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

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

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

Lets start with the data…

If this was a pedigree data file… FamidIndFatherMotherZygSexAge Trait xx xx 2312MZ MZ

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

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

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

Spss…

Mx… Means.mx

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

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

Spss…

regression.mx

Spss…

Run regression.mx

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

Sex effects?  Sex Beta =  Sex coded Male = 1 Female = 0  Female Mean: = *  Male Mean: = *-2.608

How do we get the p-values?

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

So…

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

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

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

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

Twin 1 smoking initiation

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

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

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

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

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

Threshold.mx

Threshold =.075 – Huh what?

How about age/sex correction?

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

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

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

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

22 year olds Threshold = year olds Threshold = Is the age effect significant? If Beta =.03

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

Female Threshold = Male Threshold = Are males or females more likely to smoke?

Both effects together  38 year old Males:.1042= * *38  38 year old Females:.1542= * *38  22 year old Males: = * *22  22 year old Females:.0422= * *22

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

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

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.

Adding a regression  L*T +  maxth =2, ndef=2, nsib=2, nthr=4

Adding a regression

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

#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;

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

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

Transpose:

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

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