How To Find The Expected Value In Chi Square

Chi-squared tests of independence determine whether a relationship exists between two categorical variables. Do the values of i categorical variable depend on the value of the other categorical variable? If the ii variables are independent, knowing the value of one variable provides no data about the value of the other variable.

I've previously written nigh Pearson's chi-square exam of independence using a fun Star Trek instance. Are the compatible colors related to the chances of dying? Yous tin test the notion that the infamous blood-red shirts have a higher likelihood of dying. In that mail, I focus on the purpose of the test, applied information technology to this instance, and interpreted the results.

In this post, I'll accept a bit of a different approach. I'll show you the nuts and bolts of how to calculate the expected values, chi-foursquare value, and degrees of freedom. So you'll acquire how to utilize the chi-squared distribution in conjunction with the degrees of freedom to calculate the p-value.

I've used the same approach to explain how:

t-Tests piece of work.
F-tests work in one-manner ANOVA.

Of course, y'all'll usually merely let your statistical software perform all calculations. Nonetheless, understanding the underlying methodology helps y'all fully cover the assay.

Chi-Squared Example Dataset

For the Star Trek case, compatible color and status are the two categorical variables. The contingency table below shows the combination of variable values, frequencies, and percentages.

	Blue	Gilt	Red	Row total
Dead	7	9	24	40
Alive	129	46	215	390
Column total	136	55	239	N = 430
Column percentage (Dead)	5.15%	xvi.36%	x.04%

If uniform color and fatality rates are contained, we'd wait the column percentage in the lesser row to be roughly equal for all uniform colors. Later on all, if there is no connection between these variables, in that location's no reason for the fatality rates to be different.

Withal, our fatality rates are non equal. Gold has the highest fatality charge per unit at 16.36%, while Blue has the everyman at 5.15%. Red is in the middle at 10.04%. Does this inequality in our sample suggest that the fatality rates are different in the population? Does a relationship exist between uniform colour and fatalities?

Thank you to random sampling error, our sample's fatality rates don't exactly equal the population's rates. If the population rates are equal, nosotros'd likely even so see differences in our sample. So, the question becomes, after factoring in sampling error, are the fatality rates in our sample different enough to conclude that they're different in the population? In other words, we want to exist confident that the observed differences represent a relationship in the population rather than simply random fluctuations in the sample. That's where Pearson'due south chi-squared test for independence comes in!

The two hypotheses for the chi-squared test of independence are the post-obit:

Nada: The variables are independent. No relationship exists.
Culling: A human relationship between the variables exists.

Related posts: Hypothesis Testing Overview and Guide to Information Types

Computing the Expected Frequencies for the Chi-squared Examination of Independence

The chi-squared examination of independence compares our sample data in the contingency table to the distribution of values we'd expect if the null hypothesis is right. Let's construct the contingency table we'd expect to see if the naught hypothesis is true for our population.

For chi-squared tests, the term "expected frequencies" refers to the values we'd expect to encounter if the naught hypothesis is truthful. To summate the expected frequency for a specific combination of categorical variables (e.g., blue shirts who died), multiply the column total (Blue) by the row total (Expressionless), and separate past the sample size.

Row total 10 Column total / Sample Size = Expected value for ane tabular array cell

To summate the expected frequency for the Dead/Bluish cell in our dataset, do the following:

Find the row full for Dead (40)
Observe the column total for Blue (136)
Multiply those two values and split up by the sample size (430)

40 * 136 / 430 = 12.65

If the aught hypothesis is true, nosotros'd await to meet 12.65 fatalities for wearers of the Blue uniforms in our sample. Of course, we can't have a fraction of a death, only that doesn't bear upon the results.

Contingency Table with the Expected Values

I'll calculate the expected values for all 6 cells that stand for the combinations of the three uniform colors and ii statuses. I'll also include the observed values in our sample. Expected values are in parentheses.

	Blue	Gold	Crimson	Row full
Dead	7 (12.65)	9 (five.12)	24 (22.23)	xl
Alive	129 (123.35)	46 (49.88)	215 (216.77)	390
Column% (Expected Dead)	9.3%	ix.3%	9.3%

In this table, notice how the column percentages for the expected dead are all 9.three%. This equality occurs when the nada hypothesis is valid, which is the condition that the expected values represent.

Using this table, we tin can also compare the values nosotros observe in our sample to the frequencies nosotros'd expect if the null hypothesis that the variables are not related is correct.

For example, the observed frequency for Blueish/Dead is less than the expected value (7 < 12.65). In our sample, deaths of those in blue uniforms occurred less frequently than nosotros'd expect if the variables are independent. On the other paw, the observed frequency for Gold/Dead is greater than the expected value (nine > 5.12). Meanwhile, the observed frequency for Red/Dead approximately equals the expected value. This estimation matches what nosotros concluded by assessing the column percentages in the first contingency table.

Pearson'due south chi-squared examination works by mathematically comparison observed frequencies to the expected values and boiling all those differences down into ane number. Permit'southward see how it does that!

Related post: Using Contingency Tables to Calculate Probabilities

Calculating the Chi-Squared Statistic

About hypothesis tests calculate a test statistic. For example, t-tests use t-values and F-tests use F-values as their test statistics. These statistical tests compare your observed sample data to what you lot would wait if the goose egg hypothesis is true. The calculations reduce your sample data downward to 1 value that represents how dissimilar your data are from the null. Learn more than well-nigh Test Statistics.

For chi-squared tests, the test statistic is, unsurprisingly, chi-squared, or χⁱⁱ.

The chi-squared calculations involve a familiar concept in statistics—the sum of the squared differences between the observed and expected values. This concept is like to how regression models assess goodness-of-fit using the sum of the squared differences.

Hither's the formula for chi-squared.

Let's walk through information technology!

To summate the chi-squared statistic, take the difference betwixt a pair of observed (O) and expected values (E), square the departure, and carve up that squared difference by the expected value. Repeat this process for all cells in your contingency tabular array and sum those values. The resulting value is χⁱⁱ. We'll calculate information technology for our case information presently!

Of import Considerations nearly the Chi-Squared Statistic

Observe several important considerations about chi-squared values:

Goose egg represents the null hypothesis. If all your observed frequencies equal the expected frequencies exactly, the chi-squared value for each cell equals zero, and the overall chi-squared statistic equals zero. Zero indicates your sample information exactly friction match what you lot'd expect if the nothing hypothesis is right.

Squaring the differences ensures both that cell values must be non-negative and that larger differences are weighted more than smaller differences. A cell tin can never subtract from the chi-squared value.

Larger values represent a greater difference between your sample data and the null hypothesis. Chi-squared tests are one-tailed tests rather than the more familiar two-tailed tests. The exam determines whether the entire gear up of differences exceeds a significance threshold. If your χ² passes the limit, your results are statistically significant! You can reject the null hypothesis and conclude that the variables are dependent–a relationship exists.

Related post: One-tailed and Ii-tailed Hypothesis Tests

Calculating Chi-Squared for our Example Data

Let'due south calculate the chi-squared statistic for our instance information! To practice that, I'll rearrange the contingency table, making it easier to illustrate how to calculate the sum of the squared differences.

The first 2 columns point the combination of categorical variable values. The adjacent 2 are the observed and expected values that nosotros calculated before. The last column is the squared difference divided by the expected value for each row. The bottom line sums those values.

Our chi-squared test statistic is 6.17. Ok, groovy. What does that hateful? Larger values bespeak a more than substantial divergence between our observed information and the zero hypothesis. However, the number by itself is not useful because we don't know if it's unusually big. Nosotros need to identify it into a broader context to determine whether it is an extreme value.

Using the Chi-Squared Distribution to Test Hypotheses

One chi-squared test produces a single chi-squared value. However, imagine performing the following process.

First, assume the null hypothesis is valid for the population. At the population level, there is no relationship between the two categorical variables. Now, we'll repeat our study many times past drawing many random samples from this population using the same design and sample size. Adjacent, we perform the chi-squared test of independence on all the samples and plot the distribution of the chi-squared values. This distribution is known as a sampling distribution, which is a type of probability distribution.

If nosotros follow this procedure, we create a graph that displays the distribution of chi-squared values for a population where the naught hypothesis is truthful. We use sampling distributions to calculate probabilities for how unlikely our sample statistic is if the null hypothesis is correct. Chi-squared tests use the chi-square distribution.

Fortunately, we don't need to collect many random samples to create this graph! Statisticians understand the backdrop of chi-squared distributions so we can estimate the sampling distribution using the details of our design.

Our goal is to determine whether our sample chi-squared value is and then rare that it justifies rejecting the null hypothesis for the entire population. The chi-squared distribution provides the context for making that conclusion. We'll calculate the probability of obtaining a chi-squared value that is at least as loftier equally the value that our study found (6.17).

This probability has a name—the P-value! A low probability indicates that our sample data are unlikely when the null hypothesis is true.

Alternatively, you lot tin use a chi-foursquare table to determine whether our study's chi-foursquare exam statistic exceeds the critical value.

Related posts: Sampling Distributions, Understanding Probability Distributions and Interpreting P-values

Graphing the Chi-Squared Test Results for Our Case

For chi-squared tests, the degrees of liberty define the shape of the chi-squared distribution for a design. Chi-square tests use this distribution to calculate p-values. The graph below displays several chi-foursquare distributions with differing degrees of freedom.

For a tabular array with r rows and c columns, the method for calculating degrees of liberty for a chi-square test is (r-1) (c-1). For our example, nosotros have ii rows and three columns: (2-1) * (3-1) = 2 df.

Read my post about degrees of liberty to larn about this concept along with a more than intuitive mode of agreement degrees of freedom in chi-squared tests of independence.

Below is the chi-squared distribution for our study's design.

The distribution curve displays the likelihood of chi-squared values for a population where at that place is no human relationship betwixt uniform color and status at the population level. I shaded the region that corresponds to chi-foursquare values greater than or equal to our written report's value (6.17). When the null hypothesis is correct, chi-square values autumn in this area approximately 4.6% of the time, which is the p-value (0.046). With a significance level of 0.05, our sample data are unusual enough to reject the zero hypothesis.

The sample evidence suggests that a relationship between the variables exists in the population. While this test doesn't indicate cherry shirts have a higher chance of dying, there is something else going on with red shirts. Read my other post chi-squared to larn well-nigh that!

Pearson's chi-squared test for independence doesn't tell you the outcome size. To understand the strength of the human relationship, y'all'd demand to use something similar Cramér'south V, which is a measure of clan similar Pearson's correlation—except for categorical variables. That's the topic of a futurity post!