1. Chi-Square and odds ratios

2. Semester Recap We’ve covered: Descriptive Statistics
Measures of Central Tendency
Measures of Variability
Z-scores and Graphing
Association and Prediction
Correlation
Regression (simple and multiple)
Testing for Group Differences
t-tests (one, indep., and paired)
ANOVA (One-way, factorial, ANCOVA, RM ANOVA
Statistical Concepts
Statistical Significance
Type I and Type II Error
Alpha and p-values
Beta and power
Effect Sizes

3. Last weeks…
All of the statistical tests on the prior slide are known as ‘parametric’ statistics
Parametric Statistics have strict assumptions that must be met before a t-test, correlation, etc… can be used
Assumes homoscedasticity of variance (variance between two variables are similar)
Assumes a normal distribution (bell-shaped curve)
These assumptions are not easily met
For example, we’ve used physical activity in several of our examples this semester – physical activity is RARELY normally distributed…

4. Is this a normal distribution?
Nationally Representative Sample of Minutes of Daily Physical Activity
Is this a normal distribution?
What kind is it?
Positive Skew PA

5. Additionally…
Sometimes you need to use a dependent variable that is categorical (grouping)
Recall from the chart that the statistical tests we’ve discussed all require 1 continuous dependent variables
Example Research Question:
Are there more male than female athletic trainers across the United States?
I want to know if there are more men than women – this is a nominal dependent variable
No correlation, regression, or ANOVA, etc… will help me answer this simple question

6. Non-Parametric Tests Non-parametric tests can be used when:
Parametric tests statistical assumptions are not met
Categorical DV’s are used
Non-parametric statistics is an entirely different line of statistics that includes dozens of new tests
Usually a parametric test has a non-parametric ‘relative’
Chi-Square Test of Independence is similar to Pearson Correlation
Most of the time if you can’t meet the assumptions of the parametric test you wanted to do – you can find an appropriate non- parametric test

7. Non-Parametric Tests (cont)
Non-parametric tests have benefits and drawbacks like all statistical tests
Benefits:
Assumptions are easier to meet
Non-parametric tests basically just need nominal or ordinal data
Remember that interval or ratio data scales can always be converted to nominal or ordinal – the inverse is not true
More ‘robust’
Non-parametric statistics are more versatile tests
Recall small changes in our data required completely different parametric tests (e.g., the difference between a t-test and ANOVA)
Usually easier to calculate
We can realistically hand-calculate several non-parametric tests

8. Non-Parametric Tests (cont)
Non-parametric tests have benefits and drawbacks like all statistical tests
Drawbacks:
Less ‘mainstream’
Non-parametric tests are commonly used – but far less frequently than parametric tests
Less powerful
By design, non-parametric tests have less statistical power…
What’s that mean?
Unless you NEED to use a categorical dependent variable, if your data meets the statistical assumptions you should use parametric stats

9. Tonight We will discuss and use two non-parametric tests:
Chi-square test of independence
Logistic Regression (Odds Ratio)
These are the most common non-parametric tests
Both are related to each other (often used together)
Do NOT require normal distribution or homoscedasticity
Do require 2 categorical variables (nominal or ordinal)

10. Chi-Square, χ2
There are actually a few different types of Chi-Square tests, we will discuss the Chi-Square test of independence
Test determines if two variables are related (or unrelated)
‘Test of independence’
Similar to Pearson correlation
For example, we would expect
Sex (Male/Female) is NOT related to hair color (dark/light)
Men and women are just as likely to have dark or light hair
Sex (Male/Female) is related to having a heart attack (Yes/No)
Men have more heart attacks than women do
For these tests, start thinking about data in 2x2 tables…

11. Chi-Square Data ‘Picture’
Cause of Death in Men and Women
Heart Attack?
Yes No
Sex
Men 68 32
Women 42 58
A 2x2 table provides a nice summary of the data
In this example, ‘Sex’ is the IV and ‘Heart Attack’ is the DV
Does male/female increase risk of heart attack?
This table provides frequency of occurrence
Can also convert to percentage – you will get the same result

12. SPSS View Data Structure: Key variables are categorical
Can look at the data labels or values:
Males = 1
Females = 2
HeartAttack Yes = 1
HeartAttack No = 2
Look at Labels and Values:

13. SPSS View Data Structure: Key variables are categorical
Can look at the data labels or values:
Males = 1
Females = 2
HeartAttack Yes = 1
HeartAttack No = 2

14. Chi-Square Data ‘Picture’
Cause of Death in Men and Women
Heart Attack?
Yes No
Sex
Men 68 32
Women 42 58
You should fill out the margins of the table (how many men, women, total n, heart attacks, other causes, etc…
Do on board

15. How the Chi-Square works…
Cause of Death in Men and Women
Heart Attack?
Yes No
Sex
Men 68 32
Women 42 58
The χ2 test has a null hypothesis that there is no difference in the frequency of men/women having heart attacks
If the two variables are unrelated (independent), we would expect men and women to have the about same number
But, we need a statistical test to know if this difference is RSE

16. How to run Chi-Square

17. It is CRITICAL you put the variables in the correct spots
Typically the IV goes in the Row
And DV goes in the Column
Then click on ‘statistics’
It doesn’t really change the answer – but it makes it easier for you to understand the results

19. What else is there?
The ‘cells’ tab will allow you to request percentages in each cell, to go along with the frequencies
The ‘format’ tab will allow you to change the organization of your table
E.g., put ‘Females’ on the top row, or put ‘No Heart Attack’ in the left column

20. SPSS Output SPSS provides two initial tables:
1) Case Processing Summary: Ignore, repeat info of…
2) CrossTabs Table (our 2x2):
= Frequency. Could have asked for Percentages

21. Chi-Square Tests
We only care about the ‘Pearson Chi-Square’ – yeah, it’s that same guy from correlation…
Important info is the χ2 = , df, p, and n
χ2 = is just like the t-statistic or r, or F ratio...

22. Chi-Square df
df is calculated by (number of columns – 1) multiplied by (number of rows – 1)
2 rows – 1 = 1
2 columns – 1 = 1
1 x 1 = 1 df
All 2x2 tables have 1 df, more variables will change this

23. Reporting the results:
A chi-square test of independence was used compare the frequency of heart attacks between men and women. Sex was significantly related to having a heart attack. Men tended to have more heart attacks than women (χ2(1) = 13.66, p < ).
Questions on Chi-Square?

24. Logistic Regression
Researchers do often report just the Chi-Square test results. However, it is also common for them to incorporate logistic regression/or odds ratios
Quick definition of Logistic Regression:
Type of regression equation that uses a categorical DV
Such as heart attack yes/no from our example
It allows you to include any type of IV (categorical or continuous) – and any number of IV’s
In this sense, it is very similar to simple or multiple linear reg.
Instead of providing you with a slope – it provides an odds ratio for each IV

25. What are the odds…?
Odds
The probability of an event happening divided by the probability of the event not happening
Students often get confused here…
A die has 6 sizes, each with a difference number
On one die, the odds of rolling a 1 is…?
1 side has a 1 on it, 5 sides do NOT
Odds of rolling a 1:
1/5, or 20%

26. What’s an odds ratio…? Odds Ratio
The odds of an event happening in one group divided by the odds of an event happening in another group
It is literally the ratio of two odds
It acts like effect size for a chi-square
The chi-square tells you if there is a difference – the odds ratio tells you how big/strong that difference is
If the chi-square is significant – the odds ratio is also statistically significant
Literal interpretation is how much more likely an event is to happen in one group versus another
It’s easier to see in our example…

27. Cause of Death in Men and Women
Back to our example
What are the odds of a heart attack in men?
68/32, or 2.125
What are the odds of a heart attack in women?
42/58, or 0.724
Cause of Death in Men and Women
Heart Attack?
Yes No
Sex
Men 68 32
Women 42 58

28. Cause of Death in Men and Women
Back to our example
What is the ratio of these odds, or odds ratio?
2.125 / = 2.9 = OR
Interpretation:
Men are 2.9 times more likely to have a heart attack than women (we know it’s significant because of the χ2)
Cause of Death in Men and Women
Heart Attack?
Yes No
Sex
Men 68 32
Women 42 58

29. More on odds ratios
Interpreting odds ratios can trip up some students:
For example, 2.9 is the odds ratio for men vs. women
Men are 3 times more likely than women
Being a man is a ‘risk factor’ for heart attack
What is the odds ratio for women vs. men?
0.724 / = 0.34
Women are one-third as likely to have a heart attack than men
Being a woman is ‘protective’ of a heart attack
Odds Ratios:
> 1.0 indicate an increased risk
< 1.0 indicate a decreased risk
= 1.0 indicate the SAME risk

30. Another Example: Lung Cancer
Cause of Death in Men and Women
Lung Cancer?
Yes No
Sex
Men 6 64
Women 16
201
First, notice that way more women had lung cancer
But – there are way more women in this sample
I’ll run a chi-square in SPSS to see if there is a difference…

31. Lung Cancer Chi-Square results
χ2 = 2.451, df = 1, p = 0.456, n = 287
Is there a difference in the frequency of lung cancer between men and women?
A chi-square test revealed that there was no significant difference in the odds of lung cancer between men and women (χ2 (1) = 2.451, p = ).
Let’s calculate the odds ratio…

32. Odds Ratio: Lung Cancer
What are the odds of cancer in men?
6/64 = 0.094
In women?
16/201 = 0.079
What is the OR?
Odds Ratio = / = 1.19
It appears men might be slightly more likely than women, but this could be due to RSE
Cause of Death in Men and Women
Lung Cancer?
Yes No
Sex
Men 6 64
Women 16
201

33. More on Odds Ratios
1) Take care in setting up your 2x2 table – this can make it really easy to calculate the odds and understand your chi-square or really hard
2) As you can see we are hand-calculating an odds ratio. You can get SPSS to do this for you.
If you have only 1 IV (like these example), it’s called “Risk” in the CrossTabs option
Like simple linear regression

35. New Heart Attack Output
When you do this, ignore the bottom rows of the box (they are more confusing than helpful)
You get the OR and the 95% CI
This is the same result we got from hand calculating it, and we knew it was significant because of the chi- square test

36. 95% CI’s and Odds Ratios Is this odds ratio statistically significant?
How can you tell?

37. More on Odds Ratios
2) As you can see we are hand-calculating an odds ratio. You can get SPSS to do this for you.
With 1 IV, use crosstabs
With multiple IV’s, use ‘logistic regression’

38. I won’t ask you to do this – but you should know it’s there

39. More on Odds Ratios
2) As you can see we are hand-calculating an odds ratio. You can get SPSS to do this for you.
With 1 IV, use crosstabs
With multiple IV’s, use ‘logistic regression’
3) Odds Ratios can be tricky without the 2x2 table
They tell you how much more likely something is to happen for a certain group (like males vs females)
But, you should always include the information in the 2x2 table so people know what you’re talking about
Example…

40. Odds of Winning the Powerball
Odds of winning the Powerball lotto with 1 ticket:
1 / 175,223,510 =
Odds of winning the Powerball with 10 tickets:
10/175,223,510 =
Odds Ratio = 10.0
You are 10 times as likely to win with 10 tickets!!!!!
Still better odds to die driving to buy your ticket – or getting hit by lightning – than winning with 10 tickets.
Odds Ratios are great statistics – but should not be used to model VERY, VERY rare things

41. Referent Group 4) Odds ratios are a ratio between two groups
The odds ratio was 2.9 or 0.34 for our heart attack example – depending on which group was the referent
Referent Group = The ‘baseline’ group in the odds ratio. The group that is in the denominator.
2.125 / = 2.9
Women were the referent group – meaning men were 2.9 times more likely than women
Must mention this when talking about your odds ratio – you can’t just say, ‘the odds ratio was 2.9’.
WHO was 2.9 times more likely than WHO?
This is especially important when using them for more than 2 groups…

42. Example with multiple groups
Odds ratios from a current study I’m working on
Shows compares the odds of obesity in children meeting recommendations for PA, screen time, sleep
Notice that WHO I choose to be the referent group influences all of the odds ratios

43. QUESTIONS on Chi-Square? Odds Ratios?

44. Upcoming… In-class activity Homework:
Cronk – Read Section 7.2 on Chi-Square of Independence
Holcomb Exercises 55, 56, 57 and 59
Exercises are on Chi-Square, 59 is on Odds Ratio