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# Intro to the chi-square test

Understanding when to use a chi square test

# Chi-square Test

### When do you use a chi-squared test?

The chi-squared test is a statistical test that you can use to test if there are differences across groups of categorical data. There are two general settings where the chi-squared test is appropriate. In the first setting, you are interested in knowing whether two categorical variables are related. In the second setting, you only have one categorical variable and a specific hypothesis about the distribution of this variable across the categories.

#### Comparing two categorical variables

Example 1. Do males and females differ in their propensity to like videogames? Here, gender is the first categorical variable, and liking videogames (yes or no) is the second.

Example 2. Do the three products I currently manufacture malfunction at different rates? In this example, product is the first categorical variable and malfunctioning (yes or no) is the second.

#### One categorical variable and a specific hypothesis

Example 3. Your boss thinks that 10% of your website traffic occurs each day between Monday and Friday and that Saturday and Sunday receive 25% of your traffic per day. You are skeptical about this claim and want to test it using data on website traffic from last week.

Example 4. Your clothing supplier only stocks green, red, and blue shirts and claims that 20% of the shirts are green, 30% are blue, and 50% are red. He sends you a random selection of 100 shirts and you want to test if this claim is true using these shirts.

#### The null hypothesis

For the first two examples, we begin by hypothesizing no relationship between the two variables (this is called our Null Hypothesis). For the first example the null hypothesis is that males and females are equally likely to like videogames and in the second setting the null hypothesis is that all three products have the same failure rate. Using the chi-squared test, you will be able to say whether there is evidence in favour of the null hypothesis or against the null hypothesis.

For the third and fourth examples, the null hypothesis is that the stated distributions are correct. That is, in the first example the null hypothesis is that your website traffic occurs according to the way your boss told you, and in the second example the null hypothesis is that you received shirts in the proportions specified by your supplier. Again, we can use the chi-squared test to say whether there is evidence supporting the null hypothesis or against the null hypothesis.

#### How to calculate the chi-squared statistic: two variable setting

In this section, we will walk you through how to calculate the chi-squared statistic step-by-step, using hypothetical data from Example 2 that we introduced earlier. For example, suppose we observed the following:

Product A Product B Product C
Number that failed 25 40 25
Number that did not fail 75 160 125
Total 100 200 150

As with all hypothesis testing, the first step is to start by creating a null hypothesis that corresponds to no difference/relationship between the groups. If the null hypothesis is indeed true, then what is our best guess of what the data would look like? For this example, the null hypothesis would imply that the failure rates across products are the same, and our best guess at the common failure rate is denoted by the failure rate when combining all the data across the groups (which is a weighted average). The next step is to create a table of counts we’d expect using the common failure rate and see how far off the actual data is from this table. If each cell count is very different, then this will provide more evidence against the null hypothesis and our test statistic becomes large. On the other hand, if the numbers in the cells are very similar then this will translate into little evidence against our null hypothesis. Let’s see the test in action.

Remember: our null hypothesis is that all three products have the same likelihood of failure. To start thinking about whether this is likely to be true or not, let’s add some column percentages to the table and a totals column on the right:

Product A Product B Product C Total
Number that failed 25 (25%) 40 (20%) 25 (16.7%) 90 (20%)
Number that did not fail 75 (75%) 160 (80%) 125 (83.3%) 360 (80%)
Total 100 200 150 450

From our observed data, the failure rate is highest in Product A (25%) and lowest in Product C (16.7%), but it could be the case that these proportions only differed because not enough data was observed (i.e., “due to chance alone”). To see if this is the case, we pretend that the malfunction rate is the same across the three products. Our best guess at the malfunction rate under the assumption of no difference is the average malfunction rate calculated in the totals column (20%). Using this number we can calculate the proportion of each product expected to fail under a common failure rate of 20%:

Expected Frequencies (multiply the average failure rate by the total number of each product)

Product A Product B Product C Total
Number that failed 20%*100 20%*200 20%*150 90 (20%)
Number that did not fail 80%*100 80%*200 80%*150 360 (80%)
Total 100 200 150 450

Expected Frequencies (calculated)

Product A Product B Product C Total
Number that failed 20 40 30 90 (20%)
Number that did not fail 80 160 120 360 (80%)
Total 100 200 150 450

Now here is where the magic takes place! We need to compare, for each cell in the table, the number of products that were expected to fail to the number of products that actually did fail (the observed frequencies).

Observed vs. Expected Frequencies

Product A Product B Product C Total
Number that failed 25 vs. 20 40 vs. 40 25 vs. 30 90 (20%)
Number that did not fail 75 vs. 80 160 vs. 160 125 vs. 120 360 (80%)
Total 100 200 150 450

Intuitively, if the null hypothesis is true, then the expected frequencies should be very close to the observed frequencies for all cells in the table. The chi-squared test uses the magnitudes of the differences between the observed and expected frequencies as part of the calculation. To perform the test, we perform calculations to create the chi-squared test statistic and use a table to decide if the statistic offers evidence against the null hypothesis. We’ll present the test statistic here, and then spend a little time breaking down what it all means. Our test statistic, denoted by $\chi^2$ is

where $E_i$ denotes the expected frequency for cell $i$, and $O_i$ denotes the observed frequency for cell $i$. The larger the $\chi^2$ value, the more evidence there is against the null hypothesis.

So what does it all mean? Remember, we take the difference between the expected and observed frequencies to begin to quantify how different each observed frequency is from what we would expect under the null hypothesis. Now, the reason we take the square of these differences is because we will eventually be summing up all the differences, and we need to give equal weight to a negative difference (e.g., 20-25=-5) as we do to a positive difference (30-25=5) since both of these differences are of the same absolute magnitude. We then divide this squared difference by the expected frequency so that a difference of 5 units is counted as a big difference when the expected frequency is 10, but this same absolute difference is counted as a small relative difference when the expected frequency is 1000.

Using our example data, the calculation looks like this:

This table shows each cell’s contribution to the chi-squared statistic. The cells with the highest contribution to the statistic had observed values that were the most different from what we’d expect under the null hypothesis.

Product A Product B Product C
Number that failed 1.25 0 0.8333
Number that did not fail 0.3125 0 0.2083
Sum across all cells: 2.64

The last step is to determine if the calculated chi-squared value of 2.64 is large enough to provide evidence against the null hypothesis. To do that, we compare this value to a statistical distribution known as the chi-squared distribution. We make this comparison because if the null hypothesis is true, then the observed test statistic, 2.64, should be drawn from a chi-squared distribution. So if we compared 2.64 with the chi-squared distribution, it would be reasonable to say that a random draw from the distribution could equal 2.64. The chi-squared distribution also requires that you provide the corresponding “degrees of freedom”. This is equal $(r-1)\times(c-1)$, where $r$ is the number of rows in the table, and $c$ is the number of columns in the table. Thus, for this example $r=3$ and $c=3$ so that $df=(3-1)X(2-1)=2$. We can ask Plotly what is the chance of observing a chi-squared statistic larger than or equal to 2.64 in the situation where there is no difference between the products. This probability (know as the p-value) is 26.7%, meaning that it is likely that you would observe this data under the null hypothesis. Thus, we would say that there is no evidence against the null hypothesis that the failure rates between the products are the same. If the probability had been much smaller, say either 5% or 10%, then there would have been more evidence against the null hypothesis. By convention, people use cut-offs of 1%, 5% or 10% to denote enough evidence to reject the null hypothesis in favour of the alternative hypothesis. For example, if we calculated a probability of 4.3% (rather than 26.7%) we could have then concluded that there is enough evidence against the null hypothesis and evidence supporting an alternative hypothesis that the failure rates between the three products differ.

#### How to calculate the chi-squared statistic: one variable setting

In this section, we will walk you through how to calculate the chi-squared statistic step-by-step, using hypothetical data from Example 3 that we introduced earlier. For example, suppose we observed the following number of visitors to your website, and recall your boss’s hypothesis about the proportion of website traffic throughout the week:

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Boss’s hypothesis 25% 10% 10% 10% 10% 10% 25%
Number of unique visitors 750 688 1200 1100 300 313 650

We will use the chi-squared test to determine if the boss’s hypothesis regarding traffic website is in-line with what we can observe from the data. First let’s add a totals column to the right and also include the row percentage for each day of the week:

Sunday Monday Tuesday Wednesday Thursday Friday Saturday Total
Boss’s hypothesis 25% 10% 10% 10% 10% 10% 25% 100%
Number of unique visitors 750 (15.0%) 688 (13.8%) 1200 (24.0%) 1100 (22.0%) 300 (6.0%) 313 (6.3%) 650 (13%) 5001

If you compare the hypothetical proportions to the proportions based on the observed data, it doesn’t look like your boss is correct - most of the website traffic was observed mid-week (on Tuesday and Wednesday), not on the weekend like your boss hypothesized. To perform a chi-squared test in this setting, the next step is to calculate what number of visitors we would have expected each day, had your boss been correct. To do that, we simply multiply the boss’s daily expected proportions by the total number of visitors (5001) and add this information to our table:

Expected Frequencies (multiply the boss’s hypothetical proportions by the total number of visitors)

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Boss’s hypothesis 25% 10% 10% 10% 10% 10% 25%
Number of unique visitors 750 688 1200 1100 300 313 650 5001
Expected number of visitors 25%*5001 10%*5001 10%*5001 10%*5001 10%*5001 10%*5001 25%*5001

Expected Frequencies (calculated)

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Boss’s hypothesis 25% 10% 10% 10% 10% 10% 25%
Number of unique visitors 750 688 1200 1100 300 313 650
Expected number of visitors 1250.25 500.1 500.1 500.1 500.1 500.1 1250.25

To test the hypothesis, we compare the actual number of visitors to the number expected under your boss’s hypothesis. Similar to Example 2, if the null hypothesis is true the expected frequencies should be very close to the observed frequencies for each day of the week. Using our example data, the calculation looks like this:

This table shows each cell’s contribution to the chi-squared statistic. The cells with the highest contribution to the statistic had observed values that were the most different from what we’d expect under the null hypothesis.

Sunday Monday Tuesday Wednesday Thursday Friday Saturday
Cell chi-squared value 200.16 70.60 979.52 719.62 80.06 70.00 288.18

Thus, we can see that the number of visitors on Tuesday and Wednesday were the most discordant from what your boss expected. We’re almost there - we just have to determine if the calculated chi-squared value of 2408.14 is large enough to provide evidence against the null hypothesis. Remember that we also need the degrees of freedom in order to determine the corresponding p-value. When there is only one categorical variable, the degrees of freedom is just equal to the number of categories minus 1. Here, $df=7-1=6$. When we ask Plotly the probability of observing a chi-squared statistic of 2408.14 or larger using 6 degrees of freedom, we find that there is less than a 0.01% chance! Thus, there is very strong evidence against your boss’s hypothesis about website traffic.

### Be careful…

The chi-squared statistic won’t perform well if you don’t have enough data! One rule of thumb is that every cell should have an expected frequency of at least 5. The test might perform poorly if your data has many cells with small frequencies and in this case you should collect more data (if you can!) before performing the test.

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