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Background

I have the following contingency table

+------+-----------+-----------+
|      |    X      |  Not X    |
+------+-----------+-----------+-------+
|  Y   |    60     |     20    |  80   |
+------+-----------+-----------+-------+
| notY |    15     |     40    |  55   |
+------+-----------+-----------+-------+
       |    75     |     60    |  135  |
       +-----------+-----------+-------+

where I've included the row and column marginals as well as the table total.

I would like to know if the rows and columns are independent or not. I could just use a Pearson's Chi-Square Test for Independence to get the p-value, which in R is

our_table <- matrix(c(60,20,15,40),nrow=2,byrow=T)
pvalue <- chisq.test(our_table, correct=F)$p.value

This gives a p-value of $4.17\times10^{-8}$, which seems reasonable since this table does not look independent.

My Goal

As an exercise, I attempted to calculate the exact p-value for my contingency table by looking at every possible contingency table constrained to have 135 observations and see how many are as or more extreme than mine. Note: I don't fix the row or column marginals, just as in the Chi-Squared Test for Independence.

Pearson's Chi-Square is just an approximation which was derived from a (generalized) likelihood ratio test[1] so I will be working with this test directly.

Likelihood Ratio

The marginal probabilities for the rows and columns are $$\pi_{i\cdot}=\sum_{j=1}^2\pi_{ij}$$ and $$\pi_{\cdot j}=\sum_{i=1}^2\pi_{ij}$$ respectively, where $\pi_{ij}$ is the probability of an observation being in the i-th row and j-th column.

The null hypothesis is that the rows and columns are independent, i.e. $$\text{H}_0:\;\; \pi_{ij}=\pi_{i\cdot}\pi_{\cdot j},\;\; \forall i,j\;\;,\tag{1}$$ and under the null hypothesis, the MLEs for the $\pi_{ij}$ are given by[2] $$\hat{\pi}_{ij}=\hat{\pi}_{i\cdot}\hat{\pi}_{\cdot j} =\frac{n_{i\cdot}}{n}\frac{n_{\cdot j}}{n} \tag{2}$$ where $n=135$ is the total number of observations, $n_{i\cdot}=(75,60)$ are the column marginal counts and $n_{\cdot j}=(80,55)$ are the row marginal counts.

Under the alternative hypothesis, the cell probabilities are free, but must all sum to 1. Under this hypothesis, the MLEs for the cell probabilities are: $$\tilde{\pi}_{ij}=\frac{n_{ij}}{n}$$

The likelihood ratio is then given by $$\Lambda = \frac{\frac{n!}{n_{11}!n_{12}!n_{21}!n_{22}!} \hat{\pi}_{11}^{n_{11}} \hat{\pi}_{12}^{n_{12}} \hat{\pi}_{21}^{n_{21}} \hat{\pi}_{22}^{n_{22}}} {\frac{n!}{n_{11}!n_{12}!n_{21}!n_{22}!} \tilde{\pi}_{11}^{n_{11}} \tilde{\pi}_{12}^{n_{12}} \tilde{\pi}_{21}^{n_{21}} \tilde{\pi}_{22}^{n_{22}}} = \prod_{i=1}^2\prod_{j=1}^2 \left(\frac{\hat{\pi}_{ij}}{\tilde{\pi}_{ij}}\right)^{n_{ij}} \;\;.$$ Simplifying and taking the logartihm, we have $$\ln \Lambda = \sum_{i=1}^2\sum_{j=1}^2 n_{ij} \ln\left[\frac{n_{i\cdot}n_{\cdot j}}{n_{ij}n}\right]\;\;.\tag{3}$$ and for our table, $\ln\Lambda = -15.53$.

p-value

(edited)

I have two nuisance parameters, $\pi_{1 \cdot}$ and $\pi_{\cdot 1}$. According to @Scortchi, the "the p-value is defined as the maximum of the sum of the probabilities of the more extreme tables over all values of both these nuisance parameters". In other words, $$\text{p-value}= \;\max_{\pi_{1 \cdot}, \pi_{\cdot 1}} \;P\left(\ln \Lambda < -15.53 | \pi_{1 \cdot}, \pi_{\cdot 1} \right)$$ where the number -15.53 was calculated above and $\ln \Lambda$ is given by Eq (3).

This can be rewritten as $$\text{p-value}=\,\max_{\pi_{1 \cdot}, \pi_{\cdot 1}}\;\left\{ \sum_{n_{11},n_{12},n_{21},n_{22}} \;\; \frac{n!}{n_{11}!n_{12}!n_{21}!n_{22}!} \pi_{11}^{n_{11}} \pi_{12}^{n_{12}} \pi_{21}^{n_{21}} \pi_{22}^{n_{22}} \right\}$$ where the summation is over all possible contingency tables which have $\ln \Lambda < -15.53$, and where the $\pi_{ij}$ are given by Eq (1).

The Python script below will calculate the above p-value, but only if you tell it what the nuisance parameters are.

Correct me if I'm wrong, but I don't think there's a way to analytically get the nuisance parameters. We could use a computer to numerically maximize the p-value. I wonder if that would be a convex optimization...

I didn't implement the optimization but I tested out a couple different choices of nuisance parameters and the highest p-value I got was from $\pi_{1 \cdot}=0.8, \pi_{\cdot 1}=0.5$, with p-value = $4.72\times10^{-8}$, which is in good agreement with our original p-value of $4.17\times10^{-8}$.


Script

(edited)

import numpy as np
from math import log, factorial, exp

our_table = np.array([[60,20],[15,40]])

# I found these particular choices for the 2 nuisance parameters to be a good
# choices to maximize p-value (implementing a proper numerical optimization would
# be better of course)
row_probs = np.array([0.8,0.2])
col_probs = np.array([0.5,0.5])
cell_probs = np.outer(row_probs, col_probs)

num_rows = our_table.shape[0]
num_cols = our_table.shape[1]
n = our_table.sum(axis=(0,1))


def binomial(N,k):
    '''N choose k'''
    f = factorial
    return f(N) / (f(k) * f(N-k))


def log_likelihood_ratio(table):
    '''Find log likelihood ratio under the setup of a chi-squared test for
    independence

    table:  array representing a contingency table
    '''
    row_marginals = table.sum(axis=1)
    col_marginals = table.sum(axis=0)
    n = table.sum(axis=(0,1)) # (same as n in the enclosing scope)

    # Now we take the double-summation in the formula for log likelihood ratio
    log_lik_ratio=0
    for row in range(num_rows):
        for col in range(num_cols):
            # if table[i,j] is zero, it is understand that this term is zero
            if table[row,col] != 0:
                log_lik_ratio += table[row,col] * log((row_marginals[row] * col_marginals[col]) \
                                / (n * table[row,col]))

    return log_lik_ratio


def prob_of_table(table, cell_probs):
    '''Given the particular settings of our nuisance parameters, calculate
    the probability of getting a given table, which is just given by a
    multinomial distribution'''
    n = table.sum(axis=(0,1)) # (same as n in the enclosing scope)
    num_rows = our_table.shape[0]
    num_cols = our_table.shape[1]

    # Now build up the log of the multinomial porbability one cell at a time
    # I use logarithms to make this computationally feasible
    log_prob = log(factorial(n))
    for row in range(num_rows):
        for col in range(num_cols):
            log_prob += table[row,col] * log(cell_probs[row,col]) \
                    - log(factorial(table[row,col]))

    return exp(log_prob)


our_log_lik_ratio = log_likelihood_ratio(our_table)

# Find out how many contingency table have equal or smaller likelihoods ratios
# variable to hold cumulative probability for tables more extreme than ours
# by probability, we mean the likelihood of a table under given nuisance parameters
cumul_prob = 0.0

# 3 nested for loops for all possible ways to split points into cells (3 degrees of freedom here)
for count11 in range(0, n+1):
    for count12 in range(0, n-count11+1):
        for count21 in range(0, n-count11-count12+1):
            count22 = n - count11 - count12 - count21
            cont_table = np.array([[count11, count12], [count21, count22]])
            #current log likelihood ratio
            cur_log_lik_ratio = log_likelihood_ratio(cont_table)
            print('__')
            if cur_log_lik_ratio <= our_log_lik_ratio:
                cumul_prob += prob_of_table(cont_table, cell_probs)
                print(cont_table, ' has log_likelihood of ', cur_log_lik_ratio, ' which is <= ', our_log_lik_ratio,'\n')


p_value = cumul_prob
print('For the table, ', our_table, ', the p-value is: ', p_value)

[1]: See Mathematical Statistics and Data Analysis, 3rd ed. by John Rice, Section 13.4.

[2]: Rice, p. 522.

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  • 1
    $\begingroup$ You're counting the more extreme tables rather than summing their probabilities under the null $\endgroup$ – Scortchi Sep 25 '14 at 9:42
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    $\begingroup$ Note also the null is composite. $\endgroup$ – Scortchi Sep 25 '14 at 13:43
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    $\begingroup$ @Garret: In the case of a fixed total but no fixed margins you have two nuisance parameters to consider. So the p-value is defined as the maximum of the sum of the probabilities of the more extreme tables over all values of both these nuisance parameters. $\endgroup$ – Scortchi Sep 26 '14 at 7:18
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    $\begingroup$ Garrett - now I've seen what you're trying to achieve - note that both the usual chi squared tests for an $m\times n$ table (homogeneity and independence) condition on both margins. $\endgroup$ – Glen_b Sep 26 '14 at 8:22
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    $\begingroup$ @Glen_b, are you referring to the fact that the Pearson's Chi-Squared statistic approximately follows a $\chi ^2$ distribution with df=1? If so, my understanding was that the test for independence only fixes the total. The df=1 comes from $\text{H}_0$ having 2 df, $\text{H}_A$ having 3 df, and so the $\chi ^2$ distribution has 3-2=1 df. $\endgroup$ – Garrett Sep 26 '14 at 8:52
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First, rather than counts of more (or as) extreme tables that need to be summed, it's their probabilities under the null hypothesis. Second the null hypothesis is composite, with $\pi_{1.}$ and $\pi_{.1}$ as nuisance parameters: the p-value is defined conservatively as the maximum of the sum of the probabilities of the more extreme tables over all values of both these nuisance parameters. There's no analytic solution, but the approximation of both the likelihood ratio and Pearson's score statistic to a chi-squared distribution with one degree of freedom is often close enough. If you want an exact test then, even for a multinomial sampling scheme, conditioning on either row or column totals (and thus removing one nuisance parameter) can be justified/urged on the grounds that either is ancillary; conditioning on both (and thus removing both nuisance parameters) can be justified/urged on the grounds that together they're approximately ancillary. In a Bayesian analysis you have priors for the nuisance parameters and simply integrate them out to get the posterior density of the odds ratio. Berger & Boos devised some tests that carry out the maximization over a confidence interval for the nuisance parameters, and still maintain the correct size.

simulated distribution of Pearson's test statistic under different conditioning schemes

# specify marginal totals
n1. = 80
n2. = 55
n.1 = 75
n.2 = n1. + n2. - n.1
n.. = n1. + n2.

# specify nuisance parameters
pi1. = 0.6
pi.1 = 0.5

# specify no. simulations to perform throughout
no.sims <- 100000

# define function to calculate Pearson's test statistic
pearson.test.stat <- function(x){chisq.test(matrix(x,c(2,2)), correct=F)$statistic}

# simulate conditioning on overall total
multinom.obs <- rmultinom(no.sims,n..,c(pi.1*pi1., (1-pi.1)*pi1., pi.1*(1-pi1.), (1-pi.1)*(1-pi1.)))
apply(multinom.obs , pearson.test.stat, MAR=2) -> multinom.sim.stats

# simulate conditioning on row totals
matrix(NA, 4, no.sims) -> binom.row.obs
binom.row.obs[1, ] <- rbinom(no.sims,n1., pi.1)
binom.row.obs[2, ] <- n1. - binom.row.obs[1, ]
binom.row.obs[3, ] <- rbinom(no.sims,n2., pi.1)
binom.row.obs[4, ] <- n2. - binom.row.obs[3, ]
apply(binom.row.obs, pearson.test.stat, MAR=2) -> binom.row.sim.stats

# simulate conditioning on column totals
matrix(NA, 4, no.sims) -> binom.col.obs
binom.col.obs[1, ] <- rbinom(no.sims,n.1, pi1.)
binom.col.obs[2, ] <- n.1 - binom.col.obs[1, ]
binom.col.obs[3, ] <- rbinom(no.sims,n.2, pi1.)
binom.col.obs[4, ] <- n.2 - binom.col.obs[3, ]
apply(binom.col.obs, pearson.test.stat, MAR=2) -> binom.col.sim.stats

# simulate conditioning on both row & column totals
matrix(NA, 4, no.sims) -> hypergeom.obs
rhyper(no.sims, n.1, n.2, n1.) -> hypergeom.obs[1, ]
n1. - hypergeom.obs[1, ] -> hypergeom.obs[2, ]
n.1 - hypergeom.obs[1, ] -> hypergeom.obs[3, ]
n.. - hypergeom.obs[1, ] - hypergeom.obs[2, ] - hypergeom.obs[3, ] -> hypergeom.obs[4, ]
apply(hypergeom.obs, pearson.test.stat, MAR=2) -> hypergeom.sim.stats

Berger & Boos (1994), "P-values maximized over a confidence set for the nuisance parameter", JASA, 89, 427

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