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CAUTION: GitHub flavoured markdown doesn't support the rendering of mathematics at this moment. Hence the mathematics in this README are not rendered properly. The information below is also available as a vignette within the package. The mathematics will be rendered in the vignette. To read the vignette one needs to install the package first.
Installation instructions
This package requires the INLA package. You need to install it with install.packages("INLA", repos = "https://www.math.ntnu.no/inla/R/stable"). If this fails you can use devtools::install_github("INBO-BMK/INLA"). Note that the latter is just a read-only mirror which is infrequently updated. Hence installing INLA from https://www.math.ntnu.no/inla is highly recommended.
When INLA is installed, we can install multimput with devtools::install_github("INBO-BMK/multimput", build_vignettes = TRUE). To view the vignette use vignette("Impute", package = "multimput")
A docker image with all the required dependencies is available from https://hub.docker.com/r/inbobmk/multimput/. Use docker pull inbobmk/multimput to get it.
Very short intro to multiple imputation
- Create the imputation model
- Generate imputations for the missing observation
- Aggregate the imputed data
- Model the aggregated imputed data
Short intro to multiple imputation
The imputations are based on a model (Y \sim X \beta^*) which the user has to specify. For a missing value (i) with covariates (x_i), we draw a random value (y_i) from the distribution of (\hat{y}_i). In case of a linear model, we sample a normal distribution (y_i \sim N(\hat{y}_i, \sigma_i)). An imputation set (l) holds an impute value (y_i) for each missing value.
With the missing values replaced by imputation set (l), the dataset is complete. So we can apply the analysis that we wanted to do in the first place. This can, but don't has to, include aggregating the dataset prior to analysis. The analysis results in a set of coefficients ({\gamma_a}_l) and their standard error ({\sigma_a}_l). Of course, this set will depend on the imputed values of the imputation set (l). Another imputation set has different imputed values and will hence lead to different coefficients.
Therefore the imputation, aggregation and analysis is repeated for (L) different imputation sets, resulting in (L) sets of coefficients and their standard errors. They are aggregated by the formulas below. The coefficient will be the average of the coefficient in all imputation sets. The standard error of a coefficient is the square root of a sum of two parts. The first part is the average of the squared standard error in all imputation sets. The second part is the variance of the coefficient among the imputation sets, multiplied by a correction factor (1 + \frac{1}{L}).
[\bar{\gamma}a = \frac{\sum{l = 1}^L{\gamma_a}_l}{L}] [\bar{\sigma}a = \sqrt{\frac{\sum{l = 1}^J {{\sigma_a^2}l}}{L} + (1 + \frac{1}{L}) \frac{\sum{l = 1}^L({\gamma_a}_l - \bar{\gamma}_a) ^ 2}{L - 1}}]
The dataset
First, let's generate a dataset and set some observations missing. generateData() creates a balanced dataset with repeated visits of a number of sites. Each site is visited several years and multiple times per year. Have a look at the help-file of generateData() for more details on the model.
library(multimput)
set.seed(123)
prop.missing <- 0.5
dataset <- generateData(
n.year = 10, n.period = 6, n.site = 50,
n.run = 1
)
dataset$Observed <- dataset$Count
which.missing <- sample(nrow(dataset), size = nrow(dataset) * prop.missing)
dataset$Observed[which.missing] <- NA
dataset$fYear <- factor(dataset$Year)
dataset$fPeriod <- factor(dataset$Period)
dataset$fSite <- factor(dataset$Site)
str(dataset)
#> 'data.frame': 3000 obs. of 10 variables:
#> $ Year : int 1 2 3 4 5 6 7 8 9 10 ...
#> $ Period : int 1 1 1 1 1 1 1 1 1 1 ...
#> $ Site : int 1 1 1 1 1 1 1 1 1 1 ...
#> $ Mu : num 11.36 9.31 11.33 10.98 9.79 ...
#> $ Run : int 1 1 1 1 1 1 1 1 1 1 ...
#> $ Count : num 8 2 10 21 2 2 13 8 10 4 ...
#> $ Observed: num NA NA 10 21 NA 2 13 8 10 4 ...
#> $ fYear : Factor w/ 10 levels "1","2","3","4",..: 1 2 3 4 5 6 7 8 9 10 ...
#> $ fPeriod : Factor w/ 6 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...
#> $ fSite : Factor w/ 50 levels "1","2","3","4",..: 1 1 1 1 1 1 1 1 1 1 ...Variables in dataset
Year
The year of the observation as an integer
fYear
The year of the observation as a factor
Period
The period of the observation as an integer
fPeriod
The period of the observation as a factor
Site
The ID of the site as an integer
fSite
The ID of the site as a factor
Mu
The expected value of a negative binomial distribution
Count
A realisation of a negative binomial distribution with expected value Mu
Observed
The Count variable with missing data
library(ggplot2)
ggplot(dataset, aes(x = Year, y = Mu, group = Site)) +
geom_line() +
facet_wrap(~Period) +
scale_y_log10()
Create the imputation model
We will create several models, mainly to illustrate the capabilities of the multimput package. Hence several of the models are not good for a real life application.
# a simple linear model
imp.lm <- lm(Observed ~ fYear + fPeriod + fSite, data = dataset)
# a mixed model with Poisson distribution
# fYear and fPeriod are the fixed effects
# Site are independent and identically distributed random intercepts
library(lme4)
imp.glmm <- glmer(
Observed ~ fYear + fPeriod + (1 | fSite),
data = dataset,
family = poisson
)
library(INLA)
# a mixed model with Poisson distribution
# fYear and fPeriod are the fixed effects
# Site are independent and identically distributed random intercepts
# the same model as imp.glmm
imp.inla.p <- inla(
Observed ~ fYear + fPeriod + f(Site, model = "iid"),
data = dataset,
family = "poisson",
control.predictor = list(compute = TRUE)
)
# the same model as imp.inla.p but with negative binomial distribution
imp.inla.nb <- inla(
Observed ~ fYear + fPeriod + f(fSite, model = "iid"),
data = dataset,
family = "nbinomial",
control.predictor = list(compute = TRUE)
)
# a mixed model with negative binomial distribution
# fPeriod is a fixed effect
# f(Year, model = "rw1") is a global temporal trend
# modelled as a first order random walk
# delta_i = Year_i - Year_{i-1} with delta_i \sim N(0, \sigma_{rw1})
# f(YearCopy, model = "ar1", replicate = Site) is a temporal trend per Site
# modelled as an first order autoregressive model
# Year_i_k = \rho Year_{i-1}_k + \epsilon_i_k with \epsilon_i_k \sim N(0, \sigma_{ar1})
dataset$YearCopy <- dataset$Year
imp.better <- inla(
Observed ~
f(Year, model = "rw1") +
f(YearCopy, model = "ar1", replicate = Site) +
fPeriod,
data = dataset,
family = "nbinomial",
control.predictor = list(compute = TRUE)
)Apply the imputation model
Most models have a predict method. In such a case impute() requires both a model and a data argument. Note that this implies that one can apply an imputation on any dataset as long as the dataset contains the necessary variables.
inla do the prediction simultaneously with the model fitting. Hence the model contains all required information and the data is not used.
n.imp is the number of imputations. The default is n.imp = 19.
raw.lm <- impute(imp.lm, data = dataset)
raw.glmm <- impute(imp.glmm, data = dataset)
raw.inla.p <- impute(imp.inla.p)
raw.inla.nb <- impute(imp.inla.nb)
raw.better <- impute(imp.better)
raw.better.9 <- impute(imp.better, n.imp = 9)Aggregate the imputed dataset
Suppose that we are interested in the sum of the counts over all sites for each combination of year and period. Then we must aggregate the imputations on year and period. The resulting object will only contain the imputed response and the grouping variables. The easiest way to have a variable like year both a continuous and factor is to add both Year and fYear to the grouping.
aggr.lm <- aggregate_impute(
raw.lm,
grouping = c("fYear", "fPeriod", "Year"),
fun = sum
)
aggr.glmm <- aggregate_impute(
raw.glmm,
grouping = c("fYear", "fPeriod", "Year"),
fun = sum
)
aggr.inla.p <- aggregate_impute(
raw.inla.p,
grouping = c("fYear", "fPeriod", "Year"),
fun = sum
)
aggr.inla.nb <- aggregate_impute(
raw.inla.nb,
grouping = c("fYear", "fPeriod", "Year"),
fun = sum
)
aggr.better <- aggregate_impute(
raw.better,
grouping = c("fYear", "fPeriod", "Year"),
fun = sum
)
aggr.better.9 <- aggregate_impute(
raw.better.9,
grouping = c("fYear", "fPeriod", "Year"),
fun = sum
)Model the aggregated imputed dataset
Simple example
model_impute() will apply the model.fun to each imputation set. The covariates are defined in the rhs argument. So model.fun = lm in combination with rhs = "0 + fYear + fPeriod" is equivalent to lm(ImputedResponse ~ 0 + fYear + fPeriod, data = ImputedData).
The tricky part of this function the extractor argument. This is a user defined function which must have an argument called model. The function should return a data.frame or matrix with two columuns. The first column hold the estimate of a parameter of the model, the second column their standard error. Each row represents a parameter.
extractor.lm <- function(model){
summary(model)$coefficients[, c("Estimate", "Std. Error")]
}
model_impute(
aggr.lm,
model.fun = lm,
rhs = "0 + fYear + fPeriod",
extractor = extractor.lm
)
#> Estimate SE
#> fYear1 915.6794 143.7884
#> fYear2 1039.5812 144.8056
#> fYear3 1319.8559 137.0545
#> fYear4 1248.5774 136.5014
#> fYear5 1034.5023 142.1748
#> fYear6 1543.1000 160.2901
#> fYear7 1641.6491 135.1280
#> fYear8 1325.5156 141.3124
#> fYear9 1184.6325 131.2830
#> fYear10 1092.5166 146.4315
#> fPeriod2 394.9609 136.6279
#> fPeriod3 536.5071 136.9948
#> fPeriod4 354.1306 129.3417
#> fPeriod5 -125.4260 133.4333
#> fPeriod6 -421.7390 131.9829Return only the parameters associated with fYear
The extractor function requires more work from the user. This cost is compensated by the high degree of flexibility. The user doesn't depend on the predefined extractor functions. This is illustrated by the following examples.
extractor.lm2 <- function(model){
cf <- summary(model)$coefficients
cf[grepl("fYear", rownames(cf)), c("Estimate", "Std. Error")]
}
model_impute(
aggr.lm,
model.fun = lm,
rhs = "0 + fYear + fPeriod",
extractor = extractor.lm2
)
#> Estimate SE
#> fYear1 915.6794 143.7884
#> fYear2 1039.5812 144.8056
#> fYear3 1319.8559 137.0545
#> fYear4 1248.5774 136.5014
#> fYear5 1034.5023 142.1748
#> fYear6 1543.1000 160.2901
#> fYear7 1641.6491 135.1280
#> fYear8 1325.5156 141.3124
#> fYear9 1184.6325 131.2830
#> fYear10 1092.5166 146.4315Predict a smoother for predefined values
Note that we pass extra arguments to the extractor function through the extractor.args argument. This has to be a list. We recommend to use a named list to avoid confusion.
library(mgcv)
new.set <- expand.grid(
Year = pretty(dataset$Year, 20),
fPeriod = dataset$fPeriod[1]
)
extractor.lm3 <- function(model, newdata){
predictions <- predict(model, newdata = newdata, se.fit = TRUE)
cbind(
predictions$fit,
predictions$se.fit
)
}
model.gam <- model_impute(
aggr.lm,
model.fun = gam,
rhs = "s(Year) + fPeriod",
extractor = extractor.lm3,
extractor.args = list(newdata = new.set)
)
model.gam <- cbind(new.set, model.gam)
model.gam$LCL <- qnorm(0.025, mean = model.gam$Estimate, sd = model.gam$SE)
model.gam$UCL <- qnorm(0.975, mean = model.gam$Estimate, sd = model.gam$SE)
ggplot(model.gam, aes(x = Year, y = Estimate, ymin = LCL, ymax = UCL)) +
geom_ribbon(alpha = 0.1) +
geom_line()
Compare the results using different imputation models
Modelling aggregated data with glm.nb
Suppose that we are interested in a yearly relative index taking into account the average seasonal pattern. With a complete dataset (without missing values) we could model it like the example below: a generalised linear model with negative binomial distribution because we have counts that are likely overdispersed. fYear models the yearly index and fPeriod the average seasonal pattern. The 0 + part removes the intercept for the model. This simple trick gives direct estimates for the effect of fYear.
Only the effects of fYear are needed for the index. Therefore the extractor functions selects only the parameters who's row name contains fYear. In case that we want the first year to be used as a reference (index year 1 = 100%), we can subtract the estimate for this year from all estimates. The result are the indices relative to the first year, but still in the log scale. Note that the estimated index for year 1 will be 0 and (log(100%) = 0).
library(MASS)
aggr.complete <- aggregate(
dataset[, "Count", drop = FALSE],
dataset[, c("fYear", "fPeriod")],
FUN = sum
)
model.complete <- glm.nb(Count ~ 0 + fYear + fPeriod, data = aggr.complete)
summary(model.complete)
#>
#> Call:
#> glm.nb(formula = Count ~ 0 + fYear + fPeriod, data = aggr.complete,
#> init.theta = 31.78765186, link = log)
#>
#> Deviance Residuals:
#> Min 1Q Median 3Q Max
#> -2.71783 -0.47073 0.00869 0.47030 2.62272
#>
#> Coefficients:
#> Estimate Std. Error z value Pr(>|z|)
#> fYear1 6.72514 0.09016 74.590 < 2e-16 ***
#> fYear2 6.83730 0.09005 75.929 < 2e-16 ***
#> fYear3 7.06993 0.08985 78.684 < 2e-16 ***
#> fYear4 7.09550 0.08983 78.985 < 2e-16 ***
#> fYear5 6.94750 0.08995 77.237 < 2e-16 ***
#> fYear6 7.13813 0.08980 79.487 < 2e-16 ***
#> fYear7 7.23272 0.08974 80.597 < 2e-16 ***
#> fYear8 7.16345 0.08979 79.784 < 2e-16 ***
#> fYear9 7.05221 0.08987 78.475 < 2e-16 ***
#> fYear10 7.10071 0.08983 79.046 < 2e-16 ***
#> fPeriod2 0.36211 0.08027 4.511 6.44e-06 ***
#> fPeriod3 0.41944 0.08024 5.227 1.72e-07 ***
#> fPeriod4 0.34524 0.08027 4.301 1.70e-05 ***
#> fPeriod5 -0.02515 0.08045 -0.313 0.755
#> fPeriod6 -0.47112 0.08076 -5.833 5.44e-09 ***
#> ---
#> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
#>
#> (Dispersion parameter for Negative Binomial(31.7877) family taken to be 1)
#>
#> Null deviance: 541101.695 on 60 degrees of freedom
#> Residual deviance: 60.596 on 45 degrees of freedom
#> AIC: 852.28
#>
#> Number of Fisher Scoring iterations: 1
#>
#>
#> Theta: 31.79
#> Std. Err.: 5.96
#>
#> 2 x log-likelihood: -820.275
extractor.logindex <- function(model){
coef <- summary(model)$coefficients
log.index <- coef[grepl("fYear", rownames(coef)), c("Estimate", "Std. Error")]
log.index[, "Estimate"] <- log.index[, "Estimate"] - log.index["fYear1", "Estimate"]
log.index
} Now that we have a relevant model and extractor function, we can apply them to the aggregate imputed datasets.
model.glmm <- model_impute(
object = aggr.glmm,
model.fun = glm.nb,
rhs = "0 + fYear + fPeriod",
extractor = extractor.logindex
)
model.p <- model_impute(
object = aggr.inla.p,
model.fun = glm.nb,
rhs = "0 + fYear + fPeriod",
extractor = extractor.logindex
)
model.nb <- model_impute(
object = aggr.inla.nb,
model.fun = glm.nb,
rhs = "0 + fYear + fPeriod",
extractor = extractor.logindex
)
model.better <- model_impute(
object = aggr.better,
model.fun = glm.nb,
rhs = "0 + fYear + fPeriod",
extractor = extractor.logindex
)
model.complete <- extractor.logindex(model.complete)
colnames(model.complete) <- c("Estimate", "SE")
covar <- data.frame(
Year = sort(unique(dataset$Year))
)
# combine all results and add the Year
parameters <- rbind(
cbind(covar, model.glmm, Model = "glmm"),
cbind(covar, model.p, Model = "poisson"),
cbind(covar, model.nb, Model = "negative binomial"),
cbind(covar, model.better, Model = "better"),
cbind(covar, model.complete, Model = "complete")
)
# calculate the confidence intervals in the log scale
parameters$LCL <- qnorm(0.025, mean = parameters$Estimate, sd = parameters$SE)
parameters$UCL <- qnorm(0.975, mean = parameters$Estimate, sd = parameters$SE)
# convert estimate and confidence interval to the original scale
parameters[, c("Estimate", "LCL", "UCL")] <- exp(parameters[, c("Estimate", "LCL", "UCL")])
ggplot(parameters, aes(x = Year, y = Estimate, ymin = LCL, ymax = UCL)) +
geom_hline(yintercept = 1, linetype = 3) +
geom_ribbon(alpha = 0.2) +
geom_line() +
facet_wrap(~Model)
Modelling aggregated data with inla
The example below does something similar. Two things are different: 1) instead of glm.nb we use inla to model the imputed totals. 2) we model the seasonal pattern as a random intercept instead of a fixed effect.
extractor.inla <- function(model){
fe <- model$summary.fixed[, c("mean", "sd")]
log.index <- fe[grepl("fYear", rownames(fe)), ]
log.index[, "mean"] <- log.index[, "mean"] - log.index["fYear1", "mean"]
log.index
}
model.p <- model_impute(
object = aggr.glmm,
model.fun = inla,
rhs = "0 + fYear + f(fPeriod, model = 'iid')",
model.args = list(family = "nbinomial"),
extractor = extractor.inla
)
model.p <- model_impute(
object = aggr.inla.p,
model.fun = inla,
rhs = "0 + fYear + f(fPeriod, model = 'iid')",
model.args = list(family = "nbinomial"),
extractor = extractor.inla
)
model.nb <- model_impute(
object = aggr.inla.nb,
model.fun = inla,
rhs = "0 + fYear + f(fPeriod, model = 'iid')",
model.args = list(family = "nbinomial"),
extractor = extractor.inla
)
model.better <- model_impute(
object = aggr.better,
model.fun = inla,
rhs = "0 + fYear + f(fPeriod, model = 'iid')",
model.args = list(family = "nbinomial"),
extractor = extractor.inla
)
m.complete <- inla(
Count ~ 0 + fYear + f(fPeriod, model = "iid"),
data = aggr.complete,
family = "nbinomial"
)
model.complete <- extractor.inla(m.complete)
colnames(model.complete) <- c("Estimate", "SE")
# combine all results and add the Year
parameters <- rbind(
cbind(covar, model.glmm, Model = "glmm"),
cbind(covar, model.p, Model = "poisson"),
cbind(covar, model.nb, Model = "negative binomial"),
cbind(covar, model.better, Model = "better"),
cbind(covar, model.complete, Model = "complete")
)
# calculate the confidence intervals in the log scale
parameters$LCL <- qnorm(0.025, mean = parameters$Estimate, sd = parameters$SE)
parameters$UCL <- qnorm(0.975, mean = parameters$Estimate, sd = parameters$SE)
# convert estimate and confidence interval to the original scale
parameters[, c("Estimate", "LCL", "UCL")] <- exp(parameters[, c("Estimate", "LCL", "UCL")])
ggplot(parameters, aes(x = Year, y = Estimate, ymin = LCL, ymax = UCL)) +
geom_hline(yintercept = 1, linetype = 3) +
geom_ribbon(alpha = 0.2) +
geom_line() +
facet_wrap(~Model)