Handling missing values with R
Julie Josse
1) Regression with NA (quantitative) for ozone
First of all you will need to install the following packages
install.packages("VIM")
install.packages("naniar")
install.packages("missMDA")
install.packages("Amelia")
install.packages("mice")
install.packages("missForest")
install.packages("FactoMineR")
install.packages("Tidyverse")Air pollution is currently one of the most serious public health worries worldwide. Many epidemiological studies have proved the influence that some chemical compounds, such as sulphur dioxide (SO2), nitrogen dioxide (NO2), ozone (O3) can have on our health. Associations set up to monitor air quality are active all over the world to measure the concentration of these pollutants. They also keep a record of meteorological conditions such as temperature, cloud cover, wind, etc.
We have at our disposal 112 observations collected during the summer of 2001 in Rennes. The variables available are
- maxO3 (maximum daily ozone)
- maxO3v (maximum daily ozone the previous day)
- T12 (temperature at midday)
- T9
- T15 (Temp at 3pm)
- Vx12 (projection of the wind speed vector on the east-west axis at midday)
- Vx9 and Vx15 as well as the Nebulosity (cloud) Ne9, Ne12, Ne15
Here the final aim is to analyse the relationship between the maximum daily ozone (maxO3) level and the other meteorological variables. To do so we will perform regression to explain maxO3 in function of all the other variables. This data is incomplete (there are missing values). Indeed, it occurs frenquently to have machines that fail one day, leading to some information not recorded. We will therefore perform regression with missing values via multiple imputation.
- Importation of the data.
ozo <- read.table("ozoneNA.csv", header = TRUE,
sep=",", row.names = 1)
WindDirection <- ozo[, 12]
don <- ozo[, 1:11] #### keep the continuous variables
summary(don)## maxO3 T9 T12 T15
## Min. : 42.00 Min. :11.30 Min. :14.30 Min. :14.90
## 1st Qu.: 71.00 1st Qu.:16.00 1st Qu.:18.60 1st Qu.:18.90
## Median : 81.50 Median :17.70 Median :20.40 Median :21.40
## Mean : 91.24 Mean :18.22 Mean :21.46 Mean :22.41
## 3rd Qu.:108.25 3rd Qu.:19.90 3rd Qu.:23.60 3rd Qu.:25.65
## Max. :166.00 Max. :25.30 Max. :33.50 Max. :35.50
## NA's :16 NA's :37 NA's :33 NA's :37
## Ne9 Ne12 Ne15 Vx9
## Min. :0.000 Min. :0.000 Min. :0.00 Min. :-7.8785
## 1st Qu.:3.000 1st Qu.:4.000 1st Qu.:3.00 1st Qu.:-3.0000
## Median :5.000 Median :5.000 Median :5.00 Median :-0.8671
## Mean :4.987 Mean :4.986 Mean :4.60 Mean :-1.0958
## 3rd Qu.:7.000 3rd Qu.:7.000 3rd Qu.:6.25 3rd Qu.: 0.6919
## Max. :8.000 Max. :8.000 Max. :8.00 Max. : 5.1962
## NA's :34 NA's :42 NA's :32 NA's :18
## Vx12 Vx15 maxO3v
## Min. :-7.8785 Min. :-9.000 Min. : 42.00
## 1st Qu.:-3.6941 1st Qu.:-3.759 1st Qu.: 70.00
## Median :-1.9284 Median :-1.710 Median : 82.50
## Mean :-1.6853 Mean :-1.830 Mean : 89.39
## 3rd Qu.:-0.1302 3rd Qu.: 0.000 3rd Qu.:101.00
## Max. : 6.5778 Max. : 3.830 Max. :166.00
## NA's :10 NA's :21 NA's :12head(don)## maxO3 T9 T12 T15 Ne9 Ne12 Ne15 Vx9 Vx12 Vx15 maxO3v
## 20010601 87 15.6 18.5 NA 4 4 8 0.6946 -1.7101 -0.6946 84
## 20010602 82 NA NA NA 5 5 7 -4.3301 -4.0000 -3.0000 87
## 20010603 92 15.3 17.6 19.5 2 NA NA 2.9544 NA 0.5209 82
## 20010604 114 16.2 19.7 NA 1 1 0 NA 0.3473 -0.1736 92
## 20010605 94 NA 20.5 20.4 NA NA NA -0.5000 -2.9544 -4.3301 114
## 20010606 80 17.7 19.8 18.3 6 NA 7 -5.6382 -5.0000 -6.0000 94dim(don)## [1] 112 11- Load the libraries.
library(VIM)
library(FactoMineR)
library(missMDA)1.1) Descriptive statistics with missing values
Q1 When could it be a good idea to delete rows or columns with missing values to work with a complete data set? Could you do it here?
dim(na.omit(don))## [1] 13 11Deleting rows or columns is possible as long as there is enough data left and the missing values are of the MCAR type so that the sample is a subsample of the original data. We will obtain unbiased estimators but with more variance. Deleting observations with missing data for ozone data leads to a table with 13 rows.
First, we perfom some descriptive statistics (how many missing? how many variables, individuals with missing?) and try to inspect and vizualize the pattern of missing entries and get hints on the mechanism. For this purpose, we use the R package VIM (Visualization and Imputation of Missing Values - Mathias Templ) as well as Multiple Correspondence Analysis (FactoMineR package). You should install the package VIM, then you can check the documentation by executing
?VIMThe other package that can be used is the package naniar developped by Nick Tierney’s and which is based on ggplot. Naniar provides principled, tidy ways to summarise, visualise, and manipulate missing data with minimal deviations from the workflows in ggplot2 and tidy data.
library(naniar)
gg_miss_var(don)The function VIM aggr calculates and represents the number of missing entries in each variable and for certain combinations of variables (which tend to be missing simultaneously).
res<-summary(aggr(don, sortVar=TRUE))$combinations
##
## Variables sorted by number of missings:
## Variable Count
## Ne12 0.37500000
## T9 0.33035714
## T15 0.33035714
## Ne9 0.30357143
## T12 0.29464286
## Ne15 0.28571429
## Vx15 0.18750000
## Vx9 0.16071429
## maxO3 0.14285714
## maxO3v 0.10714286
## Vx12 0.08928571head(res[rev(order(res[,2])),])## Combinations Count Percent
## 1 0:0:0:0:0:0:0:0:0:0:0 13 11.607143
## 45 0:1:1:1:0:0:0:0:0:0:0 7 6.250000
## 10 0:0:0:0:0:1:0:0:0:0:0 5 4.464286
## 35 0:1:0:0:0:0:0:0:0:0:0 4 3.571429
## 41 0:1:0:0:1:1:1:0:0:0:0 3 2.678571
## 28 0:0:1:0:0:0:0:0:0:0:0 3 2.678571We can see that the combination which is the most frequent is the one where all the variables are observed (13 values). Then, the second one is the one where T9, T12 and T15 are simultaneously missing (7 rows) (1 is missing, 0 is observed - there is a 1 for the second, third and fourth variables). The graph on the right panel represents the pattern, with blue for observed and red for missing.
The VIM function matrixplot creates a matrix plot in which all cells of a data matrix are visualized by rectangles. Available data is coded according to a continuous color scheme (gray scale), while missing/imputed data is visualized by a clearly distinguishable color (red). If you use Rstudio the plot is not interactive (there are the warnings), but if you use R directly, you can click on a column of your choice: the rows are sorted (decreasing order) of the values of this column. This is useful to check if there is an association between the value of a variable and the missingness of another one.
matrixplot(don, sortby = 2)
##
## Click in a column to sort by the corresponding variable.
## To regain use of the VIM GUI and the R console, click outside the plot region.#Here the variable selected is variable 2. Q2 Do you observe any associations between the missing entries ? When values are missing on a variable does it correspond to small or large values on another one ?
We observe that the temperature variables T9, T12 and T15 tend to be missing together (probably indicating that thermometers failed) [as well as the Ne9, Ne12 and Ne15 variables.] We see more “red” values. We do not see more black or white values which should imply that when T9 is missing it would have corresponded to high or low values in another variable which should suggest MAR missing values for instance. Here everything points to MCAR values.
The VIM function marginplot creates a scatterplot with additional information on the missing values. If you plot the variables (x,y), the points with no missing values are represented as in a standard scatterplot. The points for which x (resp. y) is missing are represented in red along the y (resp. x) axis. In addition, boxplots of the x and y variables are represented along the axes with and without missing values (in red all variables x where y is missing, in blue all variables x where y is observed).
marginplot(don[,c("T9","maxO3")])
We can see that the distribution of T9 is the same when maxO3 is oberved and when maxO3 is missing. If the two boxplots (red and blue) would have been very different it would imply that when maxO3 is missing the values of T9 can be very high or very low which lead to suspect the MAR hypothesis.
R1 Create a categorical dataset with “o” when the value of the cell is observed and “m” when it is missing, and with the same row and column names as in the original data. Then, you can perform Multiple Correspondence Analysis with the MCA function of the FactoMineR package.
?MCAMCA can be seen as the counterpart of PCA for categorical data and here is used to study associations between missing and observed entries. MCA is a straightforwardly tool to visualise the missing data pattern even if the number of variable is large. It shows if missing values simultaneously occur in several variables or if missing values occur when some other variables are observed
data_miss <- data.frame(is.na(don))
data_miss <- apply(X=data_miss, FUN=function(x) if(x) "m" else "o", MARGIN=c(1,2))
# data_miss <- as_shadow(don) with the naniar package.
res.mca <- MCA(data_miss, graph = F)
plot(res.mca, invis = "ind", title = "MCA graph of the categories", cex = 0.5)
Other features from the package naniar
Summary with missing values
pct_miss(don) # percentage of missing value in the data.## [1] 23.7013n_miss(don) # number of missing values in the ## [1] 292n_complete(don) # without missing value## [1] 940n_miss(don$maxO3) # number of missing value for maxO3 ## [1] 16A matrix with missing and non missing:
as_shadow(don)## # A tibble: 112 x 11
## maxO3_NA T9_NA T12_NA T15_NA Ne9_NA Ne12_NA Ne15_NA Vx9_NA Vx12_NA
## <fct> <fct> <fct> <fct> <fct> <fct> <fct> <fct> <fct>
## 1 !NA !NA !NA NA !NA !NA !NA !NA !NA
## 2 !NA NA NA NA !NA !NA !NA !NA !NA
## 3 !NA !NA !NA !NA !NA NA NA !NA NA
## 4 !NA !NA !NA NA !NA !NA !NA NA !NA
## 5 !NA NA !NA !NA NA NA NA !NA !NA
## 6 !NA !NA !NA !NA !NA NA !NA !NA !NA
## 7 !NA !NA !NA !NA !NA !NA NA !NA !NA
## 8 !NA !NA !NA !NA !NA !NA NA !NA !NA
## 9 !NA !NA !NA !NA !NA NA !NA !NA !NA
## 10 !NA !NA NA !NA !NA NA NA !NA !NA
## # ... with 102 more rows, and 2 more variables: Vx15_NA <fct>,
## # maxO3v_NA <fct>The initial matrix concatenated with the matrix with missing and non missing:
bind_shadow(don)## # A tibble: 112 x 22
## maxO3 T9 T12 T15 Ne9 Ne12 Ne15 Vx9 Vx12 Vx15 maxO3v
## <int> <dbl> <dbl> <dbl> <int> <int> <int> <dbl> <dbl> <dbl> <int>
## 1 87 15.6 18.5 NA 4 4 8 0.695 -1.71 -0.695 84
## 2 82 NA NA NA 5 5 7 -4.33 -4 -3 87
## 3 92 15.3 17.6 19.5 2 NA NA 2.95 NA 0.521 82
## 4 114 16.2 19.7 NA 1 1 0 NA 0.347 -0.174 92
## 5 94 NA 20.5 20.4 NA NA NA -0.5 -2.95 -4.33 114
## 6 80 17.7 19.8 18.3 6 NA 7 -5.64 -5 -6 94
## 7 79 16.8 15.6 14.9 7 8 NA -4.33 -1.88 -3.76 80
## 8 79 14.9 17.5 18.9 5 5 NA 0 -1.04 -1.39 99
## 9 101 16.1 19.6 21.4 2 NA 4 -0.766 -1.03 -2.30 79
## 10 106 18.3 NA 22.9 5 NA NA 1.29 -2.30 -3.94 101
## # ... with 102 more rows, and 11 more variables: maxO3_NA <fct>,
## # T9_NA <fct>, T12_NA <fct>, T15_NA <fct>, Ne9_NA <fct>, Ne12_NA <fct>,
## # Ne15_NA <fct>, Vx9_NA <fct>, Vx12_NA <fct>, Vx15_NA <fct>,
## # maxO3v_NA <fct>Replacing values with NA: replace_with_na recodes various values with a missing value (NA). For example, we might know that all values of “N/A”, “N A”, and “Not Available”, or -99, or -1 are supposed to be missing.
Plot with missing values
Missing values in each variable per category of another variable the wind direction:
library(dplyr)
don %>%
group_by(ozo$WindDirection) %>%
miss_var_summary()## # A tibble: 44 x 5
## `ozo$WindDirection` variable n_miss pct_miss n_miss_cumsum
## <fct> <chr> <int> <dbl> <int>
## 1 North T15 12 38.7 39
## 2 North T9 11 35.5 16
## 3 North T12 11 35.5 27
## 4 North Ne12 9 29.0 55
## 5 North Ne9 7 22.6 46
## 6 North Ne15 7 22.6 62
## 7 North Vx9 6 19.4 68
## 8 North maxO3 5 16.1 5
## 9 North Vx15 5 16.1 75
## 10 North maxO3v 4 12.9 79
## # ... with 34 more rowsBelow with using bind_shadow function, we show the mean, sd, variance, and min and max values of T9 for when maximum daily ozone level is present, and when it is missing.
don %>%
bind_shadow() %>%
group_by(maxO3_NA) %>%
summarise_at(.vars = "T9",
.funs = c("mean", "sd", "var", "min", "max"),
na.rm = TRUE)## # A tibble: 2 x 6
## maxO3_NA mean sd var min max
## <fct> <dbl> <dbl> <dbl> <dbl> <dbl>
## 1 !NA 18.3 3.06 9.36 11.3 25.3
## 2 NA 17.9 2.61 6.83 13.3 21As VIM package has matrix plot, similarly naniar has the var_miss() function. It provides a summary of whether the data is missing (in black) or not. It also provides the percentage of missing values in each column.
vis_miss(don, sort_miss = TRUE) 
The function geom_miss_point() is close to the margin plot function of VIM but within the ggplot framework.
library(ggplot2)
ggplot(don,
aes(x = T9,
y = maxO3)) +
geom_miss_point() +
facet_wrap(~ozo$WindDirection)+
theme_dark()
Below, we can plot the distribution of Temperature at 9, plotting for values of temperature when Ozone is missing, and when it is not missing.
ggplot( bind_shadow(don),
aes(x = T9,
fill = maxO3_NA)) +
geom_density(alpha=0.5)
1.2) PCA with missing values
Then, before modeling the data, we perform a PCA with missing values to explore the correlation between variables. Use the R package missMDA dedicated to perform principal components methods with missing values and to impute data with PC methods.
- Perform PCA with missing values using the imputePCA functions, with the number of components determined by the estim_ncpPCA. Then plot the variables circle.
library(missMDA)?estim_ncpPCA
?imputePCAThe package missMDA allows the use of principal component methods for an incomplete data set. To achieve this goal in the case of PCA, the missing values are predicted using the iterative PCA algorithm for a predefined number of dimensions. Then, PCA is performed on the imputed data set. The single imputation step requires tuning the number of dimensions used to impute the data.
nb <- estim_ncpPCA(don,method.cv = "Kfold", verbose = FALSE) # estimate the number of components from incomplete data
#(available methods include GCV to approximate CV)
nb$ncp #2## [1] 2plot(0:5, nb$criterion, xlab = "nb dim", ylab = "MSEP")
res.comp <- imputePCA(don, ncp = nb$ncp) # iterativePCA algorithm
res.comp$completeObs[1:3,] # the imputed data set## maxO3 T9 T12 T15 Ne9 Ne12 Ne15 Vx9
## 20010601 87 15.6000 18.50000 20.47146 4 4.000000 8.000000 0.6946
## 20010602 82 18.5047 20.86986 21.79932 5 5.000000 7.000000 -4.3301
## 20010603 92 15.3000 17.60000 19.50000 2 3.984066 3.812104 2.9544
## Vx12 Vx15 maxO3v
## 20010601 -1.710100 -0.6946 84
## 20010602 -4.000000 -3.0000 87
## 20010603 1.950563 0.5209 82imp <- cbind.data.frame(res.comp$completeObs,WindDirection)
res.pca <- PCA(imp, quanti.sup = 1, quali.sup = 12, ncp = nb$ncp, graph=FALSE)
plot(res.pca, hab=12, lab="quali");
plot(res.pca, choix="var")
head(res.pca$ind$coord) #scores (principal components)## Dim.1 Dim.2
## 20010601 -0.6604580 -1.2048271
## 20010602 -1.2317545 1.0465411
## 20010603 0.7984643 -2.7299508
## 20010604 2.5423205 -1.7435774
## 20010605 -0.4047517 0.8406578
## 20010606 -2.6701824 1.6934864The incomplete data set can be imputed using the function imputePCA performing the iterative PCA algorithm, specifying the number of dimensions through the argument ncp=2. At convergence the algorithm provides both an estimation of the scores and loadings as well as a completed data set. The imputePCA function outputs the imputed data set. The completed data set is in the object completeObs. The imputePCA function also outputs the fitted matrix \(\hat X\) in the object fitted.
Q3 Could you guess how cross-validation is performed to select the number of components?
The cross-validation is performed with the Kfold methodFor the Kfold. A percentage pNA of missing values is inserted and predicted with a PCA model using ncp.min to ncp.max dimensions. This process is repeated nbsim times. The number of components which leads to the smallest MSEP (Mean Standard Error of Prediction) is retained.
Through the argument method.cv, the function estim_ncpPCA proposes several cross-validation procedures to choose this number. The default method is the generalised cross-validation method (method.cv=“gcv”). It consists in searching the number of dimensions which minimises the generalised cross-validation criterion, which can be seen as an approximation of the leave-one-out cross-validation criterion. The procedure is very fast, because it does not require adding explicitly missing values and predicting them for each cell of the data set. However, the number of dimensions minimising the criterion can sometimes be unobvious when several local minimum occur. In such a case, more computationally intensive methods, those performing explicit cross-validation, can be used, such as Kfold (method.cv=“Kfold”) or leave-one-out (method.cv=“loo”).
The Kfold cross-validation suggests to retain 2 dimensions for the imputation of the data set.
1.3) Multiple imputation
Generate multiple data sets
We perform multiple imputation either assuming 1) Joint Modeling (one joint probabilistic model for the variables all together) - We use the R package Amelia, which is by default consider Gaussian distribution 2) Condional Modeling (one model per variable) approach - We use the R package mice which by default consider one model of linear regression per variable 3) a PCA based model - We use the R package missMDA
For each approach we generate 100 imputed data sets.
library(Amelia)?ameliares.amelia <- amelia(don, m = 5) ## -- Imputation 1 --
##
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
## 41 42 43 44 45 46 47 48 49
##
## -- Imputation 2 --
##
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
## 41 42 43 44 45
##
## -- Imputation 3 --
##
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
## 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60
## 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80
## 81
##
## -- Imputation 4 --
##
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37
##
## -- Imputation 5 --
##
## 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
## 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40
## 41 42 43 44#names(res.amelia$imputations)
#res.amelia$imputations$imp1# the first imputed data setlibrary(mice)imp.mice <- mice(don, m = 100, defaultMethod = "norm.boot") # the variability of the parameters is obtained - Now generate 100 imputed data sets with the MIPCA method and 2 components. Store the result in a variable called res.MIPCA.
?MIPCA
?plot.MIPCAres.MIPCA <- MIPCA(don, ncp = 2, nboot = 100) # MI with PCA using 2 dimensions The function MIPCA gives as output the data set imputed by the iterative PCA algorithm (in res.imputePCA) and the other data sets generated by the MIPCA algorithm (in res.MI). The number of data sets generated by this algorithm is controlled by the nboot argument, equal to 100 by default. The other arguments of this function are the same as those for the imputePCA function.
Inspect the imputed values
Exploratory analysis is very important and even at this stage of the analysis.
We will inspect the imputed values created to know if the imputation method should require more investigation or if we can continue and analyze the data. A common practice consists in comparing the distribution of the imputed values and of the observed values. Check the compare.density function.
compare.density(res.amelia, var = "T12")
Q Do both distributions need to be close? Could the missing values differ from the observed ones both in spread and in location?
Note that a difference between these distributions does not mean that the model is unsuitable. Indeed, when the missing data mechanism is not MCAR, it could make sense to observe differences between the distribution of imputed values and the distribution of observed values. However, if differences occur, more investigations would be required to try to explain them.
The quality of imputation can also be assessed with cross-validation using the overimpute function. Each observed value is deleted and for each one 100 values are predicted (using the same MI method) and the mean and 90% confidence intervals are computed for these 100 values. Then, we inspect whether the observed value falls within the obtained interval. On the graph, the y=x line is plotted (where the imputations should fall if they were perfect), as well as the mean (dots) and intervals (lines) for each value. Around ninety percent of these confidence intervals should contain the y = x line, which means that the true observed value falls within this range. The color of the line (as coded in the legend) represents the fraction of missing observations in the pattern of missingness for that observation (ex: blue=0-2 missing entries).
overimpute(res.amelia, var = "maxO3")
- Comment the quality of the imputation.
We can also examine the variability by projecting as supplementary tables the imputed data sets on the PCA configuration (plot the results of MI with PCA).
plot(res.MIPCA,choice= "ind.supp")
plot(res.MIPCA,choice= "var")
The plots represent the projection of the individuals (top) and variables (bottom) of each imputed data set as supplementary elements onto the reference configuration obtained with the iterative PCA algorithm. For the individuals, a confidence area is constructed for each, and if one has no missing entries, its confidence area is restricted to a point. All the plots show that the variability across different imputations is small and a user can interpret the PCA results with confidence.
Perform regression
MI aims to apply a statistical method on an incomplete data set. We now apply a regression model on each imputed data set of the amelia method and MIPCA methods.
resamelia <- lapply(res.amelia$imputations, as.data.frame)
# A regression on each imputed dataset
fitamelia<-lapply(resamelia, lm,
formula="maxO3~ T9+T12+T15+Ne9+Ne12+Ne15+Vx9+Vx12+Vx15+maxO3v")
# fitamelia <- lapply(resamelia, with,
# lm(maxO3 ~ T9+T12+T15+Ne9+Ne12+Ne15+Vx9+Vx12+Vx15+maxO3v))imp.mice <- mice(don, m=100,defaultMethod="norm.boot") # the variability of the parameters is obtained
lm.mice.out <- with(imp.mice, lm(maxO3 ~ T9+T12+T15+Ne9+Ne12+Ne15+Vx9+Vx12+Vx15+maxO3v))res.MIPCA <- lapply(res.MIPCA$res.MI, as.data.frame)
fitMIPCA<-lapply(res.MIPCA,lm, formula="maxO3~T9+T12+T15+Ne9+Ne12+Ne15+Vx9+Vx12+Vx15+maxO3v")- Aggregate the results of Regression with Multiple Imputation according to Rubin’s rule for MI with amelia and with PCA with the pool function from the mice package
poolamelia<-pool(as.mira(fitamelia))
summary(poolamelia)## estimate std.error statistic df p.value
## (Intercept) 19.9420015 19.4901617 1.0231830 7.078897 0.32625157
## T9 1.0070129 4.2031210 0.2395869 3.161697 0.81466268
## T12 2.4735036 4.1893532 0.5904261 3.383789 0.56577415
## T15 -0.2473683 1.8644070 -0.1326794 5.808150 0.89662884
## Ne9 -2.5547392 1.1369961 -2.2469200 12.094691 0.04407947
## Ne12 -2.3052677 3.6854195 -0.6255102 3.750581 0.54325910
## Ne15 0.6871875 2.2297828 0.3081858 4.390821 0.76318418
## Vx9 0.4595960 2.2721088 0.2022773 4.249724 0.84306258
## Vx12 0.4610013 3.0938075 0.1490078 3.694674 0.88400324
## Vx15 0.4372643 1.4972484 0.2920453 6.784286 0.77520100
## maxO3v 0.3124507 0.1236261 2.5273857 6.256804 0.02641370poolMIPCA<-pool(as.mira(fitMIPCA))
summary(poolMIPCA)## estimate std.error statistic df p.value
## (Intercept) 12.2231485 18.40710622 0.6640451 61.82679 0.508868203
## T9 1.1201262 1.28505469 0.8716564 46.74630 0.386410286
## T12 1.6440663 1.03302947 1.5914999 50.14252 0.116055786
## T15 0.6877634 0.95080846 0.7233459 47.30412 0.471905654
## Ne9 -1.2553566 1.15348210 -1.0883191 57.43342 0.280230483
## Ne12 -1.6173184 1.41326639 -1.1443832 62.90186 0.256408058
## Ne15 0.3069931 1.17113893 0.2621321 59.25107 0.793998434
## Vx9 0.7566807 1.07550651 0.7035575 66.90143 0.484069200
## Vx12 0.9674316 1.19089898 0.8123541 62.56397 0.419374622
## Vx15 0.3065800 1.18920510 0.2578025 58.31779 0.797324535
## maxO3v 0.2524263 0.09034403 2.7940564 69.19377 0.006728215#pool.mice <- pool(lm.mice.out)
#summary(pool.mice)- Write a function that removes the variables with the largest pvalues step by step (each time a variable is removed the regression model is performed again) until all variables are significant.
don2 <- don
reg <- lm(maxO3 ~. , data = don2)
while(any(summary(reg)$coeff[-1, 4]>0.05)){
don2 <- don2[,!(colnames(don2)%in%names(which.max(summary(reg)$coeff[-1, 4])))]
reg <- lm(maxO3 ~. , data = don2)
}We combine the results and performed the regression with missing values
# Submodel to compare
fitMIPCA<-lapply(res.MIPCA,lm, formula="maxO3~ T12+Ne9+Vx12+maxO3v")
poolMIPCA<-pool(as.mira(fitMIPCA))
summary(poolMIPCA)## estimate std.error statistic df p.value
## (Intercept) 9.7948175 14.53718673 0.6737767 61.07589 5.024488e-01
## T12 2.9627531 0.61038949 4.8538731 67.15423 6.108510e-06
## Ne9 -1.9143282 1.06362944 -1.7998075 60.08612 7.576516e-02
## Vx12 1.7828147 0.72646723 2.4540882 72.60097 1.635728e-02
## maxO3v 0.3279477 0.08035646 4.0811612 77.85290 1.076020e-04#lm.mice.out <- with(imp.mice, lm(maxO3 ~ T12+Ne9+Vx12+maxO3v))
#pool.mice <- pool(lm.mice.out)
#summary(pool.mice)
fitamelia<-lapply(resamelia,lm, formula="maxO3~ T12+Ne9+Vx12+maxO3v")
poolamelia<-pool(as.mira(fitamelia))
summary(poolamelia) ## estimate std.error statistic df p.value
## (Intercept) 12.1506142 11.72848055 1.035992 18.58387 3.090306e-01
## T12 3.2340389 0.51325416 6.301048 23.78404 8.008505e-07
## Ne9 -3.2861788 0.95822637 -3.429439 10.79985 1.885150e-03
## Vx12 1.2487496 0.69891023 1.786710 15.37911 8.476774e-02
## maxO3v 0.3077342 0.06916046 4.449568 28.12829 1.235728e-041.4) Ecological example
Studies in community ecology aim to understand how and why individuals of different species co-occur in the same location at the same time. Hence, ecologists usually collect and store data on species distribution as tables containing the abundances of different species in several sampling sites. Additional information such as measures of environmental variables or species traits can also be recorded to examine the effects of abiotic features (characteristics, i.e. due to physico-chemical action and no biological action) and biotic features. Several projects compile data from preexisting databases. Due to the wide heterogeneity of measurement methods and research objectives, these huge data sets are often characterized by a high number of missing values. Hence, in addition to ecological questions, such data sets also present some important methodological and technical challenges for multivariate analysis. The GLOPNET data set contains 6 traits measured for 2494 plant species: LMA (leaf mass per area), LL (leaf lifes-pan), Amass (photosynthetic assimilation), Nmass (leaf nitrogen), Pmass (leaf phosphorus), Rmass (dark respiration rate). The last four variables are expressed per leaf dry mass. GLOPNET is a compilation of several existing data sets and thus contains a large proportion of missing values. All traits were log-normally distributed and log-transformed before analysis.
Ecolo <- read.csv("ecological.csv", header = TRUE, sep=";",dec=",")## Delete species with only missing values for contiuous variables
ind <- which(rowSums(is.na(Ecolo[,-1])) == 6)
biome <- Ecolo[-ind,1] ### Keep a categorical variable
Ecolo <- Ecolo[-ind,-1] ### Select continuous variables
dim(Ecolo)## [1] 2494 6## proportion of missing values
sum(is.na(Ecolo))/(nrow(Ecolo)*ncol(Ecolo)) # 55% of missing values## [1] 0.5338145## Delete species with missing values
dim(na.omit(Ecolo)) # only 72 remaining species!## [1] 72 653.38% of the entries in the GLOPNET data set are missing. Only 72 species have complete information for the 6 traits and the proportion of missing values varied between 4.97 % (LMA) to 89.01 % (Rmass).
# Visualize the pattern
library(VIM)
#aggr(Ecolo)
aggr(Ecolo,only.miss=TRUE,numbers=TRUE,sortVar=TRUE)
##
## Variables sorted by number of missings:
## Variable Count
## Rmass 0.89013633
## LL 0.69967923
## Pmass 0.69847634
## Amass 0.69125902
## Nmass 0.17361668
## LMA 0.04971933res <- summary(aggr(Ecolo,prop=TRUE,combined=TRUE))$combinations
#res[rev(order(res[,2])),]
mis.ind <- matrix("o",nrow=nrow(Ecolo),ncol=ncol(Ecolo))
mis.ind[is.na(Ecolo)] <- "m"
dimnames(mis.ind) <- dimnames(Ecolo)
library(FactoMineR)
resMCA <- MCA(mis.ind)
plot(resMCA,invis="ind",title="MCA graph of the categories")
### Impute the incomplete data set
library(missMDA)
### nb <- estim_ncpPCA(Ecolo,method.cv="Kfold",nbsim=100) ### Time consuming!
res.comp <- imputePCA(Ecolo,ncp=2)
#Perform a PCA on the completed data set
imp <- cbind.data.frame(res.comp$completeObs,biome)
res.pca <- PCA(imp,quali.sup=7,graph=FALSE)
plot(res.pca, hab=7, lab="quali")
plot(res.pca, hab=7, lab="quali",invisible="ind")
plot(res.pca, choix="var")
# Compare with PCA on the data imputed by the mean
PCA(Ecolo)
## **Results for the Principal Component Analysis (PCA)**
## The analysis was performed on 2494 individuals, described by 6 variables
## *The results are available in the following objects:
##
## name description
## 1 "$eig" "eigenvalues"
## 2 "$var" "results for the variables"
## 3 "$var$coord" "coord. for the variables"
## 4 "$var$cor" "correlations variables - dimensions"
## 5 "$var$cos2" "cos2 for the variables"
## 6 "$var$contrib" "contributions of the variables"
## 7 "$ind" "results for the individuals"
## 8 "$ind$coord" "coord. for the individuals"
## 9 "$ind$cos2" "cos2 for the individuals"
## 10 "$ind$contrib" "contributions of the individuals"
## 11 "$call" "summary statistics"
## 12 "$call$centre" "mean of the variables"
## 13 "$call$ecart.type" "standard error of the variables"
## 14 "$call$row.w" "weights for the individuals"
## 15 "$call$col.w" "weights for the variables"This first axis corresponding to the “leaf economic spectrum” separates species with potential for quick returns for investment with high values for Nmass, Amass, Rmass and Pmass and low values for LL and LMA (right part) from species with slow returns on the left part. Scores for the traits are very consistent between methods, to a lesser extent for the Mean. This representation can be used to add external information: grouping species by major biomes illustrates the universality of the leaf economic spectrum but also some specificities (e.g., Desert and Boreal forest mainly contain species of the quick-return end).
The graphical representation obtained by the Mean imputation highlights a very particular shape indicating that results are not reliable.
2) Categorical/mixed/multi-block data with missing values
2.1) Single imputation of categorical data with MCA/ MCA with missing values
We use the survey data set health concerning students’ health. 320 students answered 20 questions on their consumption of products (drugs, alcohol), on their psychological state and their sleeping condition. In addition, we have information regarding their gender, age and accommodation. The aim is to study the principal dimensions of variability of this data and to see if there are relationships between alcohol consumption and psychological state for instance. Then, after grouping individuals with the same profile, one can “label” them and see if there are relationships with the socio-economic questions.
Missing values are inserted to illustrate the methods.
library(FactoMineR)
health <- read.csv("sante.tex",sep=";",header=T)
dim(health)## [1] 327 20summary(health)## Pbsleep Asleep Fatigue Nightmare
## Never:136 Never : 66 Never : 41 Never :172
## Often: 70 QuiteOften: 86 QuiteOften:159 QuiteOften: 28
## Rare :121 Rare :142 Rare : 91 Rare :119
## Veryoften : 33 Veryoften : 36 Veryoften : 8
##
##
##
## Fatiguecste Insomnia Sex Age
## Never : 73 Never :250 Boy :120 18yrsorless:147
## QuiteOften: 98 QuiteOften: 16 Girls:207 19yrs : 99
## Rare :139 Rare : 54 20yrs : 49
## Veryoften : 17 Veryoften : 7 21yrsetplus: 32
##
##
##
## Liveparents Housing Absenteeism Tabac
## No :186 Campus : 31 Exceptionally:112 Frequent : 81
## Yes:141 Flat_alone : 59 Never :129 Never :208
## FurnishedRoom: 18 Often : 21 Occasional: 38
## Hostel : 10 Sometimes : 57
## Other : 8 Veryoften : 8
## Parents :141
## Roomate : 60
## Alcohol Druunk Cannabis Lonely Depress
## Frequent : 27 Never:122 Frequent : 22 Never:128 Never:119
## Never : 42 Yes :205 Never :165 Often: 55 Often: 54
## Ocasional:258 Occasional:140 Rare :144 Rare :154
##
##
##
##
## Desperate Aggressiv Hallucination
## Never:184 Never:182 Never :314
## Often: 52 Often: 31 RareorOften: 13
## Rare : 91 Rare :114
##
##
##
## healthNA <-health
healthNA[5:10,4:6] <- NA
healthNA[55:60,12:14] <- NAFirst, we can explore the pattern of missing using MCA (by default it codes a missing values as a new category):
res.mcaNA <- MCA(healthNA, quali.sup = c(7:11))
