Quantile Normalization in R with the {TidyDensity} Package (2024)

In data analysis, especially when dealing with multiple samples or distributions, ensuring comparability and removing biases is crucial. One powerful technique for achieving this is quantile normalization. This method aligns the distributions of values across different samples, making them more similar in terms of their statistical properties.

Quantile normalization is a statistical method used to adjust the distributions of values in different datasets so that they have similar quantiles. This technique is particularly valuable when working with high-dimensional data, such as gene expression data or other omics datasets, where ensuring comparability across samples is essential.

The quantile_normalize() function is a new addition to the TidyDensity package, designed to simplify the process of quantile normalization within R. Let’s delve into how this function works and how you can integrate it into your data analysis pipeline.

The quantile_normalize() function takes a numeric matrix as input, where each column represents a sample. Here’s a breakdown of its usage:

quantile_normalize(.data, .return_tibble = FALSE)

• .data: A numeric matrix where each column corresponds to a sample that requires quantile normalization.
• .return_tibble: A logical value (default: FALSE) indicating whether the output should be returned as a tibble.

When you apply quantile_normalize() to your data, you receive a list object containing the following components:

1. Quantile-Normalized Matrix: A numeric matrix where each column has been quantile-normalized.
2. Row Means: The means of each row across the quantile-normalized matrix.
3. Sorted Data: The sorted values used during the quantile normalization process.
4. Ranked Indices: The indices of the sorted values.

The quantile_normalize() function performs quantile normalization through the following steps:

1. Sorting: Each column of the input matrix is sorted.
2. Row Mean Calculation: The mean of each row across the sorted columns is computed.
3. Normalization: Each column’s sorted values are replaced with the corresponding row means.
4. Unsorting: The columns are restored to their original order, ensuring that the quantile-normalized matrix maintains the same structure as the input.

Let’s demonstrate the usage of quantile_normalize() with a simple example:

# Load TidyDensitylibrary(TidyDensity)# Create a sample matrixset.seed(123)data <- matrix(rnorm(50), ncol = 4)head(data, 5)

 [,1] [,2] [,3] [,4][1,] -0.56047565 0.1106827 0.8377870 -0.3804710[2,] -0.23017749 -0.5558411 0.1533731 -0.6947070[3,] 1.55870831 1.7869131 -1.1381369 -0.2079173[4,] 0.07050839 0.4978505 1.2538149 -1.2653964[5,] 0.12928774 -1.9666172 0.4264642 2.1689560

# Apply quantile normalizationresult <- quantile_normalize(data)# Access the quantile-normalized matrixnormalized_matrix <- result[["normalized_data"]]# View the normalized matrixhead(normalized_matrix, 5)

 [,1] [,2] [,3] [,4][1,] -0.65451945 -0.3180877 0.84500772 -0.6545195[2,] -0.06327669 0.8450077 1.09078797 -0.9506544[3,] -1.40880292 -0.5235134 0.33150422 0.0863713[4,] 0.84500772 1.0907880 0.08637130 0.1991151[5,] -0.31808774 -0.6545195 -0.06327669 0.3315042

Let’s now look at the rest of the output components:

head(result[["row_means"]], 5)

[1] -1.4088029 -0.9506544 -0.6545195 -0.5235134 -0.3180877

head(result[["duplicated_ranks"]], 5)

 [,1] [,2] [,3] [,4][1,] 9 13 13 7[2,] 10 10 12 12[3,] 2 11 2 9[4,] 13 9 9 3[5,] 7 1 1 11

head(result[["duplicated_rank_row_indicies"]], 5)

NULL

head(result[["duplicated_rank_data"]], 5)

 [,1] [,2] [,3] [,4][1,] -0.23017749 -0.5558411 0.1533731 -0.6947070[2,] 0.07050839 0.4978505 1.2538149 -1.2653964[3,] 0.12928774 -1.9666172 0.4264642 2.1689560[4,] -0.68685285 -0.2179749 0.8215811 -0.4666554[5,] -0.44566197 -1.0260044 0.6886403 0.7799651

Now, lets take a look at the before and after quantile normalization summary:

as.data.frame(data) |> sapply(function(x) quantile(x, probs = seq(0, 1, 1/4)))

 V1 V2 V3 V40% -1.2650612 -1.9666172 -1.13813694 -1.2653963525% -0.4456620 -1.0260044 -0.06191171 -0.5604756550% 0.1292877 -0.5558411 0.55391765 -0.3804710075% 0.4609162 0.1106827 0.83778704 -0.08336907100% 1.7150650 1.7869131 1.25381492 2.16895597

as.data.frame(normalized_matrix) |> sapply(function(x) quantile(x, probs = seq(0, 1, 1/4)))

 V1 V2 V3 V40% -1.40880292 -1.40880292 -1.40880292 -1.4088029225% -0.52351344 -0.52351344 -0.52351344 -0.5235134450% -0.06327669 -0.06327669 -0.06327669 -0.0632766975% 0.33150422 0.33150422 0.33150422 0.33150422100% 1.73118725 1.73118725 1.73118725 1.73118725

Now let’s use the .return_tibble argument to return the output as a tibble:

quantile_normalize(data, .return_tibble = TRUE)

$normalized_data# A tibble: 13 × 4 V1 V2 V3 V4 <dbl> <dbl> <dbl> <dbl> 1 -0.655 -0.318 0.845 -0.655 2 -0.0633 0.845 1.09 -0.951 3 -1.41 -0.524 0.332 0.0864 4 0.845 1.09 0.0864 0.199 5 -0.318 -0.655 -0.0633 0.332 6 1.73 -0.0633 -0.133 -0.133 7 -0.524 -0.133 -0.524 -0.524 8 -0.133 1.73 1.73 1.73 9 0.332 0.0864 0.199 1.09 10 1.09 -0.951 -0.655 -0.318 11 -0.951 -1.41 -0.318 -1.41 12 0.199 0.199 -1.41 0.845 13 0.0864 0.332 -0.951 -0.0633$row_means# A tibble: 13 × 1 value <dbl> 1 -1.41 2 -0.951 3 -0.655 4 -0.524 5 -0.318 6 -0.133 7 -0.0633 8 0.0864 9 0.199 10 0.332 11 0.845 12 1.09 13 1.73 $duplicated_ranks# A tibble: 6 × 4 V1 V2 V3 V4 <int> <int> <int> <int>1 9 13 13 72 10 10 12 123 2 11 2 94 13 9 9 35 7 1 1 116 3 6 7 6$duplicated_rank_row_indices# A tibble: 6 × 1 row_index <int>1 22 43 54 95 106 12$duplicated_rank_data# A tibble: 6 × 4 V1 V2 V3 V4 <dbl> <dbl> <dbl> <dbl>1 -0.230 -0.556 0.153 -0.6952 0.0705 0.498 1.25 -1.27 3 0.129 -1.97 0.426 2.17 4 -0.687 -0.218 0.822 -0.4675 -0.446 -1.03 0.689 0.7806 0.360 -0.625 -0.0619 -0.560