Chi-Square Test
Chi-Square Test
When comparing categorical variables, your data won't always have 2x2 dimensions as in the Fisher's Test Topic Note. For tables with greater dimensions we can use the Chi-square test to test hypotheses of association. The Chi-Square or \(\chi^2\) test is defined by:
with:
Explanation of Terms
- \(\chi^2\) chi-square statistic
- \(O_i\) observed value at cell i
- \(E_i\) expected value at cell i
- \(d.f.\) degrees of freedom
- \(r\) number of rows
- \(c\) number of columns
Chi-Square Distribution
This \(\chi^2\) value is then compared to a probability distriubtion. In our Binomial Test Topic Note, we described that probability distributions help us determine the probability of an event given some parameter. In the Binomial Test Topic Note, that parameter was the number of successes, but here it is our degrees of freedom. We won't cover the math of this probability distribution function here, but if your curious check out the wikipedia page on chi-sqare distributions. Let's use R to examine how probability changes with varying degrees of freedom and \(\chi^2\) values:
# examine the chi square cdf
chi.sq.df <- data.frame(
probability = 0,
chi_square = 0,
df = 0
)
for(i in 2:9){
chi.sq.df.i <- data.frame(
probability = dchisq(1:10,i),
chi_square = 1:10,
df = rep(as.character(i),10)
)
chi.sq.df <- rbind(chi.sq.df,
chi.sq.df.i)
}
chi.sq.df <- chi.sq.df[-1,]
ggplot(chi.sq.df,
aes(x = chi_square,
y = probability,
color = df)) +
geom_line()+
theme_bw()
Now, what you might have guessed from the equation above, but we are testing the following hypotheses:
- \(H_0\) or null hypothesis that there is no association between the two categorical values
- \(H_a\) or alternative hypothesis that there is an association between the two categorical values
Visualizing Data
Here we will test the association between losing a patient to follow up and their country of origin. Let's try visualizing this first:
# load meta data
meta <- read.table("./data/gbm_cptac_2021/data_clinical_patient.txt",
header = T,
sep="\t")
# generate a table of losing a patient to follow up
# by their country of origin
# removing zero count countries
table <- as.data.frame.matrix(
table(meta$LOST_TO_FOLLOW_UP,meta$COUNTRY_OF_ORIGIN)
) %>%
select(c(China,Poland,Russia, `United States`))
table
China Poland Russia United States
No 21 14 16 16
Yes 9 2 3 5
Now let's see this plotted!
#plot our data
table %>%
t() %>%
reshape2::melt() %>%
ggplot(.,aes(x=Var1,y=value,fill=Var2))+
geom_bar(stat = "identity",position="fill") +
scale_y_continuous(labels = scales::percent)+
theme_bw()+
scale_fill_manual(values=c("aquamarine3","lightpink3")) +
labs(
x="",
y="Frequency",
fill="Lost To Follow Up?"
)
Running the Chi-Square Test
# run the chi-square test
chisq.test(table)
Pearson's Chi-squared test
data: table
X-squared = 2.4197, df = 3, p-value = 0.49
Warning message:
In chisq.test(table) : Chi-squared approximation may be incorrect
Explanation of Results
Here we note that:
- Our chi-square test statistic is
2.4197 - The degrees of freedom is
3 - Our p-value is
0.49and is greater than 0.05:- indicating there is not enough evidence to reject the null hypothesis (a.k.a there is no association between the two categorical values)
- Our test assumptions might not be correct!
Assumptions
The Chi-Square test is not free of assumptions:
- categories shouldn't have an expected frequency less than one
- 20% of the categories should not have an expected frequency less than 5
In the table above we note that some frequency values are indeed less than 5! Now how does our test change if our values are increased ten-fold?
chisq.test(table*10)
Pearson's Chi-squared test
data: table * 10
X-squared = 24.197, df = 3, p-value = 2.272e-05
Here we see that not only did the warning disappear but our p-value is below the canonical 0.05! So be aware that the Chi-Square test is very dependent on sample size.