# Predictive Mean Matching Imputation (Theory & Example in R)

**Predictive mean matching** is the new gold standard of imputation methodology!

Forget about all these outdated and crappy methods such as mean substitution or regression imputation…

In the following article, I’ll show you why predictive mean matching is **heavily outperforming** all the other imputation methods for missing data.

I have split the article in **several sections**. You may jump to the specific topic you are interested in:

## Predictive Mean Matching Explained

Before we can dive into the R programming example, let’s first define what predictive mean matching exactly is. The predictive mean matching algorithm can be split into **6 steps** (See also Paul Allison or Vink et al., 2014):

- Estimate a linear regression model:
- Use the variable we want to impute as \(Y\).
- Use a set of good predictors as \(X\) (Guidelines for the selection of \(X\) can be found in van Buuren, 2012, p. 128).
- Use only the observed values of \(X\) and \(Y\) to estimate the model.

- Draw randomly from the posterior predictive distribution of \(\hat{\beta}\) and produce a new set of coefficients \(\beta^{*}\).
- This bayesian step is needed for all multiple imputation methods to create some
**random variability**in the imputed values. - More details, e.g., in Yuan, 2005.

- This bayesian step is needed for all multiple imputation methods to create some
- Calculate predicted values for
**observed and missing**\(Y\). - Use \(\hat{\beta}\) to calculate predicted values for observed \(Y\).
- Use \(\beta^{*}\) to calculate predicted values for missing \(Y\).
- For each case where \(Y\) is missing, find the closest predicted values among cases where \(Y\) is observed.
**Example:**- \(Y_i\) is missing. Its predicted value is 10 (based on \(\beta^{*}\)).
- Our data consists of five observed cases of \(Y\) with the values 6, 3, 22, 7, and 12.
- In step 3, we predicted the values 7, 2, 20, 9, and 13 for these five observed cases (based on \(\hat{\beta}\)).
- The predictive mean matching algorithm selects the closest observed values (typically three cases) to our missing value \(Y_i\). Hence, the algorithm selects the values 7, 9, and 13 (the closest values to 10).
- Draw randomly one of these three close cases and impute the missing value \(Y_i\) with the observed value of this close case.
**Example continued:**- The algorithm draws randomly from 6, 7, and 12 (the observed values that correspond to the predicted values 7, 9, and 13).
- The algorithm chooses 12 and substitutes this value to \(Y_i\).
- In case of
**multiple imputation**(which I strongly advise), steps 1-5 are repeated several times. - Each repetition of steps 1-5 creates a new imputed data set.
- With multiple imputation, missing data is typically imputed 5 times.

## Predictive Mean Matching in R (Example)

I have to admit, the predictive mean matching algorithm is not so easy to understand, when you read it the first time. However, **in practice it’s easy to apply**!

In the following, I’m going to show you how to apply predictive mean matching in R.

Let’s create some random data for our example:

##### Example data ##### set.seed(918273) # Seed N <- 3000 # Sample size y <- round(runif(N, -10, 10)) # Target variable Y x1 <- y + round(runif(N, 0, 50)) # Auxiliary variable 1 x2 <- round(y + 0.25 * x1 + rnorm(N, - 3, 15)) # Auxiliary variable 2 x3 <- round(0.1 * x1 + rpois(N, 2)) # Auxiliary variable 3 x4 <- as.factor(round(0.02 * y + runif(N))) # Auxiliary variable 4 (categorical variable) y[rbinom(N, 1, 0.2) == 1] <- NA # Insert 20% missing data in Y data <- data.frame(y, x1, x2, x3, x4) # Store data in dataset head(data) # First 6 rows of our data |

##### Example data ##### set.seed(918273) # Seed N <- 3000 # Sample size y <- round(runif(N, -10, 10)) # Target variable Y x1 <- y + round(runif(N, 0, 50)) # Auxiliary variable 1 x2 <- round(y + 0.25 * x1 + rnorm(N, - 3, 15)) # Auxiliary variable 2 x3 <- round(0.1 * x1 + rpois(N, 2)) # Auxiliary variable 3 x4 <- as.factor(round(0.02 * y + runif(N))) # Auxiliary variable 4 (categorical variable) y[rbinom(N, 1, 0.2) == 1] <- NA # Insert 20% missing data in Y data <- data.frame(y, x1, x2, x3, x4) # Store data in dataset head(data) # First 6 rows of our data

Our data consists of five variables: A target variable \(Y\) with 20% missing values and four auxiliary variables \(X_1\), \(X_2\), \(X_3\), and \(X_4\). The first six rows of our example data look as follows:

**Table 1: First Six Rows of Our Data with Missing Values in Y**

Let’s move on to the **application of predictive mean matching** to our example data. First, we need to install the R package mice (detailed explanations can be found in van Buuren and Groothuis-Oudshoorn, 2011).

##### Install mice package in R##### install.packages("mice") # Install mice package library("mice") # Load mice package |

##### Install mice package in R##### install.packages("mice") # Install mice package library("mice") # Load mice package

With the following code, we can impute our missing data via single imputation.

The **function mice** is used to impute the data; m = 1 specifies single imputation; and method = “pmm” specifies predictive mean matching as imputation method.

The **function complete** stores the imputed data in a new data object (in our example, we call it data_imp_single).

##### Impute data via predictive mean matching (single imputation)##### imp_single <- mice(data, m = 1, method = "pmm") # Impute missing values data_imp_single <- complete(imp_single) # Store imputed data # head(data_imp_single) # First 6 rows of our imputed data |

##### Impute data via predictive mean matching (single imputation)##### imp_single <- mice(data, m = 1, method = "pmm") # Impute missing values data_imp_single <- complete(imp_single) # Store imputed data # head(data_imp_single) # First 6 rows of our imputed data

As I mentioned previously, single imputation is almost never a good idea, since it underestimates your standard errors. However, by **specifying m = 5 within the function mice**, you can easily switch to multiple imputation (i.e. 5 imputed data sets).

The specifications **“repeated” and include = TRUE** tell the complete function how to store our multiply imputed data in the object data_imp_multi_all. Have a look at the documentation of complete in order to explore different ways of storing your imputed data (by typing ?complete into your R Studio console).

With the remaining code we do some simple data cleaning and store the dataset in the object data_imp_multi.

##### Predictive mean matching (multiple imputation)##### imp_multi <- mice(data, m = 5, method = "pmm") # Impute missing values multiple times data_imp_multi_all <- complete(imp_multi, # Store multiply imputed data "repeated", include = TRUE) data_imp_multi <- data.frame( # Combine imputed Y and X1-X4 (for convenience) data_imp_multi_all[ , 1:6], data[, 2:5]) head(data_imp_multi) # First 6 rows of our multiply imputed data |

##### Predictive mean matching (multiple imputation)##### imp_multi <- mice(data, m = 5, method = "pmm") # Impute missing values multiple times data_imp_multi_all <- complete(imp_multi, # Store multiply imputed data "repeated", include = TRUE) data_imp_multi <- data.frame( # Combine imputed Y and X1-X4 (for convenience) data_imp_multi_all[ , 1:6], data[, 2:5]) head(data_imp_multi) # First 6 rows of our multiply imputed data

This is how our multiply imputed data set looks like:

**Table 2: First Six Rows with Multiply Imputed Values**

- \(Y.0\) is equal to the original \(Y\) with missing values.
- \(Y.1\)–\(Y.5\) are the five imputed versions of \(Y\).
- \(X1\)–\(X4\) are the four auxiliary variables that we used as predictors for the imputation.

By comparing rows 4 and 6, i.e. the rows with NAs, you can see the effect of multiple imputation. While \(Y.0\) contains missings, \(Y.1\)–\(Y.5\) are filled with imputed values. Note that the **imputed values differ** between the five imputed data sets (row 4: -9, -5, 4, -6, 1; row 6: -3, 9, 0, 1, -3).

You might say: “*Isn’t that bad?! It seems like the imputed values are very different to each other.*”

Yes, they are different – But that is exactly what we expect (and what we want) when we do multiple imputation. The difference between the imputed values reflects the **uncertainty of our imputation**. When we calculate standard errors (i.e. confidence intervals) for our point estimates, we include this uncertainty into the calculation, which leads to more correct standard errors.

## Predictive Mean Matching vs. Stochastic Regression Imputation

In a previous post, I discussed pros and cons of stochastic regression imputation. Regression imputation has many advantages, but I have also shown two serious drawbacks:

- Stochastic regression imputation might lead to
**implausible values**(e.g. negative incomes). - Stochastic regression imputation has
**problems with heteroscedastic data**.

In my previous post, I raved about predictive mean matching, as this method is supposed to fix all the problems of regression imputation.

Of course I do not want to deprive you of the proof! Therefore, I’m going to use exactly the same data as in the regression imputation example – but this time I’m also imputing via predictive mean matching.

### Plausibility of Imputed Values

Let’s have a closer look at drawback 1 – the **plausibility of imputed values**. Consider the following example data:

# Income data set.seed(91919) # Set seed N <- 1000 # Sample size income <- round(rnorm(N, 0, 500)) # Create some synthetic income data income[income < 0] <- income[income < 0] * (- 1) x1 <- income + rnorm(N, 1000, 1500) # Auxiliary variables x2 <- income + rnorm(N, - 5000, 2000) income[rbinom(N, 1, 0.1) == 1] <- NA # Create 10% missingness in income data_inc_miss <- data.frame(income, x1, x2) |

# Income data set.seed(91919) # Set seed N <- 1000 # Sample size income <- round(rnorm(N, 0, 500)) # Create some synthetic income data income[income < 0] <- income[income < 0] * (- 1) x1 <- income + rnorm(N, 1000, 1500) # Auxiliary variables x2 <- income + rnorm(N, - 5000, 2000) income[rbinom(N, 1, 0.1) == 1] <- NA # Create 10% missingness in income data_inc_miss <- data.frame(income, x1, x2)

We impute once via stochastic regression imputation…

imp_inc_sri <- mice(data_inc_miss, method = "norm.nob", m = 1) data_inc_sri <- complete(imp_inc_sri) |

imp_inc_sri <- mice(data_inc_miss, method = "norm.nob", m = 1) data_inc_sri <- complete(imp_inc_sri)

… and once via predictive mean matching (for the sake of simplicity, both times via single imputation).

imp_inc_pmm <- mice(data_inc_miss, method = "pmm", m = 1) data_inc_pmm <- complete(imp_inc_pmm) |

imp_inc_pmm <- mice(data_inc_miss, method = "pmm", m = 1) data_inc_pmm <- complete(imp_inc_pmm)

Comparing the plausibility of both imputations, we can see that stochastic regression imputation leads to **10 implausible values** (i.e. incomes below 0):

data_inc_sri$income[data_inc_sri$income < 0] # 10 values below 0 (implausible) # [1] -57.8859452 -58.9457388 -104.1099599 -147.9968026 -70.9192935 -280.2621111 # [2] -0.5694554 -91.3508690 -76.8628876 -468.5756297 |

data_inc_sri$income[data_inc_sri$income < 0] # 10 values below 0 (implausible) # [1] -57.8859452 -58.9457388 -104.1099599 -147.9968026 -70.9192935 -280.2621111 # [2] -0.5694554 -91.3508690 -76.8628876 -468.5756297

Predictive mean matching, on the other hand, does not lead to implausible values:

data_inc_pmm$income[data_inc_pmm$income < 0] # No values below 0 |

data_inc_pmm$income[data_inc_pmm$income < 0] # No values below 0

In respect to the plausibility of imputed values, predictive mean matching clearly outperforms stochastic regression imputation.

Reason: Predictive mean matching only **imputes values that were actually observed** for other units. The range of imputed values therefore always lies between the minimum and the maximum of the observed values.

### Imputation of Heteroscedastic Data

How well does predictive mean matching work for heteroscedastic data (i.e. drawback 2)? Let’s check it with the following example data:

# Heteroscedastic data set.seed(654654) # Set seed N <- 1:5000 # Sample size a <- 0 b <- 1 sigma2 <- N^2 eps <- rnorm(N, mean = 0, sd = sqrt(sigma2)) y <- a + b * N + eps # Heteroscedastic variable x <- 30 * N + rnorm(N[length(N)], 1000, 200) # Correlated variable y[rbinom(N[length(N)], 1, 0.3) == 1] <- NA # 30% missings data_het_miss <- data.frame(y, x) |

# Heteroscedastic data set.seed(654654) # Set seed N <- 1:5000 # Sample size a <- 0 b <- 1 sigma2 <- N^2 eps <- rnorm(N, mean = 0, sd = sqrt(sigma2)) y <- a + b * N + eps # Heteroscedastic variable x <- 30 * N + rnorm(N[length(N)], 1000, 200) # Correlated variable y[rbinom(N[length(N)], 1, 0.3) == 1] <- NA # 30% missings data_het_miss <- data.frame(y, x)

As before, we first impute by stochastic regression imputation…

imp_het_sri <- mice(data_het_miss, method = "norm.nob", m = 1) data_het_sri <- complete(imp_het_sri) |

imp_het_sri <- mice(data_het_miss, method = "norm.nob", m = 1) data_het_sri <- complete(imp_het_sri)

… and then by predictive mean matching:

imp_het_pmm <- mice(data_het_miss, method = "pmm", m = 1) data_het_pmm <- complete(imp_het_pmm) |

imp_het_pmm <- mice(data_het_miss, method = "pmm", m = 1) data_het_pmm <- complete(imp_het_pmm)

A good way for evaluating the two imputation methods for heteroscedastic data is a simple correlation plot:

par(mfrow = c(1, 2)) # Both plots in one graphic plot(x[!is.na(data_het_sri$y)], # Plot of observed values data_het_sri$y[!is.na(data_het_sri$y)], main = "", xlab = "X", ylab = "Y") points(x[is.na(y)], data_het_sri$y[is.na(y)], # Plot of missing values col = "red") title("Stochastic Regression Imputation", # Title of plot line = 0.5) abline(lm(y ~ x, data_het_sri), # Regression line col = "#1b98e0", lwd = 2.5) legend("topleft", # Legend c("Observed Values", "Imputed Values", "Regression Y ~ X"), pch = c(1, 1, NA), lty = c(NA, NA, 1), col = c("black", "red", "#1b98e0")) plot(x[!is.na(data_het_pmm$y)], # Plot of observed values data_het_pmm$y[!is.na(data_het_pmm$y)], main = "", xlab = "X", ylab = "Y") points(x[is.na(y)], data_het_pmm$y[is.na(y)], # Plot of missing values col = "red") title("Predictive Mean Matching", # Title of plot line = 0.5) abline(lm(y ~ x, data_het_pmm), col = "#1b98e0", lwd = 2.5) legend("topleft", # Legend c("Observed Values", "Imputed Values", "Regression Y ~ X"), pch = c(1, 1, NA), lty = c(NA, NA, 1), col = c("black", "red", "#1b98e0")) mtext("Imputation of Heteroscedastic Data", # Main title of plot side = 3, line = - 1.5, outer = TRUE, cex = 2) |

par(mfrow = c(1, 2)) # Both plots in one graphic plot(x[!is.na(data_het_sri$y)], # Plot of observed values data_het_sri$y[!is.na(data_het_sri$y)], main = "", xlab = "X", ylab = "Y") points(x[is.na(y)], data_het_sri$y[is.na(y)], # Plot of missing values col = "red") title("Stochastic Regression Imputation", # Title of plot line = 0.5) abline(lm(y ~ x, data_het_sri), # Regression line col = "#1b98e0", lwd = 2.5) legend("topleft", # Legend c("Observed Values", "Imputed Values", "Regression Y ~ X"), pch = c(1, 1, NA), lty = c(NA, NA, 1), col = c("black", "red", "#1b98e0")) plot(x[!is.na(data_het_pmm$y)], # Plot of observed values data_het_pmm$y[!is.na(data_het_pmm$y)], main = "", xlab = "X", ylab = "Y") points(x[is.na(y)], data_het_pmm$y[is.na(y)], # Plot of missing values col = "red") title("Predictive Mean Matching", # Title of plot line = 0.5) abline(lm(y ~ x, data_het_pmm), col = "#1b98e0", lwd = 2.5) legend("topleft", # Legend c("Observed Values", "Imputed Values", "Regression Y ~ X"), pch = c(1, 1, NA), lty = c(NA, NA, 1), col = c("black", "red", "#1b98e0")) mtext("Imputation of Heteroscedastic Data", # Main title of plot side = 3, line = - 1.5, outer = TRUE, cex = 2)

**Graphic 1: Stochastic Regression Imputation vs. Predictive Mean Matching for Heteroscedastic Data**

The plot makes it obvious:

Stochastic regression imputation overestimates the variance of smaller \(X\) and underestimates the variance of larger \(X\) – not good.

In contrast, predictive mean matching reflects the structure of our observed values almost perfectly – very good.

In other words: Even though stochastic regression imputation is much more popular among practitioners, there are **strong arguments for using predictive mean matching** instead!

## PMM via Multiple Imputation in Stata (Video)

So far, I have only shown you how to apply predictive mean matching in R. However, the imputation method is implemented in **many different software packages**, such as SAS or Stata. Personally, I’m not an expert for these software packages, but there are many good instructions out there.

For instance, one tutorial I can recommend was published by the Stata YouTube channel. The speaker, Chuck Huber, explains step by step how to handle missing data by **predictive mean matching in Stata**.

## I Would Like to Hear From You!

In this article, I have shown you why I’m such a big fan of predictive mean matching.

However, I would like to hear about **your opinion**!

Are you using predictive mean matching or do you prefer other imputation methods? Have I missed any pros or cons in my article?

Let me know in the comments (questions are also very welcome)!

## References

van Buuren, S. (2012). *Flexible Imputation of Missing Data*. Chapman & Hall/CRC Interdisciplinary Statistics.

van Buuren, S., and Groothuis-Oudshoorn, K. (2011). mice: Multivariate Imputation by Chained Equations in R. *Journal of Statistical Software*. 45(3).

Vink, G., Frank, L. E., Pannekoek, J., and van Buuren, S. (2014). Predictive mean matching imputation of semicontinuous variables. *Statistica Neerlandica*. 68(1). 61-90.

Yuan, C. Y. (2005). *Multiple Imputation for Missing Data: Concepts and New Development*. SAS Institute Inc., Rockville, MD.

## Appendix

The header graphic of this page shows a correlation plot of two highly correlated variables. The black line indicates the regression slope. The black stars indicate missing values, which are replaced by a close observed value.

set.seed(123) # Seed N <- 50000 # Sample size x <- rnorm(N) # X variable y <- x + rnorm(N, 0, 1.5) # Y variable par(bg = "#353436") # Background color par(mar = c(0, 0, 0, 0)) # Remove space around plot plot(x, y, # Plot observed cases col = "#1b98e0", pch = 16, cex = 0.0001) abline(lm(y ~ x), # Plot regression slope lwd = 5, col = "#353436") points(x[1:30], y[1:30], # Plot missing cases col = "#353436", pch = 8, lwd = 2, cex = 3) |

set.seed(123) # Seed N <- 50000 # Sample size x <- rnorm(N) # X variable y <- x + rnorm(N, 0, 1.5) # Y variable par(bg = "#353436") # Background color par(mar = c(0, 0, 0, 0)) # Remove space around plot plot(x, y, # Plot observed cases col = "#1b98e0", pch = 16, cex = 0.0001) abline(lm(y ~ x), # Plot regression slope lwd = 5, col = "#353436") points(x[1:30], y[1:30], # Plot missing cases col = "#353436", pch = 8, lwd = 2, cex = 3)

**4.9**/

**5**(

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## 4 Comments. Leave new

Please I help on how Gibbs sampling works in MICE

Hi Cynthia,

if you want to learn more about Gibbs sampling in MICE you could have a look at the book Flexible Imputation of Missing Data by Stef van Buuren. You can find a free PDF of the book here: http://pzs.dstu.dp.ua/DataMining/preprocessing/bibl/fimd.pdf

On page 86, he is explaining how Gibbs sampling is implemented in MICE.

Regards,

Joachim

hi,

According to your 6-step analysis, in the 3rd step,

–Use β∗ to calculate predicted values for missing Y,

I think you are saying that if the Y in ith sample (i.e., yi) is missing, then we could use x1i, x2i,…xni and β∗ to predicate the missing yi, so we need x1i,x2i…xni to predicate yi,

however, I find that , when I use mice in R, I still could get the predicated yi, even if the x1i, x2i,…xni are although missing. could you give me an explaination? thanks!

my data is like

Y X1 X2 X3 …. Xn

1 ob ob ob ob ob

2 ob ob ob ob ob

3 NA ob ob ob NA

…..

i NA NA NA NA NA (why the yi could be predicated?)

…

Hi Denghai,

this is because the MICE package is using a Multivariate Imputation by Chained Equations, i.e. missing values in predictors are also imputed. The following quote can be found in the help documentation of MICE:

“For predictors that are incomplete themselves, the most recently generated imputations are used to complete the predictors prior to imputation of the target column.”

Regards,

Joachim