1. Means, Thresholds and Moderation Sarah Medland – Boulder 2008 Corrected Version Thanks to Hongyan Du for pointing out the error on the regression examples

2. This morning  Fitting a mean and regression with continuous data  Modelling Ordinal data  Fitting the regression model with ordinal data

3. Lets start with the data…  File: Wednesday.dat Contains 6 of the variables from Dorret’s example ntrid zygMZDZ age1 sekse1 AQ1 age2 sekse2 AQ2

4. Lets start with the data…

5. If this was a pedigree data file… FamidIndFatherMotherZygSexAge Trait 210001xx 220002xx 2312MZ118.1291 2412MZ118.1295

6. How can we make this data file?  Assume we have data with 3 variables:

7. How do we make this data?  SPSS SORT CASES BY Family Individual. CASESTOVARS /ID = Family /INDEX = Individual /GROUPBY = VARIABLE.  SAS?  R?

8. Means…  In spss sas etc we calculate the mean  In Mx and other ML programs we estimate the mean

9. Spss…

11. Mx… Means.mx

13.  Spss assumes this is a sample  Mx assumes this is a population  Slightly different algebra

14. How about regression?  Y=X*B +C  Regression speak AutismQuotient = Sex*Beta1 + Age*Beta2 + Intercept  BG speak AutismQuotient = Sex Effect + Age Effect + Grand Mean

15. Spss…

16. regression.mx

19. Spss…

21. Run regression.mx

23. What does this mean?  Age Beta =.549 For every 1 unit increase in Age the mean shifts.549 Grand mean =93.564 Mean Age =18.2  So the mean for 20 year olds is predicted to be: 104.544 = 93.564 + 20*.549

24. Sex effects?  Sex Beta = -2.608  Sex coded Male = 1 Female = 0  Female Mean: 93.564 = 93.564 + 0*-2.608  Male Mean: 90.656 = 93.564 + 1*-2.608

25. How do we get the p-values?

26. Set the elements to equal 0  Do this one at a time!

27. So…

28. Why bother with Mx?  Because most stat packages can’t handle non-independent data… Non-independence reduces the variance Biases t and F tests

29. Why bother with Mx?  Because we want complete flexibility in the model specification… As you see later today

30. Why bother with Mx?  Because very few packages can handle ordinal data adequately…

31. Binary data  File: two_cat.dat  NI=5  Labels Zyg twin1 twin2 Age Sex  Trait – smoking initiation Never Smoked/Ever Smoked (Recoded from yesterday) Data is sorted to speed up the analysis

32. Twin 1 smoking initiation

35. Mean =.47 SD =.499 Non Smokers =53%

36. Raw data distribution Mean =.47 SD =.499 Non Smokers =53% Threshold=.53 Standard normal distribution Mean = 0 SD =1 Non Smokers =53% Threshold =.074

37. Threshold =.074 – Huh what?  How can I work this out Excell  =NORMSINV()

38. Why do we rescale the data this way?  Convenience Variance always 1 Mean is always 0 We can interpret the area under a curve between two z-values as a probability or percentage

39. Why do we rescale the data this way? You could use other distributions but you would have to specify the fit function

40. Threshold.mx

41. Threshold =.075 – Huh what?

42. How about age/sex correction?

44. What does this mean?  Age Beta =.007 For every 1 unit increase in Age the threshold shifts.007

45. What does this mean?  Beta =.007  Threshold is -.1118  38 is +1.38 SD from the mean age The threshold for 38 year olds is:.1544= -.1118 +.007*38  22 is -1.38 SD from the mean age The threshold for 38 year olds is:.0422= -.1118 +.007*22

46. 22 year olds Threshold =.0422 38 year olds Threshold =.1544 Is the age effect significant?

47. How to interpret this  The threshold moved slightly to the right as age increases  This means younger people were more likely to have tried smoking than older people But this was not significant

48. 22 year olds Threshold =.548 38 year olds Threshold = 1.028 Is the age effect significant? If Beta =.03

49. How about the sex effect  Beta = -.05  Threshold = -.1118  Sex coded Male = 1, Female = 0  So the Male threshold is: -.1618= -.1118 + 1*-.05  The Female threshold is: -.1118 =-.1118 + 0*-.05

50. Female Threshold =-.1118 Male Threshold =-.1618 Are males or females more likely to smoke?

51. Both effects together  38 year old Males:.1042=-.1118 + 1*-.05 +.007*38  38 year old Females:.1542=-.1118 + 0*-.05 +.007*38  22 year old Males: -.0078=-.1118 + 1*-.05 +.007*22  22 year old Females:.0422=-.1118 + 0*-.05 +.007*22

52. Mx Threshold Specification: 3+ Cat. Threshold matrix : T Full 2 2 Free 1st threshold Twin 1 Twin 2 increment

53. Mx Threshold Model : ThresholdsL*T / Threshold matrix : T Full 2 2 Free 1st threshold Twin 1 Twin 2 increment Mx Threshold Specification: 3+ Cat.

54. Mx Threshold Model : ThresholdsL*T / Threshold matrix : T Full 2 2 Free 1st threshold Twin 1 Twin 2 increment 2nd threshold Mx Threshold Specification: 3+ Cat.

55. Adding a regression  L*T + G@(D*B);  maxth =2, ndef=2, nsib=2, nthr=4

56. Adding a regression

58. Multivariate Threshold Models Specification in Mx Thanks Kate Morley for these slides

59. #define nsib 2! Number of variables * number of siblings = 2 #define maxth 2! Maximum number of thresholds #define nvar 2! Number of variables #define ndef 1 ! Number of definition variables #define nthr 4 ! nsib x nvar #NGROUPS 8 G1: MZ Females Data NInput=8 Ordinal File=data.dat Labels famID zyg covar_a covar_b var1_a var2_a var1_b var2_b Select if zyg = 1 / SELECT covar_a covar_b var1_a var2_a var1_b var2_b / DEFINITION_VARIABLE covar_a covar_b / BEGIN MATRICES; X Lower nvar nvar Free! Genetic paths Y Lower nvar nvar Free! Common environmental paths Z Lower nvar nvar Free! Unique environmental paths H Full 1 1 T Full maxth nthr Free! Thresholds B Full nvar ndef Free ! Regression betas L lower maxth maxth! For converting incremental to cumulative thresholds G Full maxth 1! For duplicating regression betas across thresholds K Full ndef nsib! Contains definition variables END MATRICES;

60. Threshold model for multivariate, multiple category data with definition variables: We will break the algebra into two parts: 1 - Definition variables; 2 - Uncorrected thresholds; and go through it in detail. Part 1Part 2

61. Threshold correction Twin 1 Variable 1 Threshold correction Twin 1 Variable 2 Twin 1 Twin 2 Definition variables Threshold correction Twin 2 Variable 2 Threshold correction Twin 2 Variable 1

62. Transpose:

64. Thresholds 1 & 2 Twin 1 Variable 1 Thresholds 1 & 2 Twin 1 Variable 2 Thresholds 1 & 2 Twin 2 Variable 1 Thresholds 1 & 2 Twin 2 Variable 2

65. =

66.  http://davidmlane.com/hyperstat/z _table.html