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I want to test if two observations of nominal data accord to the same distribution. I am using the chi squared statistics to perform a chi squared homogeneity test and normalize the result with Cramer's $\phi $.

Unfortunately, all the examples for performing a chi squared homogeneity test I could find (e.g. here) perform the test with two one-dimensional observations. The example after the link above, for example, compares boys and girls dependent on their viewing preferences. This makes two observations in the form $[x_0, \ldots, x_n] $. However, I want to test observations in the form $[[x_0, \ldots, x_n], \ldots, [z_0, ..., z_m]] $. I don't know, if "multidimensional" is the right term in this context. I figured it might be.

How do you calculate $\chi^2 $ for determining the homogeneity of multidimensional observations? Please note that I know how to calculate $\chi^2 $ on a multi-row contingency table. Instead I want to know how to perform a for the homogeneity of two contingency tables.

The sum of the squared differences, divided by the expected value, yields the $\chi^2 $ value. Note, however, that the expected tables are not just the expected values from each individual table (quick proof for the first cell of table 1: $\frac{146 \cdot 165}{263} = 91.60 $). Instead their calculation seems to be somehow dependant on each other.

I want to test if two observations of nominal data accord to the same distribution. I am using the chi squared statistics to perform a chi squared homogeneity test and normalize the result with Cramer's $\phi$.

Unfortunately, all the examples for performing a chi squared homogeneity test I could find (e.g. here) perform the test with two one-dimensional observations. The example after the link above, for example, compares boys and girls dependent on their viewing preferences. This makes two observations in the form $[x_0, \ldots, x_n]$. However, I want to test observations in the form $[[x_0, \ldots, x_n], \ldots, [z_0, ..., z_m]]$. I don't know, if "multidimensional" is the right term in this context. I figured it might be.

How do you calculate $\chi^2$ for determining the homogeneity of multidimensional observations? Please note that I know how to calculate $\chi^2$ on a multi-row contingency table. Instead I want to know how to perform a for the homogeneity of two contingency tables.

The sum of the squared differences, divided by the expected value, yields the $\chi^2$ value. Note, however, that the expected tables are not just the expected values from each individual table (quick proof for the first cell of table 1: $\frac{146 \cdot 165}{263} = 91.60$). Instead their calculation seems to be somehow dependent on each other.

Example:

Table 1                                   Table 2
        outcome0 outcome1 outcome2 sum                outcome0 outcome1 outcome2
action0       95       31       20 146        action0       21       69       98
action1       70       29       18 117        action1       54       35       11
    sum      165       60       38 263

Question: Are these observations following the same distribution?

Scipy allows to determine the expected values (and $\chi^2 $) of multi-dimensional observations. The expected values are returned as the last element of its call:

Expected table 1                      Expected table 2
        outcome0 outcome1 outcome2            outcome0 outcome1 outcome2
action0    69.44    47.45    42.53    action0    76.04    51.96    46.58
action1    45.12    30.83    27.63    action1    49.40    33.76    30.26

The sum of the squared differences, divided by the expected value, yields the $\chi^2 $ value. Note, however, that the expected tables are not just the expected values from each individual table (quick proof for the first cell of table 1: $\frac{146 \cdot 165}{263} = 91.60 $). Instead their calculation seems to be somehow dependant on each other.

I'd like to know how the expected values depend on each other, so that I can calculate it myself.

I want to test if two observations of nominal data accord to the same distribution. I am using the chi squared statistics to perform a chi squared homogeneity test and normalize the result with Cramer's $\phi $.

Unfortunately, all the examples for performing a chi squared homogeneity test I could find (e.g. here) perform the test with two one-dimensional observations. The example after the link above, for example, compares boys and girls dependent on their viewing preferences. This makes two observations in the form $[x_0, \ldots, x_n] $. However, I want to test observations in the form $[[x_0, \ldots, x_n], \ldots, [z_0, ..., z_m]] $. I don't know, if "multidimensional" is the right term in this context. I figured it might be.

Could you explain to me, how a chi squared homogeneity test can be performed with multi-row observations? I know it works, because scipy provides unique values for two dimensional input lists.

import scipy.stats as sps
observation1 = [[95, 31, 20], [70, 29, 18]]
observation2 = [[21, 69, 98], [54, 35, 11]]
data = [observation1, observation2]
print sps.chi2_contingency(data)

The code above yields (159.18016188570166, 4.772222443744986e-31, 7, array([[[ 69.44008748, 47.45072645, 42.53205358], [ 45.11526642, 30.82876539, 27.63310068]], [[ 76.04085626, 51.96125177, 46.57502446], [ 49.40378984, 33.75925639, 30.25982128]]])) where the first value is chi squared. Flattening the observations yields different values, so there must be a difference.

How do you calculate $\chi^2 $ for determining the homogeneity of multidimensional observations?

I want to test if two observations of nominal data accord to the same distribution. I am using the chi squared statistics to perform a chi squared homogeneity test and normalize the result with Cramer's $\phi $.

Unfortunately, all the examples for performing a chi squared homogeneity test I could find (e.g. here) perform the test with two one-dimensional observations. The example after the link above, for example, compares boys and girls dependent on their viewing preferences. This makes two observations in the form $[x_0, \ldots, x_n] $. However, I want to test observations in the form $[[x_0, \ldots, x_n], \ldots, [z_0, ..., z_m]] $. I don't know, if "multidimensional" is the right term in this context. I figured it might be.

Could you explain to me, how a chi squared homogeneity test can be performed with multi-row observations? I know it works, because scipy provides unique values for two dimensional input lists.

import scipy.stats as sps
observation1 = [[95, 31, 20], [70, 29, 18]]
observation2 = [[21, 69, 98], [54, 35, 11]]
data = [observation1, observation2]
print sps.chi2_contingency(data)

The code above yields (159.18016188570166, 4.772222443744986e-31, 7, array([[[ 69.44008748, 47.45072645, 42.53205358], [ 45.11526642, 30.82876539, 27.63310068]], [[ 76.04085626, 51.96125177, 46.57502446], [ 49.40378984, 33.75925639, 30.25982128]]])) where the first value is chi squared. Flattening the observations yields different values, so there must be a difference.

How do you calculate $\chi^2 $ for determining the homogeneity of multidimensional observations? Please note that I know how to calculate $\chi^2 $ on a multi-row contingency table. Instead I want to know how to perform a for the homogeneity of two contingency tables.