Understanding Two Way Contingency Table Statistics

A two way contingency table is used to analyze the relationship between two categorical variables.

The table is typically presented in a grid format with the rows representing one variable and the columns representing the other.

Each cell in the table contains a frequency count of the observations that fall within that category.

For example, let's say we have a table that shows the relationship between the favorite color of students (red, blue, or green) and their favorite subject (math, science, or English).

Additional reading: What Are the Two Main Types of Generative Ai Models

Theoretical probabilities of getting an outcome (i,j) are denoted as πij. These probabilities can be tabulated in a table.

Marginal probabilities are also important in two-way contingency tables. The marginal probability of getting an outcome X=i is denoted as πi+, which is the sum of πij for all j. Similarly, the marginal probability of getting an outcome Y=j is denoted as π+ j, which is the sum of πij for all i.

The sum of all πij is equal to 1, denoted as π++.

Conditional probabilities can also be represented in a contingency table. The conditional probability of getting an outcome X=i given that the outcome Y=j is denoted as πi|j = πij / π+j.

Check this out: Contingency Table Probability

The chi-square test statistic is calculated using the formula X^2 = ∑(O_j - E_j)^2 / E_j, where O_j are the observed frequencies, E_j are the expected frequencies, and m is the total sample size.

This formula shows us that the larger the differences between the expected and observed frequencies, the larger the chi-square test statistic value will be.

The chi-square test statistic is based on the difference between the expected and observed frequencies.

The expected frequencies are calculated as E_j = m * π_0,j, where m is the total sample size and π_0,j is a value related to the null hypothesis.

The total sample size, m, is the sum of all observed frequencies, ∑O_j.

The larger the chi-square test statistic value, the more evidence we have against the null hypothesis.

Additional reading: Chi Square 2x2 Contingency Table Exmaple

To test the independence of two variables in a two-way contingency table, we need to consider the null and alternative hypotheses. The null hypothesis, denoted as $\mathcal{H}_0$, states that the variables are not associated, while the alternative hypothesis, denoted as $\mathcal{H}_1$, states that the variables are associated.

The Chi-Square test can be applied to test $\mathcal{H}_0$, where the $k$ mutually exclusive categories are taken to be the $IJ$ cross-classified possible pairings of $X$ and $Y$. Under $\mathcal{H}_0$, we have that $E_{ij} = n_{++} \pi_{ij} = n_{++} \pi_{i+} \pi_{+j}$.

We can reject $\mathcal{H}_0$ at significance level $\alpha$ if the Chi-Square test statistic $X^2$ is greater than or equal to the $\chi^{2,\star}_{(I-1)(J-1), \alpha}$ quantile of the $\chi^2_{(I-1)(J-1)}$-distribution.

For your interest: Chi-squared Contingency Table

The likelihood of a hypothesis is a crucial concept in hypothesis testing. It's a measure of how likely it is that the observed data would occur under the assumption of a particular hypothesis. In the context of hypothesis testing, the likelihood is often used to determine the probability of observing the data given a particular hypothesis.

The likelihood function is typically denoted as L(π), where π represents the parameters of the hypothesis. For example, in the case of a multinomial distribution, the likelihood function is given by L(π) = ∏(nij log(πij)), where nij is the number of observations in the ith row and jth column, and πij is the probability of observing an observation in that cell.

The likelihood function can be maximized using various methods, such as the method of Lagrange multipliers. This involves finding the values of the parameters that maximize the likelihood function subject to certain constraints.

In the case of a multinomial distribution, the likelihood function can be written as L(π) = ∏(nij log(πij)) = ∑(nij log(πij)) + ∑(nij log(πi+)) + ∑(nij log(π+j)), where πi+ and π+j are the marginal probabilities of the ith row and jth column, respectively.

The maximum likelihood estimates (MLEs) of the parameters can be obtained by maximizing the likelihood function. For example, in the case of a multinomial distribution, the MLEs of the parameters are given by πij = nij/n++, where n++ is the total number of observations.

Here are some examples of likelihood functions for different distributions:

Note that the likelihood function is not the same as the probability of the hypothesis. The likelihood function is a measure of the probability of the data given the hypothesis, whereas the probability of the hypothesis is a measure of the probability of the hypothesis being true.

A hypothesis test is a way to determine whether there's a significant relationship between two variables. The goal is to decide whether to accept or reject a null hypothesis, which states that there's no association between the variables.

The null hypothesis is a statement of no effect or no difference, while the alternative hypothesis suggests an effect or difference. For example, if we're testing the independence of two variables, the null hypothesis would be that the variables are not associated.

The Chi-Square test statistic is a measure of the difference between the expected and observed frequencies. The larger the differences, the larger the test statistic value, which provides evidence against the null hypothesis.

In a Chi-Square test of independence, the test statistic follows a χ² distribution with (I-1)(J-1) degrees of freedom, where I and J are the number of categories for each variable. The degrees of freedom come from the number of cells in the table minus the number of parameters being estimated.

Here's a summary of the hypotheses for a Chi-Square test of independence:

The null hypothesis is typically denoted as H0, and the alternative hypothesis as H1. The Chi-Square test statistic is used to determine whether to reject the null hypothesis.

When you reject the null hypothesis of independence in a two-way contingency table, you want to know which parts of the table are driving the rejection. One way to do this is by analyzing the residuals.

Residuals are the differences between the observed values and the expected values under the null hypothesis. Basic residuals are defined as e_ij = n_ij - E_ij, where e_ij is the residual for cell ij, n_ij is the observed frequency, and E_ij is the expected frequency.

You can examine the sign and magnitude of basic residuals, but detecting a systematic structure in the signs can be particularly interesting. However, evaluating the importance of a particular cell based on these residuals can be misleading.

To get a better sense of the contribution of each cell, you can use Pearson's residuals, which are defined as e_ij^P = (n_ij - E_ij) / sqrt(E_ij). These residuals directly measure the contribution towards the X^2 statistic.

For more insights, see: Contingency Table vs Frequency Table

Here's a brief comparison of basic residuals and Pearson's residuals:

Pearson's residuals are particularly useful because they are asymptotically normally distributed under certain conditions. However, they can be sensitive to the sampling scheme used.

To get residuals that are invariant to the sampling scheme and asymptotically normally distributed, you can use adjusted residuals, which are defined as e_ij^s = (e_ij^P) / sqrt(hat_v_ij). Here, hat_v_ij is an estimate of the variance of the residuals, and e_ij^s is the adjusted residual.

Adjusted residuals are commonly used for both Poisson and multinomial sampling schemes when assessing independence. They are a good choice when you want residuals that are robust to the sampling scheme used.

Deviance residuals are another type of residual that can be used to assess the contribution of each cell to the rejection of the null hypothesis. They are defined as e_ij^d = sign(n_ij - E_ij) sqrt(2 n_ij | log(n_ij / E_ij)).

Deviance residuals are equal to the square root of the cell components of the G^2 statistic. They can be useful when you want to get a sense of the contribution of each cell to the rejection of the null hypothesis.

The sample correlation coefficient, denoted as rXY, is a measure of the linear relationship between two variables. It can be calculated using the formula provided, which takes into account the frequency of observations in each category.

In the context of a contingency table, the sample correlation coefficient can be expressed in terms of the frequency counts and marginal totals. This allows for a more efficient calculation of the correlation coefficient.

One of the simplest measures of association is the phi coefficient, denoted as φ. It is applicable only to 2 × 2 contingency tables and varies from 0 to 1 or -1, depending on the strength and direction of the association.

The tetrachoric correlation coefficient is another measure of association that is specifically designed for 2 × 2 tables. It assumes that the underlying variables are normally distributed and provides a convenient measure of the Pearson product-moment correlation.

The tetrachoric correlation coefficient should not be confused with the Pearson correlation coefficient, which is calculated by assigning arbitrary values to the two levels of each variable. This is mathematically equivalent to the φ coefficient.

A crosstab shows the frequencies of two variables, with each cell representing the number of times a characteristic combination occurs. For example, in a crosstab with sex and handedness, the cell for "female and left-handed" might show that this combination occurs 4 times.

To interpret a crosstab, look at the marginal totals, which show the number of individuals in each category. In the example above, the marginal total for "male" is 52, indicating that 52 individuals are male. The grand total, or the total number of individuals represented in the contingency table, is also important, as it shows the total number of individuals in the study.

Here are some key things to look for when interpreting a crosstab:

* Look at the marginal totals to get a sense of the distribution of each variable.Check if there are any cells with zero frequency, which can indicate a problem with the data.

A different take: A Contingency Table Shows the Frequencies for

A contingency table is a powerful tool for displaying the relationship between two variables. This type of table is also known as a crosstab.

The numbers of individuals in each category are called cell frequencies. For example, in the 2 × 2 contingency table shown in Example 6, there are 43 males who are right-handed and 44 females who are right-handed.

The marginal totals are the sums of the cell frequencies in each row or column. In the same example, the marginal total for males is 52 and the marginal total for females is 48.

The grand total is the sum of all the cell frequencies in the table. For the 2 × 2 contingency table in Example 6, the grand total is 100.

The strength of the association between the two variables can be measured by the odds ratio. The population odds ratio can be estimated by the sample odds ratio. In Example 4, the sample odds ratio for cell (2,2) of Table 2.15 is calculated as 1.12037 and 2.359... for different types of sample odds ratios.

There are several statistical tests that can be used to assess the significance of the difference between the proportions in each column. These include Pearson's chi-squared test, the G-test, Fisher's exact test, Boschloo's test, and Barnard's test.

In a contingency table, if the proportions of individuals in the different columns vary significantly between rows (or vice versa), it is said that there is a contingency between the two variables. This means that the two variables are not independent.

Check this out: A Contingency Table Shows the Frequencies for Categorical Variables.

Sampling schemes are an essential part of understanding crosstabs and interpretation. A generic 2x2 contingency table is a great starting point, but the counts within it are variables to be sampled from a particular distribution.

The sampling distribution can have a significant impact on the results, so it's crucial to understand the implications of different schemes. Each scheme represents a different data collection mechanism, determined by the experiment being performed.

A multinomial sampling scheme occurs when the total sample size is fixed. This is represented by the probability vector π, which is the joint distribution of X and Y. The probability of a particular outcome is calculated using the formula P((N11, N12, N21, N22) = (n11, n12, n21, n22) | N++ = n++) = (n++)! / (∏i,j nij!) ∏i,j πij^nij.

In a multinomial sampling scheme, each patient belongs to one of the four compartments, and the setup of the experiment results in a multinomial sampling procedure. This is illustrated in Table 2.2, which shows the hypothetical data from a crossover trial.

There are different types of sampling schemes, including the product binomial sampling scheme. In this scheme, the marginal row sizes are fixed, and the counts within each row are sampled independently. The probability of a particular outcome is calculated using the binomial distribution.

Here are the different types of sampling schemes:

Understanding the sampling scheme used in an experiment is essential for accurate interpretation of the results. It's not just about the data itself, but also about how it was collected.

Crosstabs and Market Research are a perfect match. In fact, crosstabs are very often used in market research because they can be used to compare customers or products very well. For example, one of the following questions can be answered:

Crosstabs help us understand how different variables relate to each other, making it easier to identify patterns and trends in customer behavior. By analyzing crosstabs, market researchers can gain valuable insights that inform business decisions and drive growth.

In market research, crosstabs are used to compare customers or products very well. They help us understand how different variables relate to each other, making it easier to identify patterns and trends in customer behavior.

Continuity correction is a technique used to reduce the upwardly biased approximation error of the Pearson's X^2 test statistic.

This correction is known as Yates's correction, and it's applied by default in R when using the chisq.test function.

Yates suggested this correction in 1934 to account for the fact that discrete counts of categorical variables are approximated by the continuous chi-square distribution, which is more pronounced for small sample sizes.

The correction reduces the Pearson's X^2 statistic value, which in turn increases the corresponding p-value.

The continuity correction is applied by subtracting 0.5 from the absolute difference between observed and expected values before squaring them in the calculation of the X^2 statistic.

This correction is useful to know because it's a default setting in R, and understanding its purpose can help you interpret the results of your chi-square tests more accurately.

A fresh viewpoint: Create Contingency Table in R

Choosing the right scores for your analysis can be a crucial step in understanding your data. Different scoring systems can lead to different results.

Scores are a powerful tool in the analysis of ordinal contingency tables, and they can greatly impact the outcome of your analysis. The choice of scores can be particularly sensitive to the choice of scoring system when the margins of the table are highly unbalanced.

Some cells may have considerably larger frequencies than others, which can make the choice of scores even more critical. There is no direct way to measure the sensitivity of an analysis to the choice of scores used.

The most common scores used are: