1. Megan Templin, MPH, M.S. H. James Norton, PhD Dickson Advanced Analytics Research Division Carolinas Healthcare System

2. Chi-square Test

3. Jeopardy: Statistics for 100
The test that should be performed to answer: …“The general problem may be stated as follows: having given the number of instances in which things are both thus and so, in which they are thus but not so, in which they are so but not thus, and in which they are neither thus nor so, it is required …to determine the quantitative relativity between the thusness and the soness of the things.”

4. Jeopardy: Statistics for 100
What is the chi-square test?
Thus
Not Thus So
Not So
Bulletin of the Philosophical Society of Washington, (1888).

5. Jeopardy: Statistics for 800
The book this is a quote from:…“Grown-ups love figures. When you tell them you have made a new friend, they never ask you about essential matters. They never ask you, “What games does he love best? …Instead they demand, “How old is he? How much money does his father make?” Only from these figures do they think they have learned anything about him.”

6. Jeopardy: Statistics for 800
What is “The Little Prince?”

7. What is the chi-square test?
A statistical test used to compare groups on the percentage with a condition or outcome
Also known as contingency table

8. Assumptions of Random sample Adequate sample size
Frequency data
Random sample
Adequate sample size
Adequate expected cell counts
Independent observations

9. Chi-square hypothesis
Null hypothesis:
There is no difference in the proportions of the two treatment groups
There is not an association between the two variables
The outcome (survival) is INDEPENDENT of the treatment group
H0: P1=P2
Alternative hypothesis:
There is a difference in the proportions of the two treatment groups
There is an association between the two variables
The outcome (survival) is DEPENDENT on the treatment group (the 2 variables are not independent and therefore survival depends on which group you are in)
HA:P1≠P2

10. Chi-square Example
Suppose we are interested in studying a drug that we hope will increase the 2-year survival of patients following an M.I.
We have 46 patients in each of 2 groups (treatment vs. placebo).
Furthermore, suppose we lose to follow-up one patient in the experimental drug group.

11. Example (continued) Null hypothesis: H0: P1=P2
There is no difference in the proportions of the two treatment groups.
The outcome (survival) is INDEPENDENT of treatment group
Alternative hypothesis:
HA:P1≠P2
There is a difference in the proportions of the two treatment groups;
The outcome (survival) is DEPENDENT of the treatment group (survival and treatment are not independent of each other and therefore survival depends on which treatment group you are in)

12. Example (continued) Outcome 91 Drug Experimental Drug Control Survived38 29 67
Died 7 17 24 45 46
Marginal totals 91
Grand Total
Marginal totals

13. Example (continued) In the literature we could report this data as:
% surviving at 2 years with experimental drug
= = 84%
% surviving at 2 years with control = = 63%

14. Example (continued) 67 24 45 46 91 Drug Outcome 38 29 7 17 Exp. Drug
Control 38 29 7 17
Outcome 67
Survived 24
Died 45 46 91
The 4 cells of this table are OBSERVED values
We must calculate what is known as EXPECTED values one would expect if the 2 variables were INDEPENDENT

15. Example (continued) 67 24 45 46 91 Drug Outcome 38 29 7 17 Exp. Drug
Control 38 29 7 17
Outcome 67
Survived 24
Died 45 46 91
To calculate the critical value you need to first calculate the degrees of freedom and the expected value for each cell.
Expected values: Chi-square:
= # in row x # in column
total number

16. Example (continued)

17. Chi-Square Table

18. Chi-Square Table 5.024 6.635 So, 0.025 > p > 0.01
Computers give exact calculation, p=0.02
Conclude: The percentage of people who survived at 2 years was significantly
higher in the experimental drug group (84%) than in the control group (63%).

19. Pediatric Trauma Center
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total N
Blunt
3,175
1,676
4,851
Penetrating
618
321
939
Burn 82 66
148
Asphyxia 12 26 38
3,887
2,089
5,976
The null hypothesis says that there is not an association between injury type and trauma center (they are independent)
The alternative hypothesis says there is an association between injury type and trauma center (they are dependent)

20. Pediatric Trauma Center Pediatric Trauma Center
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total
Column % N
Blunt
81.7%
80.2%
4,851
Penetrating
15.9%
15.3%
939
Burn
2.1%
3.1%
148
Asphyxia
0.3%
1.2% 38
3,887
2,089
5,976
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total
Row % N
Blunt
65.5%
34.5%
4,851
Penetrating
65.8%
34.2%
939
Burn
55.4%
44.6%
148
Asphyxia
31.6%
68.4% 38
3,887
2,089
5,976

21. Pediatric Trauma Center
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total N
Blunt
3,175
1,676
4,851
Penetrating
618
321
939
Burn 82 66
148
Asphyxia 12 26 38
3,887
2,089
5,976
Degree of Freedom:
df = (# rows - 1) x (# columns - 1)
Injury Type: =3
Type of Center: =1
df = 3 x 1 = 3

22. Pediatric Trauma Center
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total N
Blunt
3,175
1,676
4,851
Penetrating
618
321
939
Burn 82 66
148
Asphyxia 12 26 38
3,887
2,089
5,976
Expected values:
= # in row x # in column
total number

23. Pediatric Trauma Center
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total N
Blunt
3,175
1,676
4,851
Penetrating
618
321
939
Burn 82 66
148
Asphyxia 12 26 38
3,887
2,089
5,976
Expected values:
= # in row x # in column
total number
Blunt injury – Adult Trauma Center:
= 4,851 x 3, = 3,155.3
5,976
Observed n = 3,175
Expected n = 3,155.3

24. Pediatric Trauma Center
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total N
Blunt
3,175
1,676
4,851
Penetrating
618
321
939
Burn 82 66
148
Asphyxia 12 26 38
3,887
2,089
5,976
Expected values:
= # in row x # in column
total number
Burn injury – Peds. Trauma Center:
= x 2, = 51.7
5,976
Observed n = 66
Expected n = 51.7

25. Pediatric Trauma Center
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Observed and Expected Values
Injury Type
Adult Trauma Center
Pediatric Trauma Center
Total N
Blunt
3,175 (3155.3)
1,676 (1695.7)
4,851
Penetrating
618 (610.8)
321 (328.2)
939
Burn
82 (96.3)
66 (51.7)
148
Asphyxia
12 (24.7)
26 (13.3) 38
3,887
2,089
5,976

26. Walther, A. E. , MD; Pritts, T. A. , MD, PhD; Falcone, R. A
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Chi-Square Calculation

27. Walther, A. E. , MD; Pritts, T. A. , MD, PhD; Falcone, R. A
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Degrees of freedom = 3
Critical value = 25.33

28. Walther, A. E. , MD; Pritts, T. A. , MD, PhD; Falcone, R. A
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Degrees of freedom = 3
Critical value = 25.33

29. Walther, A. E. , MD; Pritts, T. A. , MD, PhD; Falcone, R. A
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Degrees of freedom = 3
Critical value = 25.33
Chi-square value of and 3 degrees of freedom is greater than Therefore, the p-value is less than For adolescent patients, there is a statistically significant difference in the distribution of injury types between adult and pediatric trauma centers

30. Our calculated significance: P-value: < 0.005
Walther, A. E., MD; Pritts, T. A., MD, PhD; Falcone, R. A., MD, MPH; Hanseman, D. J., PhD; Robinson, B. R. H., MD. (2014). Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers. Journal of Trauma Acute Care Surgery, vol. 77 (n. 1), pp
Our calculated significance:
P-value: < 0.005
Therefore, reject the null hypothesis and
state there is an association between type of trauma center and injury type.

31. Our calculations: Df: 3 Critical value: 25.33 P-value: < 0.005
Teen Trauma without the Drama: Outcomes of Adolescents Treated at Ohio Adult versus Pediatric Trauma Centers.
SAS Output:
Our calculations:
Df: 3
Critical value:
P-value: < 0.005

32. Key points
The further apart observed values are from expected values then the larger the chi-square value. This goes back to our null hypothesis that the two things we are studying are independent…in other words, the less independent two things are, the further apart the observed and expected values will be.
If the chi-square equals 0, then proportions perfectly match the hypothesis.
When the chi-square value is large, it leads to a smaller p-value and rejection of the null hypothesis.
You need a larger sample size for categorical comparisons compared to the sample size need for numeric comparisons (example, age in categories vs. numeric age)
Tests ASSOCIATION, NOT CAUSE & EFFECT!!
If the two variables being compared are dependent (paired) samples, McNemar’s test should be used.
If the grand total is <40 (some say<20) or if the expected value of any cell is <5 (some say <1), then use Fisher’s Exact test.

33. Sample Size Calculations for Chi-Square
To calculate the number of patients needed in an experimental and a control group for a given probability of obtaining a significant result (two-sided test), you need to know (or educated guess):
Smaller proportion
Difference in proportions
Alpha
Power
From: Career Medicine – Holland, JF,& Frei E.I. Chapter 8

34. Sample Size Example 1
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 20% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.80?

35. Sample Size Example 1
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 20% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.80?
Smaller proportion:
20%
2. Difference in proportions:
20%
3. Alpha:
.05
4. Power:
.80

36. Sample Size Example 1
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 20% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.80?
Smaller proportion:
20%
2. Difference in proportions:
20%
3. Alpha:
.05
4. Power:
.80
Answer:
At least 80 per group

37. Sample Size Example 2
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 20% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.90?

38. Sample Size Example 2
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 20% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.90?
Smaller proportion:
20%
2. Difference in proportions:
20%
3. Alpha:
.05
4. Power:
.90

39. Sample Size Example 2
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 20% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.90?
Smaller proportion:
20%
2. Difference in proportions:
20%
3. Alpha:
.05
4. Power:
.90
Answer:
At least 105 per group

40. Sample Size Example 3
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 30% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.80?

41. Sample Size Example 3
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 30% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.80?
Smaller proportion:
30%
2. Difference in proportions:
10%
3. Alpha:
.05
4. Power:
.80

42. Sample Size Example 3
Suppose the failure rate of a drug is 40%. You would like to decrease that rate to 30% with a new treatment and design a study comparing the two drugs. Using the table, how many subjects are needed in each group for an alpha level of 0.05 and power of 0.80?
Smaller proportion:
30%
2. Difference in proportions:
10%
3. Alpha:
.05
4. Power:
.80
Answer:
At least 360 per group