Algorithm 1.

Federated Genotype-Expression-Image Data Integration Model.

Input: Data pairs of the I institutions, (X1, y1), …, (Xi, yi), …, (XI, yI) and the sample numbers of each group, $\left(N_1^{\left(1\right)},N_1^{\left(2\right)},N_1^{\left(3\right)}\right),...,\left(N_i^{\left(1\right)},N_i^{\left(2\right)},N_i^{\left(3\right)}\right),...,\left(N_I^{\left(1\right)},N_I^{\left(2\right)},N_I^{\left(3\right)}\right)$

Output: p-value of the studying Genotype-Expression-Image trio

Initialize: w(1), w(2), w(3), w(C) = 0

1:  for g = {1, 2, 3, C} do

2:    while convergence and maximum number of iterations are not reached do

3:        Get an image patch xi from X.

4:        Each institution computes the gradient: $\nabla S_i^{(g)}\left(w^{(g)}\right)={[X_i^{(g)}]}^T\left(X_i^{(g)}{w}^{(g)}-y_i^{g(g)}\right).$

5:        Global center computes and sends global gradient to each institution: $\nabla S^{\left(g\right)}\left(w^{\left(g\right)}\right)=\sum _{i=1}^I\nabla S_i^{\left(g\right)}\left(w^{\left(g\right)}\right).$

6:        Each institution updates the coefficient with the global gradient: w(g) ← w(g) − η∇S(g) (w(g)).

7:       end while

8:       Each institution calculates the sum of squared residual: $S_i^{\left(g\right)}\left(w^{\left(g\right)};X_i^{\left(g\right)},y_i^{\left(g\right)}\right).$

9:       Global center gathers the global sum of squared residual: $S^{\left(g\right)}=\sum _{i=1}^I{S}_i^{\left(g\right)}$.

10:       Global center gathers the global sample numbers: $N^{\left(g\right)}=\sum _{i=1}^I{N}_i^{\left(g\right)}$.

11:  end for

12:  Global center calculates F value with equation (1) and then computes and sends p-value to all institutions.