1. Chi-square: Comparing Observed and Expected Counts
Scientific Practice
Chi-square: Comparing Observed and Expected Counts

2. Where We Are/Where We Are Going
Most of what we have done so far has looked at variables and how those might vary in relation to something else…
BP when a subject takes a drug (eg paired t-test)
what was the effect of the drug on the mean BP?
Lung function in association with another variable (correlation/regression)
how does carbon dioxide affect Minute Volume?
A different approach involves putting subjects into ‘bins’ based on overall outcomes
eg ‘the patient died’
we can compare the bin-size of the actual (observed) data against what we expected

3. Proportions: Observed vs Expected
Here’s an example from Intuitive Biostatistics…
On average, 10% of patients die following a particularly risky operation
this month, 16 out of 75 died
is this a ‘real change’ or just ‘coincidence’?
it’s obviously a ‘real change’; however, we can
estimate the probability of seeing a proportion of 16/75 (21.3%) if the background average still 10%
We need to compare what we observed (16/75) with what we expected (7.5/75, 10%)
can show this as a table…

4. Proportions: Observed vs Expected
#Observed #Expected (10%)
Alive
Dead
Total
Step 1 : The Null Hypothesis
there has been no change in the proportion dying
Step 2 : Generate Test Statistic, Chi-square (χ2)
first work out what ‘expected’ under Null Hypo
then, χ2 = Σ ((Observed – Expected)2 / Expected)
the more the data do not fit the expected pattern, the bigger χ2 gets

5. Proportions: Observed vs Expected
Step 2 : Generate Test Statistic, Chi-square (χ2)
χ2 = Σ ((Observed – Expected)2 / Expected)
χ2 = (( )2 / 67.5)
+((16-7.5)2 / 7.5) = 10.7
Step 3 : Calculate the probability
use a table of critical values of χ2
cols = levels of significance (p-value, or alpha)
rows = degrees of freedom
α = 0.05
df = categories-1
2-1 = 1

6. Proportions: Observed vs Expected

7. Proportions: Observed vs Expected
Critical value for χ2 = 3.84
Our value for χ2 = 10.70
So, p < 0.05
if fact, critical value for p = is 9.14,
ours is so p <
Step 4 : Interpret the probability
the probability of seeing our proportion (21.3%) dying if the underlying trend of 10% still applied is less than 0.25% (p < )
so reject the Null Hypo in favour of Alt Hypo…
that some factor other than chance responsible

8. Comparing Two Proportions
The above example just looked at one category with two ‘states’ (dead and alive)
But what if we wanted to include male and female in our investigation?
eg suspect more men than women dying
χ2 can be used to look do this via a 2 x 2 table
Example, disease progression in AZT/placebo patients…
Disease progression No prog Total
AZT
Placebo
Total

9. Comparing Two Proportions
Disease progression No prog Total
AZT
Placebo
Total
Step 1 : The Null Hypothesis
no difference in disease progression in AZT vs placebo
Step 2 : Generate the Test Statistic
first, need to work out what we would expect if the Null Hypo were true…

10. Comparing Two Proportions
Disease progression No prog Total
AZT
Placebo
Total
Disease progression is 205/936 = 21.9%
so, if Null Hypo is true, we’d expect…
21.9/100 x 475 = 104 AZT to progress
21.9/100 x 461 = 101 Placebo to progress
that leaves 475 – 104 = 371 AZT to not prog
and 461 – 101 = 360 Placebo to not prog
(can do this with row total x col total / grand total)

11. Comparing Two Proportions
OBSERVED
Disease progression No prog Total
AZT
Placebo
Total
EXPECTED
AZT
Placebo

12. Comparing Two Proportions
Step 2 : Generate Test Statistic, Chi-square (χ2)
χ2 = Σ ((Observed – Expected)2 / Expected)
χ2 = ((76-104)2 / 104)
+(( )2 / 371)
+(( )2 / 101)
+(( )2 / 360)
χ2 =
= 19.53

13. Comparing Two Proportions
Step 3 : Calculate the probability
use a table of critical values of χ2
cols = levels of significance (p-value, or alpha)
rows = degrees of freedom
α = 0.05
df = (rows-1) x (cols-1)
(2-1) x (2-1) = 1
Critical value for χ2 = 3.84
Our value for χ2 = 19.53
So, p < 0.05
if fact, critical value for p = is 10.83,
ours is so p < 0.001

14. Comparing Two Proportions
Step 4 : Interpret the probability
if the Null Hypo is true (no difference in disease progression in AZT vs placebo)…
…there is < 0.1% chance of us observing the pattern of progression we actually saw.
so reject the Null Hypo in favour of Alt Hypo…
that AZT produces different progression rates
a reduction! Phew!

15. Chi-square in Minitab C1 = AZT, C2 = Placebo
Row 1 = Progression, Row 2 = No prog

16. Summary
Where counts can be put in ‘bins’ then tests of proportions can be carried out
One such test is the chi-square (χ2) test
Involves working out what we expected to see if the Null Hypo applied
χ2 = Σ ((Observed – Expected)2 / Expected)
The bigger (χ2) is, the more unlikely that the Null Hypo (ie the ‘expected’ pattern) is true
A common use of χ2 is the 2x2 table