# Missing Value Estimation of Explanatory and Dependent Variables in Cross-Classified Random Effects Models

Nice to meet you, everyone.
I’m new to Stan.
I am challenging the following cross-classified random effects model.

Level1（Within-Cell Model）

Y_{ijk} = \beta_{0jk}+\beta_{1}AGE_{ijk}+\displaystyle \Sigma_{m=2}^{n}\beta_{m}X_{mijk}+e_{ijk}, e_{ijk} \sim N(0,\sigma^{2})

Level2（Between-Cell Model）

\beta_{0jk}=\gamma_{0}+u_{0j}+v_{0k}, u_{0j} \sim N(0,\tau_{u}), v_{0k} \sim N(0,\tau_{v})

The Stan code is as follows.

data {
int<lower=0> N; // length of y; data size
int<lower=0> J;     //number of groups
int<lower=0> K;    //number of cohort group
vector[N] indi; // response variable
vector[N] edum; // explanatory variable; centralized
vector[N] age; // explanatory variable; centralized
vector[N] income; // explanatory variable; centralized
vector[N] sex; // explanatory variable; centralized
vector[N] work2; // explanatory variable; centralized
vector[N] work3; // explanatory variable; centralized
vector[N] work4; // explanatory variable; centralized
vector[N] marry;  // explanatory variable; centralized
int<lower=1, upper=J> year[N]; //  indeces
int<lower=1, upper=K> cohort[N];  //  indeces

real sx1;  //Standard deviation of the explanatory variable
real sx2; //Standard deviation of the explanatory variable
real sx3; //Standard deviation of the explanatory variable
real sx4; //Standard deviation of the explanatory variable
real sx5; //Standard deviation of the explanatory variable
real sx6; //Standard deviation of the explanatory variable
real sx7; //Standard deviation of the explanatory variable
real sx8; //Standard deviation of the explanatory variable
real sy; //Standard deviation of the dependent variable
real my; //Mean value of the dependent variable

int indi_nMiss; // N missing from variables; excluded have no missings
int edum_nMiss;
int marry_nMiss;
int income_nMiss;
int work2_nMiss;
int work3_nMiss;
int work4_nMiss;

int indi_index[indi_nMiss];  //  indeces for missing values
int edum_index[edum_nMiss];
int marry_index[marry_nMiss];
int income_index[income_nMiss];
int work2_index[work2_nMiss];
int work3_index[work3_nMiss];
int work4_index[work4_nMiss];

}

parameters {

vector[J] u0j;
vector[K] v0k;
real beta1;
real beta2;
real beta3;
real beta4;
real beta5;
real beta6;
real beta7;
real beta8;
real alpha;
real<lower=0> sigma; //residualSD
real<lower=0> u0j_tau_u; //standard deviation of the random effects
real<lower=0> v0k_tau_v; //standard deviation of the random effects

vector<lower=0>[indi_nMiss] indi_imp; // missing values
vector<lower=0>[edum_nMiss] edum_imp;
vector<lower=0>[marry_nMiss] marry_imp;
vector<lower=0>[income_nMiss] income_imp;
vector<lower=0>[work2_nMiss] work2_imp;
vector<lower=0>[work3_nMiss] work3_imp;
vector<lower=0>[work4_nMiss] work4_imp;

cholesky_factor_cov[8] Sigma;
vector[8] Mu;
}

transformed parameters{
// Recombine data
vector[N] mu;
matrix[N,9] Data;
Data[,2] = edum; Data[edum_index,2] = edum_imp;
Data[,3] = age;
Data[,4] = income; Data[income_index,4] = income_imp;
Data[,5] = sex;
Data[,6] = work2; Data[work2_index,6] = work2_imp;
Data[,7] = work3; Data[work3_index,7] = work3_imp;
Data[,8] = work4; Data[work4_index,8] = work4_imp;
Data[,9] = marry; Data[marry_index,9] = marry_imp;
Data[,1] = indi; Data[indi_index,1] = indi_imp;

for(i in 1:N)  mu[i] =  Data[i,2]*beta1  + Data[i,3]*beta2  +  Data[i,4]*beta3  +Data[i,5]*beta4
+Data[i,6]*beta5+Data[i,7]*beta6 +Data[i,8]*beta7 + Data[i,9]*beta8 + u0j[year[i]]+v0k[cohort[i]] + alpha;

model {

// MVN imputation
for(n in 1:N){
Data[n,2:9] ~ multi_normal_cholesky(Mu,Sigma);
}

target += normal_lpdf(Data[,1] | mu,sigma);
beta1~ normal(0,2.5*sy/sx1);
beta2 ~ normal(0,2.5*sy/sx2);
beta3 ~ normal(0,2.5*sy/sx3);
beta4 ~ normal(0,2.5*sy/sx4);
beta5 ~ normal(0,2.5*sy/sx5);
beta6 ~ normal(0,2.5*sy/sx6);
beta7 ~ normal(0,2.5*sy/sx7);
beta8 ~ normal(0,2.5*sy/sx8);
target += normal_lpdf(u0j | 0, u0j_tau_u);
target += normal_lpdf(v0k | 0, v0k_tau_v);
alpha ~ normal(my, 2.5*sy);
sigma ~ exponential(1/sy);
target += cauchy_lpdf(u0j_tau_u | 0, 2.5)
- 1 * cauchy_lccdf(0 | 0, 2.5);
target += cauchy_lpdf(v0k_tau_v | 0, 2.5)
- 1 * cauchy_lccdf(0 | 0, 2.5);

}


I use default prior for each explanatory variable. This is the same one used in rstanarm.
The R code is as follows.

missings <- lapply(wavea,function(var){which(is.na(var))}) # For each variable in data, which rows are NA
nMissings <- lapply(missings,function(var){length(var)}) # Number of NAs
names(missings) <- paste0(names(missings),'_index') # Rename indices to variable_index
names(nMissings) <- paste0(names(nMissings),'_nMiss') # Rename n_miss to variable_nMiss

datas <- list(
indi = indi,
age = age,
marry = marry,
sex = sex,
income = income,
edum = edum,
work2 = work2,
work3=work3,
work4=work4,
N = N,
J = 6,
K = 21,
year=year2,
cohort = birth3 ,
sx1 = sx1,
sx2 = sx2,
sx3 = sx3,
sx4 = sx4,
sx5 = sx5,
sx6 = sx6,
sx7 = sx7,
sx8 = sx8,
my = my,
sy = sy,
indi_nMiss=nMissings[["indi_nMiss"]],
edum_nMiss=nMissings[["edum_nMiss"]],
marry_nMiss=nMissings[["marry_nMiss"]],
income_nMiss= nMissings[["income_nMiss"]],
work2_nMiss=nMissings[["work2_nMiss"]],
work3_nMiss=nMissings[["work3_nMiss"]],
work4_nMiss= nMissings[["work4_nMiss"]],
indi_index=missings[["indi_index"]],
edum_index=missings[["edum_index"]],
marry_index=missings[["marry_index"]],
income_index=missings[["income_index"]],
work2_index=missings[["work2_index"]],
work3_index=missings[["work3_index"]],
work4_index=missings[["work4_index"]]
)

fitl <- modapc\$sample(
data = datas,
seed = 123,
iter_warmup = 1000,
iter_sampling = 3000,
chains = 4,
parallel_chains = 4,
)


In the data used here, missing values are present in many of the explanatory and dependent variables. Therefore, we have also built a model in which the explanatory variables are assumed to be multivariate normal.
The model runs, but it takes about a day for the entire chain to finish sampling. Also, R crashes when displaying the summary.

Is there still something wrong with the notation of the model?
I would appreciate it if you could help me.

1 Like

The following is the original paper I referred to.

https://www.jstor.org/stable/pdf/25046692.pdf?refreqid=excelsior%3A7170ab0f24a6880b4e174f600db824a6

This is a cross-classified random effects model applied to an APC (cohort) analysis.
The data are time-series data（1990~2019） from the World Value Survey.

Sorry, your question seems to have been left behind. Did you manage to resolve this in the meantime or do you still need help?

There definitely can be models that take very long and nothing can be done about it. How big is your dataset? How many missing values you have?

Also I think brms should support both missing values and cross-classified models so you might want to compare your implementation to using brms… (brms now also supports within-chain paralellization, so if the model is really slow because it is just big, it can help a bit)

1 Like

Thanks for the reply, Martin.

Unfortunately, my problem is not solved yet.

The size of the dataset is, n=8,319.
The size of the missing values is 2,787 (of which 420 are missing values of the dependent variable).

I see. I’ll consider using brms (I think the function for missing value estimation is brm_multiple).

brm_multiple is just one way of doing the imputation - you can also directly estimate the values within the model (similarly to the way you seem to be doing it), see: https://cran.r-project.org/web/packages/brms/vignettes/brms_missings.html#imputation-during-model-fitting

This is IMHO at least “moderately big”. I would expect such a model to take at least an hour, possibly much more, so there might be some place for improvement, but it is possible it is not huge. There are some usual techniques to let you zoom in on what could be wrong:

• Use a subset of the data. Are the inferences reasonable?
• If you exclude the missing values, this should greatly speed-up the model. Does the model work well in this case?
• Create a (smaller) simulated datset where you know the true values of the parameters. Is the model able to recover those?

That is most likely a bug, if you could share the model fit and more details about what exactly happens at https://github.com/stan-dev/rstan/issues/ it would help.