re going to learn Paired t-test 1-way ANOVA with levels Detour to introduce F-tests x ANOVA Revisit mean centering

26 Paired t-test 1-way ANOVA with levels and repeated measures Wow, sounds so fancy!

27 Paired T Test A common mistake is to use a -sample t test instead of a paired test Tire example -sample T test p=0.58 Paired T test p<0.001 Automobile Tire A Tire

28 Why so different?

29 Why so different? Mean A Mean

30 Why so different? Difference is OK

31 Why so different? Residuals are HUGE!

32 Paired T Test Adjust for the mean of each pair

33 Paired T Test Mean A Mean

34 Paired T Test Difference is same Residual variance much smaller

35 Paired T Test GLM

36 Paired T Test GLM Mean of each pair

37 Paired T Test GLM Difference Mean of each pair

38 Paired T Test GLM Difference Mean of each pair HA : A> =) HA : A >0 c = [ ]

39 Paired T Test GLM Difference Mean of each pair HA : A> =) HA : A >0 c = [ ] What if you wanted to test HA: >A?

40 SPM is slightly different A 1 1 A A A 4 4 A A = A A

41 SPM is slightly different A 1 1 A A A 4 4 A A = A A Rank deficient, but that s okay in SPM. an you construct the contrast for A>?

42 1-Way ANOVA levels asically an extended -sample t-test I will teach the cell means approach ells = ollections of data within each factor level (or level combination) -level 1-way ANOVA has ** cells e.g. types of intervention

43 1-Way ANOVA levels L1 1 L1 L1 L 1 L L L 1 L L 1 A = A 1 1 A

44 1-Way ANOVA levels L1 1 L1 L1 L 1 L L L 1 L L 1 A = an you come up with a contrast to test if L1 > L? 1 A

45 1-Way ANOVA levels L1 1 L1 L1 L 1 L L L 1 L L 1 A = an you come up with a contrast to test if average of L1 and L > L 1 A

46 General rule for contrasts I m only willing to make this claim for cell means models (what we ve been doing) Positive parts of contrast (if present) should sum to 1 Negative parts of contrast (if present) should sum to -1 For previous example, we could have used [1 1 -] Stats would have been the same, but contrast estimates would differ

47 F-tests Simultaneously test multiple contrasts E.g. are any of the levels different from each other? L1 = L = L Rejection of the null implies at least 1 difference L1&L differ or L1&L differ or L&L differ

48 F-tests Uses a contrast matrix Tips for constructing matrix Start with your alternative Get a 0 on the end of your system of equations reak it up

49 F-tests Uses a contrast matrix Tips for constructing matrix Start with your null Get a 0 on the end of your system of equations reak it up H0 :L1 =L =L

50 F-tests Uses a contrast matrix Tips for constructing matrix Start with your null Get a 0 on the end of your system of equations reak it up H0 :L1 L =L L =0

51 F-tests Uses a contrast matrix Tips for constructing matrix Start with your null Get a 0 on the end of your system of equations reak it up to get HA HA : L1 L 6= 0 or L L 6= 0

52 F-tests Uses a contrast matrix Tips for constructing matrix Start with your null Get a 0 on the end of your system of equations reak it up to get HA reate contrast matrix HA : L1 L 6= 0 or L L 6= 0

53 F-tests Uses a contrast matrix Tips for constructing matrix Start with your null Get a 0 on the end of your system of equations reak it up to get HA reate contrast matrix

54 T-test vs F-test T- stat using contrast c t = ˆ dvar c ˆ

55 T-test vs F-test T- stat using contrast t = c(x0 X) 1 X 0 Y p c(x0 X) 1 c 0ˆ

56 T-test vs F-test T- stat using contrast t = c(x0 X) 1 X 0 Y p c(x0 X) 1 c 0ˆ F-stat : still has mean/variance structure but with matrices

57 T-test vs F-test T- stat using contrast t = c(x0 X) 1 X 0 Y p c(x0 X) 1 c 0ˆ F-stat : still has mean/variance structure but with matrices f df 1,df = c 0 ˆ 0 c 0 (X 0 X) 1 crˆ 1 c 0 ˆ

58 T-test vs F-test T- stat using contrast t = c(x0 X) 1 X 0 Y p c(x0 X) 1 c 0ˆ F-stat : still has mean/variance structure but with matrices f df 1,df = c 0 ˆ 0 c 0 (X 0 X) 1 crˆ 1 c 0 ˆ r x p contrast matrix

59 T-test vs F-test T- stat using contrast t = c(x0 X) 1 X 0 Y p c(x0 X) 1 c 0ˆ F-stat : still has mean/variance structure but with matrices f df 1,df = c 0 ˆ 0 c 0 (X 0 X) 1 crˆ 1 c 0 ˆ r=# row in c

60 T-test vs F-test T- stat using contrast t = c(x0 X) 1 X 0 Y p c(x0 X) 1 c 0ˆ F-stat : still has mean/variance structure but with matrices f df 1,df = c 0 ˆ 0 c 0 (X 0 X) 1 crˆ 1 c 0 ˆ df1 = r, df = N-p ~ F(df1, df)

61 T-test vs F-test T- stat using contrast t = c(x0 X) 1 X 0 Y p c(x0 X) 1 c 0ˆ F-stat : still has mean/variance structure but with matrices f df 1,df = c 0 ˆ 0 c 0 (X 0 X) 1 crˆ 1 c 0 ˆ Does this boil down to a scalar or a matrix?

62 When an F-test is a T-test If r=1 (contrast matrix has one row) Your F-test is equivalent to a -sided t-test

63 Uses for F-test ut down on multiple comparisons Instead of running separate t-tests, you can run a single f-test If F-test is significant you can look at separate t- tests to see what s going on If F-test isn t significant, none of your separate contrasts is significant

64 Uses for F-test ut down on multiple comparisons Instead of running separate t-tests, you can run a single f-test If F-test is significant you can look at separate t- tests to see what s going on If F-test isn t significant, none of your separate contrasts is significant Or is it!?!?!

65 F-test vs T-test illustration onsider the following F-test H0 : 1 = =0

66 F-test vs T-test illustration onsider the following F-test H0 : 1 = =0 1 0 c = 0 1

67 F-test vs T-test illustration onsider the following F-test H0 : 1 = =0 1 0 c = 0 1 We will compare to the separate t-tests

68 F-test vs T-test illustration onsider the following F-test H0 : 1 = =0 1 0 c = 0 1 We will compare to the separate t-tests Hope: If the F-test is significant, at least one of the t-tests is significant

69 b b 1

70 b t-test for beta1 significant b 1

71 b t-test for beta significant b 1

72 b Outside circle, F-test is significant b 1

73 b All tests significant b 1

74 b F and t for beta1 significant b 1

75 b None significant b 1

76 b F isn t significant but t is!! b 1

77 b F is significant and t s aren t! b 1

78 Do those last cases ever really happen? Yes! That s why I looked into it J More likely to occur if you re riding the p=0.05 edge Still not a bad idea to run F-tests

79 x ANOVA levels of gender levels of intervention Yet another extension of the -sample t-test Just focus on the cells How many cells are there? That s how many regressors you must have

80 F, L1 1 F, L1 F, L 1 F, L F, L 1 F, L M,L1 1 M,L1 M,L 1 M,L M,L 1 M,L 1 A x ANOVA = A A onstruct the contrast for F>M

81 F, L1 1 F, L1 F, L 1 F, L F, L 1 F, L M,L1 1 M,L1 M,L 1 M,L M,L 1 M,L 1 A x ANOVA = A A How about L1 = L = L?

82 F, L1 1 F, L1 F, L 1 F, L F, L 1 F, L M,L1 1 M,L1 M,L 1 M,L M,L 1 M,L 1 A x ANOVA = A A How about the interaction?

83 x ANOVA The interaction, what does it mean? That males and females differ for at least one level No interaction L1 L L

84 x ANOVA The interaction, what does it mean? That males and females differ for at least one level Interaction L1 L L

85 x ANOVA The interaction, what does it mean? That males and females differ for at least one level (there are multiple ways this can occur) Interaction L1 L L

86 x ANOVA What does the null look like? M-F differences the same for all levels

87 x ANOVA What does the null look like? M-F differences the same for all levels H0 :F, L1 M,L1 =F, L M,L =F, L M,L

88 x ANOVA What does the null look like? M-F differences the same for all levels H0 :F, L1 M,L1 =F, L M,L =F, L M,L What is this in terms of the betas??

89 x ANOVA What does the null look like? M-F differences the same for all levels H0 :F, L1 M,L1 =F, L M,L =F, L M,L Get zero on end and split to get HA HA :( 1 4 ) ( 6 ) 6= 0 or ( 5 ) ( 6 ) 6= 0

90 x ANOVA What does the null look like? M-F differences the same for all levels H0 :F, L1 M,L1 =F, L M,L =F, L M,L Get zero on end and split to get HA HA :( 1 4 ) ( 6 ) 6= 0 or ( 5 ) ( 6 ) 6= 0 an you construct the contrast matrix?

91 Time for a curve ball What if I parameterized the model differently? an you still figure out the contrasts? I ll show you a trick to get contrasts with ease

92 How to grab contrasts off design matrix Ask yourself the following questions Are there the appropriate number of regressors e.g. x ANOVA needs 6 regressors Is this matrix full rank? compute rank(x) in matlab, rank(x) = # columns If matrix isn t full rank, hopefully there s a column you can dump to make it full rank. Just ignore this column Within cell do the rows of the design matrix match? e.g. all female subjects in L1 If yes to all, the design matrix row is the contrast for that cell

93 1 Way ANOVA - Factor Effects

94 1 Way ANOVA - Factor Effects

95 1 Way ANOVA - Factor Effects

96 1 Way ANOVA - Factor Effects

97 1 Way ANOVA - Factor Effects

98 1 Way ANOVA - Factor Effects Now you can combine these to compare groups

99 1 Way ANOVA - Factor Effects HA : G1 G > 0 =) ( 1 + ) ( 1 + ) > 0 =) > 0

100 Is there time for mean centering regressors?

101 Mean centering regressors Is it magic? No Model fit is the same Just changes the interpretation of some of the parameters in your model

102 Review Age (years) RT (s) RT = Age +

103 General observation Y = X + The intercept is simply the value of Y when X is 0 No matter what X is

104 Mean centering age RT = Age + vs RT = Age demeaned +

105 Mean centering age RT = Age + vs RT = (Age mean(age)) +

106 Mean centering age RT = Age + vs RT = (Age mean(age)) + 0 RT when Age = 0 0 RT when Age-mean(Age) = 0 RT when Age = mean(age) (or just mean RT)

107 Mean centering age RT = Age + vs RT = (Age mean(age)) + 0 RT when Age = 0 0 These are not the same RT when Age-mean(Age) = 0 RT when Age = mean(age) (or just mean RT)

108 Mean centering age RT = Age + vs RT = (Age mean(age)) + The same! a 1 year difference in mean centered age is the same as a 1 year difference in age ages: 10, 11, 8, 9, 10, 1 ages mean centered: 0, 1, -, -1, 0,

109 Mean centering age RT = Age + vs RT = (Age mean(age)) + The same! a 1 year difference in mean centered age is the same as a 1 year difference in age ages: 10, 11, 8, 9, 10, 1 ages mean centered: 0, 1, -, -1, 0,

110 Mean centering age RT = Age + vs RT = (Age mean(age)) + The same! a 1 year difference in mean centered age is the same as a 1 year difference in age ages: 10, 11, 8, 9, 10, 1 ages mean centered: 0, 1, -, -1, 0,

111 Mean centering age RT = Age + vs RT = (Age mean(age)) + The same! a 1 year difference in mean centered age is the same as a 1 year difference in age ages: 10, 11, 8, 9, 10, 1 ages mean centered: 0, 1, -, -1, 0,

112 Mean centering X is basically moving the Y-axis Age (years) RT (s)

113 Age (years) RT (s) Mean centering X is basically moving the Y-axis

114 Mean centered age(years) RT (s) Mean centering X is basically moving the Y-axis

115 Interaction model group x age interaction Multiple ways to set it up I ll show most intuitive

116 Interaction model Hypothesis Does the RT/age slope vary by group? Regressors Mean for G1 Mean for G Slope for G1 Slope for G

117 Interaction model 6 4 Y 1 Y Y Y 4 Y 5 Y =

118 Interaction model 6 4 Y 1 Y Y Y 4 Y 5 Y = What is the contrast to test the interaction??

119 Interaction model 6 4 Y 1 Y Y Y 4 Y 5 Y = Does it make sense to test the [ ] contrast?

120 Interaction model : mean centering You can mean center age across all subjects Does not change inference for the interaction 6 4 Y 1 Y Y Y 4 Y 5 Y 6 hanges interpretation of first betas 7 5 =

121 Interaction model : mean centering Do not mean center within group There are some exceptions With this model, you may be interested in group difference adjusted for age. If you mean center within group, the age adjustment won t occur 6 4 Y 1 Y Y Y 4 Y 5 Y =

122 Interaction model : mean centering Do not mean center within group 6 4 There are some exceptions With this model, you may be interested in group difference adjusted for age. If you mean center within group, the age adjustment won t occur Y 1 Y Y Y 4 Y 5 Y =

123 Interaction model : mean centering Do not mean center within group 6 4 There are some exceptions With this model, you may be interested in group difference adjusted for age. If you mean center within group, the age adjustment won t occur Y 1 Y Y Y 4 Y 5 Y =

124 I have a handy pdf you can refer to ng/

125 All you need to know What does mean centering do to the parameter associated with the regressor that was centered? Does the model fit better after mean centering? What does change in the model?

126 Other model This is a great resource for the more complicated models s/rik_anova.pdf

127 How do these modeling things relate to SPM output? beta*.img eta image (one for each beta) con*.img ontrast estimates (numbered in order you specified your contrasts) spm_t*.img orresponding t-stats for each contrast ees*.img Extra sums of squares (relates to F-test)

128 How do these things relate to SPM output? spm_f_*.img F statistic output ResMS.img Estimate of σ

129 Questions?