Last updated: 2019-01-17

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Investigate the non-convergence problem in susie_z.

devtools::install_github('zouyuxin/susieR')
Skipping install of 'susieR' from a github remote, the SHA1 (7cd1fa2d) has not changed since last install.
  Use `force = TRUE` to force installation
library(susieR)

Simulation under null effect

dscout_null = readRDS('output/dscout_gaussian_null.rds')
dscout_null = dscout_null[!is.na(dscout_null$sim_gaussian_null.output.file),]
dscout_null = dscout_null[!is.na(dscout_null$susie_z.output.file),]

The maximum number of iteration is 7. All models converge. There is no false discovry.

Simulation with one effect

We simulate data with only one non-zero effect.

We simulate data under PVE = 0.01, 0.05, 0.1, 0.2, 0.5, 0.8, 0.95. There are 20 replicates.

dscout_gaussian_z = readRDS('output/dscout_gaussian_z.rds')
dscout_gaussian_z = dscout_gaussian_z[!is.na(dscout_gaussian_z$sim_gaussian.output.file),]
dscout_gaussian_z = dscout_gaussian_z[!is.na(dscout_gaussian_z$susie_z.output.file),]
dscout_gaussian_z$NotConverge = dscout_gaussian_z$susie_z.niter == 100
dscout_gaussian_z_1 = dscout_gaussian_z[dscout_gaussian_z$sim_gaussian.effect_num == '1', ]
dscout_gaussian_z_1 = dscout_gaussian_z_1[dscout_gaussian_z_1$susie_z.L == '5' , ]

When the PVE is greater than 0.5, some replicates fail to converge in 100 iterations.

converge.summary = aggregate(NotConverge ~ sim_gaussian.pve, dscout_gaussian_z_1, sum)
colnames(converge.summary) = c('pve', 'NotConverge')
converge.summary
   pve NotConverge
1 0.01           0
2 0.05           0
3 0.10           0
4 0.20           0
5 0.50           4
6 0.80           9
7 0.95           6

Now we change the initial susie object to the truth:

dscout_gaussian_init = readRDS('output/dscout_gaussian_init.rds')
dscout_gaussian_init = dscout_gaussian_init[!is.na(dscout_gaussian_init$sim_gaussian.output.file),]
dscout_gaussian_init = dscout_gaussian_init[!is.na(dscout_gaussian_init$susie_z_init.output.file),]
dscout_gaussian_init$NotConverge = dscout_gaussian_init$susie_z_init.niter == 100
dscout_gaussian_init_1 = dscout_gaussian_init[dscout_gaussian_init$sim_gaussian.effect_num == '1', ]
converge.summary = aggregate(NotConverge ~ sim_gaussian.pve, dscout_gaussian_init_1, sum)
colnames(converge.summary) = c('pve', 'NotConverge')
converge.summary
   pve NotConverge
1 0.01           0
2 0.05           0
3 0.10           0
4 0.20           0
5 0.50           0
6 0.80           0
7 0.95           0

All models converge.

Simulation with 5 effects

We simulate data with 5 true effects. We simulate data under PVE = 0.01, 0.05, 0.1, 0.2, 0.5, 0.8, 0.95. There are 20 replicates.

dscout_gaussian_z_5 = dscout_gaussian_z[dscout_gaussian_z$sim_gaussian.effect_num == '5', ]
dscout_gaussian_z_5 = dscout_gaussian_z_5[dscout_gaussian_z_5$susie_z.L == '5' , ]

When the PVE is greater than 0.5, some replicates fail to converge in 100 iterations.

converge.summary = aggregate(NotConverge ~ sim_gaussian.pve, dscout_gaussian_z_5, sum)
colnames(converge.summary) = c('pve', 'NotConverge')
converge.summary
   pve NotConverge
1 0.01           0
2 0.05           0
3 0.10           0
4 0.20           0
5 0.50           1
6 0.80           5
7 0.95           5

Now we change the initial susie object to the truth,

dscout_gaussian_init_5 = dscout_gaussian_init[dscout_gaussian_init$sim_gaussian.effect_num == '5', ]
converge.summary = aggregate(NotConverge ~ sim_gaussian.pve, dscout_gaussian_init_5, sum)
colnames(converge.summary) = c('pve', 'NotConverge')
converge.summary
   pve NotConverge
1 0.01           0
2 0.05           0
3 0.10           0
4 0.20           0
5 0.50           1
6 0.80           5
7 0.95           5

There are still non-convergence cases with the true initialization.

Examples

One effect

Let’s see an example data with one true effect. The PVE is 0.5, the residual variance is 0.47^2.

X = readRDS('data/susie_X.rds') # X is from susieR package, N3finemapping, X is coloumn mean centered.
R = readRDS('data/susie_R.rds')
data = readRDS('data/sim_gaussian_75.rds')
n = data$n
beta = numeric(data$p)
beta[data$beta_idx] = data$beta_val
z = data$ss$effect/data$ss$se
susie_plot(z, y = 'z', b=beta)

Expand here to see past versions of unnamed-chunk-13-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

Using z scores,

fit_z = susie_z(z, R=R, max_iter = 20, L=5, track_fit = T)
Warning in susie_ss(XtX = R, Xty = z, n = 2, var_y = 1, L = L,
estimate_prior_variance = TRUE, : IBSS algorithm did not converge in 20
iterations!
susie_get_objective(fit_z)
[1] 409.0831
susie_plot(fit_z, y = 'PIP', b = beta)

Expand here to see past versions of unnamed-chunk-14-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

Now we initialize at the truth (L=1), the model converges to a lower objective value.

s_init = susie_init_coef(data$beta_idx, data$beta_val, data$p)
fit_z_init = susie_z(z, R=R, s_init = s_init)
susie_get_objective(fit_z_init)
[1] 299.4515
susie_plot(fit_z_init, y = 'PIP', b = beta)

Expand here to see past versions of unnamed-chunk-15-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

I mentioned in the write-up, we approximate the BF(\(z_j(\sigma)\)) (eqn 7.61) using BF(\(z_j(\hat{\sigma}_j)\)). When the data contain strong association signals (\(\hat{\sigma}_j > \sigma\)), the z score approximation under-estimates the Bayes Factor.

The 7.61 in the write-up: \[ \begin{align} BF(z_j(\sigma); w) &= \sqrt{\frac{1}{1+w^2}} \exp\left( \frac{1}{2} z_j(\sigma)^2 \frac{w^2}{1+w^2} \right) \quad \quad \text{where }w^2 = (n-1)\frac{\sigma_{0}^{2}}{\sigma^2} \end{align} \]

We check the under-estimate now:

fit_z_1 = susie_ss(XtX = R, Xty = z, n = 2, var_y = 1, L = 1,
                   estimate_prior_variance = FALSE,
                   estimate_residual_variance = FALSE, max_iter = 1)
X.s = apply(X, 2, function(x) x/(sd(x)*sqrt(n-1)))
z_true = (t(X.s) %*% data$sim_y) / as.numeric(data$sigma)

fit_z_1_true = susie_ss(XtX = R, Xty = z_true, n = 2, var_y = 1, L = 1,
                        estimate_prior_variance = FALSE,
                        estimate_residual_variance = FALSE,
                        prior_weights = NULL, null_weight = NULL,
                        coverage = 0.95, min_abs_corr = 0.5,
                        verbose = FALSE, track_fit = TRUE, max_iter = 1)
par(mfrow=c(1,3))
{plot(fit_z_1$lbf_mtx, fit_z_1_true$lbf_mtx, xlab = expression(logBF(z[j](hat(sigma)[j]))), ylab = expression(logBF(z[j](sigma))), main='log BF')
abline(0,1)}
{plot(fit_z_1$alpha, fit_z_1_true$alpha, xlab = expression(alpha(hat(sigma)[j])), ylab = expression(alpha(sigma)), main=expression(alpha))
abline(0,1)}
{plot(abs(fit_z_1$mu), abs(fit_z_1_true$mu), xlab = expression(abs(mu(hat(sigma)[j]))), ylab = expression(abs(mu(sigma))), main=expression(abs(mu)))
abline(0,1)}

Expand here to see past versions of unnamed-chunk-18-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

It is clear from the plot that the Bayes Factor is under-estimated. The estimated \(\alpha\), \(\mu\) are smaller.

Checked the estimated prior variance,

fit_z_10_prior = susie_z(z=z, R=R, L=5)
Warning in susie_ss(XtX = R, Xty = z, n = 2, var_y = 1, L = L,
estimate_prior_variance = TRUE, : IBSS algorithm did not converge in 100
iterations!
susie_get_prior_variance(fit_z_10_prior)
[1] 24449.0562  4961.9053  2624.2558  2017.4113   163.6263
fit_z_10_true_prior = susie_z(z = z_true, R=R, L=5)
susie_get_prior_variance(fit_z_10_true_prior)
[1] 604.074353   3.315658   0.000000   0.000000   0.000000
susie_plot(fit_z_10_true_prior, y='PIP', b=beta)

Expand here to see past versions of unnamed-chunk-20-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

Using the true z, \(z_j(\sigma)\), the estimated prior variance for some L becomes zero. Using the estimated z, \(z_j(\hat{\sigma}_j)\), the estimated prior variance for all L are non-zero.

Five effects

Let’s see an example data with 5 true effects. The PVE is 0.8, the residual variance is 0.74^2.

data = readRDS('data/sim_gaussian_475.rds')
n = data$n
beta = numeric(data$p)
beta[data$beta_idx] = data$beta_val
z = data$ss$effect/data$ss$se
susie_plot(z, y = "z", b=beta)

Expand here to see past versions of unnamed-chunk-21-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

Using z scores,

fit_z = susie_z(z, R=R, max_iter = 20, L=5, track_fit = T)
Warning in susie_ss(XtX = R, Xty = z, n = 2, var_y = 1, L = L,
estimate_prior_variance = TRUE, : IBSS algorithm did not converge in 20
iterations!
susie_get_objective(fit_z)
[1] 525.9303
susie_plot(fit_z, y = 'PIP', b = beta)

Expand here to see past versions of unnamed-chunk-22-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

Now we initialize at the truth, the model fails to converge.

s_init = susie_init_coef(data$beta_idx, data$beta_val, data$p)
fit_z_init = susie_z(z, R=R, s_init = s_init)
Warning in susie_ss(XtX = R, Xty = z, n = 2, var_y = 1, L = L,
estimate_prior_variance = TRUE, : IBSS algorithm did not converge in 100
iterations!
susie_get_objective(fit_z_init)
[1] 1089.854
susie_plot(fit_z_init, y = 'PIP', b = beta)

Expand here to see past versions of unnamed-chunk-23-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

Changing the z score to \(z_j(\sigma)\), the model converges.

z_true = (t(X.s) %*% data$sim_y) / as.numeric(data$sigma)
fit_z_5_true = susie_z(z = z_true, R=R, L=5)
susie_get_objective(fit_z_5_true)
[1] 1119.521
susie_get_prior_variance(fit_z_5_true)
[1] 1467.02402  640.34084  185.85349   53.12493    0.00000
susie_plot(fit_z_5_true, y = 'PIP', b = beta)

Expand here to see past versions of unnamed-chunk-24-1.png:
Version Author Date
a6cee3a zouyuxin 2019-01-17

Session information

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.14.2

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] susieR_0.6.3

loaded via a namespace (and not attached):
 [1] Rcpp_1.0.0        compiler_3.5.1    git2r_0.24.0     
 [4] workflowr_1.1.1   R.methodsS3_1.7.1 prettyunits_1.0.2
 [7] R.utils_2.7.0     remotes_2.0.2     tools_3.5.1      
[10] testthat_2.0.1    digest_0.6.18     pkgbuild_1.0.2   
[13] pkgload_1.0.2     lattice_0.20-38   evaluate_0.12    
[16] memoise_1.1.0     rlang_0.3.1       Matrix_1.2-15    
[19] cli_1.0.1         curl_3.3          yaml_2.2.0       
[22] expm_0.999-3      withr_2.1.2       stringr_1.3.1    
[25] knitr_1.20        desc_1.2.0        fs_1.2.6         
[28] devtools_2.0.1    grid_3.5.1        rprojroot_1.3-2  
[31] glue_1.3.0        R6_2.3.0          processx_3.2.1   
[34] rmarkdown_1.11    sessioninfo_1.1.1 callr_3.1.1      
[37] magrittr_1.5      whisker_0.3-2     backports_1.1.3  
[40] ps_1.3.0          htmltools_0.3.6   usethis_1.4.0    
[43] assertthat_0.2.0  stringi_1.2.4     crayon_1.3.4     
[46] R.oo_1.22.0

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