# Test of gene-environment interaction between a SNP and a continuous non-genetic covariate from case-parent trio data.

### Description

Performs permutation test of gene-environment interaction based on the associated penalized maximum likelihood estimates obtained by fitting a generalized additive model to case-parent trio data.

### Usage

1 2 3 |

### Arguments

`object` |
A returned object from |

`data` |
Trio data to be passed into |

`nreps` |
Desired number of permutation replicates. |

`fix.sp` |
When |

`level` |
Desired significance level for the test. |

`early.stop` |
When |

`output` |
A character string specifying the name of the output file that writes the values of
the test statistics calculated for the actual and simulated data set.
When |

`return.data` |
When |

`return.object` |
When |

`...` |
Arguments passed to |

### Details

Suppose *k_1* and *k_2* are the numbers of knots used to represent
the interaction functions *f_1* and *f_2*, respectively, via
cubic regression spline functions.
Let *\bm{c}_1 = (c_{11},...,c_{1K_1-1})^\prime* and
*\bm{c}_2 = (c_{21},...,c_{2K_2-1})^\prime*
are the spline coefficient vectors for *f_1* and *f_2*
that satisfy model identifiability constraints.

The function `test.trioGxE`

calculates test statistic *T*,

* T = t(\hat{\bm{c}}){\rm V}^{-1}(\bm{c})\hat{\bm{c}}, *

where *\bm{c}=(\bm{c}_1^{\prime},\bm{c}_2^{\prime})^{\prime}*
and *V_c* is a square matrix of size *(k_1+k_2-2)*,
formed by extracting the rows and columns, corresponding to the spline
coefficients from the Bayesian posterior variance-covariance matrix
calculated in `trioGxE`

.

If the actual data were fitted under the co-dominant penetrance mode
(i.e., `object$penmod="codominant"`

),
the test statistic *T* represents an *overall* test of GxE, where

*{\rm H}_0: \bm{c}=0.*

Depending on the context, an investigator may also want to perform individual tests:
*{\rm H}_{01}: \bm{c}_1 = \bm{0}* and *{\rm H}_{02}: \bm{c}_2 = \bm{0}*.
For example, when the null hypothesis is rejected, the user may want to know which of
the two interaction function is not zero (i.e., which curve is not flat).
For the *individual* tests, `test.trioGxE`

calculates the
permutation p-values based on the Monte-Carlo distributions of the individual
test statistics *T_1* and *T_2*, where

*T_h = t(\hat{\bm{c}}_h){\rm V}^{-1}(\bm{c}_h)\hat{\bm{c}}_h, h=1,2.*

Under the dominant, log-additive (multiplicative) or recessive penetrance model,
*T* can be viewed as an individual test
since *\bm{c}_2=\bm{0}*, *\bm{c}_1=\bm{c}_2* and *\bm{c}_1=\bm{0}*,
respectively, under the dominant, log-additive and recessive models.
For example, under the dominant penetrance model, *T\equiv{T_1}* because
*\bm{c}_2=\bm{0}*, and *T_2=0*.

As the analysis is conditional on parental genotypes,
the distribution of the test statistic under *{\rm H}_0* is calculated by
shuffling the column that holds the values of the non-genetic covariate within mating types.
This can be justifiable based on the fact that under no interaction,
the SNP and the non-genetic covariate are independent of each other within
a random affected trio when they are independent within a trio from the general population
(Umbach and Weinberg, 2000).

The distribution of the test statistics can be obtained in two ways:
either under fixed smoothing parameters (`fixed.sp=TRUE`

)
or under varying smoothing parameters (`fixed.sp=FALSE`

).
Under the fixed smoothing parameters, the penalized iteratively re-weighted least squares
procedure is performed for each simulated data set under the same smoothing parameter values.
Under varying smoothing parameters, smoothing parameters are estimated for each
simulated data set.
Therefore, the test under `fixed.sp=FALSE`

accounts for the extra uncertainty introduced by
the smoothing parameter estimation.

To save computation time, the user can use ‘early-termination’ option (Besag and Clifford, 1991).
Under this option, sampling is terminated when the number of the simulated data sets reaches
`nreps*{level}`

*<* `nreps`

when the evidence is not strong enough to reject the null hypothesis at the given significance level (`level`

).
For example, if the user specifies `nreps=1000`

and `level=0.05`

, the test terminates when the number of
data sets that have test statistic values that are more extreme than or as extreme as the observed test statistic value reaches 50.

### Value

`GxE.test` |
Either a 3- or 1-column matrix.
When the actual data was fitted under a co-dominant penetrance mode
(i.e., |

`p.value` |
If |

### Author(s)

Ji-Hyung Shin <shin@sfu.ca>, Brad McNeney <mcneney@sfu.ca>, Jinko Graham <jgraham@sfu.ca>

### References

Umbach, D. and Weinberg, C. (2000). The use of case-parent triads to study joint effects of genotype
and exposure.
*Am J Hum Gen*, **66**:251-61.

Besag, J. and P. Clifford (1991). Sequential Monte Carlo p-values. *Biometrika*, **78**:301-304.

### See Also

`trioGxE`

, `plot.trioGxE`

, `trioSim`

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | ```
data(hypoTrioDat)
example.fit <- trioGxE(hypoTrioDat, pgenos = c("parent1","parent2"), cgeno = "child",
cenv = "attr",penmod="codominant", k=c(5,5))
# A toy example with 'few' permutation replicates
example.test <- test.trioGxE(example.fit, nreps=10, early.stop = FALSE,
output=NULL)
## Not run:
## More proper examples of permutation tests with 5000 replicates
## Example1: does not generate an output file containing test statistic values
example.test1 <- test.trioGxE(example.fit, nreps=5000, early.stop = TRUE,
output=NULL)
## Example 2: generates an output file 'myoutput.out' containing test statistic values
example.test2 <- test.trioGxE(example.fit, nreps=5000, early.stop = TRUE,
output="myoutput.out")
## End(Not run)
``` |