Multiple Correspondence Analysis: A Political Example

MULTIPLE CORRESPONDENCE ANALYSIS

Correspondence analysis and multiple correspondence analysis are techniques from multivariate analysis. You can use the techniques to find clusters in a data set. Correspondence and multiple correspondence analysis are similar to principal component analysis, in that the analysis attempts to reduce the dimensions (number of columns or rows) of a set of intercorrelated variables so that the smaller dimensioned (number of columns or rows) variables explain most of the variation in the original variables. However, correspondence and multiple correspondence analysis are for categorical variables rather than the numerical variables of principal component analysis. Correspondence analysis was developed by Jean Paul Benzecri and measures similarities of patterns in contingency tables.

The Mathematics Behind Correspondence Analysis

Correspondence analysis is used in the analysis of just two categorical variables. In correspondence analysis, the reduced variables are found by applying singular value decomposition to a transformation of the contingency table created from the two original variables. The transformation replaces the value in each cell of the contingency table by the original value minus the product of the row total and the column total divided by the overall total, with the difference divided by the square root of the product of the row total and the column total. The resulting cells then contain the signed square roots of the terms used in the calculation of the chi square test for independence for a contingency table, divided by the square root of the overall total.

The Mathematics Behind Multiple Correspondence Analysis

For multiple correspondence analysis, more than two categorical variables are reduced. The reduced set of variables is found by applying correspondence analysis to one of two matrices. The first matrix is a matrix made up of observations in the rows and indicator variables in the columns, where the indicator variables take on the value one if the observation has a quality measured by the variable and zero if the individual does not. For example, say there are three variables, ‘gender’, ‘hair color’, and ‘skin tone’. Say that the categories for gender are ‘female’, ‘male’, ‘prefer not to answer’; for hair color, ‘red’, ‘blond’, ‘brown’, ‘black’; and for skin tone, ‘light’, ‘medium’, and ‘dark’, then, there would be three columns associated with gender and, for a given person, only one would contain a one, the others would contain zeros; there would be four columns associated with hair color and, for a given person, only one would contain a one, others would contain zeros; and there would be three columns associated with skin tone and, for a given person, only one would contain a one, the others would contain zeros.

The second type of matrix is a Burt table. A Burt table is multiple sort of contingency table. The contingency tables between the variables make up blocks of the matrix. From the example above, the first block is the contingency table of gender by gender, and is made of up of a diagonal matrix with the counts of males, females, and those who did not want to answer on the diagonal. The second block, going horizontally, is the contingency table of gender by hair color. The third block, going horizontally, is the contingency table of gender by skin tone. The second block, going vertically, is the contingency table of hair color by gender. The rest of the blocks are found similarly.

Plotting for Clustering

Once the singular value decomposition is done and the reduced variables are found, the variables are usually plotted to look for clustering of attributes. (For the above example, some of the attributes are brown hair, male, red hair, light skin tone, each of which would be one point on the plot.) Usually just the first two dimensions of the reduced matrix are plotted, though more dimensions can be plotted. The dimensions are ordered with respect to how much of the variation in the input matrix the dimension explains, so the first two dimensions are the dimensions that explain the most variation. With correspondence analysis, the reduced dimensions are with respect to the contingency table. Both the attributes of the rows and the attributes of the columns are plotted on the same plot. For multiple correspondence analysis, the reduced dimensions are with respect to the matrix used in the calculation. If one uses the indicator variable matrix, one can plot just the attributes for the columns or one can also plot labels for the rows on the same plot (or plot just the labels for the rows). If one uses the Burt table, one can only plot attributes for the columns. For multiple correspondence analysis, the plots for the columns are the same by either method, thought the scaling may be different.

Interpreting the Plot

The interpretation of the row and column variables in correspondence anaylsis is done separately. Row variables are compared as to level down columns and column variables are compared as to level across rows. However if a row variable is near a column variable in the plot, then both are represented at similar relative levels in the contingency table.

The interpretation of the relationship between the variables in the indicator or Burt table is a bit subtle. Within an attribute, the levels of the attribute are compared to each other on the plot. Between attributes, points that are close together are seen at similar levels.

An Example Using the Deficit by Political Party Data Set

Below is a multiple correspondence plot for which only the column reduction is plotted. We look at the size of the deficit (-) / surplus (+) using the political affiliation of the President and the controlling parties of the Senate and the House of Representatives. The size of the onbudget budget deficit (-) / surplus(+) as a percentage of gross domestic product for the years 1947 to 2008 was classified into four classes. The largest deficit over the years was -6.04 percent and the largest surplus was 4.11 percent. The class breaks were -6.2, -4, -2, 0, and 4.2, which gives four classes. (There was only one observation with a surplus greater than 2, so years of surplus were all classed together.) Each of the President, Senate, and House variables were classified into two classes, either Democrat or Republican. Since the budget is created in the year before the budget is in effect, the deficit (-) / surplus (+) percentages are associated with the political parties in power the year before the end of the budget year.

Multiple correspondence analysis plots of deficit / surplus as a percentage if GDP by party of president, senate, and house

The first principal axis appears to measure distance between the relative counts for the parties of the senate, and house. On the plot, the greatest distances on the first principal axis are between the Republican and Democratic senates and the Republican and Democratic houses. Looking at the tables below, the lowest counts were for the Republican senates and houses, which means the highest counts were for the Democratic senates and houses. For the deficit classes, classes (0,4.2] is the smallest class in the table and, like the Republican senates and houses, is to the left on the plot. Still there is not much difference between the deficit classes on the first principal axis (or the parties of the presidents).

The second principal axis appears to show how the differing levels of deficit (or surplus) are associated with the political parties of the presidents, senates, and houses. Looking at the senates and houses, there is not a lot of difference between the Democrats and the Republicans on the second principal axis, both cluster around the deficit class (-4,-2]. For the presidents, however, there is a great difference. Democratic presidents are near the deficit classes (-2,0] and (0,4.2] on the second principal axis while Republican presidents are between the deficit classes (-4,-2] and (-6.2,-4].

Below are two classifications of the data.

Deficit Classes
(-6.2,-4] (-4,-2] (-2,0] (0,4.2]
11 22 21 8
Political Parties
Party President Senate House
Democrat 27 42 48
Republican 35 20 14

In this example, we have treated the deficit, which is a numeric variable, as a categorical variable. In the post before this post, an example of principal component analysis, the same data is analyzed, with all of the variables treated as numeric rather than categorical.

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Principal Component Analysis: An Example

Principal component analysis is a technique from multivariate analysis for data sets with numeric (as opposed to categorical) variables. If you are interest in how variables in a data set relate or you are dealing with multicollinearity, you would probably find principle component analysis useful. According to the Wikipedia entry on principal component analysis, principal component analysis was invented by Karl Pearson in 1901. The purpose of principal component analysis is to reduce the number of dimensions of a set of measurements.

What Principal Component Analysis Involves

Say we have a data set of four measurements on 62 observations. The data set then has four dimensions in the columns. Using principal component analysis, we can find four vectors composed of weighted linear combinations of the original four measurements which contain the same information as in the original four measurement vectors, however, which combinations are linearly independent. (If a set of vectors are linearly independent, then the cross-product matrix of the vectors is diagonal.) Usually, the first few linear combinations explain most of the variation in the original data set. Say, in our example, that the first two linear combinations explain 85% of the variation in the data set. Then, we could approximate the original data using the first two vectors. The vectors are called either the first two principal components or the scores of the first two principal components. The weights used to find the scores are called the loadings of the principal components.

The principal components can be used in regression analysis. In the example above, the first two principal components could be used in place of the four original measurements to get rid of problems from multicollinearity. The principal components can also be used in cluster analysis. By plotting the first two scores against each other and labeling the points with the row labels, one can see if different observations cluster together. By plotting the first two loadings against each other, we can see if there is clustering within the measurement variables. The principal components can be used to see underlying relationships within the data. The loadings can have meaning in terms of understanding the data.

The Mathematics Behind Principal Component Analysis

Given a data set, the loadings are the eigenvectors of either the variance-covariance matrix or the correlation matrix of the data set. The two methods do not give the same result. Usually, the correlation matrix is used because in the correlation matrix the variables are normalized for size and spread. The scores are found by multiplying the original data – the normalized data if the correlation matrix is used – on the right by the eigenvectors. In the matrix of eigenvectors, the eigenvectors are in the columns.

Eigenvalues have an important role in principal component analysis. Eigenvectors are sorted by the sizes of the corresponding eigenvalues, from largest to smallest. An eigenvalue using a correlation matrix is proportional to the proportion of the variation in the data set explained by the principal component associated with the eigenvalue. For a full rank correlation matrix, in most programs, the sum of the eigenvalues equals the dimension of the correlation matrix, which equals the number of columns in the data set. The magnitudes of the elements of the eigenvectors are relative with respect to the magnitudes of the eigenvalues. The magnitudes of either, separately, are not determinate. In the above explanation, we assume the eigenvectors are normalized such that the inner product of each eigenvector equals one, which is usual.

An Example Using the Deficit by Political Party Data Set

We will use the data in The Deficit by Political Party Data Set* to demonstrate the power of principal component analysis as a clustering tool. We look at the size of the on budget deficit (-) (surplus (+)) as a percentage of the gross domestic product using the political affiliation of the President and the controlling parties of the Senate and the House of Representatives. Since the budget is created in the year before the budget is in effect, the deficit (-) (surplus (+)) percentages are associated with the political parties in power the year before the end of budget year. We will only look at years between the extreme deficits of the World War II years and the current extreme deficits.

In order to code the categorical values for political party numerically, we let Democrats take in the value of ‘1’ and Republicans take on the value of ‘-1’. Our data set has four variables and 62 observations. The variables are the deficit (-) (surplus (+)) as a percentage of the gross domestic product for the years 1947 to 2008, the political party of the President for the years 1946 to 2007, the political party controlling the Senate for the years 1946 to 2007, and the political party controlling the House for the years 1946 to 2007. Below are two plots of the rotation of the four variable dimensions projected onto the two dimensional plane using the first two eigenvectors, each multiplied by the square root of the eigenvector’s eigenvalue, one for the covariance method of calculation and the other for the correlation method.

First two eigenvectors plotted against each other - with and without normalization

Normally, in principal component analysis, a biplot is created containing plots of both the loadings and the scores. Because three of our variables are dichotomous, the plots of the scores are not very interesting and I have not included the plots. For the covariance method, the first two principal components account for 86% of the variation, and for the correlation method, the first two principal components account for 79% of the variation.

We can see that the parties of the Presidents and the deficits are strongly correlated with each other in either plot, while the Senates and Houses are more weakly correlated with each other in the correlation plot but strongly correlated in the covariance plot. The deficits and the parties of the Presidents do not seem to be strongly correlated with the parties of the Senates or Houses in either plot. In the covariance plot, the lengths of the vectors differ much more than in the correlation plot, since the measurements are not standardized. Basically, the two plots give the much same information.

Below is a table of the eigenvectors and eigenvalues for the correlation method.

Eigenvectors and Eigenvalues for the Correlation Method
variable eigenvector 1 eigenvector 2 eigenvector 3 eigenvector 4
deficit 0.380 0.583 0.712 0.090
president 0.378 0.587 -0.701 0.145
senate -0.539 0.487 -0.024 -0.687
house -0.650 0.279 0.029 0.706
eigenvalue 1.683 1.462 0.506 0.350

We can see from the table that the first eigenvector is a contrast between the deficit / President and the Senate / House.  Democrats have a positive value and Republicans a negative value in the normalized matrix of measurements. So, for the deficit and the party of the President, deficits that are positive in the normalized data are positively associated with Democratic presidents and deficits that are negative in the normalized data are positively associated with Republican presidents. For the House and Senate, Republican Senates are associated with Republican Houses and Democratic Senates are associated with Democratic Houses. Together, positive normalized budget deficits and Democratic presidents align with Republican Senates and Houses and contrast with Democratic Senates and Houses.

The second eigenvector lumps together, weakly, the deficit, President, Senate, and House. Here, Democrats, at all levels, are aligned with positive values of the normalized deficit and for Republicans, negative deficits are associated with Republicans at all levels.  Also, Democrats are aligned with Democrats and Republicans are aligned with Republicans.

The third eigenvector is mainly a contrast between the deficit and the party of the President. Positive normalized deficits are contrasted with Democratic presidents and aligned with Republican presidents.

The fourth eigenvector is mainly a contrast between the parties of the Senate and House. Democratic Senates are contrasted with Democratic Houses and aligned with Republican Houses and vice versa. The last two eigenvectors only account for a small portion of the variation.

*The Deficit by Political Party Data Set

The deficit by political party data set contains data on the total and on balance deficits, the gross domestic product, and the political parties controlling the presidency, the senate, and the house for the years 1940 to 2011. The data for the deficits and the gross domestic product are from the Office of Management and Budget and can be found at http://www.whitehouse.gov/omb/budget/Historicals. Before 1978, the budget year ran from July 1st to June 30th. From 1978 onward, the budget year ran from October 1st to September 31st. The tables contain a one quarter correction between the years 1977 and 1978, which I have ignored. The data for 2011 is estimated. The year with which the deficit and gross domestic product are associated is the year at the end of the budget year for which the deficit and gross domestic product are calculated.

The data for the controlling political parties of the president, senate, and house were taken from the website of Dave Manuel, http://www.davemanuel.com/history-of-deficits-and-surpluses-in-the-united-states.php, a good resource on the history of the deficit.

Some Tools for Politicians – Researchers – Pundits

Word clouds tell you what people are saying about candidates:

You can see that tweets containing Hillary or Clinton and Bernie or Sanders have more words than tweets containing just Cruz or just Trump.  Tweets are centered on Indiana, which was just having their primary.  Each word cloud was based on 100 tweets.

Word clouds of tweets containing Hillary, Clinton, Bernie, Sanders, Cruz, and Trump

You can use Box plots to provide visual information about a Data Set:

For example, You might find it is Useful to know How the different Mixes of Political Parties performed with respect to the Budget Deficit, using the Majority Party of the House and Senate and the Political Party of the President.

Box plots of deficit / surplus as a percentage if GDP by party of president, senate, and house

Bar graphs compare categories:

Comparing the different Tools governments use to Raise Money might be of interest to you for setting or evaluating policy on taxes.

iowa-2009 taxes by class

Principal component plots show how numeric data clusters:

You can see from the plot that the parties of the Senate and House cluster together – Republican  Houses and Senates are similar and Democratic Houses and Senates are similar – and the  party of the President clusters with the Budget Deficit. You could find this information very useful.  These plots do not include the Obama years or the years of WWII.

Principal component plots show how numeric data clusters

Multiple correspondence plots show how categorical data clusters:

Multiple correspondence analysis plots of deficit / surplus as a percentage if GDP by party of president, senate, and house

You can see that the House and Senate cluster together for both Republicans and Democrats while Democratic presidents cluster with small deficits than Republican presidents. This might be of interest for your campaign or analysis.

Log linear modeling allows comparisons between classes:

Expected Cell Counts Normalized to Percentages by Row
For the On Budget Deficit (-) (Surplus (+)) per GDP
by the Controlling Political Parties
of the Presidency, Senate, and House
Budget Deficit from 1947 to 2008Political Parties Lagged One Year
Rows Sum to 100 Except for Rounding
President Senate House (-6.2,4] (-4,-2] (-2,0] (0,4.2]
D D D 0% 30% 55% 15%
R
R D
R 0% 29% 42% 29%
R D D 33% 40% 21% 7%
R 26% 42% 17% 15%
R D 33% 40% 21% 7%
R 26% 42% 17% 15%

In the table, each row is the combination of the  Presidential, Senate, and House parties. You find in the rows the log linear modeled percentage of years in which the President, Senate, and House combination falls in each Budget Deficit class. The Party of the Senate does not affect the outcome and the Parties of the President and House do – Republican Houses do a bit better –  Democratic Presidents do much better – which might be of interest to you.

Odds ratios can be used to compare parties:

Iowa 2013 Legislature
Odds Legislator is Male
House Senate
Democrats 3 : 2 10 : 3
Republicans 16 : 2 10 : 2

You could be interested in the odds a legislator is a male, by party.
For both the House and Senate in Iowa, Democrats have lower odds of being male than Republicans. Depending on your political philosophy, this could be bad or good.