The chi-square test of independence is used to analyze the frequency table (i.e. contengency table) formed by two categorical variables. The chi-square test evaluates whether there is a significant association between the categories of the two variables. This article describes the basics of chi-square test and provides practical examples using R software.
# Import the data file_path An image of the data is displayed below:
Data format correspondence analysis
The data is a contingency table containing 13 housetasks and their distribution in the couple:
Contingency table can be visualized using the function balloonplot() [in gplots package]. This function draws a graphical matrix where each cell contains a dot whose size reflects the relative magnitude of the corresponding component.
To execute the R code below, you should install the package gplots: install.packages(“gplots”).
library("gplots") # 1. convert the data as a table dt 
Chi-Square Test of Independence in R
Note that, row and column sums are printed by default in the bottom and right margins, respectively. These values can be hidden using the argument show.margins = FALSE.
It’s also possible to visualize a contingency table as a mosaic plot. This is done using the function mosaicplot() from the built-in R package garphics:
library("graphics") mosaicplot(dt, shade = TRUE, las=2, main = "housetasks")
Chi-Square Test of Independence in R
Note that the surface of an element of the mosaic reflects the relative magnitude of its value.
From this mosaic plot, it can be seen that the housetasks Laundry, Main_meal, Dinner and breakfeast (blue color) are mainly done by the wife in our example.
There is another package named vcd, which can be used to make a mosaic plot (function mosaic()) or an association plot (function assoc()).
# install.packages("vcd") library("vcd") # plot just a subset of the table assoc(head(dt, 5), shade = TRUE, las=3)
Chi-Square Test of Independence in R
Chi-square test examines whether rows and columns of a contingency table are statistically significantly associated.
For each cell of the table, we have to calculate the expected value under null hypothesis.
For a given cell, the expected value is calculated as follow:
The Chi-square statistic is calculated as follow:
This calculated Chi-square statistic is compared to the critical value (obtained from statistical tables) with \(df = (r - 1)(c - 1)\) degrees of freedom and p = 0.05.
If the calculated Chi-square statistic is greater than the critical value, then we must conclude that the row and the column variables are not independent of each other. This implies that they are significantly associated.
Note that, Chi-square test should only be applied when the expected frequency of any cell is at least 5.
Chi-square statistic can be easily computed using the function chisq.test() as follow:
chisq Pearson's Chi-squared test data: housetasks X-squared = 1944.5, df = 36, p-value < 2.2e-16
In our example, the row and the column variables are statistically significantly associated (p-value = 0).
The observed and the expected counts can be extracted from the result of the test as follow:
# Observed counts chisq$observed Wife Alternating Husband Jointly Laundry 156 14 2 4 Main_meal 124 20 5 4 Dinner 77 11 7 13 Breakfeast 82 36 15 7 Tidying 53 11 1 57 Dishes 32 24 4 53 Shopping 33 23 9 55 Official 12 46 23 15 Driving 10 51 75 3 Finances 13 13 21 66 Insurance 8 1 53 77 Repairs 0 3 160 2 Holidays 0 1 6 153
# Expected counts round(chisq$expected,2) Wife Alternating Husband Jointly Laundry 60.55 25.63 38.45 51.37 Main_meal 52.64 22.28 33.42 44.65 Dinner 37.16 15.73 23.59 31.52 Breakfeast 48.17 20.39 30.58 40.86 Tidying 41.97 17.77 26.65 35.61 Dishes 38.88 16.46 24.69 32.98 Shopping 41.28 17.48 26.22 35.02 Official 33.03 13.98 20.97 28.02 Driving 47.82 20.24 30.37 40.57 Finances 38.88 16.46 24.69 32.98 Insurance 47.82 20.24 30.37 40.57 Repairs 56.77 24.03 36.05 48.16 Holidays 55.05 23.30 34.95 46.70
As mentioned above the total Chi-square statistic is 1944.456196.
If you want to know the most contributing cells to the total Chi-square score, you just have to calculate the Chi-square statistic for each cell:
The above formula returns the so-called Pearson residuals (r) for each cell (or standardized residuals)
Cells with the highest absolute standardized residuals contribute the most to the total Chi-square score.
Pearson residuals can be easily extracted from the output of the function chisq.test():
round(chisq$residuals, 3) Wife Alternating Husband Jointly Laundry 12.266 -2.298 -5.878 -6.609 Main_meal 9.836 -0.484 -4.917 -6.084 Dinner 6.537 -1.192 -3.416 -3.299 Breakfeast 4.875 3.457 -2.818 -5.297 Tidying 1.702 -1.606 -4.969 3.585 Dishes -1.103 1.859 -4.163 3.486 Shopping -1.289 1.321 -3.362 3.376 Official -3.659 8.563 0.443 -2.459 Driving -5.469 6.836 8.100 -5.898 Finances -4.150 -0.852 -0.742 5.750 Insurance -5.758 -4.277 4.107 5.720 Repairs -7.534 -4.290 20.646 -6.651 Holidays -7.419 -4.620 -4.897 15.556
Let’s visualize Pearson residuals using the package corrplot:
library(corrplot) corrplot(chisq$residuals, is.cor = FALSE)
Chi-Square Test of Independence in R
For a given cell, the size of the circle is proportional to the amount of the cell contribution.
The sign of the standardized residuals is also very important to interpret the association between rows and columns as explained in the block below.
The contribution (in %) of a given cell to the total Chi-square score is calculated as follow:
\[ contrib = \frac \]# Contibution in percentage (%) contrib Wife Alternating Husband Jointly Laundry 7.738 0.272 1.777 2.246 Main_meal 4.976 0.012 1.243 1.903 Dinner 2.197 0.073 0.600 0.560 Breakfeast 1.222 0.615 0.408 1.443 Tidying 0.149 0.133 1.270 0.661 Dishes 0.063 0.178 0.891 0.625 Shopping 0.085 0.090 0.581 0.586 Official 0.688 3.771 0.010 0.311 Driving 1.538 2.403 3.374 1.789 Finances 0.886 0.037 0.028 1.700 Insurance 1.705 0.941 0.868 1.683 Repairs 2.919 0.947 21.921 2.275 Holidays 2.831 1.098 1.233 12.445
# Visualize the contribution corrplot(contrib, is.cor = FALSE)
Chi-Square Test of Independence in R
The relative contribution of each cell to the total Chi-square score give some indication of the nature of the dependency between rows and columns of the contingency table.
It can be seen that:
From the image above, it can be seen that the most contributing cells to the Chi-square are Wife/Laundry (7.74%), Wife/Main_meal (4.98%), Husband/Repairs (21.9%), Jointly/Holidays (12.44%).
These cells contribute about 47.06% to the total Chi-square score and thus account for most of the difference between expected and observed values.
This confirms the earlier visual interpretation of the data. As stated earlier, visual interpretation may be complex when the contingency table is very large. In this case, the contribution of one cell to the total Chi-square score becomes a useful way of establishing the nature of dependency.
The result of chisq.test() function is a list containing the following components:
The format of the R code to use for getting these values is as follow:
# printing the p-value chisq$p.value # printing the mean chisq$estimateThis analysis has been performed using R software (ver. 3.2.4).
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