Common models and contrasts. Tuesday, Lecture 2 Jeane5e Mumford University of Wisconsin - Madison

Transcription

1 ommon models and contrasts Tuesday, Lecture Jeane5e 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 Mulply out the right hand side

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

7 1-sample t-test Y Y = Y N 0 Mulply out the right hand side

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

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

10 I m going to write this out

11 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?} +

12 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

13 Two-sample t-test Group 1 Group 6 4 Y 1 Y Y Y 4 Y 5 Y = apple 1 +

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

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

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

17 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?

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

19 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

20 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

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 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 +

24 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 +

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

26 Take away Did you feel pre5y confident with the last example? Yes = Yay! No = Ask quesons!

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

28 ut those are all group models! Time series models look very similar Just add convoluon! Doesn t really apply to paired t-test

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

30 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

31 Why so different?

32 Why so different? Mean A Mean

33 Why so different? Difference is OK

34 Why so different? Residuals are HUGE!

35 Paired T Test Adjust for the mean of each pair

36 Paired T Test Mean A Mean

37 Paired T Test Difference is same Residual variance much smaller

38 Paired T Test GLM

39 Paired T Test GLM Mean of each pair

40 Paired T Test GLM Difference Mean of each pair

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

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

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

44 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>?

45 1-Way ANOVA levels asically an extended -sample t-test I will teach the cell means approach ells = ollecons of data within each factor level (or level combinaon) -level 1-way ANOVA has ** cells e.g. types of intervenon

46 1-Way ANOVA levels L1 1 L1 L1 L 1 L L L 1 L L 1 A = A 1 1 A

47 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

48 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

49 General rule for contrasts I m only willing to make this claim for cell means models (what we ve been doing) Posive parts of contrast (if present) should sum to 1 Negave 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 esmates would differ I ll show you on board

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

51 F-tests Uses a contrast matrix Tips for construcng matrix Start with your alternave Get a 0 on the end of your system of equaons reak it up

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

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

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

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

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

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

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 : sll has mean/variance structure but with matrices

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 : sll 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

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 : sll 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

62 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 : sll 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

63 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 : sll 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)

64 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 : sll 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?

65 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

66 Uses for F-test ut down on mulple 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

67 Uses for F-test ut down on mulple 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!?!?!

68 F-test vs T-test illustraon onsider the following F-test H0 : 1 = =0

69 F-test vs T-test illustraon onsider the following F-test H0 : 1 = =0 1 0 c = 0 1

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

71 F-test vs T-test illustraon 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

72 β β 1

73 β t-test for beta1 significant β 1

74 β t-test for beta significant β 1

75 β Outside circle, F-test is significant β 1

76 β All tests significant β 1

77 β F and t for beta1 significant β 1

78 β None significant β 1

79 β F isn t significant but t is!! β 1

80 β F is significant and t s aren t! β 1

81 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 Sll not a bad idea to run F-tests

82 x ANOVA levels of gender levels of intervenon 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

83 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

84 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?

85 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 interacon?

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

87 x ANOVA The interacon, what does it mean? That males and females differ for at least one level Interacon L1 L L

88 x ANOVA The interacon, what does it mean? That males and females differ for at least one level (there are mulple ways this can occur) Interacon L1 L L

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

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

91 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??

92 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

93 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?

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

95 How to grab contrasts off design matrix Ask yourself the following quesons 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

96 1 Way ANOVA - Factor Effects

97 1 Way ANOVA - Factor Effects

98 1 Way ANOVA - Factor Effects

99 1 Way ANOVA - Factor Effects

100 1 Way ANOVA - Factor Effects

101 1 Way ANOVA - Factor Effects Now you can combine these to compare groups

102 1 Way ANOVA - Factor Effects HA : G1 G > 0 =) ( 1 + ) ( 1 + ) > 0 =) > 0

103 Is there me for mean centering regressors?

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

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

106 General observaon Y = X + The intercept is simply the value of Y when X is 0 No ma5er what X is

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

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

109 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)

110 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)

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 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,

113 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,

114 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,

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

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

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

118 Interacon model group x age interacon Mulple ways to set it up I ll show most intuive

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

120 Interacon model 6 4 Y 1 Y Y Y 4 Y 5 Y =

121 Interacon model 6 4 Y 1 Y Y Y 4 Y 5 Y = What is the contrast to test the interacon??

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

123 Interacon model : mean centering 6 4 You can mean center age across all subjects Y 1 Y Y Y 4 Y 5 Y 6 Does not change inference for the interacon hanges interpretaon of first betas =

124 Interacon model : mean centering Do not mean center within group There are some excepons 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 =

125 Interacon model : mean centering Do not mean center within group 6 4 There are some excepons 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 =

126 Interacon model : mean centering Do not mean center within group 6 4 There are some excepons 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 =

127 I have a handy pdf you can refer to h5p://mumford.fmripower.org/ mean_centering/

128 All you need to know What does mean centering do to the parameter associated with the regressor that was centered? Does the model fit be5er awer mean centering? What does change in the model?

129 All you need to know TW, this is exactly what we talked about with parametric regressors yesterday!

130 Other model This is a great resource for the more complicated models h5p:// publicaons/rik_anova.pdf

131 How do these modeling things relate to SPM output? beta*.img eta image (one for each beta) con*.img ontrast esmates (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)

132 How do these things relate to SPM output? spm_f_*.img F stasc output ResMS.img Esmate of σ

133 Quesons?