Chi-Square Test of Indpendence
Priyank Goyal
04/04/2020
Chi-Square Test of Independence in R
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.
Data format Contingency Tables
# Import the data
file_path <- "http://www.sthda.com/sthda/RDoc/data/housetasks.txt"
housetasks <- read.delim(file_path, row.names = 1)
head(housetasks)## 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 53The data is a contingency table containing 13 housetasks and their distribution in the couple:
rows are the different tasks
values are the frequencies of the tasks done :
by the wife only
alternatively
by the husband only
or jointly
Graphical Display of contingency Tables
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.
install gplots
library(gplots)## Warning: package 'gplots' was built under R version 3.6.1##
## Attaching package: 'gplots'## The following object is masked from 'package:stats':
##
## lowess# 1. convert the data as a table
dt <- as.table(as.matrix(housetasks))
# 2. Graph
balloonplot(t(dt), main ="housetasks", xlab ="", ylab="",
label = FALSE, 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")
The argument shade is used to color the graph
The argument las = 2 produces vertical labels
Blue color indicates that the observed value is higher than the expected value if the data were random
Red color specifies that the observed value is lower than the expected value if the data were random
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.
## Compute Chi-square test in R
chisq <- chisq.test(housetasks)
chisq##
## Pearson's Chi-squared test
##
## data: housetasks
## X-squared = 1944.5, df = 36, p-value < 2.2e-16In our example, the row and the column variables are statistically significantly associated (p-value = 0).
# 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 46.70Nature of dependence between row and column variables
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:
Cells with the highest absolute standardized residuals contribute the most to the total Chi-square score.
round(chisq$residuals, 3)## Wife Alternating Husband Jointly
## Laundry 12.266 -2.298 -5.878 -6.609
## Main_meal 9.836 -0.484 -4.917 -6.084
## Dinner 6.537 -1.192 -3.416 -3.299
## Breakfeast 4.875 3.457 -2.818 -5.297
## Tidying 1.702 -1.606 -4.969 3.585
## Dishes -1.103 1.859 -4.163 3.486
## Shopping -1.289 1.321 -3.362 3.376
## Official -3.659 8.563 0.443 -2.459
## Driving -5.469 6.836 8.100 -5.898
## Finances -4.150 -0.852 -0.742 5.750
## Insurance -5.758 -4.277 4.107 5.720
## Repairs -7.534 -4.290 20.646 -6.651
## Holidays -7.419 -4.620 -4.897 15.556library(corrplot)## Warning: package 'corrplot' was built under R version 3.6.2## corrplot 0.84 loadedcorrplot(chisq$residuals, is.cor = FALSE)
For a given cell, the size of the circle is proportional to the amount of the cell contribution.
Positive residuals are in blue. Positive values in cells specify an attraction (positive association) between the corresponding row and column variables.
In the image above, it’s evident that there are an association between the column Wife and the rows Laundry, Main_meal.
There is a strong positive association between the column Husband and the row Repair
Negative residuals are in red. This implies a repulsion (negative association) between the corresponding row and column variables. For example the column Wife are negatively associated (~ “not associated”) with the row Repairs. There is a repulsion between the column Husband and, the rows Laundry and Main_meal
# Contibution in percentage (%)
contrib <- 100*chisq$residuals^2/chisq$statistic
round(contrib, 3)## 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)
It can be seen that:
The column “Wife” is strongly associated with Laundry, Main_meal, Dinner
The column “Husband” is strongly associated with the row Repairs
The column jointly is frequently associated with the row Holidays
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.
Access to values returned by chisq.test()function
The result of chisq.test() function is a list containing the following components:
statistic: the value the chi-squared test statistic.
parameter: the degrees of freedom
p.value: the p-value of the test
observed: the observed count
expected: the expected count
# printing the p-value
chisq$p.value## [1] 0# printing the mean
chisq$estimate## NULL