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Linear Models of Association

As you saw in the previous section, trios.ped includes 100 parent-offspring trios. All individuals have been phenotyped and genotyped except for those in family 10. By default, qtdt will test for association by fitting a simple linear model to the data. qtdt input files are specified on the command line: use the -d datafile and -p pedfile options to specify the appropriate file names. Try it now, by typing qtdt -d trios.dat -p trios.ped.

You should see a brief copyright notice followed by references to the literature:

QTDT - Quantitative TDT 2.1
(c) 1998-2000 Goncalo Abecasis 
This program implements tests described by
Abecasis et al, Am J Hum Genet 66:279-292 (2000)
Allison, in Am J Hum Genet 60:676-690 (1997) [TDTQ5]
Fulker et al, in Am J Hum Genet 64:259-267 (1999)
Monks et al, ASHG meeting (1998)
Rabinowitz, in Hum Hered 47:342-350 (1997)

This will be followed by a brief description of available parameters, of which we will now examine the first few lines:

The following parameters are in effect:
                QTDT Data File :       trios.dat (-dname)
            QTDT Pedigree File :       trios.ped (-pname)
          QTDT IBD Status File :        qtdt.ibd (-iname)
            Missing Value Code :         -99.999 (-xname)

The first four parameters describe the input files and are identical to the pedstats parameters. When a parameter is not listed on the command line, qtdt uses common default values. For example, phenotype scores of -99.999 will be assumed to denote missing values.

                    Covariates :  USER SPECIFIED (-c{p|s|u})
             Association Model :      ORTHOGONAL (-a[a|f|m|o|p|r|t|w|-])

The next two options define the linear part of the model, which will include one of the predefined models for association and covariates. Covariates can include parental phenotypes (p), sex (s) and user-specified covariates defined in the pedigree file (u). Possible models for association include the orthogonal model described by Abecasis et al (o), as well as models described by Allison (a), Rabinowitz (r), Fulker (f) and Monks (m).

For now we will ignore the remaining command-line options, and look at the part of the output describing the models to be evaluated:

The following models will be evaluated...
  NULL MODEL
     Means = Mu + B
  FULL MODEL
     Means = Mu + B + W

This section provides a succinct summary of the models to be fitted, including any user specified covariates. In this case, the model includes a mean term (mu), a between family component of association (b) and, in the full model used under the alternative hypothesis, a within family component of association (w). For a description of b and w see the paper by Abecasis et al (2000).

In the final section of the output, each trait is tested for association against each allele at each marker locus. The significance of the association is also listed (if <= 0.10). In this case 99 probands were evaluated at each marker locus, but only 75 were informative. Transmission disequilibrium tests require individuals to have heterozygous ancestors in the pedigree, so offspring of homozygous parents are uninformative.

Testing trait:                        Trait_1
=============================================
Testing marker:                         SNP_1
---------------------------------------------
 Allele   df(0)   Rsq(0)   df(T)   Rsq(T)       F       p
    1 :      97     0.09      96     0.17    8.39  0.0047  (  75/99 probands)
    2 :      97     0.09      96     0.17    8.39  0.0047  (  75/99 probands)
Testing trait:                        Trait_2
=============================================
Testing marker:                         SNP_1
---------------------------------------------
 Allele   df(0)   Rsq(0)   df(T)   Rsq(T)       F       p
    1 :      97     0.01      96     0.04    3.23  0.0753  (  75/99 probands)
    2 :      97     0.01      96     0.04    3.23  0.0753  (  75/99 probands)

Note: In newer versions of QTDT you might get slightly different 
results because founder individuals (which are not informative for scoring
the effect of transmitted alleles) are included in the analysis to help estimate
covariate effects and the grand mean. You can disable this behaviour with the
--exclude-founder-phenotypes option.

The results suggest an association between Trait_1 and SNP_1. Other models for association in families with a single offspring include the Allison model (TDTQ5) and the Rabinowitz model. To fit the Allison model to this data set, run qtdt -p trios.ped -d trios.dat -aa or, to fit the Rabinowitz model, run qtdt -p trios.ped -d trios.dat -ar. The results should be quite similar.

You will probably have noticed that allele 1 and allele 2 always provide the same evidence for association. This is a characteristic of SNP data. To reduce the amount of output, and to speed up analysis, you can specify that only allele 1 should be analysed (-1 option). Try running qtdt -p trios.ped -d trios.dat -1.

In the next section we will see how qtdt handles families with multiple offspring.