gxe.test {logicDT}

Gene-environment interaction test

Description

Using a fitted logicDT model, a general GxE interaction test can be performed.

Usage

gxe.test(model, X, y, Z, perm.test = TRUE, n.perm = 10000)

Arguments

model

A fitted logicDT model with 4pL models in its leaves.

X

Binary predictor data for testing the interaction effect. This can be equal to the training data.

y

Response vector for testing the interaction effect. This can be equal to the training data.

Z

Quantitative covariable for testing the interaction effect. This can be equal to the training data.

perm.test

Should additionally permutation testing be performed? Useful if likelihood ratio test asymptotics cannot be justified.

n.perm

Number of random permutations for permutation testing

Details

The testing is done by fitting one shared 4pL model for all tree branches with different offsets, i.e., allowing main effects of SNPs. This shared model is compared to the individual 4pL models fitted in the logicDT procedure using a likelihood ratio test which is asymptotically \chi^2 distributed. The degrees of freedom are equal to the difference in model parameters. For regression tasks, alternatively, a F-test can be utilized.

The shared 4pL model is given by

Y = \tilde{f}(x, z, b, c, d, e, \beta_1, \ldots, \beta_{G-1}) + \varepsilon = c + \frac{d-c}{1+\exp(b \cdot (x-e))} + \sum_{g=1}^{G-1} \beta_g \cdot 1(z = g) + \varepsilon

with z \in \lbrace 1, \ldots, G \rbrace being a grouping variable, \beta_1, \ldots, \beta_{G-1} being the offsets for the different groups, and \varepsilon being a random error term. Note that the last group G does not have an offset parameter, since the model is calibrated such that the curve without any \beta's fits to the last group.

The likelihood ratio test statistic is given by

\Lambda = -2(\ell_{\mathrm{shared}} - \ell_{\mathrm{full}})

for the log likelihoods of the shared and full 4pL models, respectively. In the regression case, the test statistic can be calculated as

\Lambda = N(\log(\mathrm{RSS}_{\mathrm{shared}}) - \log(\mathrm{RSS}_{\mathrm{full}}))

with \mathrm{RSS} being the residual sum of squares for the respective model.

For regression tasks, the alternative F test statistic is given by

f = \frac{\frac{1}{\mathrm{df}_1}(\mathrm{RSS}_{\mathrm{shared}} - \mathrm{RSS}_{\mathrm{full}})} {\frac{1}{\mathrm{df}_2} \mathrm{RSS}_{\mathrm{full}}}

with

\mathrm{df}_1 = \mathrm{Difference\ in\ the\ number\ of\ model\ parameters} = 3 \cdot n_{\mathrm{scenarios}} - 3,

\mathrm{df}_2 = \mathrm{Degrees\ of\ freedom\ of\ the\ full\ model} = N - 4 \cdot n_{\mathrm{scenarios}},

and n_{\mathrm{scenarios}} being the number of identified predictor scenarios/groups by logicDT.

Alternatively, if linear models were fitted in the supplied logicDT model, shared linear models can be used to test for a GxE interaction. For continuous outcomes, the shared linear model is given by

Y = \tilde{f}(x, z, \alpha, \beta_1, \ldots, \beta_{G}) + \varepsilon = \alpha \cdot x + \sum_{g=1}^{G} \beta_g \cdot 1(z = g) + \varepsilon.

For binary outcomes, LDA (linear discriminant analysis) models are fitted. In contrast to the 4pL-based test for binary outcomes, varying offsets for the individual groups are injected to the linear predictor instead of to the probability (response) scale.

If only few samples are available and the asymptotics of likelihood ratio tests cannot be justified, alternatively, a permutation test approach can be employed by setting perm.test = TRUE and specifying an appropriate number of random permutations via n.perm. For this approach, computed likelihoods of the shared and (paired) full likelihood groups are randomly interchanged approximating the null distribution of equal likelihoods. A p-value can be computed by determining the fraction of more extreme null samples compared to the original likelihood ratio test statistic, i.e., using the fraction of higher likelihood ratios in the null distribution than the original likelihood ratio.

Value

A list containing

p.chisq

The p-value of the chi-squared test statistic.

p.f

The p-value of the F test statistic.

p.perm

The p-value of the optional permutation test.

ll.shared

Log likelihood of the shared parameters 4pL model.

ll.full

Log likelihood of the full logicDT model.

rss.shared

Residual sum of squares of the shared parameters 4pL model.

rss.full

Residual sum of squares of the full logicDT model.