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However, note that when, as in column two, values are tied in rank, they should instead be assigned the mean of the values corresponding to the ranks they would normally represent if they were different. In the case of column 2, they represent ranks iii and iv. So we assign the two tied rank iii
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These rank values are set aside to use later. Go back to the first set of data. Rearrange that first set of column values so each column is in order going lowest to highest value. (First column consists of 5,2,3,4. This is rearranged to 2,3,4,5. Second Column 4,1,4,2 is rearranged to 1,2,4,4, and
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identical in statistical properties. To quantile-normalize a test distribution to a reference distribution of the same length, sort the test distribution and sort the reference distribution. The highest entry in the test distribution then takes the value of the highest entry in the reference
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Min. :2.000 Min. :2.000 Min. :2.000 1st Qu.:2.750 1st Qu.:2.750 1st Qu.:2.750 Median :3.833 Median :4.083 Median :3.833 Mean :3.833 Mean :3.833 Mean :3.833 3rd Qu.:4.917 3rd Qu.:5.167 3rd Qu.:4.917 Max. :5.667 Max. :5.167 Max. :5.667
39:) of the distributions. So the highest value in all cases becomes the mean of the highest values, the second highest value becomes the mean of the second highest values, and so on.
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distribution, the next highest entry in the reference distribution, and so on, until the test distribution is a perturbation of the reference distribution.
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A 5 4 3 becomes A 2 1 3 B 2 1 4 becomes B 3 2 4 C 3 4 6 becomes C 4 4 6 D 4 2 8 becomes D 5 4 8
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The new values have the same distribution and can now be easily compared. Here are the summary statistics for each of the three columns:
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entries the mean of 4.67 for rank iii and 5.67 for rank iv, which is 5.17. And so we arrive at the following set of normalized values:
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normalize two or more distributions to each other, without a reference distribution, sort as before, then set to the average (usually,
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A (2 + 1 + 3)/3 = 2.00 = rank i B (3 + 2 + 4)/3 = 3.00 = rank ii C (4 + 4 + 6)/3 = 4.67 = rank iii D (5 + 4 + 8)/3 = 5.67 = rank iv
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column 3 consisting of 3,4,6,8 stays the same because it is already in order from lowest to highest value.) The result is:
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185:"A comparison of normalization methods for high density oligonucleotide array data based on variance and bias"
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A 5.67 5.17 2.00 B 2.00 2.00 3.00 C 3.00 5.17 4.67 D 4.67 3.00 5.67
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A 5.67 4.67 2.00 B 2.00 2.00 3.00 C 3.00 4.67 4.67 D 4.67 3.00 5.67
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Generally a reference distribution will be one of the standard statistical distributions such as the
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Amaratunga, D.; Cabrera, J. (2001). "Analysis of Data from Viral DNA Microchips".
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A iv iii i B i i ii C ii iii iii D iii ii iv
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A iv iii i B i i ii C ii iii iii D iii ii iv
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For each column determine a rank from lowest to highest and assign number i-iv
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of the distribution. However, any reference distribution can be used.
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A 5 4 3 B 2 1 4 C 3 4 6 D 4 2 8
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Bolstad, B. M.; Irizarry, R. A.; Astrand, M.; Speed, T. P. (2003).
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A quick illustration of such normalizing on a very small dataset:
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Now take the ranking order and substitute in new values
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150:Journal of the American Statistical Association
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255:Equivalence (mathematics)
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